MathTools
Math routines for the
TI-89/92+/V200
Linear Algebra,
Polynomials, Calculus, Statistics, Special Functions, Vector Analysis, and more
Version: 2.4.3
Author: Bhuvanesh Bhatt (bbhatt1@towson.edu)
Introduction
A Computer Algebra System (CAS) is a kind of
mathematical software that enables you to use a computer to manipulate
mathematical expressions. A CAS has symbolics, numerics, graphics, programming,
and typesetting capabilities, and usually also data import/export. CAS’s are
used in many fields such as mathematics, physics, chemistry, engineering,
computer science, computational biology, economics, and education. The TI-68k
devices (TI-89, TI-92 Plus, and Voyage 200) are the most advanced calculators
available today. Once you have learned a topic, you can use the TI-68k CAS to
automate some or all of your work, so that you can concentrate on more advanced
and/or abstract topics without worrying about calculational errors. The TI-68k
are also invaluable when learning a topic, since they allow you to explore the
topic in many ways.
One problem with the TI-68k
is that a lot of commonly used functions are not built-in. MathTools aims to
bridge that gap, mainly in the area of mathematics. One of my goals in
maintaining MathTools and its extensive documentation is to introduce people to
the beauty of mathematics. MathTools mainly consists of TI-Basic functions, but
there are also TI-Basic programs, math programs and utilities written in C, and
a Flash application.
MathTools is far from complete. Some categories that
need more work are factorization, Gröbner bases, GMP, differential equations,
and logic. Some of these have performance problems, while others have missing
functionality. However, MathTools has good organization and is quite powerful,
with well over 340 routines.
Installation
·
Without
GMP and the C utilities, MathTools takes up approximately 230 KB space.
·
I
recommend using TI Connect 1.3. Extract the appropriate group file (.89g or
.9xg), right-click on it, and in the context menu click Send To ® Connected TI Device ® Archive. If you get a garbage collection
message on the calculator, reply “OK” to it, and when it is done, click “Retry”
in TI Connect to send the rest of MathTools.
·
You
can choose the functions you want to install, as long as you also install all
subroutines used by those functions. The dependencies are not given recursively
in this documentation, for example, the Eigenval help entry shows that it needs
mZeros and Sort, but Sort also needs ListSwap. Everything except the Flash app
should be installed in the mathtool folder.
·
To
install the MathTools Flash application, which requires AMS 2.04 or higher:
u
If
you are using TI Connect, connect the cable to the calculator, right click on
the Flash app file and from the context menu select Send To ® Connected TI Device ® Archive.
u If you are using the Graph-Link software, connect the cable to your calculator, click on Link ® Send Flash Software ® Applications and Certificates, select the Flash app in the dialog, click on Add, then click on OK.
Functionality
§ Eigenvalues/Eigenvectors, Decompositions, and PseudoInverse
§ Arbitrary-precision arithmetic (GMP)
§ Calculus
§ Differential and Recurrence Equations
§ Lists
§ Logic
§ Strings
Links to other sections
Screenshots
MathTools routines, with dependencies, examples, and notes
§
Linear algebra
w Eigenvalues/Eigenvectors, Decompositions, and PseudoInverse
§
Cholesky(mat)
computes the Cholesky decomposition of a square symmetric positive-definite
matrix mat and returns a lower-triangular matrix L such that mat=L×LT
Example: Cholesky(Hilbert(3)) Þ [[1,0,0][1/2,Ö(3)/6,0][1/3,Ö(3)/6,Ö(5)/30]]
Notes: Cholesky returns a lower-triangular matrix instead of an
upper-triangular one, because trying to transpose the lower-triangular matrix
results in conj() being applied to all symbolic elements, which looks ugly.
(PowerExp can be used to remove most but not all of the conj()'s.) Complex
elements are not allowed, since a symmetric positive-definite matrix must be
real.
§
Eigenval(matrix)
returns the eigenvalues of matrix, including repeated ones
Needs: mZeros, Sort
Example: Eigenval([[1,1,-1][0,0,2][0,-1,3]]) Þ {1,1,2}
Note: Eigenval tries to return exact eigenvalues whenever possible, but it
calls the built-in eigVl() function for approximate matrices. Non-symmetric
matrices can have eigenvalues with nonzero imaginary parts.
