Contributing
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Learning from Open Source
Who is this for?
For HP Prime calculator users who are interested in adding functionality to their device, writing their own apps can be a valuable option.
Command Search
#link discussion
!
Syntax
value!
Description
Factorial. Returns the factorial of a positive integer. For non-integers, ! = Γ(x + 1). This calculates the Gamma function.
Example
6! returns 720
#link discussion
%
Syntax
%(x, y)
Description
x percent of y. Returns (x/100)*y.
Example
%(20,50) returns 10
#link discussion
%CHANGE
Syntax
%CHANGE(x, y)
Description
Percent change from x to y. Returns 100*(y-x)/x.
%CHANGE(20,50) returns 150
#link discussion
%TOTAL
Syntax
%TOTAL(x, y)
Description
Percent total; the percentage of x that is y. Returns 100*y/x.
%TOTAL(20,50) returns 250.
#link discussion
*
Syntax
Object1×Object2
Description
Multiplication.
Returns the result of multiplying Object1 and Object2. The objects may be numerical values or expressions that return numerical results. The objects may also be lists or matrices of appropriate dimensions.
Example
3*2 returns 6
#link discussion
+
Syntax
Object1 + Object2
Description
Addition.
Returns the result of adding Object2 to Object 1. The objects may be numerical values or expressions that return numerical results. The objects may also be lists or matrices of appropriate dimensions.
Example
3+2 returns 5
#link discussion
-
Syntax
Object1 - Object2
Description
Subtraction.
Returns the result of subtracting Object2 from Object 1. The objects may be numerical values or expressions that return numerical results. The objects may also be lists or matrices of appropriate dimensions.
Example
3-2 returns 1
#link discussion
.*
Syntax
.*(Lst||Mtrx,Lst||Mtrx)
Description
Performs an element-by-element multiplication of 2 lists or 2 matrices.
Example
[[1,2],[3,4]] .* [[3,4],[5,6]] returns [[3,8],[15,24]]
#link discussion
.+
Syntax
matrix .+ real/complex or real/complex .+ matrix
Description
Adds the real/complex to each element of the matrix
Example
[1,2].+3 returns [4,5]
#link discussion
.-
Syntax
matrix .- real/complex or real/complex .- matrix
Description
Substract the real/complex to each element of the matrix (or the reverce as appropriate)
Example
[3,4].-2 returns [1,2]
#link discussion
./
Syntax
./(Lst||Mtrx,Lst||Mtrx)
Description
Performs an element-by-element division of 2 lists or 2 matrices.
Example
[[1,2],[3,4]] ./ [[3,4],[5,6]] returns [[1/3,1/2],[3/5,2/3]]
#link discussion
.^
Syntax
.^(Mtrx,Intg(n))
Description
Calculates the power of each element of the matrix.
Example
[[1,2],[3,4]] .^ 3 returns [[1,8],[27,64]]
#link discussion
/
Syntax
Object1/Object2
Description
Division.
Returns the result of dividing Object1 by Object2. The objects may be numerical values or expressions that return numerical results. The objects may also be lists or matrices of appropriate dimensions.
Example
3÷2 returns 1.5
#link discussion
:=
Syntax
variable := object
Description
Assigns object to variable.
Example
A := 3 stores the value 3 in the variable A
F1 := 3-X makes F1(X)=3-X
M5 := [1, 2] stores a vector in M5
#link discussion
<
Syntax
Value1 < Value2
Description
Strict less-than-inequality test.
Returns 1 if the left side of the inequality is less than the right side, and 0 otherwise.
Note that more than two objects can be compared.
Example
6 < 8 < 11 returns 1 (because it is true)
6 < 8 < 3 returns 0 (as it is false)
#link discussion
<=
Syntax
Value1 ≤ Value2
Description
Less than or equal to.
Tests whether or not Value1 is less than Value 2. Returns 1 if true, 0 if false.
Example
2 ≤ 1 returns 0
Alternative: <=
#link discussion
<>
Syntax
Value1 ≠ Value2
Description
≠: Not equal to.
Tests if Value1 is not equal to Value 2. Returns 1 if true, 0 if false.
Example
3 ≠ 5 returns 1
Alternatives: <>
#link discussion
=
Syntax
Value1 == Value2
Description
==: equal to.
Tests is Value1=Value2. Returns 1 if true, 0 if false.
Example
3==2 returns 0
#link discussion
==
Syntax
Value1 == Value2
Description
==: equal to.
Tests is Value1=Value2. Returns 1 if true, 0 if false.
Example
3==2 returns 0
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>
Syntax
Value1 > Value2
Description
>: Greater than.
Tests whether or not Value1 is greater than Value 2. Returns 1 if true, 0 if false.
Example
2 > 1 returns 1
#link discussion
>=
Syntax
Value1 ≥ Value2
Description
≥: Greater than or equal to.
Tests whether or not Value 1 is either greater than or equal to Value2. Returns 1 if true, 0 if false.
Example
3 ≥ 4 returns 0
Alternative: >=
#link discussion
^
Syntax
Value1^Value2
Description
Exponentiation.
Returns the result of raising Value1 to the power of Value2.
Example
2^3 returns 8
#link discussion
`
Syntax
QUOTE(expression)
Description
Returns the expression unchanged and un-evaluated.
This function is mostly used with the STO▶ command in order to store a function in a function variable. For example if you want to store SIN(X) in F1.you cannot do SIN(X)►F1 as SIN(X) would be evaluated and a numerical result would be stored into F1. QUOTE(SIN(X))►F1 will store SIN(X) in F1.
#link discussion
a2q
Syntax
a2q(Mtrx,VectVar)
Description
a2q(A,X)=the quadratic form q associated to A, X=vector of variables.
Example
a2q([[1,2],[4,4]],[x,y]) returns x^2+6*x*y+4*y^2
#link discussion
abcuv
Syntax
abcuv(Polya,Polyb,Polyc,[Var])
Description
Returns [u,v] suchthat au+bv=c for 3 polynomials a, b, and c.
Example
abcuv(x^2+2*x+1,x^2-1,x+1) returns [1/2,-1/2]
#link discussion
about
Syntax
about(Var(a))
Description
Returns the hypothesis made with assume on the variable a.
Example
about(n) returns n
#link discussion
ABS
Syntax
ABS(expr) or ABS(matrix)
Description
For numerical arguments, returns the absolute value of the expression. For matrix arguments, returns the returns the Frobenius (Euclidean) norm of the array.
Example
ABS(-3.14) returns 3.14 and ABS([[1,2],[3,4]]) returns 5.47722557505
#link discussion
abscissa
Syntax
abscissa(Pnt or Vect)
Description
Returns the abscissa of a point or a vector.
Example
abscissa(point(1+2*i)) returns 1
#link discussion
ACOS
Syntax
ACOS(Value)
Description
ACOS: the inverse cosine function.
This Shift-key combination returns the inverse cosine of Value. The output depends on the Angle Measure setting.
Example
ACOS(-1) returns 3.14159265359
#link discussion
acos2asin
Syntax
acos2asin(Expr)
Description
Replaces arccos(x) by π/2-arcsin(x) in the argument Expr.
Example
acos2asin(acos(x)+asin(x)) returns π/2-asin(x)+asin(x)
#link discussion
acos2atan
Syntax
acos2atan(Expr)
Description
Replaces arccos(x) by π/2-arctan(x/√(1-x^2)) in the argument.
Example
acos2atan(2*acos(x)) returns 2*(π/2-atan(x/(√(1-x^2))))
#link discussion
ACOSH
Syntax
ACOSH(value)
Description
Inverse hyperbolic cosine.
Example
ACOSH(1.54308063482) returns 1
#link discussion
ACOT
Syntax
ACOT(value)
Description
Arc cotangent. The function derived from the inverse of the Cotangent function.
Example
ACOT(1) returns 45 in degree mode
#link discussion
ACSC
Syntax
ACSC(value)
Description
Arc cosecant. The function derived from the inverse of the Cosecant function.
Example
ACSC(1) returns 90 in degree mode
#link discussion
ADDCOL
Syntax
ADDCOL(matrixname, vector, column_number)
Description
Add Column. Inserts values from vector into a column before column_number in the specified matrix. The size of vector must be the same as the number of rows in the matrix matrixname.
#link discussion
additionally
Syntax
additionally(Expr)
Description
Make an additional assumption about a variable.
Example
assume(n,integer);additionally(n>5) returns DOM_INT,n
#link discussion
ADDROW
Syntax
ADDROW(matrixname, vector, row_number)
Description
Add Row. Inserts values from vector into a row before row_number in the specified matrix. The size of vector must be the same as the number of columns in the matrix matrixname.
#link discussion
affix
Syntax
affix(Point) or affix(Vector)
Description
Returns the coordinates of a point or both the x- and y-lengths of a vector as a complex number.
Example
affix(point(3,2)) returns 3+2*i
if GA is a point at (1, -2), then affix(GA) returns 1-2*i.
#link discussion
algvar
Syntax
algvar(Expr)
Description
List of the variables by ascending algebraic extension order.
Example
algvar(√x+y) returns [[y],[x]]
#link discussion
ALOG
Syntax
ALOG(value)
Description
The common antilogarithm. This is more accurate than 10^x due to limitations of the power function.
Example
ALOG(2) returns 100
#link discussion
alog10
Syntax
alog10(Expr)
Description
Function x->10^x.
Example
alog10(3) returns 1000
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altitude
Syntax
altitude(point1, point2, point3)
Description
Given three non-collinear points, draws the altitude of the triangle defined by the three points that passes through the first point. The triangle does not have to be drawn.
Example
altitude(A, B, C) draws a line passing through point A that is perpendicular to BC.
#link discussion
AND
Syntax
Value1 AND Value2
Description
Logical AND.
Returns 1 if both value1 and value2 are non-zero; otherwise returns 0.
Example
3 AND 2 returns 1
#link discussion
angle
Syntax
angle(Vertex, Point2, Point3)
Description
Returns the measure of a directed angle. The first point is taken as the vertex of the angle as the next two points in order give the measure and orientation.
Example
angle(GA, GB, GC) returns the measure of ∡BAC
#link discussion
angleat
Syntax
angleat(Vertex, Point2, Point3, Point4)
Description
Used in Symbolic view. Given the three points of an angle and a fourth point as a location, displays the measure of the angle defined by the first three points, with a label, at the location in the Plot view given by the fourth point. The first point is the vertex of the angle.
Example
angleat(GA, GB, GC, point(0,0)) displays “aA=” at the origin, followed by the measure of ∡BAC.
#link discussion
angleatraw
Syntax
angleatraw(Pnt(A)),Pnt(B),Pnt(C),(Pnt or Cplx(z0)))
Description
angleatraw(A,B,C,z0) displays at point(z0), the value of the measure of the angle (AB,AC).
#link discussion
Ans
Syntax
ANS
Description
ANS: Last answer.
Returns the result of the last calculation made in Home view to its full precision. The variable ANS is different from the numbers in Home's history. A value in ANS is stored internally with the full precision of the calculated result, whereas the displayed numbers match the display mode.
#link discussion
append
Syntax
append((Lst||Seq|| Set,Elem)
Description
Append an element to a list.
Example
append([1,2,3],4) returns [1,2,3,4]
#link discussion
apply
Syntax
apply(Fnc(f),Lst(l))
Description
Apply the function f at the elements of the list l (option matrix for a matrix).
Example
apply(x->x^3,[1,2,3]) returns [1,8,27]
#link discussion
approx
Syntax
approx(Expr,[Int])
Description
Numerical evaluation of the first argument (we can give the number of digits as second argument).
approx(expression) works also and does the same thing.
Example
approx(2/3) returns 0.666666666667
#link discussion
ARC
Syntax
ARC(G, x, y, r, [[∠1, ∠2],[color]])
Description
Draws a circle on GROB G, centered at (x, y), with radius r. If ∠1 and ∠2 are specified, draws an arc from ∠1 to ∠2 using the current angle mode.
#link discussion
ARC_P
Syntax
ARC_P(G, x, y, r, [[∠1, ∠2],[color]])
Description
Draws a circle on GROB G, centered at (x, y), with radius r. If ∠1 and ∠2 are specified, draws an arc from ∠1 to ∠2 using the current angle mode.
#link discussion
arcLen
Syntax
arcLen(Expr, Real1, Real2)
Description
Returns the length of the arc of a curve between two points on the curve. The curve is an expression, the independent variable is declared, and the two points are defined by values of the independent variable.
This command can also accept a parametric definition of a curve. In this case, the expression is a list of 2 expressions (the first for x and the second for y) in terms of a third independent variable.
Example
arcLen(x^2, x, -2, 2) returns 9.29….
arcLen({sin(t), cos(t)}, t, 0, π/2) returns 1.57…
#link discussion
area
Syntax
area(Circle) or area(Polygon) or area(Expr, x=value1..value2)
Description
Returns the area of a circle or polygon. Can also return the area under a curve between two points.
Example
If GA is defined to be the unit circle, then area(GA) returns π.
area(4-x^2/4, x=-4..4) returns 64/3 or 21.333…
#link discussion
areaat
Syntax
areaat(Polygon, Pnt||Cplx(z0))
Description
Displays at point(z0), with a legend, algebraic area of a circle or of a (star) polygon (e.g. triangle, square, ...).
#link discussion
areaatraw
Syntax
areaatraw(Polygon, Pnt||Cplx(z0))
Description
Displays at point(z0), algebraic area of a circle or of a (star-)polygon (e.g. triangle, square, ...).
#link discussion
ARG
Syntax
ARG(x+yi)
Description
The ARG function finds the angle determined by a complex number.
Example
ARG(3+3i) returns 45 in degree mode.
#link discussion
ASC
Syntax
ASC("string")
Description
Returns a vector containing the ASCII codes of string.
Example
ASC("AB") returns [65, 66]
#link discussion
ASEC
Syntax
ASEC(value)
Description
Arc secant. The function derived from the inverse of the Secant function.
Example
ASEC(1) returns 0 in degree mode
#link discussion
ASIN
Syntax
ASIN(Value)
Description
ASIN: the inverse sine function.
This Shift-key combination returns the inverse sine of Value. The output depends on the Angle Measure setting.
Example
ASIN(1) returns 1.57079632679
#link discussion
asin2acos
Syntax
asin2acos(Expr)
Description
Replaces arcsin(x) by π/2-arccos(x) in the argument.
Example
asin2acos(acos(x)+asin(x)) returns π/2-acos(x)+acos(x)
#link discussion
asin2atan
Syntax
asin2atan(Expr)
Description
Replaces arcsin(x) by arctan(x/√(1-x^2)) in the argument.
Example
asin2atan(2*asin(x)) returns 2*atan(x/(√(1-x^2)))
#link discussion
ASINH
Syntax
ASINH(value)
Description
Inverse hyperbolic sine.
Example
ASINH(1.17520119365) returns 1
#link discussion
assume
Syntax
assume(Expr)
Description
Make an assumption on a variable.
Example
assume(a>0) returns a
#link discussion
ATAN
Syntax
ATAN(Value)
Description
ATAN: the inverse tangent function.
This Shift-key combination returns the inverse tangent of Value. The output depends on the Angle Measure setting.
Example
ATAN(0) returns 0
#link discussion
atan2acos
Syntax
atan2acos(Expr)
Description
Replaces arctan(x) by π/2-arccos(x/√(1+x^2)) in the argument.
Example
atan2acos(atan(2*x) returns π/2-acos((2*x)/√(1+(2*x)^2))
#link discussion
atan2asin
Syntax
atan2asin(Expr)
Description
Replaces arctan(x) by arcsin(x/√(1+x^2)) in the argument Expr.
Example
atan2asin(atan(y/x) returns asin((y/x)/√(1+(y/x)^2))
#link discussion
ATANH
Syntax
ATANH(value)
Description
Inverse hyperbolic tangent.
Example
ATANH(.761594155956) returns 1
#link discussion
atrig2ln
Syntax
atrig2ln(Expr)
Description
Rewrites the expression Expr containing inverse trigonometric functions with equivalent logarithmic functions.
Example
atrig2ln(atan(x)) returns (i*ln((i+x)/(i-x)))/2
#link discussion
barycenter
Syntax
barycenter([Point1, Weight1], [Point2, Weight2],…,[Pointn, Weightn])
Description
Calculates the hypothetical center of mass of a set of points, each with a given weight (a real number). Each point, weight pair is enclosed in square brackets as a vector.
Example
barycenter([–3,1],[3,1],[4,2]) returns point(2,0)
#link discussion
basis
Syntax
basis(Lst(vector1,..,vectorn))
Description
Extract a basis from a spanning set of vectors.
Example
basis([[1,2,3],[4,5,6],[7,8,9],[10,11,12]]) returns [[-3,0,3],[0,-3,-6]]
#link discussion
BEGIN
Syntax
BEGIN commands; END;
Description
Defines a set of commands to be executed in a block.
Example
SQM1
EXPORT SQM1(X)
BEGIN
RETURN X^2-1;
END;
This program defines a user function named SQM1(X). From the Home view, entering SQM1(8) returns 63.
#link discussion
Beta
Syntax
Beta(Expr,Expr)
Description
Returns Gamma(x)*Gamma(y)/Gamma(x+y).
Example
Beta(3,2) returns 1/12
#link discussion
BINOMIAL
Syntax
BINOMIAL(n, p, k)
Description
Binomial probability density function.
Computes the probability of k successes out of n trials, each with a probability of success, p.
Note that n and k are integers with k≤n.
Example
BINOMIAL(4, 0.5, 2) returns 0.375
#link discussion
BINOMIAL_CDF
Syntax
BINOMIAL_CDF(n, p, k, [k2])
Description
Cumulative binomial distribution function.
Returns the probability of k or fewer successes out of n trials, with a probability of success, p for each trial.
Note that n and k are integers with k≤n.
With the optional fourth arguments k2, returns the cumulative probability for the two k-values; that is, the probability of between k and k2 successes.
Example
BINOMIAL_CDF(20, 0.5, 6) returns 0.05765914917
BINOMIAL_CDF(20, 0.5, 6, 12) returns 0.847717285156
#link discussion
BINOMIAL_ICDF
Syntax
BINOMIAL_ICDF(n, p, q)
Description
Inverse cumulative binomial distribution function. Returns the number of successes, k, out of n trials, each with a probability of p, such that the probability of k or fewer successes is q.
Example
BINOMIAL_ICDF(4, 0.5, 0.6875) returns 2
#link discussion
bisector
Syntax
bisector(Point1, Point2, Point3)
Description
Given three points, creates the bisector of the angle defined by the three points whose vertex is at the first point. The angle does not have to be drawn in the Plot view.
Example
bisector(GA, GB, GC) draws the bisector of ∡BAC.
bisector(0,-4i,4) draws the line given by y=–x
#link discussion
BITAND
Syntax
BITAND(int1[, int2..,intn])
Description
Bitwise logical AND. Takes n integers as input and returns their bitwise logical AND.
Example
BITAND(20, 13) returns 4
#link discussion
BITNOT
Syntax
BITNOT(int)
Description
Bitwise logical NOT. Takes one integer as input and returns its bitwise not.
#link discussion
BITOR
Syntax
BITOR(int1[, int2..,intn])
Description
Bitwise logical OR. Takes n integers as input and returns their bitwise logical OR.
Example
BITOR(9, 26) returns 27
#link discussion
BITSL
Syntax
BITSL(int1[, int2])
Description
Bitwise shift left. Takes one or two integers as input and returns the result of shifting the bits in the first integer to the left by the number of places indicated by the second integer. If there is no second integer, then the bits in the first integer are shifted to the left one place.
Example
BITSL(28, 2) returns 112
BITSL(5) returns 10
#link discussion
BITSR
Syntax
BITSR(int1[, int2])
Description
Bitwise shift right. Takes one or two integers as input and returns the result of shifting the bits in the first integer to the right by the number of places indicated by the second integer. If there is no second integer, then the bits in the first integer are shifted to the right one place.
Example
BITSR(112, 2) returns 28
BITSR(10) returns 5
#link discussion
BITXOR
Syntax
BITXOR(int1[, int2..,intn])
Description
Bitwise logical exclusive OR (XOR). Takes n integers as input and returns their bitwise XOR.
Example
BITXOR(9, 26) returns 19
#link discussion
black
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
#link discussion
BLIT
Syntax
BLIT([trgtG], [dx1, dy1], [dx2, dy2], srcG, [sx1, sy1], [sx2, sy2], [c])
Description
Copies the region of graphic srcG between point (sx1, sy1) and (sx2, sy2) into the region of trgtG between points (dx1, dy1) and (dx2, dy2). Does not copy pixels from srcG that are color c.
The defaults for the optional arguments are:
trgtG=G0
srcG=G0
sx1, sy1=srcGRB top left corner
sx2, sy2=srcGRB bottom right corner
dx1, dx2=trgtGRB top left corner
dx2, dy2=calculated so destination area is the same as source area
c=all pixel colors
#link discussion
BLIT_P
Syntax
BLIT_P([trgtG], [dx1, dy1], [dx2, dy2], srcG, [sx1, sy1], [sx2, sy2], [c])
Description
Copies the region of graphic srcG between point (sx1, sy1) and (sx2, sy2) into the region of trgtG between points (dx1, dy1) and (dx2, dy2). Does not copy pixels from srcG that are color c.
The defaults for the optional arguments are:
trgtG=G0
srcG=G0
sx1, sy1=srcGRB top left corner
sx2, sy2=srcGRB bottom right corner
dx1, dx2=trgtGRB top left corner
dx2, dy2=calculated so destination area is the same as source area
c=all pixel colors
#link discussion
blue
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
#link discussion
BREAK
Syntax
BREAK [n];
Description
Exits from expression local loop structure.
Example
FOR A FROM 1 TO 10 DO
B:= (A+3) MOD 5
IF B==1 THEN BREAK;
END;
END;
If n is specified, allow to exit n loop structures.
#link discussion
canonical_form
Syntax
canonical_form(Trinom(a*x^2+b*x+c),[Var])
Description
Canonical form of a second degree polynomial.
Example
canonical_form(2*x^2-12*x+1) returns 2*(x-3)^2-17
#link discussion
CAS
Syntax
CAS(expression) or CAS.function(...) or CAS.variable[(...)]
Description
Evalute an expression or variable using the CAS.
Note that outputs in numerical mode are transformed into strings or lists of expressions for symbolic matrices.
#link discussion
CASE
Syntax
CASE IF test1 THEN commands1 END IF test2 THEN commands2 END ... IF testN THEN commandsN END [DEFAULT] [commandsD] END;
Description
Starts a "CASE...END" branch structure.
Example
Evaluates test1. If true, executes commands1 and ends the CASE. Otherwise, evaluates test2. If true, executes commands2. Continues evaluating tests until a true is found. If no true test is found, executes commandsD, if provided.
#link discussion
cat
Syntax
cat(SeqObj)
Description
Evaluates the arguments, then concatenates them into a string.
Example
cat("aaa","c",12*3) returns "aaac36"
#link discussion
CEILING
Syntax
CEILING(value)
Description
Least integer greater than or equal to value.
Example
CEILING(3.2) returns 4 and CEILING(-3.2) returns -3
#link discussion
center
Syntax
center(Circle)
Description
Returns the center of a circle
Example
center(circle(x^2+y2–x–y)) returns point(1/2,1/2)
#link discussion
cFactor
Syntax
cFactor(Expr)
Description
Factorisation of the expression in C (on the Gauss integers if there are more than 2 variables).
Example
cFactor(x^2*y+y) returns (x+i)*(x-i)*y
#link discussion
CHAR
Syntax
CHAR(list or vector) or CHAR(integer)
Description
Returns the string corresponding to the ASCII character codes in vector, or the single character associated with integer.
Example
CHAR(65) returns "A" and CHAR({82, 77, 72}) returns "RMH"
#link discussion
charpoly
Syntax
charpoly(Mtrx,[Var])
Description
List of the coefficients of the characteristic polynomial of a matrix or characteristic polynomial of a matrix with the second argument as variable.
Example
charpoly([[1,2],[3,4]]) returns poly1[1,-5,-2]
#link discussion
CHECK
Syntax
CHECK(n)
Description
Checks (selects) the corresponding symbolic definition field in the current app. The integer n must be between 0 and 9 for most apps. For Statistics 1-Var and Statistics 2-Var apps, n must be between 1 and 5.
Example
CHECK(3) would check F3 if the current app is Function. Then a checkmark would appear next to F3 in Symbolic view, F3 would be plotted in Plot view, and evaluated in Numeric view
#link discussion
chinrem
Syntax
chinrem([Lst||Expr,Lst||Expr],[Lst||Expr,Lst||Expr])
Description
Chinese remainder for polynomials written as matrices.
Example
chinrem([[1,2,0],[1,0,1]],[[1,1,0],[1,1,1]]) returns [[2,2,1] [1,1,2,1,1]].
#link discussion
CHISQUARE
Syntax
CHISQUARE(n, x)
Description
Chi-square probability density function. Computes the probability density of the Chi-squared distribution at x, given n degrees of freedom.
Example
CHISQUARE(2, 3.2) returns 0.100948258997
#link discussion
CHISQUARE_CDF
Syntax
CHISQUARE_CDF(n, k)
Description
Cumulative χ² (Chi-squared) distribution function. Returns the lower-tail probability of the χ² probability density function for the value x, given n degrees of freedom.
Example
CHISQUARE_CDF(2, 6.1) returns 0.952641075609
#link discussion
CHISQUARE_ICDF
Syntax
CHISQUARE_ICDF(n, p)
Description
Inverse cumulative χ² (Chi-squared) distribution function. Returns the value x such that the χ² lower-tail probability of x, with n degrees of freedom, is p.
Example
CHISQUARE_ICDF(2, 0.952641075609) returns 6.1
#link discussion
cholesky
Syntax
cholesky(Mtrx)
Description
For a numerical symmetric matrix A, returns L matrix such that A=L*tran(L).
Example
cholesky([[3,1],[1,4]]) returns [[3*√3/3,0],[√3/3,11/3*√33/11]]
#link discussion
CHOOSE
Syntax
CHOOSE(var, “title”, “item1”, “item2”,[…"item14"]) or CHOOSE(var,"title",{"item1"..."itemN")
Description
Displays a choose box with the given title and containing items with the strings "item1", etc. If the user choose an object, var will be updated to contain the number of the selected object (an integer, 1, 2, 3, …); otherwise, stores zero in var if the user exits without choosing.
Returns true (non zero) if the user selects an object, otherwise return false (0).
#link discussion
chrem
Syntax
chrem(LstIntg(a,b,c....),LstIntg(p,q,r,....))
Description
Chinese remainders for integers.
Example
chrem([2,3],[7,5]) returns [-12,35]
#link discussion
Ci
Syntax
Ci(Expr)
Description
Cosine integral int(cos(t)/t,t=-∞..x).
Example
Ci(1.0) returns 0.337403922901
#link discussion
circumcircle
Syntax
circumcircle(Point1, Point2, Point3)
Description
Draws the circumcircle of a triangle; that is, the circle circumscribed about a triangle.
Example
circumcircle(GA, GB, GC) draws the circle circumscribed about ΔABC
#link discussion
coeff
Syntax
coeff(Expr,[Var], [Term])
Description
Returns the list of coefficients of a polynomial with respect to the second argument or the coefficient of the term whose degree is Term.
Example
coeff(x^3+2) returns [1,0,0,2]
coeff(2*y^2-3,y,0) returns -3
#link discussion
col
Syntax
col(Mtrx(A),Intg(n)||Interval(n1..n2))
Description
Returns the column n or the sequence of the columns n1...n2 of the matrix A, or optional argument of count,count_eq,count_inf,count_sup.
Example
col([[1,2,3],[4,5,6],[7,8,9]],1) returns [2,5,8]
#link discussion
colDim
Syntax
coldim(Mtrx)
Description
Number of columns of a matrix.
Example
coldim([[1,2,3],[4,5,6]]) returns 3
#link discussion
collect
Syntax
collect(Expr or {Expr1, Expr2,...,Exprn})
Description
Collects likes terms in a polynomial expression (or of a list of polynomial expressions).
Example
collect(x+2*x+1-4) returns 3*x-3
#link discussion
COLNORM
Syntax
COLNORM(matrix)
Description
Column Norm. Finds the maximum value (over all columns) of the sums of the absolute values of all elements
Example
COLNORM([[1,2],[3,4]]) returns 6
#link discussion
COMB
Syntax
COMB(n, r)
Description
Combinations. Returns the number of combinations (without regard to order) of n things taken r at a time: n!/(r!(n-r))
Example
COMB(5,2) returns 10
#link discussion
comDenom
Syntax
comDenom(Expr,[Var(var)])
Description
Returns the expression after reduction at the same denominator: the numerator and the denominator are developed [according to the powers of the variable var].
Example
comDenom(1/x+1/y^2+1) returns (x*y^2+x+y^2)/(x*y^2)
#link discussion
common_perpendicular
Syntax
common_perpendicular(Line(D1),Line(D2))
Description
Draws the common perpendicular of the lines D1 and D2.
#link discussion
companion
Syntax
companion(Poly,Var)
Description
Companion matrix of a polynomial (an=1).
Example
companion(x^2+5x-7,x) returns [[0,7],[1,-5]]
#link discussion
compare
Syntax
compare(Obj(arg1),Obj(arg2))
Description
Returns 1 if type(arg1)<type(arg2) or if type(arg1)=type(arg2) and arg1<arg2, else returns 0.
Example
compare(1,2) returns 1
#link discussion
complexroot
Syntax
complexroot(Poly(P),Real(l),[Cplx(a)],[Cplx(b)])
Description
Returns the list of the vertices of the squares (side<=l) containing roots of P [inside the rectangle with opposed vertices a and b] with their mulitiplicity.
Example
complexroot(x^5-2*x^4+x^3+i,0.1) returns [[[(-21-12*i)/32,(-18-9*i)/32],1],[[(6-15*i)/16,(-6-21*i)/(16-16*i)],1],[[(27+18*i)/(16+16*i),(24-3*i)/16],1],[[(6+27*i)/(16+16*i),(9+6*i)/8],1],[[(-15+6*i)/(16+16*i),(-3+12*i)/16],1]]
#link discussion
CONCAT
Syntax
CONCAT(value1, value2, [..value16])
Description
Concatenation. Concatenates (joins) items into a list.
Example
CONCAT({1,2,3}, 4) returns {1,2,3,4} and CONCAT(1,2,3,4) returns {1,2,3,4}
#link discussion
COND
Syntax
COND(matrix)
Description
Condition Number. Finds the 1-norm (column norm) of a square matrix.
Example
COND([[1,2],[3,4]]) returns 21
#link discussion
conic
Syntax
conic(Expr)
Description
Plots the graph of a conic section defined by an expression in x and y.
Example
conic(x^2+y^2-81) draws a circle with center at (0,0) and radius of 9
#link discussion
CONJ
Syntax
CONJ(x+yi)
Description
Complex Conjugate. Reverses the sign of the imaginary part of a complex number.
Example
CONJ(3+4i) returns 3-4i
#link discussion
contains
Syntax
contains((Lst(l) or Set(l)),Elem(e))
Description
Tests if a set contains an expression (returns the index+1 or 0).
Example
contains(%{0,1,2,3%},2) returns 3
#link discussion
content
Syntax
content(Poly,[Var])
Description
Returns the gcd of the coefficients of the polynomial Poly.
Example
content(2*x^2+10*x+6) returns 2
#link discussion
CONVERT
Syntax
CONVERT(Value_Unit1, 1_Unit2)
Description
Converts Value in Unit1 to the corresponding value in compatible Unit2.
Example
CONVERT(20_m, 1_ft) returns 65.6167979003_ft
Alternative: 20_m ► _ft
#link discussion
convexhull
Syntax
convexhull(Lst)
Description
Convex hull of a list of 2D points.
Example
convexhull(0,1,1+i,1+2i,-1-i,1-3i,-2+i) returns 1-3*i,1+2*i,-2+i,-1-i
#link discussion
coordinates
Syntax
coordinates(Pnt or Cplx or Vect)
Description
Returns the list (resp matrix) of the abscissa and of the ordinate of a point or a vector (resp of points or vectors).
Example
coordinates(point(1+2*i)) returns [1,2]
#link discussion
CopyVar
Syntax
CopyVar(Var(var1),Var(var2))
Description
Copy the storage without evaluation of var1 into var2.
#link discussion
correlation
Syntax
correlation(Lst||Mtrx,[Lst])
Description
Returns the correlation of the elements of its argument.
Example
correlation([[1,2],[1,1],[4,7]]) returns 33/(6*√31)
#link discussion
COS
Syntax
COS(Value)
Description
Returns the cosine of Value. Value is interpreted as either degrees or radians, depending on the setting of Angle Measure in Home Modes or Symbolic Setup.
Example
in radian mode, COS(π) returns -1.
#link discussion
cos2sintan
Syntax
cos2sintan(Expr)
Description
Replaces cos(x) by sin(x)/tan(x) in the argument.
Example
cos2sintan(cos(x)) returns sin(x)/tan(x)
#link discussion
COSH
Syntax
COSH(value)
Description
Hyperbolic cosine.
Example
ASINH(1.17520119365) returns 1
#link discussion
COT
Syntax
COT(value)
Description
Cotangent. The Cotangent function; that is, cos(x)/sin(x).
Example
COT(45) returns 1 in degree mode
#link discussion
count
Syntax
count(Fnc(f),(Lst||Mtrx)(l),[Opt(row||col)])
Description
Returns f(l[0])+f(l[1])+...+f(l[size(l)-1]).
Example
count((x)->x,[2,12,45,3,7,78]) returns 147
#link discussion
covariance
Syntax
covariance(Lst||Mtrx,[Lst])
Description
Returns the covariance of the elements of its argument.
Example
covariance([[1,2],[1,1],[4,7]]) returns 11/3
#link discussion
covariance_correlation
Syntax
covariance_correlation(Lst||Mtrx,[Lst])
Description
Returns the list of the covariance and the correlation of the elements of its argument.
Example
covariance_correlation([[1,2],[1,1],[4,7]]) returns [11/3,33/(6*√31)]
#link discussion
cpartfrac
Syntax
cpartfrac(RatFrac)
Description
Performs partial fraction decomposition in C of a fraction.
Example
cpartfrac((x)/(4-x^2)) returns 1/((x-2)*-2)+1/((x+2)*-2)
#link discussion
crationalroot
Syntax
crationalroot(Poly(P))
Description
Returns the list of complex rational roots of P without indicating the multiplicity.
Example
crationalroot(2*x^3+(-5-7*i)*x^2+(-4+14*i)*x+8-4*i) returns [(3+i)/2,2*i,1+i]
#link discussion
CROSS
Syntax
CROSS(vector1, vector2)
Description
Cross Product. Finds the cross product of vector1 with vector2.
Example
CROSS([1,2],[3,4]) returns [0, 0, -2]
#link discussion
CSC
Syntax
CSC(value)
Description
Cosecant. The Cosecant function; that is, 1/sin(x)
Example
CSC(90) returns 0 in degree mode
#link discussion
cSolve
Syntax
csolve(Eq,Var)
Description
Returns the solutions, including comlex solutions, of Eq, for Var. If Eq is an expression, solves Eq=0.
Example
csolve(x^4=1,x) returns {1,-1,-i,i}
#link discussion
cumSum
Syntax
cumSum(Lst(l)||Seq||Str)
Description
Returns the list (or the sequence or the string) lr where the elements are the cumulative sum of the list l:lr[k]=sum(l[j],j=0..k) (or lr=sum(l[j],j=0..k )$(k=0..size(l)-1)).
Example
cumSum([0,1,2,3,4]) returns [0,1,3,6,10]
#link discussion
curl
Syntax
curl(Lst(A,B,C),Lst(x,y,z))
Description
Returns the curl of a vector. curl([A,B,C],[x,y,z])=[dC/dy-dB/dz,dA/dz-dC/dx,dB/dx-dA/dy].
Example
curl([2*x*y,x*z,y*z],[x,y,z]) returns [z-x,0,z-2*x]
#link discussion
cyan
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
#link discussion
cyclotomic
Syntax
cyclotomic(Expr)
Description
Generates a vector representing the nth cyclotomic polynomial.
Example
cyclotomic(20) returns [1,0,-1,0,1,0,-1,0,1]
#link discussion
cZeros
Syntax
cZeros(Expr,[Var]) or cZeros(ListExpr, ListVar)
Description
Returns the roots, including complex roots, of Expr (that is, the solution of Xpr=0) or the matrix where the lines are the solutions of the system : Expr1=0,Expr2=0....
Example
cZeros(x^4-1) returns [1,-1, i, -i]
#link discussion
C→PX
Syntax
C→PX(x, y) or C→PX({x, y})
Description
Transform cartesian coordinates into pixel coordinates. Returns a list.
#link discussion
degree
Syntax
degree(Poly)
Description
Returns the degree of the polynomial Poly.
Example
degree(x^4+x) returns 3
#link discussion
DELCOL
Syntax
DELCOL(matrixname ,column_number)
Description
Delete Column. Deletes the column column_number from the matrix matrixname.
#link discussion
delcols
Syntax
delcols(Mtrx(A),Interval(n1..n2)||n1)
Description
Returns the matrix where the columns n1..n2 (or n1) of the matrix A are deleted.
Example
delcols([[1,2,3],[4,5,6],[7,8,9]],1..1) returns [[1,3],[4,6],[7,9]]
#link discussion
DELROW
Syntax
DELROW(matrixname, row_number)
Description
Delete Row. Deletes the row row_number from the matrix matrixname.
#link discussion
delrows
Syntax
delrows(Mtrx(A),Interval(n1..n2)||n1)
Description
Returns the matrix where the rows n1..n2 (or n1) of the matrix A are deleted.
Example
delrows([[1,2,3],[4,5,6],[7,8,9]],1..1) returns [[1,2,3],[7,8,9]]
#link discussion
deltalist
Syntax
deltalist(Lst)
Description
Returns the list of the difference of two terms in succession.
Example
deltalist([1,4,8,9]) returns [3,4,1]
#link discussion
denom
Syntax
denom(a/b)
Description
Simplified Denominator. For the integers a and b, returns the denominator of the fraction a/b after simplification.
Example
denom(10/12) returns 6
#link discussion
desolve
Syntax
desolve(Eq,[TimeVar],Var)
Description
Solves a differential equation.
Example
desolve(y''+y=0,y) returns G_0*cos(x)+G_1*sin(x)
#link discussion
DET
Syntax
DET(matrix)
Description
Determinant of a square matrix.
Example
DET([[1,2],[3,4]]) returns -2
#link discussion
diag
Syntax
diag(Lst(l)||Mtrx(A))
Description
Returns either the diagonal matrix with diagonal l or the diagonal of A.
Example
diag([1,2],[3,4]) returns [1,4]
#link discussion
diff
Syntax
diff(Expr,[Var or ListVar])
Description
Returns the derivative of an expression with respect to a given variable. You can use the differentiation template in the Template menu as well.
Example
diff(x^3-x) returns 3*x^2-1
diff(sin(x)-cos(y), x) returns cos(x)
diff(sin(x)-cos(y), y) returns sin(y)
#link discussion
DIFFERENCE
Syntax
DIFFERENCE({list1}, ...{listN})
Description
Returns a list of the elements that are not common between 2 or more of the lists.
Example
DIFFERENCE({1,2,3},{2,4,8}) returns {1,3,4,8}
#link discussion
DIM
Syntax
DIM(string)
Description
Returns the number of characters in string.
Example
DIM("12345") returns 5
#link discussion
DIMGROB
Syntax
DIMGROB(G, w, h, [color]) or DIMGROB(G, w, h, list)
Description
Sets the dimensions of GROB G to w*h. Initializes the graphic G with color or with the graphic data provided in list. If the graphic is initialized using graphic data, then list is a list of integers. Each integer, as seen in base 16, describes one color every 16 bits.
Colors are in A1R5G5B5 format (ie, 1 bit for alpha channel, and 5 bits for R, G and B).
#link discussion
DIMGROB_P
Syntax
DIMGROB_P(G, w, h, [color]) or DIMGROB(G, list)
Description
Sets the dimensions of GROB G to w*h. Initializes the graphic G with color or with the graphic data provided in list. If the graphic is initialized using graphic data, then list is a list of integers. Each integer, as seen in base 16, describes one color every 16 bits.
Colors are in A1R5G5B5 format (ie, 1 bit for alpha channel, and 5 bits for R, G and B).
#link discussion
Dirac
Syntax
Dirac(Real)
Description
Function derivative of Heaviside.
Example
Dirac(1) returns 0
#link discussion
distance
Syntax
distance((Pnt or Cplx),(Pnt or Cplx or Curve))
Description
Calculates the distance between 2 points, or a point and a curve.
Example
distance(0,1+i) returns √2
#link discussion
distance2
Syntax
distance2(point1, point2) or distance2(point, curve)
Description
Returns the square of the distance between two points or between a point and a curve.
Example
distance2(1+i, 3+3i) returns 8.
if GA is the point at (0, 0) and GB is defined as plotfunc(4-x^2/4), then distance (GA, GB) returns 12.
#link discussion
distanceat
Syntax
distanceat(GeoObj(A),GeoObj(B),(Pnt or Cplx))
Description
distanceat(A,B,z0) displays at point(z0), with a legend, the distance between 2 geometrical objects.
Example
A:=point(0);B:=point(1+i);distanceat(A,B,(1+i)/2)) returns √2
#link discussion
distanceatraw
Syntax
distanceatraw(Point1, Point2, Point3) or distanceatraw(Point1, Curve, Point3)
Description
This command is used in Symbolic view. Similar to distanceat(), this commmand returns the distance between two points or between a point and a curve and places that measurement at the location of Point3 in the Plot view. The distance is unlabeled.
Example
distanceatraw(1+I, 3+3i, point(0,0)) returns 2.828…or 2√2 and places that measure at the origin in Plot view.
If GA is the point at (0, 0) and GB is defined as plotfunc(4-x^2/4), then distanceat(GA, GB, GA) returns 3.464… or 2√3 and places this measure in Plot view at (0,0).
Define A:=point(0) and B:=point(1+i); then distanceatraw(A,B,(1+i)/2)) returns √2 and places this measurement at (1/2, 1/2)
#link discussion
divergence
Syntax
divergence(Lst(A,B,C),Lst(x,y,z))
Description
Returns the divergence of a vector. divergence([A,B,C],[x,y,z])=dA/dx+dB/dy+dC/dz.
Example
divergence([x^2+y,x+z+y,z^3+x^2],[x,y,z]) returns 2*x+3*z^2+1
#link discussion
divis
Syntax
divis(Poly(P) or LstPoly)
Description
Returns the list of divisors of a polynomial.
Example
divis(x^2-1) returns [1,x-1,x+1,(x-1)*(x+1)]
#link discussion
division_point
Syntax
division_point(Point1, Point2, Realk) or division_point(Cplx1, Cplx2, Cplxk)
Description
For two points A and B, and a numerical factor k, returns a point C such that C-B=k*(C-A). The two points may be referenced by name or represented by a complex number.
Example
division_point(0,6+6*i,4) returns point (8,8)
#link discussion
divpc
Syntax
divpc(Poly1,Poly2,Integer)
Description
Returns the n-degree Taylor polynomial for the quotient of 2 polynomials.
Example
divpc(x^4+x+2,x^2+1,5) returns the 5th-degree polynomial x^5+3*x^4-x^3-2*x^2+x+2
#link discussion
DO
Syntax
FOR var FROM start TO (or DOWNTO) finish [STEP increment] DO command(s) END;
Description
Sets variable var to start; then, for as long as this variable’s value is less than or equal to (or more than for a DOWNTO) finish, executes command(s) and adds (or substract for DOWNTO) 1 (or increment) to var.
Example
FOR A FROM 1 TO 10 STEP 2
DO
PRINT(A);
END;
will print 1 3 5 7 9
#link discussion
DOT
Syntax
DOT(matrix1, matrix2)
Description
Dot Product. Finds the dot product of two arrays, matrix1 and matrix2.
Example
DOT([1,2],[3,4]) returns 11
#link discussion
DRAWMENU
Syntax
DRAWMENU({text...}) or DRAWMENU(text..)
Description
Draw a menu containing the items specified
#link discussion
DrawSlp
Syntax
DrawSlp(Reala, Realb, Realm)
Description
Given three real numbers m, a, b, draws a line with slope m that passes through the point (a, b).
Example
DrawSlp(2,1,3) draws the line given by y=3x–5
#link discussion
e
Syntax
e
Description
Natural logarithm base, internally represented as 2.71828182846
#link discussion
EDITLIST
Syntax
EDITLIST(listname)
Description
Starts the List Editor and displays the specified list. If used in programming, returns to the program when user presses OK (menu key).
Example
EDITLIST(L1) edits list L1.
#link discussion
EDITMAT
Syntax
EDITMAT(matrixname)
Description
Starts the Matrix Editor and displays the specified matrix. If used in programming, returns to the program when user presses OK (menu key).
Example
EDITMAT(M1) edits matrix M1.
#link discussion
egcd
Syntax
egcd((Poly or Lst),(Poly or Lst),[Var])
Description
Returns the extended greatest common divisor of 2 polynomials.
Example
egcd((x-1)^2,x^3-1) returns [-x-2,1,3*x-3]
#link discussion
Ei
Syntax
Ei(Expr)
Description
Exponential integral int(exp(t)/t,t=-∞..x)
Example
Ei(1.0) returns 1.89511781636
#link discussion
EIGENVAL
Syntax
EIGENVAL(matrix)
Description
Displays the eigenvalues in vector form for matrix.
Example
EIGENVAL([[1,2],[3,4]]) returns [5.37228132327 -.372281323269]
#link discussion
eigenvals
Syntax
eigenvals(Mtrx)
Description
Returns the sequence of the (calculable) eigenvalues of a matrix.
Example
eigenvals([[-2,-2,1],[-2,1,-2],[1,-2,-2]]) returns 3,-3,-3
#link discussion
eigenvects
Syntax
eigenvects(Mtrx)
Description
Computes the eigenvectors of a diagonalizable matrix.
Example
eigenvects([[-2,-2,1],[-2,1,-2],[1,-2,-2]]) returns [[1,-3,-3],[-2,0,-3],[1,3,-3]]
#link discussion
EIGENVV
Syntax
EIGENVV(matrix)
Description
Eigenvectors and Eigenvalues for a square matrix. Displays a list of two arrays. The first contains the eigenvectors and the second contains the eigenvalues.
Example
EIGENVV([[1,2],[3,4]]) returns { [[eigenvectors]],[[eigenvalues]] }
#link discussion
eigVc
Syntax
eigVc(Mtrx)
Description
Computes the eigenvectors of a diagonalizable matrix.
Example
eigVc([[-2,-2,1],[-2,1,-2],[1,-2,-2]]) returns [[1,-3,-3],[-2,0,-3],[1,3,-3]]
#link discussion
eigVl
Syntax
eigVl(Mtrx(A))
Description
Returns the Jordan matrix associated to A when the eigenvalues are calculable.
Example
eigVl([[4,1],[-4,0]]) returns [[2,1],[0,2]]
#link discussion
element
Syntax
element(object, real) or element(real1..real2)
Description
Creates a point on a geometric object whose abscissa is a given value or creates a real value on a given interval.
Example
element(plotfunc(x^2),–2) creates a point on the graph of y = x^2. Initially, this point will appear at (–2,4). You can move the point, but it will always remain on the graph of its function.
element(0..5) creates a value of 2.5 initially. Tapping on this value and pressing Enter enables you to press a cursor key to increase or decrease the value in a manner similar to a slider bar. Press Enter again to close the slider bar. The value you set can be used as a coefficient in a function you subsequently plot.
#link discussion
ellipse
Syntax
ellipse(Point1, Point2, Point3) or ellipse(Point1, Point2, Realk)
Description
Draws an ellipse, given the foci and either a point on the ellipse or a scalar that is one half the constant sum of the distances from a point on the ellipse to each of the foci.
Example
ellipse(GA, GB, GC) draws the ellipse whose foci are points A and B and which passes through point C.
ellipse(GA, GB, 3) draws an ellipse whose foci are points A and B. For any point P on the ellipse, AP+BP=6.
#link discussion
ELSE
Syntax
IF test THEN command(s) [ELSE commands] END;
Description
Evaluates test. If test is true (non 0), executes command(s); otherwise, executes the comands in the ELSE clause nothing happens.
Example
IF A<1
THEN PRINT("A IS SMALLER THAN 1");
ELSE PRINT("A IS LARGER THAN 1");
END;
#link discussion
equation
Syntax
equation(curve) or equation(point)
Description
Returns the Cartesian equation of a curve in x and y, or the Cartesian coordinates of a point.
Example
equation(line(1-i,i)) returns y=-2*x+1
If GA is the point at (0, 0), GB is the point at (1, 0), and GC is defined as circle(GA, GB-GA), then equation(GC) returns x^2 + y^2 =1.
#link discussion
equilateral_triangle
Syntax
equilateral_triangle(Point1, Point2, [Var])
Description
Draws an equilateral triangle defined by one of its sides; that is, by two consecutive vertices. The third point is calculated automatically, but is not defined symbolically. If a lowercase variable is added as a third argument, then the third point is labeled with the variable name and the coordinates of the third point are stored in that variable. The orientation of the triangle is counterclockwise from the first point.
Example
equilateral_triangle(point(0,0), point(1,0)) draws the equilateral trangle through the points at (0,0), (1,0), and (1/2, √3/2).
#link discussion
erf
Syntax
erf(Real(x0))
Description
Returns the approximate value of 2/√π*int(exp(-t^2),t,0,x0)
Example
erf(1) returns 0.84270079295
#link discussion
erfc
Syntax
erfc(Real(x0))
Description
Returns the approximate value of 2/√π*int(exp(-t^2),t,x0,∞).
Example
erfc(1) returns 0.15729920705
#link discussion
euler
Syntax
euler(x);
Description
Euler’s phi (or totient) function. Takes a positive integer x and returns the number of positive integers less than or equal to x that are coprime to x.
Example
euler(6) returns 2
#link discussion
EVAL
Syntax
EVAL(expression)
Description
Evaluates the expression. Usefull in programs where parameters are passed non evaluated with QUOTE
#link discussion
evalc
Syntax
evalc(Expr)
Description
Returns a complex expression simplified with the format real+i*imag
Example
evalc(1/(x+y*i)) returns x/(x^2+y^2)+(i)*(-y)/(x^2+y^2)
#link discussion
evalf
Syntax
evalf(Expr,[Int])
Description
Numerical evaluation of the first argument (we can give the number of digits as second argument).
approx(expression) works also and does the same thing.
Example
evalf(2/3) returns 0.666666666667
#link discussion
even
Syntax
even(Intg(n))
Description
Returns 1 if the integer is even, else returns 0.
Example
even(6) returns 1
#link discussion
exact
Syntax
exact(Expr)
Description
Converts the expression to a rational or real expression.
Example
exact(1.4141) returns 14141/10000
#link discussion
exbisector
Syntax
exbisector(Point1, Point2, Point3)
Description
Given three points that define a triangle, creates the bisector of the exterior angles of the triangle whose common vertex is at the first point. The triangle does not have to be drawn in the Plot view.
Example
exbisector(GA, GB, GC) draws the bisector of the exterior angles of ΔABC whose common vertex is at point A.
exbisector(0,–4i,4) draws the line given by y=x
#link discussion
excircle
Syntax
excircle(Point1, Point2, Point3)
Description
excircle(A,B,C) draws the A-excircle of the ABC triangle.
Draws one of the excircles of a triangle, a circle tangent to one side of the triangle and also tangent to the extensions of the other two sides.
Example
excircle(GA, GB, GC) draws the circle tangent to BC and to the rays AB and AC.
#link discussion
EXECON
Syntax
EXECON("expression with &", lists or matrices)
Description
Returns a matrix or list composed of the result of the evaluation of the expression after replacement of & by each item in the input.
Example
EXECON("&1+1", {1,2,3}) returns {2,3,4}
If EXECON has only 1 list or matrix input, using & followed by a number A (between 1 and 9) will replace &A by the element i+A-1 of the input.
Example: EXECON("&2-&1", { 1, 4, 3, 5}") returns {3, -1, 2} - the difference between 2 successive elements.
If EXECON has 2 or more lists or matrices input, using & followed by a number A (between 1 and 9) will replace &1 by the element from the Ath input.
Example: EXECON("&1+&2", {1,2,3},{4,5,6}) returns {5,7,9}
If EXECON has 2 or more lists or matrices as input, using & followed by 2 numbers A and B (between 1 and 9) will reaplace &AB by the element i+B-1 of the Ath input.
Example: EXECON("&22-&1", {1,2,3},{4,5,6,7}) returns {4,4,4}
Note that for matrix input, the elements are treated as if the matrix was a vector.
#link discussion
EXP
Syntax
EXP(value)
Description
The natural exponential. This is more accurate than e^x due to limitations of the power function.
#link discussion
exp2pow
Syntax
exp2pow(Expr)
Description
Transforms an expression of the form exp(n*ln(x)) to x^n.
Example
exp2pow(exp(3*ln(x))) returns x^3
#link discussion
exp2trig
Syntax
exp2trig(Expr)
Description
Transforms the complex exponential into sine and cosine.
Example
exp2trig(exp-(i*x)) returns cos(x)-i*sin(x)
#link discussion
expand
Syntax
expand(Expr )
Description
Full distribution of multiplication and division over addition and subtraction.
Example
expand((x+y)*(z+1)) returns y*z+x*z+y+x
#link discussion
expexpand
Syntax
expexpand(Expr)
Description
Expands exponentials usinng the identity exp(a*f(x))=(exp(f(x)))^a.
Example
expexpand(exp(3*x)) returns exp(x)^3
#link discussion
EXPM1
Syntax
EXPM1(value)
Description
Exponent minus 1. This is more accurate than EXP when x is close to zero.
Example
EXPM1(.23) returns .258600009929
#link discussion
exponential_regression
Syntax
exponential_regression(Lst||Mtrx(A),[Lst])
Description
Returns the coefficients (a,b) of y=b*a^x : it is the best exponential that approximates the points where the coordinates are the rows of A (or the 2 lists).
Example
exponential_regression([[1.0,2.0],[0.0,1.0],[4.0,7.0]]) returns 1.60092225473,1.10008339351
#link discussion
EXPORT
Syntax
Variable declaration: EXPORT var_1[:=value][, more variables];
forward function declaration: EXPORT function(params);
Normal function declaration: or EXPORT function[(params)]
BEGIN
END;
Description
In a program, declares a list of exported variable or an exported function.
#link discussion
EXPR
Syntax
EXPR(string)
Description
Parses string into a number or expression.
Example
EXPR("2+3") returns 5
#link discussion
extract_measure
Syntax
extract_measure(Var)
Description
Returns the definition of a geometric object. For a point, that definition consists of the coordinates of the point. For other objects, the definition mirrors their definition in Symbolic view, with the coordinates of their defining points supplied.
Example
extract_measure(angleatraw(0,1,1+i,1)
extract_measure(distanceatraw(0,1+i,(1+i)/2)) returns √2
#link discussion
ezgcd
Syntax
ezgcd(Poly,Poly)
Description
Returns the GCD of 2 polynomials with at least 2 variables, with the ezgcd algorithm.
Example
ezgcd(x^2-+3*x-xy-3*y,x^2-y^2) returns x-y
#link discussion
f2nd
Syntax
f2nd(Frac or RatFrac)
Description
Returns the list built with the numerator and the denominator of the simplified fraction.
Example
f2nd(42/12) returns [7,2]
#link discussion
factor
Syntax
factor(Expr)
Description
Factorizes a polynomial.
Example
factor(x^4-1) returns (x-1)*(x+1)*(x^2+1)
#link discussion
factor_xn
Syntax
factor_xn(Poly)
Description
Factorizes x^n in P\the polynomial Poly (n=degree of polynomial P).
Example
factor_xn(x^4-1) returns x^4*(1-x^-4)
#link discussion
factorial
Syntax
factorial(Intg(n)|| Real(a) )
Description
factorial(n)=n!. For non-integers, factorial(a)=a! = G(a + 1). This calculates the Gamma function.
Example
factorial(4) returns 24
#link discussion
factors
Syntax
factors(Poly) or factors({Poly1, Poly2, ..., Polyn})
Description
Returns the list of prime factors of a polynomial; each factor followed by its multiplicity.
Example
factors(x^4-1) returns [x-1,1,x+1,1,x^2+1,1]
#link discussion
fcoeff
Syntax
fcoeff(Root1, Oder1, Root2, Order2, ..., Rootn, Ordern)
Description
Returns the polynomial described by a list of roots, each followed by its order.
Example
fcoeff([1,2,0,1,3,-1]) returns ((x-1)^2)*x*(x-3)^-1
#link discussion
fft
Syntax
fft(Vect or (Vect(L),Intg(a),Intg(p))
Description
Fast Fourier Transform in R or in the field Z/pZ, with a as primitive n-th root of 1 (n=size(L)).
Example
fft([1,2,3,4,0,0,0,0]) returns [10.0,-0.414213562373-7.24264068712*(i),-2.0+2.0*i,2.41421356237-1.24264068712*i,-2.0,2.41421356237+1.24264068712*i,-2.0-2.0*i]
#link discussion
FILLPOLY
Syntax
FILLPOLY([G], {coordinates...} or [Coordinates], Color, [Alpha])
Description
Fills the polygon specified by the provided Cartésian coordinates using the color provided.
If Alpha (0 to 255) is provided, the polygon is drawn with trensparency.
Example
FILLPOLY([(0,0),(1,1),(2,0),(3,-1),(2,-2)], #FF, 128)
#link discussion
FILLPOLY_P
Syntax
FILLPOLY_P([G], {coordinates...} or [Coordinates], Color, [Alpha])
Description
Fills the polygon specified by the provided pixel coordinates using the color provided.
If Alpha (0 to 255) is provided, the polygon is drawn with trensparency.
Example
FILLPOLY_P([(20,20),(120,120),(150,20),(180,150),(50,100)], #FF, 128)
#link discussion
FISHER
Syntax
FISHER(n, d, x)
Description
F (Fisher or Fisher-Snedecor) probability density function. Computes the probability density at the value x, given numerator n and denominator d degrees of freedom.
Example
FISHER(5, 5, 2) returns 0.158080231095
#link discussion
FISHER_CDF
Syntax
FISHER_CDF(n, d, x)
Description
Cumulative F (Fisher or Fisher-Snedecor) distribution function. Returns the lower-tail probability of the F probability density function for the value x, given numerator n and denominator d degrees of freedom.
Example
FISHER_CDF(5, 5, 2) returns 0.76748868087
#link discussion
FISHER_ICDF
Syntax
FISHER_ICDF(n, d, p)
Description
Inverse cumulative F (Fisher or Fisher-Snedecor) distribution function. Returns the value x such that the F lower-tail probability of x, with numerator, n and denominator, d degrees of freedom, is p.
Example
FISHER_ICDF(5, 5, 0.76748868087) returns 2
#link discussion
FLOOR
Syntax
FLOOR(value)
Description
Greatest integer less than or equal to value.
Example
FLOOR(-3.2) returns -4
#link discussion
fMax
Syntax
fMax(Expr,[Var])
Description
Returns the abscissa of the maximum of the expression.
Example
fMax(-x^2+2*x+1,x) returns 1
#link discussion
fMin
Syntax
fMin(Expr,[Var])
Description
Returns the abscissa of the minimum of the expression.
Example
fMin(x^2-2*x+1,x) returns 1
#link discussion
FNROOT
Syntax
FNROOT(expression, variable, [guess], [guess2])
Description
Function root-finder (like the Solve app). Finds the value for variable at which expression most nearly evaluates to zero. Uses guess as initial estimate.
Example
FNROOT(M*9.8/600-1, M, 1) returns 61.2244897959
#link discussion
FOR
Syntax
FOR var FROM start TO (or DOWNTO) finish [STEP increment] DO command(s) END;
Description
Sets variable var to start; then, for as long as this variable’s value is less than or equal to (or more than for a DOWNTO) finish, executes command(s) and adds (or substract for DOWNTO) 1 (or increment) to var.
Example
FOR A FROM 1 TO 10 STEP 2
DO
PRINT(A);
END;
will print 1 3 5 7 9
#link discussion
format
Syntax
format(Real,Str("f4"||"s5"||"e6"))
Description
Transforms the real into a string with the indicated format (f=float,s=scientific,e=engineering).
Example
format(9.3456,"s3") returns 9.35
#link discussion
FP
Syntax
FP(value)
Description
Returns the Fractional part of value.
Example
FP (23.2) returns .2
#link discussion
fracmod
Syntax
fracmod(Expr(Xpr),Intg(n))
Description
Returns the fraction a/b such as a/b=Xpr mod n, -√n/2<a<=√n/2 and 0<=b<√n/2
Example
fracmod(41,121) returns 2/3
#link discussion
FREEZE
Syntax
FREEZE
Description
Prevents the screen from being redrawn after the program ends. Leaves the modified display on the screen for the user to see.
#link discussion
froot
Syntax
froot(RatPoly(F))
Description
Returns the list of roots and poles of F with their mulitiplicity.
Example
froot((x^5-2*x^4+x^3)/(x-3)) returns [0,3,1,2,3,-1]
#link discussion
fsolve
Syntax
fsolve(Expr,Var,[Guess or Interval],[Method])
Description
Numerical solution of an equation or a system of equations.
Example
fsolve(cos(x)=x,x,-1..1) returns [0.739085133215]
#link discussion
function_diff
Syntax
function_diff(Fnc(f))
Description
Returns the derivative function of the function f.
Example
function_diff(sin) returns (`x`)->cos(`x`)
#link discussion
Gamma
Syntax
Gamma(Real(x0))
Description
Calculus of Gamma at a point x0 (Gamma(n+1)=n! for n integer).
Example
Gamma(5) returns 24
#link discussion
gauss
Syntax
gauss(Expr,VectVar)
Description
Splits a quadratic form as a sum/difference of square.
Example
gauss(x^2+2*a*x*y,[x,y]) returns (a*y+x)^2+(-y^2)*a^2
#link discussion
gbasis
Syntax
gbasis(ListPoly, ListVar)
Description
Returns the Groebner basis of the ideal spanned by the list of polynomials.
Example
gbasis({x^2-y^3,x+y^2},{x,y}) returns [y^4-y^3,x+y^2]
#link discussion
gcd
Syntax
gcd(Poly1, Poly2)
Description
Returns the greatest common divisor of 2 polynomials of several variables. Can also be used as integer gcd.
Example
gcd(x^2-4,x^2-5*x+6) returns x-2
gcd(45,30) returns 15
#link discussion
GETBASE
Syntax
GETBASE(#integer)
Description
Returns the base used for display for this integer.
0: system
1: bin
2: oct
3: dec
4: hex
#link discussion
GETBITS
Syntax
GETBITS(#integer)
Description
Returns the number of bits used for calculations with this integer.
#link discussion
GETKEY
Syntax
GETKEY
Description
Returns the ID of the first key in the keyboard buffer, or -1 if no key was pressed since the last call to GETKEY. Key IDs are integers from 0 to 50, numbered from top left (key 0) to bottom right (key 50).
0= Apps
1= Symb
2= Up
3= Help
4= Esc
5= Home
6= Plot
7= Left
8= Right
9= View
10= Cas
11= Num
12= Down
13= Menu
After that, the keys are number from top left (14= Vars) to bottom right (50= +)
#link discussion
GETPIX
Syntax
GETPIX([G], x, y)
Description
Returns the color of the pixel of G with coordinates (x,y).
#link discussion
GETPIX_P
Syntax
GETPIX_P([G], x, y)
Description
Returns the color of the pixel of G with coordinates (x,y).
#link discussion
GF
Syntax
GF(Intg(p), Intg(n))
Description
Creates a Galois Field of characteristic p with p^n elements.
Example
GF(5,9) returns GF(5,k^9-k^8+2*k^7+2*k^5-k^2+2*k-2,[k,K,g],undef)
#link discussion
grad
Syntax
grad(Expr, ListVars)
Description
Returns the gradient of the expression Expr.
Example
grad(2*x^2*y-x*z^3,[x,y,z]) returns [2*2*x*y-z^3,2*x^2,-x*3*z^2]
#link discussion
gramschmidt
Syntax
gramschmidt(Basis(B),ScalarProd(Sp))
Description
Returns an orthonormal base of E of base B for the scalar product Sp.
Example
gramschmidt([1,1+x],(p,q)->integrate(p*q,x,-1,1)) returns [1/(√2),(1+x-1)/(√6)/3]
#link discussion
greduce
Syntax
greduce(Poly, ListPoly, ListVar)
Description
Returns the remainder of the division of a polynomial by a Groebner basis.
Example
greduce(x*y-1,{x^2-y^2,2*x*y-y^2,y^3},{x,y}) returns (1/2)*y^2-1
#link discussion
green
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
#link discussion
half_line
Syntax
half_line(Point1, Point2)
Description
Given 2 points, draws a ray from the first point through the second point.
Example
half_line(0, 1+i) draws a ray starting at the origin and passing through the point at (1,1)
#link discussion
halftan
Syntax
halftan(Expr)
Description
Transforms sin(x), cos(x) and tan(x) as a function of tan(x/2).
Example
halftan(sin(x)) returns (2*tan(x/2))/((tan(x/2))^2+1)
#link discussion
halftan_hyp2exp
Syntax
halftan_hyp2exp(Expr)
Description
Transforms the trigonometric functions in tan(x/2) and hyperbolic functions into expontials.
Example
halftan_hyp2exp(sin(x)+sinh(x)) returns (2*tan(x/2)/((tan(x/2))^2+1)+(exp(x)-1/exp(x))/2
#link discussion
hamdist
Syntax
hamdist(Intg,Intg)
Description
Bit Hamming distance.
Example
hamdist(0x12,0x38) returns 3
#link discussion
harmonic_conjugate
Syntax
harmonic_conjugate(Line or Pnt,Line or Pnt,Line or Pnt)
Description
Returns the harmonic conjugate of 3 points or of 3 parallel or concurrent lines or the line of conjugates of a point in respect to 2 lines.
#link discussion
harmonic_division
Syntax
harmonic_division(Pnt or Line,Pnt or Line,Pnt or Line,Var)
Description
Returns the 4 points (resp lines) and affects the last argument, such as the 4 points (resp lines) are in a harmonic division.
#link discussion
has
Syntax
has(Expr,Var)
Description
Checks if a variable is in an expression.
Example
has(x+y,x) returns 1
#link discussion
head
Syntax
head(Vect or Seq or Str)
Description
Shows the first element of a vector or a sequence or a string.
Example
head(1,2,3) returns 1
#link discussion
Heaviside
Syntax
Heaviside(Real)
Description
Function equal to 0 if x<0 and 1 if x>=0
Example
Heaviside(1) returns 1
#link discussion
hermite
Syntax
hermite(Integer)
Description
Returns nth Hermite polynomial.
Example
hermite(3) returns 8*x^3-12*x
#link discussion
hessenberg
Syntax
hessenberg(Mtrx(A))
Description
Matrix reduction to Hessenberg form. Returns [P,B] such that B=inv(P)*A*P.
Example
hessenberg([[1,2,3],[4,5,6],[7,8,1]]) returns [[[1,0,0],[0,4/7,1],[0,1,0]],[[1,29/7,2],[7,39/7,8],[0,278/49,3/7]]]
#link discussion
hessian
Syntax
hessian(Expr,LstVar)
Description
Returns the hessian matrix of the expression Expr.
Example
hessian(2*x^2*y-x*z,[x,y,z]) returns [[4*y,4*x,-1],[2*2*x,0,0],[-1,0,0]]
#link discussion
hexagon
Syntax
hexagon(Point1, Point2, [Var1, Var2, Var3, Var4])
Description
Draws a regular hexagon defined by one of its sides; that is, by two consecutive vertices. The remaining points are calculated automatically, but are not defined symbolically. The orientation of the hexagon is counterclockwise from the first point.
Example
hexagon(0,6) draws a regular hexagon whose first two vertices are at (0, 0) and (6, 0).
hexagon(0,6, a, b, c, d) draws a regular hexagon whose first two vertices are at (0, 0) and (6, 0)l labels the other four vertices a, b, c, and d, and stores the coordinates into the CAS variables a, b, c, and d. You do not have to define variables for all four remaining points, but the coordinates are stored in order. For example, hexagon(0,6, a) stores just the third point into the CAS variable a.
#link discussion
hilbert
Syntax
hilbert(Intg(n))
Description
Returns the order n Hilbert matrix: Hjk=1/(j+k+1) j,k=1..n
Example
hilbert(4) returns [[1,1/2,1/3,1/4],[1/2,1/3,1/4,1/5],[1/3,1/4,1/5,1/6],[1/4,1/5,1/6,1/7]]
#link discussion
→HMS
Syntax
→HMS(value)
Description
Decimal to hours-minutes-seconds.
Change the way a number is displayed to HMS format.
→HMS(8.5) returns 8°3
#link discussion
HMS→
Syntax
HMS→(value)
Description
Hours-minutes-seconds to decimal.
Forces a number to be displayed in decimal format if it was previously displayed in DMS format
HMS→(8°30) returns 8.5
#link discussion
homothety
Syntax
homothety(Point, Realk, Object)
Description
Dilates a geometric object, with respect to a center point, by a scale factor.
Example
homothety(GA, 2, GB) creates a dilation centered at point A that has a scale factor of 2. Each point P on geometric object B has its image P’ on ray AP such that AP’=2AP.
homothety(point(0,0),1/3,point(9,9)) creates an image point at (3,3)
#link discussion
hyp2exp
Syntax
hyp2exp(ExprHyperb)
Description
Transforms the hyperbolic functions with the exponential function.
Example
hyp2exp(cosh(x)) returns (exp(x)+1/exp(x))/2
#link discussion
hyperbola
Syntax
hyperbola(Point1, Point2, Point3) or hyperbola(Point1, Point2, Realk)
Description
Draws a hyperbola, given the foci and either a point on the hyperbola or a scalar that is one half the constant difference of the distances from a point on the hyperbola to each of the foci.
Example
hyperbola(GA, GB, GC) draws the hyperbola whose foci are points A and B and which passes through point C.
hyperbola(GA, GB, 3) draws a hyperbola whose foci are points A and B. For any point P on the hyperbola, |AP-BP|=6.
#link discussion
iabcuv
Syntax
iabcuv(Intg(a),Intg(b),Intg(c))
Description
Returns [u,v] such as au+bv=c for 3 integers a,b,c
Example
iabcuv(21,28,7) returns [-1,1]
#link discussion
ibasis
Syntax
ibasis(Lst(Vect,..,Vect),Lst(Vect,..,Vect))
Description
Basis of the intersection of two vector spaces.
Example
ibasis([[1,0,0],[0,1,0]],[[1,1,1],[0,0,1]]) returns [[-1,-1,0]]
#link discussion
ibpdv
Syntax
ibpdv(Expr1,Expr2,[Var],[Real1],[Real2])
Description
Integration by parts of Expr1=u(Var)*v'(Var) with Expr2= v'(Var) (or 0) as 2nd argument. You can specify a variable of integration and also the bounds of integration (Real1 and Real2).
Example
ibpdv(x*ln(x),1) returns (-1/4)*x^2+(1/2)*(x^2)*ln(x)
#link discussion
ibpu
Syntax
ibpu(Expr1,Expr2,[Var],[Real1],[Real2])
Description
Integration by parts of Expr1=u(Var)*v'(Var) with Expr2= u(Var) (or 0) as 2nd argument. You can specify a variable of integration and also the bounds of integration (Real1 and Real2).
Example
ibpu(ln(x),ln(x),x,1,3) returns [3*ln(3),-1]
#link discussion
ichinrem
Syntax
ichinrem([a,p],[b,q]))
Description
Integer Chinese Remainder Theorem for two equations. Takes two lists [a, p] and [b, q] and returns a list of two integers, [r, n], such that x≡r mod n. In this case, x is such that x≡a mod p and x≡b mod q; also, n=p*q.
Example
ichinrem([2, 7], [3, 5]) returns [-12, 35]
#link discussion
icontent
Syntax
icontent(Poly,[Var])
Description
Returns the GCD of the integer coefficients of a polynomial.
Example
icontent(24x^3+6x^2-12x+18) returns 6
#link discussion
id
Syntax
id(Seq)
Description
The name of the identity function (R^n-> R^n)
Example
id(1,2,3) returns 1,2,3
#link discussion
IDENMAT
Syntax
IDENMAT(n)
Description
Identity matrix. Creates a square matrix of dimension n x n whose diagonal elements are 1 and off-diagonal elements are zero.
Example
IDENMAT(2) returns [[1,0],[0,1]]
#link discussion
identity
Syntax
identity(Intg(n))
Description
Returns the identity matrix of specified dimension n.
Example
identity(3) returns [[1,0,0],[0,1,0],[0,0,1]]
#link discussion
idivis
Syntax
idivis(a)
Description
Integer divisors. Returns a list of all the factors of the integer a.
Example
idivis(12) returns [1, 2, 3, 4, 6, 12]
#link discussion
iegcd
Syntax
iegcd(a,b)
Description
Extended greatest common divisor for two integers. Returns [u,v,igcd(a,b)] such that a*u+b*v=igcd(a,b).
Example
iegcd(14, 21) returns [-1, 1, 7]
#link discussion
IF
Syntax
IF test THEN command(s) [ELSE commands] END;
Description
Evaluates test. If test is true (non 0), executes command(s); otherwise, executes the comands in the ELSE clause nothing happens.
Example
IF A<1
THEN PRINT("A IS SMALLER THAN 1");
ELSE PRINT("A IS LARGER THAN 1");
END;
#link discussion
ifactor
Syntax
ifactor(a)
Description
Prime factorization. Returns the prime factorization of the integer a as a product. Can be used with STO▶.
Example
ifactor(150) returns 2*3*5^2
#link discussion
ifactors
Syntax
ifactors(a)
Description
Prime factors. Similar to ifactor, but returns a list of the factors of the integer a with their multiplicities.
Example
ifactors(150) returns [2, 1, 3, 1, 5, 2]
#link discussion
IFERR
Syntax
IFERR commands1 THEN commands2 [ELSE commands3] END;
Description
Executes sequence of commands1. If an error occurs during execution of commands1, execute sequence of commands2. Otherwise, execute sequence of commands3.
Many conditions are automatically recognized by the HP Prime as error conditions and are automatically treated as errors in programs. This command facilitates error-trapping of such errors.
#link discussion
ifft
Syntax
ifft(Vect)
Description
Inverse Fast Fourier Transform.
Example
ifft([100.0,-52.2842712475+6*i,-8.0*i,4.28427124746-6*i,4.0,4.28427124746+6*i,8*i,-52.2842712475-6*i]) returns [0.99999999999,3.99999999999,10.0,20.0,25.0,24.0,16.0,-6.39843733552e-12]
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IFTE
Syntax
IFTE(Expr, Trueclause, Falseclause)
Description
If...Then...Else...
If Expr evaluates true (1), evaluates Trueclause; if not, evaluates Falseclause.
Example
IFTE(2<3, 5-1, 2+7) returns 4
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igcd
Syntax
igcd(a, b)
Description
Greatest common divisor. Returns the integer that is the greatest common divisor of the integers a and b.
Example
igcd(24, 36) returns 12
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ihermite
Syntax
ihermite(Mtrx(A))
Description
Hermite normal form of a matrix with coefficients in Z: returns U,B such that U is invertible in Z, B upper triangular and B=U*A
Example
ihermite([[1,2,3],[4,5,6],[7,8,9]]) returns [[-3,1,0],[4,-1,0],[-1,2,-1]],[[1,-1,-3],[0,3,6],[0,0,0]]
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ilaplace
Syntax
ilaplace(Expr,[Var],[IlapVar])
Description
Inverse Laplace transform of a rational fraction.
Example
ilaplace(1/(x^2+1)^2) returns (-x)*cos(x)/2+sin(x)/2
#link discussion
IM
Syntax
IM(x+yi)
Description
Imaginary Part. Returns the imaginary part of a complex number.
Example
IM(3+4i) returns 4
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incircle
Syntax
incircle(Point1, Point2, Point3)
Description
Draws the incircle of a triangle, the circle tangent to all three sides of the triangle.
Example
incircle(GA, GB, GC) draws the incircle of ΔABC.
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INPUT
Syntax
INPUT(var,[“title”], [“label”], [“help”], [reset])
Description
or INPUT({vars},[“title”], [{“labels”}], [{“help”}], [{reset}])
Starts a dialog box with header title and one field named label (with value default), displaying help at the bottom. The dialog box includes CANCEL and OK menu keys. If the user presses the OK menu key, the variable var is updated and 1 is returned. If the user presses the CANCL menu key, var is not updated and 0 is returned.
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INSTRING
Syntax
INSTRING(string1, string2)
Description
Returns the index of the first occurrence of string2 in string1. Returns 0 if str2 is not present in str1. Note that the first character in a string is in position 1.
Example
INSTRING("vanilla", "van") returns 1
INSTRING("banana","na") returns 3
INSTRING("ab","abc") returns 0
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int
Syntax
int(Expr,[Var],[Real1,Real2])
Description
Integral (definite or indefinite). You can specify a variable of integration as well as the bounds ofr integration. You can use the integration template in the Template menu as well.
Example
int(1/x) returns ln(abs(x))
int(sin(x),x,0,π) returns 2
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inter
Syntax
inter(Curve1, Curve2)
Description
Returns the intersections of two curves as a vector.
Example
inter(8-x^2/6, x/2-1) returns [[6, 2] [-9, -11/2]], indicating that there are two intersections-one at (6,2) and the other at (-9,-5.5).
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INTERSECT
Syntax
INTERSECT({list1}, ...{listN})
Description
Returns a list of the elements common to all the lists.
Example
INTERSECT({1,2,3},{2,4,8}) returns {2}
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interval2center
Syntax
interval2center(Interval or Real)
Description
Returns the center of the interval or the object.
Example
interval2center(2..5) returns 7/2
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inv
Syntax
inv(Expr||Mtrx)
Description
Returns the inverse of an expression or matrix.
Example
inv(9/5) returns 5/9
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inversion
Syntax
inversion(Point1, Realk, Point2)
Description
Draws the inversion of a point, with respect to another point, by a scale factor.
Example
inversion(GA, 3, GB) draws point C on line AB such that AB*AC=3. In this case, point A is the center of the inversion and the scale factor is 3. Point B is the point whose inversion is created.
In general, the inversion of point A through center C, with scale factor k, maps A onto A’, such that A’ is on line CA and CA*CA’=k, where CA and CA’ denote the lengths of the corresponding segments. If k=1, then the lengths CA and CA’ are reciprocals.
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INVERT
Syntax
INVERT([G], [x1, y1], [x2, y2])
Description
Inverts the rectangle on G defined by the diagonal points (x1,y1) and (x2,y2). The effect is reverse video.
The following values are optional and their defaults are listed:
x1, y1=top left corner of G
x2, y2=bottom right corner of G
If only one x,y pair is specified, it refers to the top left corner of G.
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INVERT_P
Syntax
INVERT_P([G], [x1, y1], [x2, y2])
Description
Inverts the rectangle on G defined by the diagonal points (x1,y1) and (x2,y2). The effect is reverse video.
The following values are optional and their defaults are listed:
x1, y1=top left corner of G
x2, y2=bottom right corner of G
If only one (x,y) pair is specified, it refers to the top left corner of G.
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invlaplace
Syntax
ilaplace(Expr,[Var],[IlapVar])
Description
Returns the inverse Laplace transform of Expr.
Example
ilaplace(1/(x^2+1)^2) returns (-x/2)*cos(x)+(1/2)*sin(x)
#link discussion
invztrans
Syntax
invztrans(Expr,[Var],[InvZtransVar])
Description
Inverse z transform of a rational fraction.
Example
invztrans(1/(x^2+1)^2) returns (x*exp(x*(-i)*π/2)+x*exp(x*(i)*π/2)+4*Dirac(x)-2*exp(x*(-i)*π/2)-2*exp(x*(i)*π/2))/4
#link discussion
IP
Syntax
IP(value)
Description
Integer part. Returns the Integer part of value.
Example
IP(23.2) returns 23
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iPart
Syntax
iPart(Real||LstReal)
Description
Returns the argument without its fractional part (type=DOM_FLOAT).
Example
iPart(4.3) returns 4.0
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iquo
Syntax
iquo(a, b)
Description
Euclidean quotient. Returns the integer quotient when the integer a is divided by the integer b.
Example
iquo(63, 23) returns 2
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iquorem
Syntax
iquorem(a, b)
Description
Euclidean quotient and remainder. Returns the integer quotient and remainder when the integer a is divided by the integer b.
Example
iquorem(63, 23) returns [2, 17]
#link discussion
irem
Syntax
irem(a, b)
Description
Euclidean remainder. Returns the integer remainder when the integer a is divided by the integer b.
Example
irem(63, 23) returns 17
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is_collinear
Syntax
is_collinear(Point1, Point2, ..., Pointn)
Description
Takes a set of points as argument and tests whether or not they are collinear. Returns 1 if the points are collinear and 0 otherwise.
Example
is_collinear(point(0,0), point(5,0), point(6,1)) returns 0.
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is_concyclic
Syntax
is_concyclic(Point1, Point2, …, Pointn)
Description
Takes a set of points as argument and tests if they are all on the same circle. Returns 1 if the points are all on the same circle and 0 otherwise.
Example
is_concyclic(point(-4,-2), point(-4,2), point(4,-2), point(4,2)) returns 1
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is_conjugate
Syntax
is_conjugate(Crcle, Point1, Point2, [Point3]) or is_conjugate(Line1, Line2, Line3, {Line4])
Description
Returns 1 if the 3 (resp 4) arguments are conjugated toward a circle (resp 2 lines) and 0 otherwise.
#link discussion
is_coplanar
Syntax
is_coplanar(Point1, Point2, Point3, Point4)
Description
Tests if 4 points are in the same plane. Returns 1 if true or 0 if false.
#link discussion
is_element
Syntax
is_element(Point, Object)
Description
Tests if a point is on a geometric object. Returns 1 if it is and 0 otherwise
Example
is_element(point((√(2)/2),(√(2)/2)),circle(0,1)) returns 1
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is_equilateral
Syntax
is_equilateral(Point1, Point2, Point3)
Description
Takes three points and tests whether or not they are vertices of a single equilateral triangle. Returns 1 if they are and 0 otherwise..
Example
is_equilateral(triangle(0,2,1+i*√3)) returns 1.
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is_harmonic
Syntax
is_harmonic(Pnt or Cplx,Pnt or Cplx,Pnt or Cplx,Pnt or Cplx)
Description
Returns 1 if the 4 points are in a harmonic division and 0 otherwise.
#link discussion
is_harmonic_circle_bundle
Syntax
is_harmonic_circle_bundle(Lst(Crcle))
Description
Returns 1 if the circles build a bundle, 2 if they have the same center, 3 if they are the same and 0 otherwise.
#link discussion
is_harmonic_line_bundle
Syntax
is_harmonic_line_bundle(Lst(Line))
Description
Returns 1 if the lines have a common point, 2 if they are parallels, 3 if they are the same and 0 otherwise.
#link discussion
is_isosceles
Syntax
is_isosceles(Point1, Point2, Point3)
Description
Takes three points and tests whether or not they are vertices of a single isosceles triangle. Returns 0 if they are not. If they are, returns the number order of the common point of the two sides of equal length (1, 2, or 3). Returns 4 if the three points form an equilateral triangle.
Example
is_isosceles(point(0,0), point(4,0), point(2,4)) returns 3
is_isosceles(triangle(0,i,1+i)) returns 2
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is_orthogonal
Syntax
is_orthogonal(Line1, Line2) or is_orthogonal(Circle1, Circle2
Description
Tests whether or not two lines or two circles are orthogonal (perpendicular). In the case of two circles, tests whether or not the tangent lines at a point of intersection are orthogonal. Returns 1 if they are and 0 otherwise.
Example
is_orthogonal(line(y=x),line(y=-x)) returns 1.
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is_parallel
Syntax
is_parallel(Line1, Line2)
Description
Tests whether or not two lines are parallel. Returns 1 if they are and 0 otherwise.
Example
is_parallel(line(2x+3y=7),line(2x+3y=9) returns 1.
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is_parallelogram
Syntax
is_parallelogram(Point1, Point2, Point3, Point4)
Description
Tests whether or not a set of four points are vertices of a parallelogram. Returns 0 if they are not. If they are, then returns 1 if they form only a parallelogram, 2 if they form a rhombus, 3 if they form a rectangle, and 4 if they form a square.
Example
is_parallelogram(point(0,0), point(2,4), point(0,8), point(-2,4)) returns 2.
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is_perpendicular
Syntax
is_perpendicular(line1, Line2)
Description
Similar to is_orthogonal. Tests whether or not two lines are perpendicular. Returns 1 if they are or 0 if they are not.
Example
is_perpendicular(line(y=x),line(y=-x)) returns 1
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is_rectangle
Syntax
is_rectangle(Point1, Point2, Point3, Point4)
Description
Tests whether or not a set of four points are vertices of a rectangle. Returns 0 if they are not, 1 if they are, and 2 if they are vertices of a square.
Example
is_rectangle(point(0,0), point(4,2), point(2,6), point(-2,4)) returns 2.
With a set of only three points as argument, tests whether or not they are vertices of a right triangle. Returns 0 if they are not. If they are, returns the number order of the common point of the two perpendicular sides (1, 2, or 3).
is_rectangle(point(0,0), point(4,2), point(2,6)) returns 2.
#link discussion
is_rhombus
Syntax
is_rhombus(Pnt or Cplx,Pnt or Cplx,Pnt or Cplx,Pnt or Cplx)
Description
Returns 1 or 2 if the 4 points (or the object) build a rhombus (2 for a square) and 0 otherwise.
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is_square
Syntax
is_square(Point1, Point2, Point3, Point4)
Description
Tests whether or not a set of four points are vertices of a square. Returns 1 if they are and 0 otherwise.
Example
is_square(point(0,0), point(4,2), point(2,6), point(-2,4)) returns 1
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ISKEYDOWN
Syntax
ISKEYDOWN(Key_ID)
Description
Returns true (non-zero) if the key whose Key_ID is provided is currently pressed, and false (0) if it is not.
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ismith
Syntax
ismith(Mtrx(A))
Description
Smith normal form of a matrix with coefficients in Z : returns U,B,V such that U and V are invertible in Z, B is the diagonal, B[i,i] divide B[i+1,i+1] and B=U*A*V.
Example
ismith([[1,2,3],[4,5,6],[7,8,9]]) returns [[1,0,0],[4,-1,0],[-1,2,-1]],[[1,0,0],[0,3,0],[0,0,0]],[[1,-2,1],[0,1,-2],[0,0,1]]
#link discussion
isobarycenter
Syntax
isobarycenter(Point1, Point2, …, Pointn)
Description
Returns the hypothetical center of mass of a set of points. Works like barycenter but assumes all points have equal weight.
Example
isobarycenter(–3,3,3*√3*i) returns
point(3*√3*i/3), which is equivalent to (0,√3).
#link discussion
isopolygon
Syntax
isopolygon(Point1, Point2, Realn), where realn is an integer greater than 1.
Description
Draws a regular polygon given the first two vertices and the number of sides, where the number of sides is greater than 1. If the number of sides is 2, then the segment is drawn. You can provide CAS variable names for storing the coordinates of the calculated points in the order they were created. The orientation of the polygon is counterclockwise.
Example
isopolygon(GA, GB, 6) draws a regular hexagon whose first two vertices are the points A and B.
#link discussion
isosceles_triangle
Syntax
isosceles_triangle(Point1, Point2, Angle)
Description
Draws an isosceles triangle defined by two of its vertices and an angle. The vertices define one of the two sides equal in length and the angle defines the angle between the two sides of equal length. Like equilateral_triangle, you have the option of storing the coordinates of the third point into a CAS variable.
Example
isosceles_triangle(GA, GB, angle(GC, GA, GB) defines an isosceles triangle such that one of the two sides of equal length is AB, and the angle between the two sides of equal length has a measure equal to that of ∡ACB.
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isPrime
Syntax
isprime(a)
Description
Prime integer test. Returns true if the integer a is prime; otherwise, returns false.
Example
isprime(1999) returns true
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ITERATE
Syntax
ITERATE(expr, var, ivalue, #times)
Description
Repeatedly for #times evaluates expr in terms of var. The value for var is updated each time, starting with ivalue.
ITERATE(X^2, X, 2, 3) returns 256.
#link discussion
ithprime
Syntax
ithprime(n)
Description
Nth prime. For the integer n, returns the nth prime number less than 100,000-200,000.
Example
ithprime(5) returns 11
#link discussion
jacobi_symbol
Syntax
jacobi_symbol(Intg,Intg)
Description
Jacobi symbol.
Example
jacobi_symbol(132,5) returns -1
#link discussion
jordan
Syntax
jordan(Mtrx)
Description
Returns the list made by the matrix of passage and the Jordan form of a matrix.
Example
jordan([[0,2],[1,0]]) returns [[√2,-√2],[1,1]],[[√2,0],[0,-√2]]
#link discussion
JordanBlock
Syntax
JordanBlock(Expr(a),Intg(n))
Description
Returns a matrix n*n with a on the diagonal, 1 above, and 0 everywhere else.
Example
JordanBlock(7,3) returns [[7,1,0],[0,7,1],[0,0,7]]
#link discussion
ker
Syntax
ker(Mtrx(M))
Description
Kernel of a linear application of matrix M.
Example
ker([[1,2],[3,6]]) returns [ 2, -1 ]
#link discussion
l1norm
Syntax
l1norm(Vect)
Description
Returns the l1 norm of the vector=sum of the absolute value of its coordinates.
Example
l1norm([3,-4,2]) returns 9
#link discussion
l2norm
Syntax
l1norm(Vect)
Description
Returns the l1 norm of the vector=sum of the absolute value of its coordinates.
Example
l1norm([3,-4,2]) returns 9
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lagrange
Syntax
lagrange((Listxk, Listyk) or lagrange(Matrix)
Description
Returns the polynomial of degree n-1 such that P(xk)=yk, for k=0, 1, ..., n-1.
Example
lagrange([1,3],[0,1]) returns (1/2)* (x-1)
#link discussion
laguerre
Syntax
laguerre(Integer)
Description
Returns the nth Laguerre polynomial.
Example
laguerre(4) returns (1/24)*a^4+(-1/6)*a^3*x+5/12*a^3+1/4*a^2*x^2+(-3/2)*a^2*x+35/24*a^2+(-1/6)*a*x^3+7/4*a*x^2+(-13/3)*a*x+25/12*a+1/24*x^4+(-2/3)*x^3+3*x^2-4*x+1
#link discussion
laplace
Syntax
laplace(Expr,[Var],[LapVar])
Description
Returns the Laplace transform of Expr.
Example
laplace(exp(x)*sin(x)) returns 1/(x^2-2*x+2)
#link discussion
laplacian
Syntax
laplacian(Expr(Xpr),LstVar)
Description
Returns the Laplacian of the expression Xpr with respect to the list of variables.
Example
laplacian(exp(z)*cos(x*y),[x,y,z]) returns -x^2*cos(x*y)*exp(z)-y^2*cos(x*y)*exp(z)+cos(x*y)*exp(z)
#link discussion
lcm
Syntax
lcm(Intgr1, Intgr2) or lcm(Poly1, Poly2) or lcm(Rational1, Rational2)
Description
Returns the lowest common multiple of 2 polynomials of several variables or of 2 integers or of 2 rationals.
Example
lcm(6,4) returns 12
#link discussion
lcoeff
Syntax
lcoeff(Poly||Lst)
Description
Returns the coefficient of the term of highest degree of a polynomial (l=leading).
Example
lcoeff(-2*x^3+x^2+7*x) returns -2
#link discussion
LEFT
Syntax
LEFT(string, n)
Description
Returns the first n characters of the string.
Example
LEFT("MOMOGUMBO",3) returns "MOM"
#link discussion
legendre
Syntax
legendre(Integer)
Description
Returns the nth Legendre polynomial.
Example
legendre(4) returns (35/8)*x^4+(-15/4)*x^2+3/8
#link discussion
legendre_symbol
Syntax
legendre_symbol(Intg,Intg)
Description
Legendre symbol.
Example
legendre_symbol(132,5) returns -1
#link discussion
length
Syntax
size(Lst or Str or Seq)
Description
Returns the size of a list, a string or a sequence.
Example
size([1,2,3]) returns 3
#link discussion
lgcd
Syntax
lgcd(Seq or Lst )
Description
Returns the greatest common divisor of a list of polynomials or of integers.
Example
lgcd({45,75,20,15}) returns 5
lgcd({x^2-2*x+1,x^3-1,x-1}) returns x-1
#link discussion
limit
Syntax
limit(Expr,Var,Val)
Description
Limit of an expression as a variable approaches a value. Returns the limit (2 sided or 1-sided) of the given expression as the given variable approaches a value.
Example
limit((n*tan(x)-tan(n*x))/(sin(n*x)-n*sin(x)),x,0) returns 2
#link discussion
lin
Syntax
lin(Expr )
Description
Linearization of exponentials.
Example
lin((exp(x)^3+exp(x))^2) returns exp(6*x)+2*exp(4*x)+exp(2*x)
#link discussion
line
Syntax
line(Point1, Point2) or line(a*x+b*y+c) or line(point1, slope=realm)
Description
Draws a line. The arguments can be two points, a linear expression of the form a*x+b*y+c, or a point and a slope.
Example
line(2+i, 3+2i) draws the line whose equation is y=x-1; that is, the line through the points (2,1) and (3,2).
line(2x-3y-8) draws the line whose equation is 2x-3y=8
line(3-2i,slope=1/2) draws the line whose equation is x-2y=7; that is, the line through (3, -2) with slope m=1/2
#link discussion
LINE
Syntax
LINE([G], x1, y1, x2, y2, [color])
Description
Draws a line on GROB G between points (x1,y1) and (x2,y2).
#link discussion
LINE_P
Syntax
LINE_P([G], x1, y1, x2, y2, [color])
Description
Draws a line on GROB G between points (x1,y1) and (x2,y2).
#link discussion
linear_interpolate
Syntax
linear_interpolate(Mtrx,xmin,xmax,xstep)
Description
Makes a regular sample from a polygonal line defined by a 2 row matrix.
Example
linear_interpolate([[1,2,6,9],[3,4,6,7]],1,9,1) returns [[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0],[3.0,4.0,4.5,5.0,5.5,6.0,6.33333333333,6.66666666667,7.0]]
#link discussion
linear_regression
Syntax
linear_regression(Lst||Mtrx(A),[Lst])
Description
Returns the coefficients a and b of y=a*x+b
It is the best line approximation to the points where the coordinates are the rows of A (or the 2 lists).
Example
linear_regression([[0.0,0.0],[1.0,1.0],[2.0,4.0],[3.0,9.0],[4.0,16.0]]) returns 4.0,-2.0
#link discussion
LineTan
Syntax
LineTan(f(x), [Var], Value)
Description
Draws the tangent to y=f(x) at x=Value.
Example
LineTan(x^2-x, 1) draws the line whose equation is y=x-1, which is tangent to the graph of y=x^2-x at x=1.
#link discussion
linsolve
Syntax
linsolve(ListLinEq,ListVar)
Description
Linear equations system solver. Solves a set of linear equations for their common variable set.
Example
linsolve([x+y+z=1,x-y=2,2*x-z=3],[x,y,z]) returns [3/2,-1/2,0]
#link discussion
∏LIST
Syntax
ΠLIST(list)
Description
List Product. Calculates the product of all elements in list.
Example
ΠLIST({2,3,4}) returns 24.
#link discussion
∆LIST
Syntax
ΔLIST(list)
Description
List Difference. Creates a new list composed of the first differences of list; that is, the differences between the sequential elements in list. The new list has one fewer elements than list.
Example
ΔLIST({1, 2, 3, 5, 8}) returns {1, 1, 2, 3}
#link discussion
∑LIST
Syntax
ΣLIST(list)
Description
Sum of a list. Returns the sum of all elements in list.
Example
ΣLIST({2,3,4}) returns 9
#link discussion
list2mat
Syntax
list2mat(Lst(l),Intg(n))
Description
Returns the matrix with n columns and where terms are the list l completed eventually by 0.
Example
list2mat([1,8,4,9],1) returns [[1],[8],[4],[9]]
#link discussion
LN
Syntax
LN(Value)
Description
Returns the natural logarithm of Value. The natural logarithm is the logarithm to the base e, Euler's number.
Example
LN(1) returns 0
#link discussion
lname
Syntax
lname(Expr )
Description
List of variables in the expression.
Example
lname(exp(x)*2*sin(y)) returns [x,y]
#link discussion
lncollect
Syntax
lncollect(Expr)
Description
Collect logarithms. Applies ln(a)+n*ln(b)=ln(a*b^n) where n is an integer.
Example
lncollect(ln(x)+2*ln(y)) returns ln(x*y^2)
#link discussion
lnexpand
Syntax
lnexpand(Expr)
Description
Expands logarithms.
Example
lnexpand(ln(3*x)) returns ln(3)+ln(x)
#link discussion
LNP1
Syntax
LNP1(value)
Description
Natural log plus 1. This is more accurate than the natural logarithm function when x is close to zero.
Example
LNP1(.23) returns .207014169384
#link discussion
LOCAL
Syntax
LOCAL var_1[:=value][, more variables];
Description
Declares a local variable.
If the declaration is in a function block, these variables will be local to the function.
if the declaration is in the main program body, the variables are local to the program.
#link discussion
locus
Syntax
locus(Point,Element)
Description
Given a first point and a second point that is an element of (a point on) a geometric object, draws the locus of the first point as the second point traverses its object.
#link discussion
LOG
Syntax
LOG(Value, [Base])
Description
Returns the logarithm of Value in Base. By default, Base=10.
Example
LOG(8,2) returns 3 while LOG(8) returns 0.903089986992
#link discussion
log10
Syntax
log10(Expr)
Description
Common logarithm (base 10).
Example
log10(10) returns 1
#link discussion
logarithmic_regression
Syntax
logarithmic_regression(Lst||Mtrx(A),[Lst])
Description
Returns the coefficients a and b of y=a*ln(x)+b : it is the best logarithm that approximates the points where the coordinates are the rows of A (or the 2 lists).
Example
logarithmic_regression([[1.0,1.0],[2.0,4.0],[3.0,9.0],[4.0,16.0]]) returns 10.1506450002,-0.564824055818
#link discussion
logb
Syntax
logb(Real)
Description
Logarithm of base b.
Example
logb(5,2) returns ln(5)/ln(2) which is approximately 2.32192809489
#link discussion
logistic_regression
Syntax
logistic_regression(Lst(L),Real(x0),Real(y0) )
Description
Returns y,y',C,y'max,xmax,R : y is a logistic function (sol of y'/y=a*y+b), such that y(x0)=y0 and where [y'(x0),y'(x0+1)...] is the best approximation of L.
Example
logistic_regression([0.0,1.0,2.0,3.0,4.0],0.0,1.0) returns [-17.77/(1+exp(-0.496893925384*x+2.82232341488+3.14159265359*i)),-2.48542227469/(1+cosh(-0.496893925384*x+2.82232341488+3.14159265359*i))]
#link discussion
LQ
Syntax
LQ(matrix)
Description
LQ Factorization. Factors an m n matrix into three matrices: {[[ m n lowertrapezoidal]],[[ n n orthogonal]], [[ m m permutation]]}.
Example
LQ([[1,2],[3,4]])
#link discussion
LSQ
Syntax
LSQ(matrix1, matrix2)
Description
Least Squares. Displays the minimum norm least squares matrix (or vector).
Example
LSQ([[1,2],[3,4]],[[5],[11]]) returns [[1],[2]]
#link discussion
LU
Syntax
LU(matrix)
Description
LU Decomposition. Factors a square matrix into three matrices:
{[[lowertriangular]],[[uppertriangular]],[[permutation]]}
The uppertriangular has ones on its diagonal.
Example
LU([[1,2],[3,4]])
#link discussion
lvar
Syntax
lvar(Expr)
Description
List of variables of an object (with rational dependence).
Example
lvar(exp(x)*2*sin(y)) returns [exp(x),sin(y)]
#link discussion
magenta
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
#link discussion
MAKELIST
Syntax
MAKELIST(expression, variable, begin, end, [increment])
Description
Make List. Calculates a sequence of elements for a new list. Evaluates expression, incrementing variable from begin to end values, using increment steps (default 1). The MAKELIST function generates a series by automatically producing a list from the repeated evaluation of an expression.
Example
MAKELIST(2*X-1, X, 1, 5, 1) returns {1, 3, 5, 7, 9}
#link discussion
MAKEMAT
Syntax
MAKEMAT(expression, n, [m])
Description
Make Matrix. Creates a matrix of dimension n × m, using expression to calculate each element. If expression contains the variables I and J, then the calculation for each element substitutes the current row number for I and the current column number for J. With two arguments, this creates a vector of size n.
Example
MAKEMAT(0,3,3) returns [[0,0,0],[0,0,0],[0,0,0]]
MAKEMAT(√2,2,3) returns [[√2,√2,√2],[√2,√2,√2]]
MAKEMAT(I+J–1,2,3) returns [[1,2,3],[2,3,4]]
MAKEMAT(√2,2) returns [√2,√2]
#link discussion
MANT
Syntax
MANT(value)
Description
Mantissa. Returns the significant digits of value.
Example
MANT(21.2E34) returns 2.12
#link discussion
map
Syntax
map(Lst(l),Fnc(f))
Description
Apply the function f at the elements of the list l or at a polynomial of internal format.
Example
map([1,2,3],x->x^3) returns [1,8,27]
#link discussion
mat2list
Syntax
mat2list(Mtrx)
Description
Returns the list of the terms of the matrix.
Example
mat2list([[1,8],[4,9]]) returns [1,8,4,9]
#link discussion
matpow
Syntax
matpow(Mtrx,Intg(n))
Description
Calculates the n power of a matrix by jordanization.
Example
matpow([[1,2],[3,4]],n) returns [[(√33-3)*((√33+5)/2)^n*-6/(-12*√33)+(-(√33)-3)*((-(√33)+5)/2)^n*6/(-12*√33),(√33-3)*((√33+5)/2)^n*(-(√33)-3)/(-12*√33)+(-(√33)-3)*((-(√33)+5)/2)^n*(-(√33)+3)/(-12*√33)],[6*((√33+5)/2)^n*-6/(-12*√33)+6*((-(√33)+5)/2)^n*6/(-12*√33),6*((√33+5)/2)^n*(-(√33)-3)/(-12*√33)+6*((-(√33)+5)/2)^n*(-(√33)+3)/(-12*√33)]]
#link discussion
MAX
Syntax
MAX(value1,[value2],[..value16])
Description
Maximum. Returns the greatest of the values given, or the greatest value of a list.
Example
MAX(210,25) returns 210 and MAX({1, 8, 2}) returns 8
#link discussion
maxnorm
Syntax
maxnorm(Vect or Mtrx)
Description
Norm with the maximum of a vector (or of a matrix): maxnorm([x1,x2,..,xn])=max(|x1|,..,|xn|)
Example
maxnorm([1,2]) returns 2
#link discussion
MAXREAL
Syntax
MAXREAL
Description
Maximum real number. The largest real number the HP Prime is capable of representing. The value of MAXREAL is 9.99999999999E499. Any number larger than this is represented as this number.
#link discussion
mean
Syntax
mean(Lst||Mtrx,[Lst])
Description
Mean of a list with the second argument as weight, or of the columns of a matrix.
Example
mean([1,2,3],[1,2,3]) returns 7/3
#link discussion
median
Syntax
median(Lst||Mtrx,[Lst])
Description
Returns the median of a list with the second argument as the weight, or of the columns of a matrix.
Example
median([1,2,3,5,10,4]) returns 3.0
#link discussion
median_line
Syntax
median_line(Point1, Point2, Point3)
Description
Given three points that define a triangle, creates the median of the triangle that passes through the first point and contains the midpoint of the segment defined by the other two points.
Example
median_line(0, 8i, 4) draws the line whose equation is y=2x; that is, the line through (0,0) and (2,4), the midpoint of the segment whose endpoints are (0, 8) and (4, 0).
#link discussion
member
Syntax
member(Elem(e),(Lst(l) or Set(l)))
Description
Tests if e is in the list or the set l (=0 or k+1 with l[k]=e)
Example
member(1,[4,3,1,2]) returns 3
#link discussion
MID
Syntax
MID(string, position, [n])
Description
Extracts n characters from string starting at position. If n is not specified, then MID extracts the remainder of the string from position.
Example
MID("MOMOGUMBO",3,5) returns "MOGUM"
MID("PUDGE",4) returns "GE"
#link discussion
midpoint
Syntax
midpoint(Segment) or midpoint(Point1, Point2)
Description
Returns the midpoint of a segment. The argument can be either the name of a segment or two points that define a segment. In the latter case, the segment need not actually be drawn.
Example
midpoint(0,6+6i) returns point(3,3)
#link discussion
MIN
Syntax
MIN(value1,[value2],[..value16])
Description
Minimum. Returns the lesser of the values given, or the lesser value of a list.
Example
MIN(210,25) returns 25 and MIN({1, 8, 2}) returns 1
#link discussion
MINREAL
Syntax
MINREAL
Description
Minimum real number. The smallest real number that the HP Prime can represent. Its value is 1E-499. Any number smaller than this is represented as zero.
#link discussion
mkisom
Syntax
mkisom(Vect,(Sign(1) or -1))
Description
Matrix of an isometry given by its proper elements.
Example
mkisom(π,1) returns [[-1,0],[0,-1]] in radian mode
#link discussion
MKSA
Syntax
MKSA(Value_Unit)
Description
Converts the measurement Value_Unit to its corresponding value and unit in Unit's MKSA equivalent. MKSA stands for the Meter-Kilogram-Second-Ampere system.
Example
MKSA(32_yd) returns 29.2608_m.
#link discussion
MOD
Syntax
value1 MOD value2
Description
Modulo. Returns the remainder of value1/value2.
Example
9 MOD 4 returns 1
#link discussion
modgcd
Syntax
modgcd(Poly,Poly)
Description
Returns the GCD of 2 polynomials, with the modular algorithm.
Example
modgcd(x^4-1,(x-1)^2) returns x-1
#link discussion
MOUSE
Syntax
MOUSE[(index)]
Description
Returns the current pointer's location.
returns: two lists of the form {#x, #y, #originalx, #originaly, #type}, one for each potential pointer.
Note, if a pointer is unused, returns an empty list
#type is: #0: New, #1: Completed, #2: Drag, #3: Stretch, #4: Rotate, #5: LongClick
MOUSE(x) returns the nth element that would be returned if MOUSE was called with no arguements or -1 if the associated pointer is not down.
#link discussion
mRow
Syntax
mRow(Expr(Xpr),Mtrx(A),Intg(n1))
Description
Multiplies the row n1 of the matrix A by Xpr.
Example
mRow(12,[[1,2],[3,4],[5,6]],1) returns [[12,24],[3,4],[5,6]]
#link discussion
MSGBOX
Syntax
MSGBOX(expr,[OK_Cancel]) or MSGBOX(string,[OK_Cancel])
Description
Displays a message box with either the value of expression or string. If OK_Cancel? is true, displays OK and CANCEL menu keys, otherwise only displays the OK menu key. Default value for OK_Cancel is false.
Returns true (non-zero) if the user presses OK, false (0) if the user presses CANCEL.
#link discussion
mult_c_conjugate
Syntax
mult_c_conjugate(Expr)
Description
Returns the expression after multiplication by the complex conjugated quantity of the denominator (or of the numerator if no denominator).
Example
mult_c_conjugate(1/(3+i*2)) returns 1*(3+(-i)*2)/((3+(i)*2)*(3+(-i)*2))
#link discussion
mult_conjugate
Syntax
mult_conjugate(Expr)
Description
Returns the expression after multiplication by the conjugated quantity of the denominator (or of the numerator if no denominator).
Example
mult_conjugate(√3-√2) returns (√3-(√2))*(√3+√2)/(√3+√2)
#link discussion
nDeriv
Syntax
nDeriv(Expr(Xpr),Var(var),[Real(h)])
Description
Returns an approximation of the derivative number at a point:(Xpr(var+h)-Xpr(var-h))/(2*h) (by default h=0.001).
Example
nDeriv(f(x),x,h) returns (f(x+h)-(f(x-h)))*0.5/h
#link discussion
NEG
Syntax
-Value or -Expression
Description
Unary minus.
Changes the sign of Value or Expression. Used to enter negative numbers.
#link discussion
nextprime
Syntax
nextprime(a)
Description
Next prime. Returns the next prime number greater than the integer a.
Example
nextprime(12) returns 13
#link discussion
normal
Syntax
normal(Expr)
Description
Simplify the expression.
Example
normal(2*x*2) returns 4*x
#link discussion
NORMALD
Syntax
NORMALD([μ, σ,] x)
Description
Normal probability density function. Computes the probability density at the value x, given the mean, μ, and standard deviation, σ, of a normal distribution. With one argument, x, returns the probability density at x, assuming a mean of zero and standard deviation of 1.
Example
NORMALD(0.5) returns 0.352065326765 and NORMALD(0, 2, 0.5) returns 0.193334058402
#link discussion
NORMALD_CDF
Syntax
NORMAL_CDF(μ, σ, x)
Description
Cumulative normal distribution function. Returns the lower-tail probability of the normal probability density function for the value x, given the mean, μ, and standard deviation, σ, of a normal distribution. With one argument, x, returns the lower-tail probability of the normal probability density function for the value x, assuming a mean of zero and standard deviation of 1.
Example
NORMAL_CDF(0, 1, 2) returns 0.97724986805
#link discussion
NORMALD_ICDF
Syntax
NORMALD_ICDF(μ, ϭ, p)
Description
Inverse cumulative normal distribution function. Returns the cumulative normal distribution value associated with the lower-tail probability, p, given the mean, μ, and standard deviation, ϭ, of a normal distribution. With one argument, p, assumes a mean of zero and a standard deviation of one.
Example
NORMALD_ICDF(0, 1, 0.841344746069) returns 1
#link discussion
normalize
Syntax
normalize(Lst||Cplx)
Description
Returns the vector divided by its l2norm. It is also an option for plotfield.
Example
normalize(3+4*i) returns (3+4*i)/5
#link discussion
NOT
Syntax
NOT Value
Description
Logical NOT.
Returns 1 if Value is zero; otherwise returns 0.
Example
NOT 3 returns 0
#link discussion
nSolve
Syntax
nSolve(Expr,Var||orVar=Guess)
Description
Returns a numerical solution of an equation or a system of equations.
Example
nSolve(cos(x)=x,x) returns 0.739085133215
nSolve(cos(x)=x,x=1.3) returns 0.739085133215
#link discussion
NTHROOT
Syntax
Value1 √ Value2
Description
NTHROOT: the nth root function.
This Shift-key combination is the NTHROOT function. It returns the primary Value1 root of Value2. On the keyboard, the NTHROOT function is represented by n√ .
Example
3√ 8 returns 2
#link discussion
numer
Syntax
numer(a,b)
Description
Simplified Numerator. For the integers a and b, returns the numerator of the fraction a/b after simplification.
Example
numer(10/12) returns 5
#link discussion
odd
Syntax
odd(Intg(n))
Description
Returns 1 if the integer is odd, otherwise returns 0.
Example
odd(6) returns 0
#link discussion
odesolve
Syntax
odesolve(Expr,VectVar,VectInitCond,FinalVal,[tstep=Val,curve])
Description
Ordinary Differential Equation solver. Solves an ordinary differential equation given by Expr, with variables declared in VectrVar and initial conditions for those variables declared in VectrInit. For example, odesolve(f(t,y),[t,y],[t0,y0],t1) returns the approximate solution of y'=f(t,y) for the variables t and y with initial conditions t=t0 and y=y0.
Example
odesolve(sin(t*y),[t,y],[0,1],2) returns [1.82241255674]
#link discussion
open_polygon
Syntax
open_polygon(LstPnt||LstCplx)
Description
Returns and draws the polygonal line where its vertices are the element of l.
#link discussion
OR
Syntax
Value1 OR Value2
Description
Logical OR.
Returns 1 if either Value1 or Value2 is non-zero, otherwise returns 0.
Example
3 OR 2 returns 1
#link discussion
order_size
Syntax
order_size(Expr)
Description
Remainder (O term) of a series expansion: limit(x^a*order_size(x),x=0)=0 if a>0
#link discussion
ordinate
Syntax
ordinate(Poinnt) or ordinate(Vecctor)
Description
Returns the ordinate of a point or a vector.
Example
ordinate(point(1+2*i)) returns 2
#link discussion
orthocenter
Syntax
orthocenter(Triangle) or orthocenter(Point1, Point2, Point3)
Description
Returns the orthocenter of a triangle; that is, the intersection of the three altitudes of a triangle. The argument can be either the name of a triangle or three non-collinear points that define a triangle. In the latter case, the triangle does not need to be drawn.
Example
orthocenter(0,4i,4) returns (0,0)
#link discussion
pa2b2
Syntax
pa2b2(Intg(n))
Description
Returns [a,b] such as a^2+b^2=n (for n prime and n=1 (mod 4))
Example
pa2b2(17) returns [4,1]
#link discussion
pade
Syntax
pade(Expr(Xpr), Var(x), (Intg(n) || Poly(N)), Intg(p))
Description
Pade approximation P/Q=Xpr mod x^(n+1) or mod N with degree(P)<p
Example
pade(exp(x),x,10,6) returns (-x^5-30*x^4-420*x^3-3360*x^2-15120*x-30240)/(x^5-30*x^4+420*x^3-3360*x^2+15120*x-30240)
#link discussion
parabola
Syntax
parabola(Point, Line) or parabola(Point, Realk) or parabola(Expr)
Description
Draws a parabola, given a focus point and a directrix line, or the vertex of the parabola and a real number that represents the focal length
Example
parabola(GA, GB) draws a parabola whose focus is point A and whose directrix is line B.
parabola(GA, 1) draws a parabola whose vertex is point A and whose focal length is 1.
parabola(x-y^2+y-2) draws the graph of the parabolic equation x=y^2-y+2
#link discussion
parallel
Syntax
parallel(Point, Line)
Description
Draws a line through a given point that is parallel to a given line.
Example
parallel(A, B) draws the line through point A that is parallel to line B.
parallel(point(3–2*i), line(x+y–5)) draws the line through the point (3, –2) that is parallel to the line whose equation is x+y=5; that is, the line whose equation is y=–x+1.
#link discussion
parallelogram
Syntax
parallelogram(Point1, Point2, Point3)
Description
Draws a parallelogram given three of its vertices. The fourth point is calculated automatically but is not defined symbolically. As with most of the other polygon commands, you can store the fourth point’s coordinates into a CAS variable. The orientation of the parallelogram is counterclockwise from the first point.
Example
parallelogram(0,6,9+5i) draws a parallelogram whose vertices are at (0, 0), (6, 0), (9, 5), and (3,5). The coordinates of the last point are calculated automatically.
#link discussion
parameq
Syntax
parameq(Obj)
Description
Returns a parametric equation for the geometric object Obj. The parametric equation is true for all complex numbers that represent points on Obj.
Example
parameq(circle(0,1)) returns -exp(i*t)
#link discussion
partfrac
Syntax
partfrac(RatFrac or Opt)
Description
Performs partial fraction decomposition on a fraction.
Example
partfrac(x/(4-x^2)) returns (-1/2)/(x-2)-(1/2)/((x+2)
#link discussion
pcoeff
Syntax
pcoeff(Vect)
Description
Returns the polynomial coefficients having the roots specified in the vector Vect.
Example
pcoeff([1,0,0,0,1]) returns poly1[1,-2,1,0,0,0]
#link discussion
perimeter
Syntax
perimeter(Polygon) or perimeter(Circle)
Description
Returns the perimeter of a polygon or the circumference of a circle.
Example
perimeter(0,1,i) returns √2+2
If GA is the point at (0, 0), GB is the point at (1, 0), and GC is defined as circle(GA, GB-GA), then perimeter(GC) returns 2π.
If GA is the point at (0, 0), GB is the point at (1, 0), and GC is defined as square(GA, GB-GA), then perimeter(GC) returns 4.
#link discussion
perimeterat
Syntax
perimeterat(Polygon, Pnt||Cplx(z0))
Description
Displays at point(z0), with a legend, the perimeter of a circle or of a polygon (e.g. triangle, square, ...).
#link discussion
perimeteratraw
Syntax
perimeteratraw(Polygone, Pnt||Cplx(z0))
Description
Displays at point(z0), the perimeter of a circle or of a polygon (e.g. triangle, square, ...).
#link discussion
PERM
Syntax
PERM(n, r)
Description
Permutations. Returns the number of permutations (with regard to order) of n things taken r at a time: n!/(n-r)!
Example
PERM(5,2) returns 20
#link discussion
perpendicular
Syntax
perpendicular(Point, Line) or perpendicular(Point1, Point2, Point3)
Description
Draws a line through a given point that is perpendicular to a given line. The line may be defined by its name, two points, or an expression in x and y.
Example
perpendicular(GA, GD) draws a line perpendicular to line D through point A.
perpendicular(3+2i, GB, GC) draws a line through the point whose coordinates are (3, 2) that is perpendicular to line BC.
perpendicular(3+2i,line(x-y=1)) draws a line through the point whose coordinates are (3, 2) that is perpendicular to the line whose equation is x – y = 1; that is, the line whose equation is y=-x+5.
#link discussion
PI
Syntax
π
Description
The ratio of the circumference to the diameter of any circle. Internally represented as 3.14159265359.
#link discussion
pivot
Syntax
pivot(Mtrx(A),Intg(nl),Intg(nc))
Description
Returns the matrix from A creating zeros in the column nc, by the method of Gauss-Jordan with the element A[nl,nc] as pivot.
Example
pivot([[1,2],[3,4],[5,6]],0,1) returns [[1,2],[0,-2],[0,-4]]
#link discussion
PIXOFF
Syntax
PIXOFF([G], x, y)
Description
Sets the color of the pixel of G with coordinates (x,y) to white.
#link discussion
PIXOFF_P
Syntax
PIXOFF_P([G], x, y)
Description
Sets the color of the pixel of G with coordinates (x,y) to white.
#link discussion
PIXON
Syntax
PIXON([G], x, y, [color])
Description
Sets the color of the pixel of GROB G with coordinates (x,y).
#link discussion
PIXON_P
Syntax
PIXON_P([G], x, y, [color])
Description
Sets the color of the pixel of GROB G with coordinates (x,y).
#link discussion
plotcontour
Syntax
plotcontour(Expr(Xpr),[LstVar],[LstVal])
Description
Draws 11 contour-lines z=z_min,,...z=z_max of the surface z=Xpr, where the contour-lines are defined by the 3rd argument.
#link discussion
plotfield
Syntax
plotfield(Expr,VectVar,[Opt])
Description
plotfield(f(t,y),[t,y]) draws the slope field of the differential equation y'=f(t,y)
#link discussion
plotfunc
Syntax
plotfunc(Expr)
Description
Draws the plot of a function, given an expression in the independent variable x. Note the use of lowercase x.
Example
plotfunc(3*sin(x)) draws the graph of y=3*sin(x).
#link discussion
plotimplicit
Syntax
plotimplicit(Expr,Var1,Var2)
Description
plotimplicit(f(x,y),x,y) or plotimplicit(f(x,y),[x,y]) graph of f(x,y)=0
#link discussion
plotinequation
Syntax
plotinequation(Expr,[x=xrange,y=yrange],[xstep],[ystep])
Description
Shows the graph of the solution of inequations with 2 variables.
#link discussion
plotlist
Syntax
plotlist(Lst(l)||Mtrx(M))
Description
Draws a polygonal line through the points of abscissa 0,...,n and ordinate l=[y0,...,yn] or the line through the points of abscissa in the first M column and the ordinates in the second column.
#link discussion
plotode
Syntax
plotode(Expr,VectVar,VectInitCond)
Description
plotode(f(t,y),[t,y],[t0,y0]) draws the solution of y'=f(t,y) and y(t0)=y0 or of the system [x'=g(t,x,y),y'=h(t,x,y)] with x(t0)=x0 and y(t0)=y0.
#link discussion
plotparam
Syntax
plotparam(Cplx||Lst,Var||Lst(Var))
Description
plotparam(a(x)+i*b(x),x=x0..x1) draws the curve X=a(x),Y=b(x) x=x0..x1 or plotparam([a(u,v),b(u,v),c(u,v)],[u=u0..u1,v=v0..v1]) draws the surface X=a(u,v),Y=b(u,v),Z=c(u,v) u=u0..u1 and v=v0..v1.
#link discussion
plotpolar
Syntax
plotpolar(Expr,Var,VarMin,VarMax)
Description
plotpolar(f(x),x,a,b) draws the polar curve r=fx) for x in [a,b]
#link discussion
plotseq
Syntax
plotseq(Expr(f(Var)),Var=[a,xm,xM],Intg(p))
Description
For seeing the pth terms of the sequence u(0)=a,u(n)=f(u(n-1))
#link discussion
pmin
Syntax
pmin(Mtrx,[Var])
Description
Returns the minimal polynomial of a square matrix.
Example
pmin([[1,0],[0,1]],x) returns x-1
#link discussion
point
Syntax
point(Real1, Real2) or point(Expr1, Expr2) or point(Complex)
Description
Creates a point, given the coordinates of the point. Each coordinate may be a value or an expression involving variables or measurements on other objects in the geometric construction.
Example
point(3,4) creates a point whose coordinates are (3,4). This point may be selected and moved later.
point(abscissa(GA), ordinate(GB)) creates a point whose x-coordinate is the same as that of a point A and whose y-coordinate is the same as that of a point B. This point will change to reflect the movements of point A or point B.
#link discussion
point2d
Syntax
point2d(Var1, Var2, ..., Varn)
Description
Randomly re-distributes a set of points such that, for each point, x is in the interval [-5, 5] and y is in the interval [-5, 5]. Any further movement of one of the points will randomly re-distribute all of the points.
#link discussion
POISSON
Syntax
POISSON(μ, k)
Description
Poisson probability mass function. Computes the probability of k occurrences of an event in a time interval, given μ expected (or mean) occurrences of the event in that interval. For this function, k is a non-negative integer and μ is a real number.
Example
POISSON(4, 2) returns 0.14652511111
#link discussion
POISSON_CDF
Syntax
POISSON_CDF(μ, x)
Description
Cumulative poisson distribution function. Returns the probability of x or fewer occurrences of an event in a given time interval, given μ expected (or mean) occurrences.
POISSON_CDF(4, 2) returns 0.238103305554
#link discussion
POISSON_ICDF
Syntax
POISSON_ICDF(μ, p)
Description
Inverse cumulative poisson distribution function. Returns the value x such that the probability of x or fewer occurrences of an event in a time interval, with μ expected (or mean) occurrences of the event in the interval, is p.
Example
POISSON_ICDF(4, 0.238103305554) returns 2
#link discussion
polar
Syntax
polar(Crcle,Pnt or Cplxe(A))
Description
Returns the line of the conjugated points of A with respect to the circle.
#link discussion
polar_coordinates
Syntax
polar_coordinates(Pnt or Cplx or LstRectCoord)
Description
Returns the list of the norm and of the argument of the affix of a point (for 2D) or of a complex number or of the the list of rectangular coordinates.
Example
polar_coordinates(point(1+2*i)) returns [√5,atan(2)]
#link discussion
polar_point
Syntax
polar_point(Real(r),Real(t))
Description
Returns the point (for 2D) with the arguments r and t as polar coordinates.
#link discussion
pole
Syntax
pole(Crcle,Line)
Description
Returns the point having the line as polar with respect to the circle.
#link discussion
poly2symb
Syntax
poly2symb(Lst,Var)
Description
Returns a polynomial (orthe polynomial and its value) in Var (by default x), the polynomial being defined by the vector of coefficents in Vect .
Example
poly2symb([1,2,3],x) returns (x+2)*x+3
poly2symb([1,2,3],x=2) returns (x+2)*x+3=11
#link discussion
POLYCOEF
Syntax
POLYCOEF(vector or list)
Description
Polynomial coefficients. Returns the coefficients of the polynomial with the roots specified in vector.
POLYCOEF({-1, 1}) returns {1, 0, -1}
#link discussion
POLYEVAL
Syntax
POLYEVAL(vector or list , value)
Description
Polynomial evaluation. Evaluates a polynomial with the coefficients specified in vector, at value.
POLYEVAL({1, 0, -1}, 3) returns 8
#link discussion
polygon
Syntax
polygon(Point1, Point2, …, Pointn)
Description
Draws a polygon from a set of vertices.
Example
polygon(GA, GB, GD) draws ΔABD
#link discussion
polygonplot
Syntax
polygonplot(Mtrx)
Description
Draws the polygons joining for j fixed and for k=0..nrows, the points (xk,yk) where xk=element row k column 0 and yk=element row k column j, when the xk are sorted (we obtain ncols-1 polygons).
#link discussion
polygonscatterplot
Syntax
polygonscatterplot(Mtrx)
Description
Draws the points (xk,yk) and the polygons joining for j fixed and for k=0..nrows, the points (xk,yk) where xk=element row k column 0 and yk=element row k column j ,when the xk are sorted (we obtain ncols-1 polygons).
#link discussion
polynomial_regression
Syntax
polynomial_regression(Lst||Mtrx(A),[Lst],Intg(n))
Description
Returns the coefficients (an,...a1,a0) of y=an*x^n+..a1x+a0): it is the best polynomial that approximates the points where the coordinates are the rows of A (or the 2 lists) (n is the 2nd argument).
Example
polynomial_regression([[1.0,1.0],[2.0,4.0],[3.0,9.0],[4.0,16.0]],3) returns [-0.0,1.0,-0.0,0.0]
#link discussion
POLYROOT
Syntax
POLYROOT(vector)
Description
Polynomial roots. Returns the roots for the polynomial whose coefficients are specified in vector.
Example
POLYROOT([1, 0, -1]) returns {-1, 1}
#link discussion
POS
Syntax
POS(list, element)
Description
List Position. Returns the position of element within list. If there is more than one instance of the element, the position of the first occurrence is returned. A value of 0 is returned if there is no occurrence of the specified element.
Example
POS({0, 1, 3, 5}, 1) returns 2
#link discussion
potential
Syntax
potential(Vect(V),VectVar)
Description
Returns U such as derive(U,Vector_of_variable)=V
Example
potential([2*x*y+3,x^2-4*z,-4*y],[x,y,z]) returns 2*x^2*y/2+3*x-4*y*z
#link discussion
pow2exp
Syntax
pow2exp(Expr)
Description
Converts powers to exponentials. Essentially the inverse of exp2pow.
Example
pow2exp(a^b) returns exp(b*ln(a))
#link discussion
power_regression
Syntax
power_regression(Lst|Mtrx(A),[Lst])
Description
Returns the coefficients (m,b) of y=b*x^m: it is the best monomial that approximates the points where the coordinates are the rows of A (or the 2 lists).
Example
power_regression([[1.0,1.0],[2.0,4.0],[3.0,9.0],[4.0,16.0]]) returns 2.0,1.0
#link discussion
powerpc
Syntax
powerpc(Cercle,Pnt or Cplx)
Description
Returns the real number d^2-R^2 (d=distance between point and center, R=radius).
Example
powerpc(circle(0,1+i),3+i) returns 8
#link discussion
powexpand
Syntax
powexpand(Expr)
Description
Expresses a power in the form of a product.
Example
powexpand(2^(x+y)) yields (2^x)*(2^y)
#link discussion
powmod
Syntax
powmod(a, n, p)
Description
Power and modulo. For the integers a, n, and p, returns a^n mod p.
Example
powmod(5,2,13) returns 12
#link discussion
prepend
Syntax
prepend(Lst,Elem )
Description
Puts the element at the beginning of the list.
Example
prepend([1,2],3) returns [3,1,2]
#link discussion
preval
Syntax
preval(f(Var), Real1, Real2, [Var])
Description
Returns f(Real2)-f(Real1).
Example
preval(x^2-2,2,3) returns 5
#link discussion
prevprime
Syntax
prevprime(a)
Description
Previous prime. Returns the previous prime number before the integer a.
Example
prevprime(11) returns 7
#link discussion
primpart
Syntax
primpart(Poly,[Var])
Description
Returns the polynomial P divided by the gcd of its coefficients.
Example
primpart(2x^2+10x+6) returns x^2+5*x+3
#link discussion
PRINT
Syntax
PRINT(expr) or PRINT(string) or PRINT( )
Description
Prints either the result of expr or string to the terminal.
The terminal is a program text output viewing mechanism which is displayed only when PRINT commands are executed. When visible, you can use the up/down keys to view the text, BKSP to erase the text and any other key to hide the terminal. You can show the terminal at anytime using the ON+T combination (press and HOLD the ON key, then press the T key, then release both keys). Pressing ON stops the interaction with the terminal.
PRINT with no argument clears the terminal.
#link discussion
product
Syntax
product(Expr||Lst,[Var||Lst],[Intg(a)],[Intg(b)],[Intg(p)])
Description
Multiplicates the values of the expression when the variable goes from a to b with a step p (product expression,var,begin,end,step) by default p=1) or product of the elements of a list or product element by element of 2 lists or matrix.
Example
product(n,n,1,10,2) returns 945
#link discussion
projection
Syntax
projection(Curve, Point)
Description
Draws the orthogonal projection of a point onto a curve.
Example
projection(circle(x^2+y^2=4),point(6,6)) creates a point on the circle at (√2, √2)
#link discussion
proot
Syntax
proot(Vect||Poly)
Description
Returns all computed roots of a polynomial given by its coefficients (may not work if roots are not simple).
Example
proot([1,0,-2]) returns [-1.41421356237,1.41421356237]
#link discussion
propfrac
Syntax
propfrac(Frac or RatFrac)
Description
Simplifies and writes the fraction (or rationnal fraction) A/B as Q+R/B with R<B (or deg(R)<deg(B))
Example
propfrac(28/12) returns 2+1/3
#link discussion
Psi
Syntax
Psi(Real(a),Intg(n))
Description
Psi(a,n) returns the nth derivative of the digamma function at x=a (Psi(a,0)=Psi(a))
Example
Psi(3,1) returns π^2/6-5/4
#link discussion
ptayl
Syntax
ptayl(Poly(P(var)),Real(a),[Var])
Description
Returns the Taylor polynomial Q such as P(x)=Q(x-a)
Example
ptayl(x^2+2*x+1,1) returns x^2+4*x+4
#link discussion
PX→C
Syntax
PX→C(x, y) or PX→C({x, y})
Description
Transform pixel coordinates into cartesian coordinates. Returns a list.
#link discussion
q2a
Syntax
q2a(QuadraForm,VectVar)
Description
q2a(q(x,y),[x,y]) returns the symmetric matrix associated with the quadratic form q
Example
q2a(x^2+2*x*y+2*y^2,[x,y]) returns [[1,1],[1,2]]
#link discussion
QR
Syntax
QR(matrix)
Description
QR Factorization. Factors an mn matrix into three matrices:
{[[mm orthogonal]],[[mn uppertrapezoidal]],[[nn permutation]]}.
Example
QR([[1,2],[3,4]])
#link discussion
quadrilateral
Syntax
quadrilateral(Point1, Point2, Point3, Point4)
Description
Draws a quadrilateral from a set of four points.
Example
quadrilateral(GA, GB, GC, GD) draws quadrilateral ABCD.
#link discussion
quantile
Syntax
quantile(Lst(l),Real(p))
Description
Returns the quantile of the elements of l corresponding to p (0<p<1)
Example
quantile([0,1,3,4,2,5,6],0.25) returns [1.0]
#link discussion
quartile1
Syntax
quartile1(Lst||Mtrx,[Lst])
Description
Returns the 1st quartile of the elements (or of the columns) of the argument.
Example
quartile1([1,2,3,5,10,4]) returns 2.0
#link discussion
quartile3
Syntax
quartile3(Lst||Mtrx,[Lst])
Description
Returns the 3rd quartile of the elements (or of the columns) of the argument
Example
quartile3([1,2,3,5,10,4]) returns 5.0
#link discussion
quartiles
Syntax
quartiles(Lst||Mtrx,[Lst])
Description
Returns the min, 1st quartile, median, 3rd quartile, and max of the elements (or of the columns) of the argument.
Example
quartiles([1,2,3,5,10,4]) returns [[1.0],[2.0],[3.0],[5.0],[10.0]]
#link discussion
quo
Syntax
quo((Vect or Poly),(Vect or Poly),[Var])
Description
Returns the Euclidean quotient of 2 polynomials
Example
quo([1,2,3,4],[-1,2]) returns poly1[-1,-4,-11]
#link discussion
quorem
Syntax
quorem(Poly1, Poly2) or quorem(Vector1, Vector2)
Description
Returns the Euclidean quotient and remainder of the quotient of 2 polynomials in a vector. If the polynomials are expressed as vectors of their coefficients, then this command returns a similar vector of the quotient and a vector of the remainder.
Example
quorem(x^3+2*x^2+3*x+4,-x+2) returns [-x^2-4*x-11, 26]
quorem([1,2,3,4],[-1,2]) returns [[-1, -4, -11] [26]]
#link discussion
QUOTE
Syntax
QUOTE(expression)
Description
Returns the expression unchanged and un-evaluated.
This function is mostly used with the STO▶ command in order to store a function in a function variable. For example if you want to store SIN(X) in F1.you cannot do SIN(X)►F1 as SIN(X) would be evaluated and a numerical result would be stored into F1. QUOTE(SIN(X))►F1 will store SIN(X) in F1.
#link discussion
radical_axis
Syntax
radical_axis(Crcle,Crcle)
Description
Returns the line of points with same powerpc with respect to the 2 circles.
#link discussion
radius
Syntax
radius(Circle)
Description
Returns the radius of a circle.
Example
If GA is the point at (0, 0), GB is the point at (1, 0), and GC is defined as circle(GA, GB-GA), then radius(GC) returns 1.
#link discussion
randexp
Syntax
randexp(Real(a))
Description
Returns a random real according to the exponential distribution of parameter a>0
Example
randexp(1) returns 1.17118631006
#link discussion
RANDINT
Syntax
RANDINT([a],[b],[c])
Description
Random number. Returns a pseudo-random integer generated using a seed value, and updates the seed value.
With no argument, this function returns a random integer x from 0 to 1. With one argument, this returns a random integer x from 0 to a. With two arguments, this returns a random integer x from a to b. With three arguments, this returns a list of size a with each element being a random integer x from b to c.
Example
RANDINT(3,1,6) returns { random1, random2, random3 }
#link discussion
RANDMAT
Syntax
RANDMAT (matrixname, rows, columns)
Description
Creates a random matrix with the specified number of rows and columns, and stores the result in matrixname. The entries will be integers ranging from –99 to 99.
Example
RANDMAT(M1,2,2) returns [[n1,n2],[n3,n4]]
#link discussion
randMat
Syntax
ranm(Intg(n),[Intg(m)],[Interval or quote(DistribLaw)])
Description
Returns a list of size n or a n*m matrix that contains random integers in the range -99 through 99 with uniform distribution or contains random numbers according to the law in quote.
Example
ranm(3) returns [-20,72,-86]
#link discussion
RANDNORM
Syntax
RANDNORM([μ],[σ]) or RANDNORM(n,μ,σ)
Description
Return a random number from the normal distribution with the specified mean μ and standard deviation σ. Default values are 0 and 1.
With three arguments, returns a list of size n with each element being a random number fron the normal distribution with the specified mean μ and standard deviation σ.
Example
RANDNORM(3,0,1) returns { random1, random2, random3 }
#link discussion
randNorm
Syntax
randnorm(Real(mu),Real(sigma))
Description
Returns a random real with normal distribution N(mu,sigma)
Example
randnorm(0,1) returns -0.860967215689
#link discussion
RANDOM
Syntax
RANDOM([a],[b],[c])
Description
Random number. Returns a pseudo-random number generated using a seed value, and updates the seed value.
With no argument, this function returns a random number x with 0 ≤ x < 1. With one argument, this returns a random number x with 0 ≤ x < a. With two arguments, this returns a random number x with a ≤ x < b. With three arguments, this returns a list of size a with each element being a random number x with b ≤ x < c.
Example
RANDOM(3,0,10) returns { random1, random2, random3 }
#link discussion
randperm
Syntax
randperm(Intg(n))
Description
Returns a random permutation of [0,1,2,..,n-1]
Example
randperm(4) returns [2,1,3,0]
#link discussion
randPoly
Syntax
randpoly([Var],Intgr,[Dist])
Description
Returns a vector of coefficients of a polynomial of variable Var (or x), of degree Intgr and where the coefficients are random integers in the range -99 through 99 with uniform distribution or in an interval specified by Intrvl.
Example
randpoly(t, 8, -1..1) returns a vector of 9 random integers, all of them between -1 and 1.
#link discussion
RANDSEED
Syntax
RANDSEED([value])
Description
Sets the random number generator seed. With no input, uses current time value as seed.
Example
RANDSEED(3.14)
#link discussion
RANK
Syntax
RANK(matrix)
Description
Rank of a rectangular matrix.
Example
RANK([[1,2],[3,4]]) returns 2
#link discussion
ratnormal
Syntax
ratnormal(Expr)
Description
Rewrites Expr as an irreducible rational fraction
Example
ratnormal((x^2-1)/(x^3-1)) returns (x+1)/(x^2+x+1)
#link discussion
RE
Syntax
RE(x+yi)
Description
Real Part. Returns the real part of a complex number.
Example
RE(3+4i) returns 3
#link discussion
reciprocation
Syntax
reciprocation(Crcle,Lst(Pnt,Line))
Description
Returns the list where a point is replaced with its polar or a line is replaced with its pole, with respect to the circle C
#link discussion
RECT
Syntax
RECT([G], [x1, y1], [x2, y2], [edgecolor],[fillcolor])
Description
Draws a rectangle on G, with diagonal defined by points (x1,y1) and (x2,y2), using edgecolor for the perimeter and fillcolor for the inside.
The following values are optional and their defaults are listed:
x1, y1=top left corner of G
x2, y2=bottom right corner of G
edgecolor=white
fillcolor=edgecolor
Note: To erase a GROB, execute RECT(G). To clear the screen execute RECT().
#link discussion
RECT_P
Syntax
RECT_P([G], [x1, y1], [x2, y2], [edgeColor],[fillColor])
Description
Draws a rectangle on G, with diagonal defined by points (x1,y1) and (x2,y2), using edgeColor for the perimeter and fillColor for the inside.
The following values are optional and their defaults are listed:
x1, y1=top left corner of G
x2, y2=bottom right corner of G
edgeColor=white
fillColor=edgeColor
Note: To erase a GROB, execute RECT(G). To clear the screen, execute RECT().
#link discussion
rectangle
Syntax
rectangle(Point1, Point2, Point3) or rectangle(Point1, Point2, Realk)
Description
Draws a rectangle given two consecutive vertices and a point on the side opposite the side defined by the first two vertices or a scale factor for the sides perpendicular to the first side. As with many of the other polygon commands, you can specify optional CAS variable names for storing the coordinates of the other two vertices as points.
Example
rectangle(GA, GB, GE) draws a rectangle whose first two vertices are points A and B (one side is segment AB). Point E is on the line that contains the side of the rectangle opposite segment AB.
rectangle(GA, GB, 3, p, q) draws a rectangle whose first two vertices are points A and B (one side is segment AB). The sides perpendicular to segment AB have length 3*AB. The third and fourth points are stored into the CAS variables p and q, respectively.
#link discussion
rectangular_coordinates
Syntax
rectangular_coordinates(LstPolCoord)
Description
Returns the list of the abscissa and of the ordinate of a point given by the list of its polar coordinates.
Example
rectangular_coordinates([1,-1]) returns [cos(1),-sin(1)]
#link discussion
red
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
#link discussion
REDIM
Syntax
REDIM(matrixname, size)
Description
Redimensions the specified matrix or vector to size. For a matrix, size is a list of two integers {n1, n2}. For a vector, size is a list containing one integer {n}. Existing values in the matrix are preserved. Fill values will be zeros.
#link discussion
reduced_conic
Syntax
reduced_conic(Expr,[LstVar])
Description
Returns the origin and the matrix of a base in which the conic (given by its equation) is reduced, 0 or 1 (0 if the conic is degenerate), and the equation of the conic in this base and also its parametric equation
Example
reduced_conic(x^2+2*x-2*y+1) returns [[-1,0],[[0,1],[-1,0]],1,y^2+2*x,[[-1+(-i)*(t*t/-2+(i)*t),t,-4,4,0.1]]]
#link discussion
ref
Syntax
ref(Mtrx(M))
Description
Performs Gauss reduction of a matrix AX=b (M=A|(-b))
Example
ref([[3,1,-2],[3,2,2]]) returns [[1,1/3,-2/3],[0,1,4]]
#link discussion
reflection
Syntax
reflection(line, Object) or reflection(Point, Object)
Description
Reflects a geometric object over a line or through a point. The latter is sometimes referred to as a half-turn.
Example
eflection(line(x=3),point(1,1)) reflects the point at (1, 1) over the vertical line x=3 to create a point at (5,1).
reflection(1+I, 3-2i) reflects the point at (3, -2) through the point at (1, 1) to create a point at (-1, 4).
#link discussion
rem
Syntax
rem(Poly1, Poly2) or rem(Vector1, Vector2)
Description
Returns the Euclidean remainder of the quotient of 2 polynomials. If the polynomials are expressed as vectors of their coefficients, then this command returns a similar vector of the remainder.
Example
rem(x^3+2*x^2+3*x+4,-x+2) returns 26
rem([1,2,3,4],[-1,2]) returns [26]
#link discussion
remove
Syntax
remove(FncBool(f)||e,Lst(l))
Description
Removes the occurences e of l or the elements e such that f(e)=true
Example
remove(x->x>=5,[1,2,6,7]) returns [1,2]
#link discussion
reorder
Syntax
reorder(Expr,LstVar)
Description
Reorders the variables in E according to the order of the 2nd argument
Example
reorder(x^2+2*x+y^2,[y,x]) returns y^2+x^2+2*x
#link discussion
REPEAT
Syntax
REPEAT command(s) UNTIL test;
Description
executes command(s) UNTIL the test is true.
A:=5;
REPEAT
PRINT(A);
A:= A-1;
UNTIL A<1;
will print 5 4 3 2 1
#link discussion
REPLACE
Syntax
REPLACE(object,start,object)
Description
Replaces portion of a matrix, vector or string starting at start by object.
For a matrix, start is a list containing two numbers; for a vector or string it is a single number.
Note: for strings, you can do: REPLACE("string", "sub_string", "replace_string")
#link discussion
residue
Syntax
residue(Expr,Var(v),Cplx(a))
Description
Returns the residue in a of the expression Expr with v as variable
Example
residue(1/z,z,0) returns 1
#link discussion
resultant
Syntax
resultant(Poly,Poly,Var)
Description
Returns the inert form of the resultant for modular computation (irem/mod)
#link discussion
RETURN
Syntax
RETURN expression;
Description
Exits from a function and returns the value of expression (optional).
Example
EXPORT FACTORIAL(N)
BEGIN
IF N==1 THEN RETURN 1; ELSE RETURN N*FACTORIAL(N-1); END;
END;
#link discussion
REVERSE
Syntax
REVERSE(list)
Description
Reverse list. Reverses the order of the elements in list and returns them in a new list.
Example
REVERSE({2, 3, 4, 5}) returns {5, 4, 3, 2}.
#link discussion
revlist
Syntax
revlist(Lst(l))
Description
Returns the list l in reverse order
Example
revlist([1,2,3]) returns [3,2,1]
#link discussion
RGB
Syntax
RGB(R, G, B, [A])
Description
Returns an integer number that can be used as the color parameter for a drawing function. Based on Red, Green and Blue components values (0 to 255).
If Alpha is greater than 128, returns the color flagged as transparent. There is no alpha channel blending on Prime.
#link discussion
rhombus
Syntax
rhombus(Pnt(A)||Cplx,Pnt(B)||Cplx,Angle(a)||Pnt(P)||Lst(P,a)),[Var(C)],[Var(D)])
Description
Returns and draws the rhombus ABCD such that the angle (AB,AD)=a or such that in the plane ABP the angle(AB,AD)=angle(AB,AP)
#link discussion
RIGHT
Syntax
RIGHT(string, n)
Description
Returns the last n characters of the string.
Example
RIGHT("MOMOGUMBO",5) returns "GUMBO"
#link discussion
right_triangle
Syntax
right_triangle(Point1, Point2, Realk)
Description
Draws a right triangle given two points and a scale factor. One leg of the right triangle is defined by the two points, the vertex of the right angle is at the first point, and the scale factor multiplies the length of the first leg to determine the length of the second leg.
Example
right_triangle(GA, GB, 1) draws an isosceles right triangles with its right angle at point A, and with both legs equal in length to segment AB.
#link discussion
romberg
Syntax
romberg(Expr(f(x)),Var(x),Real(a),Real(b))
Description
Uses Romberg's method to return the approximate value of the integral of the expression over the interval a to b
Example
romberg(exp(x^2),x,0,1) returns 1.46265174591
#link discussion
ROTATE
Syntax
ROTATE(string, n)
Description
If n is not negative, takes the first n characters of string and put them on the right of string. If n is negative, takes the last n characters and put them on the left of string. If ABS(n)>dim(string), returns string.
Example
ROTATE("12345",2) returns "34512"
ROTATE("12345",-1) returns "51234"
ROTATE("12345",6) returns "12345"
#link discussion
rotation
Syntax
rotate(Point, Angle, Object)
Description
Rotates a geometric object, about a given center point, through a given angle.
Example
rotate(GA, angle(GB, GC, GD),GK) rotates the geometric object labeled K, about point A, through an angle equal to ∡CBD.
#link discussion
ROUND
Syntax
ROUND(value, [places])
Description
Rounds value to system display settings. If optional places is given, rounds value to places decimal places. If places is negative, rounds to significant digits instead.
Example
ROUND(7.8676,2) returns 7.87
#link discussion
row
Syntax
row(Mtrx(A),Intg(n)||Interval(n1..n2))
Description
Returns the row n or the sequence of the rows n1..n2 of the matrix A
Example
row([[1,2,3],[4,5,6],[7,8,9]],1) returns [4,5,6]
#link discussion
rowAdd
Syntax
rowAdd(Mtrx(A),Intg(n1),Intg(n2))
Description
Returns the matrix obtained from matrix A when the n2th row is replaced by the sum of the n1th and n2th rows
Example
rowAdd([[1,2],[3,4],[5,6]],1,2) returns [[1,2],[3,4],[8,10]]
#link discussion
rowDim
Syntax
rowDim(Mtrx)
Description
Returns the number of rows of a matrix
Example
rowDim([[1,2,3],[4,5,6]]) returns 2
#link discussion
ROWNORM
Syntax
ROWNORM(matrix)
Description
Row Norm. Finds the maximum value (over all rows) for the sums of the absolute values of all elements in a row.
Example
ROWNORM([[1,2],[3,4]]) returns 7
#link discussion
rowSwap
Syntax
rowSwap(Mtrx(A),Intg(n1),Intg(n2))
Description
Returns the matrix obtained from A by swapping the n1th row and the n2th row
Example
rowSwap([[1,2],[3,4],[5,6]],1,2) returns [[1,2],[5,6],[3,4]]
#link discussion
RREF
Syntax
RREF(matrix)
Description
Reduced-Row Echelon Form. Changes a rectangular matrix to its reduced row-echelon form.
Example
RREF([[1,-2,1],[3,4,-1]]) returns [[1,0,.2],[0,1,-.4]]
#link discussion
rsolve
Syntax
rsolve((Expr or LstExpr),(Var or LstVar),(InitVal or LstInitVal))
Description
Gives the value of a recurrent sequence or of a system of recurrent sequences
Example
rsolve(u(n+1)=2*u(n)+n,u(n),u(0)=1 returns [-n+2*2^n-1]
#link discussion
R→B
Syntax
R→B(Real [, bits [,base]])
Description
Transform a real number into an integer. Optionally specifies bits and base.
-64<Bits<65
0<=Base<=4
0: system, 1: bin, 2: oct, 3: dec, 4: hex
#link discussion
SCALE
Syntax
SCALE(matrixname, value, rownumber)
Description
Multiplies the specified row_number of the specified matrix by value.
#link discussion
SCALEADD
Syntax
SCALEADD(matrixname, value, row1, row2)
Description
Multiplies the specified row1 of the matrix name by value, then adds this result to the second specified row2 of the matrix matrixname.
#link discussion
SCHUR
Syntax
SCHUR(matrix)
Description
Schur Decomposition. Factors a square matrix into two matrices. If matrix is real, then the result is {[[orthogonal]],[[upper-quasi triangular]]}.
If Complex mode is on and the matrix is complex, then the result is
{[[unitary]],[[upper-triangular]]}.
Example
SCHUR([[1,2],[3,4]])
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SEC
Syntax
SEC(value)
Description
Secant. The Secant function; that is, 1/cos(x).
Example
SEC(0) returns 1 in degree mode
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segment
Syntax
segment(Point1, Point2)
Description
Draws a segment defined by its endpoints.
Example
segment(1+2i, 4) draws the segment defined by the points whose coordinates are (1, 2) and (4, 0).
segment(GA, GB) draws segment AB.
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select
Syntax
select(FncBool(f),Lst(l))
Description
Selects the elements e of l such that f(e)=true
Example
select(x->x>=5,[1,2,6,7]) returns [6,7]
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seq
Syntax
seq(Expr(Xpr),Intg(n)||Var(var),[Intg(a)],[Intg(b)],[Intg(p)])
Description
Returns the sequence (if 2 or 3 arguments) or the list (if 4 or 5 arguments) obtained when var goes from a to b (step p) in Xpr, or when Xpr is repeated n times.
Example
seq(2^k,k=0..8) returns 1,2,4,8,16,32,64,128,256
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seqsolve
Syntax
seqsolve((Expr or LstExpr),(Var or LstVar),(InitVal or LstInitVal))
Description
Gives the value of a recurrent sequence (u_{n+1}=f(u_n) or u_{n+2}=f(u_{n+1},u_n)...) or of a system of recurrent sequences
Example
seqsolve(2x+n,[x,n],1) returns -n-1+2*2^n
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series
Syntax
series(Expr,Equal(var=limit_point),[Order],[Dir(1,0,-1)])
Description
Returns the series expansion of an expression in the vicinity of a given equality variable. With the optional third and fourth arguments you can specify the order and direction of the series expansion. If no order is specified the series returned is
fifth order. If no direction is specified, the series is bidirectional.
Example
series((x^4+x+2)/(x^2+1),x=0,5) returns 2+x-2x^2-x^3+3x^4+x^5+x^6*order_size(x)
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SETBASE
Syntax
SETBASE(#integer[, base])
Description
Sets the base used for display of this integer.
If base is not specified the calculator default is used.
0<=Base<=4
0: system, 1: bin, 2: oct, 3: dec, 4: hex
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SETBITS
Syntax
SETBITS(#integer[, bits])
Description
Sets the number of bits used for calculations with this integer to bits.
If bits is not specified the calculator default is used.
-64<Bits<65
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shift_phase
Syntax
shift_phase(Expr)
Description
Returns the expressions where the phase of the evaluated trigonometric expressions is increased by π/2
Example
shift_phase(sin(x)) returns -cos((π+2*x)/2)
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Si
Syntax
Si(Expr)
Description
Sine integral int(sin(t)/t,t=0..x)
Example
Si(1.0) returns 0.946083070367
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SIGN
Syntax
SIGN(value) or SIGN(x+yi)
Description
Sign. Returns the sign of value. If positive, the result is 1; if negative, -1. If zero, the result is zero. For complex inputs returns the unit vector.
Example
SIGN (2) returns 1
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signature
Syntax
signature(Permut)
Description
Returns the signature of a permutation
Example
signature([2,1,4,5,3]) returns -1
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similarity
Syntax
similarity(Point, Realk, Angle, Object)
Description
Dilates and rotates a geometric object about the same center point.
Example
similarity(0, 3, angle(0,1,i),point(2,0)) dilates the point at (2,0) by a scale factor of 3 (a point at (6,0)), then rotates the result 90° counterclockwise to create a point at (0, 6)
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simplify
Syntax
simplify(Expr)
Description
Simplifies an expression.
Example
simplify(4*atan(1/5)-atan(1/239)) yields (1/4)*π
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simult
Syntax
simult(Mtrx(A),Mtrx(B))
Description
Returns the matrix where the column of index k is solution of A*X=column of index k of B (=B[0..nr-1,k..k] with nr=number of rows of B)
Example
simult([[3,1],[3,2]],[[-2],[2]]) returns [[-2],[4]]
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SIN
Syntax
SIN(Value)
Description
Returns the sine of Value. Value is interpreted as either degrees or radians, depending on the setting of Angle Measure in Home Modes or Symbolic Setup.
Example
in radians mode, SIN(π/2) returns 1
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sin2costan
Syntax
sin2costan(Expr)
Description
Rewrites Expr so that sin(x) is replaced by cos(x)*tan(x)
Example
sin2costan(sin(x)) returns tan(x)*cos(x)
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sincos
Syntax
sincos(Expr)
Description
Returns an expression with the complex exponentials rewritten in terms of sine and cosine.
Example
sincos(exp(-i*x)) returns cos(x)-i*sin(x)
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single_inter
Syntax
single_inter(Curve,Curve,[Pnt(A)||LstPnt(L)])
Description
Gives one of the intersections of 2 curves or surfaces (or the intersection near A or not in L)
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SINH
Syntax
SINH(value)
Description
Hyperbolic sine.
Example
SINH(1) returns 1.17520119364
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SIZE
Syntax
SIZE(list)
Description
List Size. Returns the number of elements in list.
Example
SIZE({0, 1, 2, 3}) returns 4
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slope
Syntax
slope(Line||Pnt||Cplx,[Pnt||Cplx])
Description
Returns the slope of the line defined in the argument
Example
slope(line(1,2i)) returns -2
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slopeat
Syntax
slopeat(Segment, Point) or slopeat(Line, Point) or slopeat(Ray, Point)
Description
Displays, with a legend, the value of the slope of the segment, ray, or line (Line may be a tangent, bisector, etc.) at the location Point in Plot view.
Example
slopeat(line(point(0,1), point(3,2)),point(-10,4)) places "sline(point(0,1),point(3,2)=1/3" at the point (-10,4) in Plot view
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slopeatraw
Syntax
slopeatraw(Line, Pnt||Cplx(z0))
Description
slopeatraw(d,z0) displays the value of the slope of the line or segment d at point(z0)
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solve
Syntax
solve(Expr,[Var] )
Description
Solves a polynomial equation or a set of polynomial equations.
Example
solve(x^2-3=1) returns {-2,2}
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SORT
Syntax
SORT(list)
Description
Sort list. Sorts the elements of list in ascending order.
Example
SORT({2, 9, 5, 3}) returns {2, 3, 5, 9}.
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SPECNORM
Syntax
SPECNORM(matrix)
Description
Spectral Norm of matrix.
Example
SPECNORM([[1,2],[3,4]]) returns 5.46498570422
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SPECRAD
Syntax
SPECRAD(matrix)
Description
Spectral radius of matrix.
Example
SPECRAD([[1,2],[3,4]]) returns 5.37228132327
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spline
Syntax
spline(Lst(lx),Lst(ly),Var(x),Intg(d))
Description
Returns the natural spline through the points given by lx and ly, variable x, degree d
Example
spline([0,1,2],[1,3,0],x,3) returns [-5*x^3/4+13*x/4+1,5*(x-1)^3/4+-15*(x-1)^2/4+(x-1)/-2+3]
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sqrfree
Syntax
sqrfree(Expr)
Description
Returns a polynomial factorized as a product of powers of coprime factors where each factor has roots of multiplicity 1
Example
sqrfree(x^4-2*x^2+1) returns (x^2-1)^2
#link discussion
sqrt
Syntax
√(Expr)
Description
Returns the square root of Expr
Example
√50 returns 5*√2
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square
Syntax
square(Point1, Point2)
Description
Draws a square, given two consecutive vertices as points.
Example
square(0, 3+2i, p, q) draws a square with vertices at (0, 0), (3, 2), (1, 5), and (-2, 3). The last two vertices are computed automatically and are saved into the CAS variables p and q.
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STARTAPP
Syntax
STARTAPP("AppName")
Description
Starts the app AppName. The App's START function will run if present. The App’s default view will be started. Note that the START function is always executed when the user presses the START menu key in the App Library. Also works for apps saved in the App Library.
#link discussion
STARTVIEW
Syntax
STARTVIEW(ViewNumber[,Redraw])
Description
Starts a view of the current app. Redraw, is optional; if Redraw, is true (non 0), it will force a refresh for the view.
The view numbers are as follows:
0=Symbolic
1=Plot
2=Numeric
3=Symbolic Setup
4=Plot Setup
5=Numeric Setup
6=App Info
7=Views key
If the current app has views defined under the Views menu, then the following view numbers are used:
8=First special view (Split Screen Plot Detail)
9=Second special view (Split Screen Plot Table)
10=Third special view (Autoscale)
11=Fourth special view (Decimal)
12=Fifth special view (Integer)
13=Sixth special view (Trig)
If ViewNumber is negative, the following global views are used:
-1=HomeScreen
-2=Modes
-3=Memory Manager
-4=App Library
-5=Matrix Catalog
-6=List Catalog
-7=Program Catalog
-8=Note Catalog
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stddev
Syntax
stddev(Lst||Mtrx,[Lst])
Description
Returns the standard deviation of the elements in a list or of the list of standard deviations
Example
stddev([1,2,3]) returns (√6)/3
#link discussion
stddevp
Syntax
stddevp(Lst||Mtrx,[Lst])
Description
Returns the population standard eviation of the elements of a list with the second argument as weight.
Example
stddevp([1,2,3]) returns 1
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STEP
Syntax
FOR var FROM start TO (or DOWNTO) finish [STEP increment] DO command(s) END;
Description
Sets variable var to start; then, for as long as this variable’s value is less than or equal to (or more than for a DOWNTO) finish, executes command(s) and adds (or substract for DOWNTO) 1 (or increment) to var.
FOR A FROM 1 TO 10 STEP 2
DO
PRINT(A);
END;
will print 1 3 5 7 9
#link discussion
sto
Syntax
sto(arg1,Var)
Description
Stores the first argument in the variable given as second argument
Example
sto("hello",b)
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STRING
Syntax
STRING(expression)
Description
Evaluates expression and returns the result as a string.
#link discussion
STRINGFROMID
Syntax
STRINGFROMID(integer)
Description
Returns the built-in string associated with the ID of the current language.
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STUDENT
Syntax
STUDENT(n, x)
Description
Student’s t probability density function. Computes the probability density of the Student’s-t distribution at x, given n degrees of freedom.
Example
STUDENT(3, 5.2) returns 0.00366574413491
#link discussion
STUDENT_CDF
Syntax
STUDENT_CDF(n, x)
Description
Cumulative Student’s t distribution function. Returns the lower-tail probability of the Student’s t probability density function at x, given n degrees of freedom.
Example
STUDENT_CDF(3, -3.2) returns 0.0246659214813
#link discussion
STUDENT_ICDF
Syntax
STUDENT_ICDF(n, p)
Description
Inverse cumulative Student’s t distribution function. Returns the value x such that the Student’s-t lower-tail probability of x, with n degrees of freedom, is p.
Example
STUDENT_ICDF(3, 0.0246659214813) returns -3.2
#link discussion
sturmab
Syntax
sturmab(Poly,Var,Cplx1, Cplx2)
Description
Returns the number of sign changes of a polynomial in the interval (Cplx1, Cplx2] or the number of complex roots in (Cplx1, Cplx2] if Cplx1 or Cplx2 is non-real.
Example
sturmab(x^3-1,x,-2,5) returns 1
#link discussion
sturmseq
Syntax
sturmseq(Poly,[Var])
Description
Returns the Sturm sequence corresponding to a polynomial or to a rational fraction
Example
sturmseq(x^3-1,x) returns [1,[[1,0,0,-1],[3,0,0],9],1]
#link discussion
SUB
Syntax
SUB(object, start, end)
Description
Extracts a portion, of a list or matrix.
For a matrix, start and end are two lists of two numbers ({row, col}) specifying the top left and bottom right of the portion to extract.
For a vector or list, start and end are two numbers specifying the indexes of the first and last objects of the portion to extract.
#link discussion
SUBGROB
Syntax
SUBGROB(srcG, [x1, y1], [x2, y2], trgtG)
Description
Sets graphic trgtG to be a copy of the area of srcG between points (x1,y1) and (x2,y2). If both (x1, y1) and (x2, y2) are not specified, then the entire graphic srcG is used. If (x1, y1) is not specified, then the top left corner of srcG is used; if (x2, y2) is not specified, then the bottom right corner of srcG is used.
trgtGRB can be any of the graphic variables except G0.
SUBGROB(G1, G4) will copy G1 in G4.
#link discussion
SUBGROB_P
Syntax
SUBGROB_P(srcG, [x1, y1], [x2, y2], trgtG)
Description
Sets graphic trgtG to be a copy of the area of srcG between points (x1,y1) and (x2,y2). If both (x1, y1) and (x2, y2) are not specified, then the entire graphic srcG is used. If (x1, y1) is not specified, then the top left corner of srcG is used; if (x2, y2) is not specified, then the bottom right corner of srcG is used.
trgtGRB can be any of the graphic variables except G0.
SUBGROB(G1, G4) will copy G1 in G4.
#link discussion
subMat
Syntax
subMat(Mtrx(A),Intg(n1),Intg(n2),Intg(n3),Intg(n4))
Description
Extracts a sub matrix with first element=A[n1,n2] and last element=A[n3,n4]
Example
subMat([[1,2],[3,4],[5,6]],1,0,2,1) returns [[3,4],[5,6]]
#link discussion
subst
Syntax
subst(Expr,Var=value)
Description
Substitutes a value for a variable in an expression.
Example
subst(x/(4-x^2),x=3) returns -3/5
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sum
Syntax
sum(Expr,Var,Real1, Real2,[Step])
Description
Returns the discrete sum of Expr with respect to the variable Var from Real1 to Real2. You can also use the summation template in the Template menu.
Example
sum(n^2,n,1,5) returns 55
#link discussion
sum_riemann
Syntax
sum_riemann(Expr,List(Var1,Var2))
Description
Returns, in the neighbourhood of n=∞, an equivalent of the sum of Expr(Var1,Var2) for Var2 from Var2=1 to Var2=Var1 when the sum is looked at as a Riemann sum associated with a continuous function defined on [0,1]
Example
sum_riemann(1/(n+k),[n,k]) returns ln(2)
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suppress
Syntax
suppress(Vect(l),Intg(n))
Description
Returns l without the element of index n
Example
suppress([0,1,2,3],2) returns [0,1,3]
#link discussion
surd
Syntax
surd(Expr,Intg(n))
Description
Returns Expr to the power of 1/n
Example
surd(8,3) returns 8^(1/3)
#link discussion
SVD
Syntax
SVD(matrix)
Description
Singular Value Decomposition. Factors an m n matrix into two matrices and a vector: {[[m m square orthogonal]],[[n n square orthogonal]], [real]}.
Example
SVD([[1,2],[3,4]])
#link discussion
SVL
Syntax
SVL(matrix)
Description
Singular Values. Returns a vector containing the singular values of matrix.
Example
SVL([[1,2],[3,4]])
#link discussion
SWAPCOL
Syntax
SWAPCOL(matrixname, column1, column2)
Description
Swap Columns. Exchanges column1 and column2 in the specified matrix matrixname.
#link discussion
SWAPROW
Syntax
SWAPROW(matrixname, row1, row2)
Description
Swap Rows. Exchanges row1 and row2 in the specified matrix matrixname.
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sylvester
Syntax
sylvester(Poly,Poly,Var)
Description
Returns the Sylvester matrix of two polynomials
Example
sylvester(x^2-1,x^3-1,x) returns [[1,0,-1,0,0],[0,1,0,-1,0],[0,0,1,0,-1],[1,0,0,-1,0],[0,1,0,0,-1]]
#link discussion
symb2poly
Syntax
symb2poly(Expr,[Var]) or symb2poly(Expr, ListVar)
Description
Returns the coefficients of a polynomial Expr with respect tothe variable Var or if the second argument is a list returns the internal format of the polynomial. Essentaiilly the inverse of poly2symb().
Example
symb2poly( (x+2)*x+3) returns [1,2,3]
#link discussion
table
Syntax
table(SeqEqual(index=value))
Description
Defines an array where the index are strings or real numbers
#link discussion
tail
Syntax
tail(Lst or Seq or Str)
Description
Returns the list (or sequence or string) without its first element
Example
tail([3,2,4,1,0]) returns [2,4,1,0]
#link discussion
TAN
Syntax
TAN(Value)
Description
Returns the tangent of Value. Value is interpreted as either degrees or radians, depending on the setting of Angle Measure in Home Modes or Symbolic Setup.
Example
in radians mode, TAN(0) returns 0
#link discussion
tan2cossin2
Syntax
tan2cossin2(Expr)
Description
Rewrites Expr with tan(x) replaced by (1-cos(2*x))/sin(2*x)
Example
tan2cossin2(tan(x)) returns (1-cos(2*x))/sin(2*x)
#link discussion
tan2sincos
Syntax
tan2sincos(Expr)
Description
Rewrites Expr with tan(x) using sin(x)/cos(x)
Example
tan2sincos(tan(x)) returns sin(x)/cos(x)
#link discussion
tan2sincos2
Syntax
tan2sincos2(Expr)
Description
Rewrites Expr with tan(x) replaced by sin(2*x)/(1+cos(2*x))
Example
tan2sincos2(tan(x)) returns sin(2*x)/(1+cos(2*x))
#link discussion
tangent
Syntax
tangent(Curve, Point)
Description
Draws the tangent(s) to a given curve through a given point. The point does not have to be a point on the curve.
Example
tangent(plotfunc(x^2), point(1,1)) draws the tangent to the graph y=x^2 through the point (1,1); that is, the line whose equation is y=2*x-1.
tangent(plotfunc(x^2), GA) draws the tangent to the graph of y=x^2 through point A. Point A can then be moved and the tangent will move with it.
tangent(circle(GB, GC-GB), GA) draws one or more tangent lines through point A to the circle whose center is at point B and whose radius is defined by segment BC.
#link discussion
TANH
Syntax
TANH(value)
Description
Hyperbolic tangent.
Example
TANH(1) returns .761594155956
#link discussion
taylor
Syntax
taylor(Expr,[Var=Value],[Order])
Description
Returns the Taylor series expansion of an expression at a point or at infinity (by default, at x=0 and with relative order=5).
Example
taylor(sin(x)/x,x=0) returns 1-(1/6)*x^2+(1/120)*x^4+x^6*order_size(x)
#link discussion
tchebyshev1
Syntax
tchebyshev1(Integer))
Description
Returns the nth Tchebyshev polynomial of the first kind.
Example
tchebyshev1(3) returns 4*x^3-3*x
#link discussion
tchebyshev2
Syntax
tchebyshev2(Integer)
Description
Returns the nth Tchebyshev polynomial of the second kind.
Example
tchebyshev2(3) returns 8*x^3-4*x
#link discussion
tcollect
Syntax
tcollect(Expr)
Description
Collects trigonometric expressions.
Example
tcollect(sin(x)+cos(x)) returns √2*cos(x-1/4*π)
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texpand
Syntax
texpand(Expr)
Description
Expands a transcendental expression; that is, an expression containing trigonometric, logarithmic, or exponential functions. texpand develops the expression in terms of sin(), cos(), ln(), and exp().
Example
texpand(sin(2*x)+exp(x+y)) returns 2*cos(x)*sin(x)+exp(x)*exp(y)
#link discussion
TEXTOUT
Syntax
TEXTOUT(text, [G], x, y, [font], [textColor], [width], [backgroundColor])
Description
Draws text on graphic G at position (x, y) using font. Paints the background before drawing the text using color backgroundColor. If width is specified, does not draw text more than width pixels wide. If backgroundColor is not specified, the background is not erased.
The sizes for font are:
0=current font (default)
1=font_10
2=font_12 (Small)
3=font_14 (Medium)
4=font_16 (Large)
5=font_18
6=font_20
7=font_22
#link discussion
TEXTOUT_P
Syntax
TEXTOUT_P(text, [G], x, y, [font], [textColor], [width], [backgroundColor])
Description
Draws text on graphic G at position (x, y) using font. Paints the background before drawing the text using color backgroundColor. If width is specified, does not draw text more than width pixels wide. If backgroundColor is not specified, the background is not erased.
The sizes for font are:
0=current font (default)
1=font_10
2=font_12 (Small)
3=font_14 (Medium)
4=font_16 (Large)
5=font_18
6=font_20
7=font_22
#link discussion
THEN
Syntax
IF test THEN command(s) [ELSE commands] END;
Description
Evaluates test. If test is true (non 0), executes command(s); otherwise, executes the comands in the ELSE clause nothing happens.
IF A<1
THEN PRINT("A IS SMALLER THAN 1");
ELSE PRINT("A IS LARGER THAN 1");
END;
#link discussion
tlin
Syntax
tlin(Expr)
Description
Returns a trigonometric expression with the products and integer powers linearized
Example
tlin(sin(x)^3) returns (3/4)*sin(x)-(1/4)*sin(3*x)
#link discussion
TO
Syntax
FOR var FROM start TO (or DOWNTO) finish [STEP increment] DO command(s) END;
Description
Sets variable var to start; then, for as long as this variable’s value is less than or equal to (or more than for a DOWNTO) finish, executes command(s) and adds (or substract for DOWNTO) 1 (or increment) to var.
FOR A FROM 1 TO 10 STEP 2
DO
PRINT(A);
END;
will print 1 3 5 7 9
#link discussion
TRACE
Syntax
TRACE(matrix)
Description
Trace of a square matrix. Finds the trace of a square matrix, equal to the sum of the diagonal elements ( also equal to the sum of the eigenvalues).
Example
TRACE([[1,2],[3,4]]) returns 5
#link discussion
translation
Syntax
translation(Vector, Object)
Description
Translates a geometric object along a given vector. The vector is given as the difference of two points (head-tail).
Example
translation(0-i, GA) translates object A down one unit.
translation(GB-GA, GC) translates object C along the vector AB.
#link discussion
transpose
Syntax
tran(Mtrx)
Description
Transposes a matrix (without conjugation)
Example
tran([[1,2,3],[1,3,6],[2,5,7]]) returns [[1,1,2],[2,3,5],[3,6,7]]
#link discussion
triangle
Syntax
triangle(Point1, Point2, Point3)
Description
Draws a triangle, given its three vertices.
Example
triangle(GA, GB, GC) draws ΔABC.
#link discussion
TRIANGLE
Syntax
TRIANGLE([G], x1, y1, x2, y2, x3, y3, c1, [c2, c3], [Alpha], ["ZString", z1, z2, z3]) or TRIANGLE([G], {x1, y1, [c1], [z1]}, {x2, y2, [c2], [z2]},{x3, y3, [c3], [z3]}, ["ZString"]) or TRIANGLE([G], [[x/y coordinate matrix]], [[color matrix]], {[[z matrix]], [zcode], [[[projection matrix]]], [zstring]) or TRIANGLE([G])
Description
Draws a triangle between specified cartesian coordinates in the graphic using the specified color and transparency (0 ≤ Alpha ≤ 255). If 3 colors are specified, blends the colors in between the vertexes.
The next form of TRIANGLE allows display of multiple triangles at a time.
This is mostly used if you have a set of vertices and want to display them all at once.
The first 2 matrices indicate the x/y coordinates and colors of each points. TRIANGLE will draw 1 quadrilateral for each set of 4 adjacent vertices and blends the colors associated with the 4 points.
If a z and projection matrix are provided, for each point, this matrix is multiplied by the [x,y,z,1] vector to create the display x,y coordinates.
If zcode is a list that contains 3 real numbers { ex, ey, ez } then x,y are further modified by doing x=ez/z*x-ex and y=ez/z*y-ey creating a perspective projection.
If zstring is provided, z clipping will happen using the z value (see below).
If zcode="N" or is a list that starts with "N", then each z is normalized to be between 0 and 255.
About ZString
TRIANGLE([G]) returns a string adapted for z clipping.
To use Z clipping, call TRIANGLE to create a Z clipping string (initialized at 255 for each pixels). You can then call TRIANGLE with appropriate z (0-255) values for each of the triangle vertexes and TRIANGLE will not draw pixels further than the already drawn pixels. ZString is automatically updated as appropriate.
Example
TRIANGLE(0,0,5,5,5,-5,#FFh,#FF00h,#FF0000h,128)
#link discussion
TRIANGLE_P
Syntax
TRIANGLE_P([G], x1, y1, x2, y2, x3, y3, c1, [c2, c3], [Alpha], ["ZString", z1, z2, z3]) or TRIANGLE_P([G], {x1, y1, [c1], [z1]}, {x2, y2, [c2], [z2]},{x3, y3, [c3], [z3]}, ["ZString"]) or TRIANGLE_P([G], [[x/y coordinate matrix]], [[color matrix]], {[[z matrix]], [zcode], [[[projection matrix]]], [zstring]) or TRIANGLE_P([G])
Description
Draws a triangle between specified pixel coordinates in the graphic using the specified color and transparency (0 ≤ Alpha ≤ 255). If 3 colors are specified, blends the colors in between the vertexes.
The next form of TRIANGLE allows display of multiple triangles at a time.
This is mostly used if you have a set of vertices and want to display them all at once.
The first 2 matrices indicate the x/y coordinates and colors of each points. TRIANGLE_P will draw 1 quadrilateral for each set of 4 adjacent vertices and blends the colors associated with the 4 points.
If a z and projection matrix are provided, for each point, this matrix is multiplied by the [x,y,z,1] vector to create the display x,y coordinates.
If zcode is a list that contains 3 real numbers { ex, ey, ez } then x,y are further modified by doing x=ez/z*x-ex and y=ez/z*y-ey creating a perspective projection.
If zstring is provided, z clipping will happen using the z value (see below).
If zcode="N" or is a list that starts with "N", then each z is normalized to be between 0 and 255.
About ZString
TRIANGLE_P([G]) returns a string adapted for z clipping.
To use Z clipping, call TRIANGLE_P to create a Z clipping string (initialized at 255 for each pixels). You can then call TRIANGLE_P with appropriate z (0-255) values for each of the triangle vertexes and TRIANGLE_P will not draw pixels further than the already drawn pixels. ZString is automatically updated as appropriate.
Example
TRIANGLE_P(0,20,150,50,100,100,#FFh,#FF00h,#FF0000h,128)
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trig2exp
Syntax
trig2exp(Expr)
Description
Replaces trigonometric functions in Expr with complex exponentials( without linearization).
Example
trig2exp(sin(x)) returns (exp(i*x)-(1/exp(i*x)))/(2*i)
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trigcos
Syntax
trigcos(Expr)
Description
Simplifies the argument Expr using the formulas sin(x)^2+cos(x)^2=1 and tan(x)=sin(x)/cos(x) (privileging cosine)
Example
trigcos(sin(x)^4+sin(x)^2) returns cos(x)^4-3*cos(x)^2+2
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trigexpand
Syntax
trigexpand(Expr)
Description
Expands trigonometric functions.
Example
trigexpand(sin(3*x)) returns (4*cos(x)^2-1)*sin(x)
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trigsin
Syntax
trigsin(Expr)
Description
Simplifies the argument Expr using the formulas sin(x)^2+cos(x)^2=1 and tan(x)=sin(x)/cos(x) (privileging sine)
Example
trigsin(cos(x)^4+sin(x)^2) returns sin(x)^4-sin(x)^2+1
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trigtan
Syntax
trigtan(Expr)
Description
Simplifies the argument Expr using the formulas sin(x)^2+cos(x)^2=1 and tan(x)=sin(x)/cos(x) (privileging tangent)
Example
trigtan(cos(x)^4+sin(x)^2) returns (tan(x)^4+tan(x)^2+1)/(tan(x)^4+2*tan(x)^2+1)
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TRN
Syntax
TRN(matrix)
Description
Transpose. Transposes matrix. If Complex mode is on and the matrix contains complex elements, then TRN finds the conjugate transpose.
Example
TRN([[1,2],[3,4]]) returns [[1,3],[2,4]]
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trunc
Syntax
trunc(Real, [Integer]) or trunc(List, [Integer])
Description
Truncates a value to n decimal places (by default n=0).Accepts complex numbers.
Example
trunc(4.3) returns 4
trunc({3.25, 8.71, 9.01},1) returns {3.2, 8.7, 9.}
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TRUNCATE
Syntax
TRUNCATE(value, [places])
Description
Truncates value to system display settings. If optional places is given, truncates value to places decimal places. If places is negative, truncates to significant digits instead.
Example
TRUNCATE(2.3678,2) returns 2.36
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tsimplify
Syntax
tsimplify(Expr)
Description
Returns an expression with transcendentals rewritten as complex exponentials
Example
tsimplify(exp(2*x)+exp(x)) returns exp(x)^2+exp(x)
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type
Syntax
type(Expr)
Description
Returns n in [1..12] that defines the type of the argument
Example
type("abc") returns DOM_STRING
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TYPE
Syntax
TYPE(object)
Description
Returns the type of the object:
0: Real
1: Integer
2: String
3: Complex
4: Matrix
5: Error
6: List
8: Function
9: Unit
14.?: cas object. the fractional part is the cas type
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UFACTOR
Syntax
UFACTOR(Value_Unit1, 1_Unit2)
Description
Unit factor conversion.
Converts a measurement using a compound unit into a measurement expressed in constituent units.
Example
a Coulomb—a measure of electric charge—is a compound unit derived from the SI base units of Ampere and second: 1 C = 1 A * 1 s. Using UFACTOR, you can express a measurement in Coulombs as a product of Amperes and time.
UFACTOR(100_C,1_A)) returns 100_A*s
UFACTOR(100_C, 1_min) returns 1.66666666667_min*A
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unapply
Syntax
unapply(Expr,Var)
Description
Returns a function defined by an expression.
Example
unapply(2*x^2,x) returns (x)->2*x^2
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UNCHECK
Syntax
UNCHECK(n)
Description
Unchecks (deselects) the corresponding symbolic definition field in the current app. The integer n must be between 0 and 9 for most apps. For Statistics 1-Var and Statistics 2-Var apps, n must be between 1 and 5.
For example, UNCHECK(3) would uncheck F3 if the current app is Function.
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UNTIL
Syntax
REPEAT command(s) UNTIL test;
Description
executes command(s) UNTIL the test is true.
A:=5;
REPEAT
PRINT(A);
A:= A-1;
UNTIL A<1;
will print 5 4 3 2 1
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USIMPLIFY
Syntax
USIMPLIFY(Value_Unitsexpr)
Description
Unit simplification.
Simplifies Value in a complex unit expression Unitsexpr to an equivalent value in a simpler unit expression.
Example
a Joule is defined as 1 kg*m^2/s^2.
USIMPLIFY(5_kg*m2/s2) returns 5_J
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valuation
Syntax
valuation(Poly(P))
Description
Returns the valuation (degree of the term of lowest degree) of the polynomial P .
Example
valuation(x^4+x^3) returns 3
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vandermonde
Syntax
vandermonde(Vect(V))
Description
Returns the Vandermonde matrix=[V^0,V^1,..]
Example
vandermonde([1,2,3]) returns [[1,1,1],[1,2,4],[1,3,9]]
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variance
Syntax
variance(Lst||Mtrx,[Lst])
Description
Returns the variance of a list with the second argument as the weight, or the list of variance of the columns of a matrix.
Example
variance([3,4,2]) returns 2/3
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vector
Syntax
vector(Pnt,Pnt || Pnt,Vect)
Description
Defines a vector(origin is 0 if 1 arg) with two points or two components or two affix (for 2D) or with a point and a vector or with a point (its extrmity and its origin is [0,0,0]).
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vertices
Syntax
vertices(Polygon or Polyedr(P))
Description
Returns the list of the vertices of the polygon or polyhedron P.
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vertices_abca
Syntax
vertices_abca(Polygon or Polyedr(P))
Description
Returns the closed list [A,B,...A] of the vertices of the polygon or polyhedron P.
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VIEW
Syntax
VIEW "text", Function()
Description
VIEW. Allows a programmer to customize the Views menu. Causes "text" to apear when VIEW key is pressed and Function to be executed when the OK menu key (or ENTER key) is pressed.
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vpotential
Syntax
vpotential(Vect(V),LstVar)
Description
Returns U such as curl(U)=V
Example
vpotential([2*x*y+3,x^2-4*z,-2*y*z],[x,y,z]) returns [0,-2*x*y*z,-x^3/3+4*x*z+3*y]
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WAIT
Syntax
WAIT(n)
Description
Halts program execution for the specified number of seconds.
If n is omitted or 0, halts execution until the user presses a key and returns the keycode (or -1 after 1 minute).
If n is -1, halts executions until the user presses a key or there is a mouse event.
If a key is pressed, the keycode is returned.
After a 1 minute timeout, returns -1
If a mouse event happends, a list of the form { type, [x, y], [dx, dy] } is returned. Normally x/y is the event position unless otherwise indicated.
Type can be:
0: Mouse Down
1: Mouse Move
2: Mouse Up (x/y is not provided)
3: Mouse Click (note, if a click is detected, there is no MouseUp)
5: Mouse Stretch. x/y is the delta since the last event. dx/dy is the delta since the ORIGINAL mouse down...
6: Mouse Rotate, x is original angle, y is new angle in 32nd of a circle.
7: Mouse Long Click, This means that the mouse stayed down for 1 second...
Example
WAIT(5) halts program execution for 5 seconds.
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when
Syntax
when(Cond,Expr1,Expr2)
Description
If condition (even symbolic) returns expr1 else returns expr2 (? is the infixed version of when).
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WHILE
Syntax
WHILE test DO command(s) END;
Description
executes command(s) WHILE the test is true.
A:=5;
WHILE A>1 DO
PRINT(A);
A:= A-1;
END;
will print 5 4 3 2 1
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white
Syntax
white(Opt)
Description
Option of the display command to display with color.
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XOR
Syntax
Value1 XOR Value2
Description
Exclusive OR.
Returns 1 if either Value1 or Value2 is non-zero but not both; otherwise, returns 0.
Example
3 XOR 2 returns 0
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XPON
Syntax
XPON(value)
Description
Exponent. Returns the exponent of value.
Example
XPON(123.4) returns 2
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yellow
Syntax
('display')=[color]
Description
For example, suppose you have drawn a circle in the Geometry app. In Symbolic view, the circle's definition might be GC:=circle(GA,GB-GA). If you wanted that circle to be, say, red, you could modify that definition to read:
Example
GC:=circle(GA,GB-GA, ('display')=red)
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zeros
Syntax
zeros(Expr,[Var])
Description
Returns the zeros (reals or complex according to the CAS settings) of the expression Expr for the variable Var (or the matrix where the lines are the solutions of the system : Expr1=0,Expr2=0...).
Example
zeros(x^2+4) returns [] in real mode and [-2*i,2*i] in complex mode
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Zeta
Syntax
Zeta(Real(a))
Description
Returns if a>1 sum(1/n^a,n,1,∞)
Example
Zeta(2) returns π^2/6
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zip
Syntax
zip(Fnc2d(f),Lst(l1),Lst(l2),[Val(default)])
Description
Returns a list whose j-th entry is f(l1[j],l2[j]): without default value its length is the minimum of the lengths of the two input lists and else the shorter list is padded with the default value.
Example
zip('+',[a,b,c,d], [1,2,3,4]) returns [a+1,b+2,c+3,d+4]
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ztrans
Syntax
ztrans(Expr,[Var],[ZtransVar])
Description
Z transform of a sequence.
Example
ztrans(a^x)
P+
Prime Plus Pre-Proccessor
P+ is a proprietary programming language designed to mitigate the challenges associated with code maintenance and comprehensibility in the HP Programming Language (PPL). PPL does not support pre-processing or multiple files, resulting in a single large file that can be unwieldy to manage. P+ serves as an intermediary language that addresses these issues by introducing support for pre-processing and facilitating code organization. P+ code is pre-processed into a single PPL file, which is subsequently executed on the HP Prime calculator since PPL is an interpreted language.