§
Eigenvec(matrix)
returns the generalized eigenvectors of matrix
Needs: DelElem, Eigenval, list2eqn, MemberQ, NullVecs
Example: Eigenvec([[1,1,-1][0,0,2][0,-1,3]]) Þ [[1,@1,0][0,2,1][0,1,1]]
§
GenEigVl(mat1,mat2)
returns the generalized eigenvalues of the matrix pencil mat1-l×mat2
Needs: Coef, Cubic, Degree, NullVecs
Examples: GenEigVl([[3,6][1,-4]],[[-10,6][0,10]]) Þ
{-8/25-Ö(194)/50×i, -8/25+Ö(194)/50×i},
GenEigVl([[1,0,0,0][0,2,0,0][0,0,3,0][0,0,0,4]],
[[1,0,0,0][0,1,0,0][0,0,0,0][0,0,0,0]])
Þ {¥, ¥, 1, 2},
GenEigVl(diag(seq(i,i,1,3)), identity(3)) Þ {1, 3, 2},
GenEigVl([[0,0][0,0]],[[0,0][0,0]]) Þ {undef, undef},
GenEigVl([[0,0][0,1]],[[0,0][0,0]])
Þ
"Error: Singular intermediate matrix P = [[1,0][0,0]]"
Notes: Generalized eigenvalues can be infinite. The generalized eigenvalue
problem is called regular if the determinant det(mat1-l×mat2) does not vanish identically for all
values of l, otherwise it is called
singular. There are n generalized eigenvalues iff the matrix rank of mat2 is n.
Application areas for generalized eigenproblems include optimization,
statistics, signal processing, control theory, spectral imaging, and solid
mechanics.
§
GSchmidt(mat)
performs Gram-Schmidt orthonormalization on mat
Example: GSchmidt([[1,2][3,4]]) Þ [[Ö(10)/10,3×Ö(10)/10]
[3×Ö(10)/10,-Ö(10)/10]]
Note: GSchmidt returns a matrix with orthogonal columns that span the same
subspace as the columns of mat, provided the columns of mat are linearly
independent. Gram-Schmidt orthonormalization is known to be numerically
unstable (see “Matrix Computations” by Golub and van Loan), but should be fine
for symbolic manipulation.
§
Jordan(matrix)
returns the Jordan decomposition {J,S} of matrix such that J=S-1×matrix×S
Needs: Eigenvec
Example: Jordan([[1,2][3,4]]) Þ {j=[[5/2-Ö(33)/2,0][0,Ö(33)/2+5/2]],
s=[[-(Ö(33)+3)/6,(Ö(33)-3)/6][1,1]]}
Note: Jordan decomposition is often used for evaluating symbolic functions of
matrices, such as matrix exponentials.
§
PInverse(mat,tol)
returns the pseudoinverse (also called the generalized or Moore-Penrose
inverse) of the matrix mat with tolerance tol
Needs: SVD
Example: PInverse([[1,2][3,4][5,6]],1E-10) Þ [[-1.33,-0.333,0.667]
[1.08,0.333,-0.417]]
(to Float-3 precision)
Note: mat can be non-square. The pseudoinverse equals the inverse when mat is a
non-singular (all singular values > 0) square matrix. The sum of squares of
the elements of mat×PInverse(mat)-I is minimized, where I is the identity
matrix, and so the pseudoinverse is useful for data fitting.
§
SingVals(mat)
returns the sorted singular values of the numeric matrix mat
Needs: IsCmplxN, Sort
Examples: SingVals([[i,7][-2,1-i]]) Þ
{2.09236143579,7.18484680575},
SingVals([[1,0][2,0]])
Þ
{0.,2.2360679775}
§
SVD(mat)
computes the singular value decomposition of matrix mat
Needs: Eigenval, Eigenvec, GSchmidt, Reverse, Sort
Example: SVD([[1,2][3,4][5,6.0]]) Þ {uu=[[-.883,-.23][-.241,-.525][.402,-.82]],
SS=[[.514,0][0,9.53][0,0]],
vv=[[.785,-.62][-.62,-.785]]}
(with
Float-3 precision)
Note: SVD returns a list containing three matrices, U, S, and V, where U×S×V is the decomposition. If mat is an exact
numeric matrix or a symbolic matrix, SVD tries to return an exact singular
value decomposition. Putting a decimal point into a numeric matrix usually
speeds up the calculation. The input matrix can be non-square. The S matrix contains the diagonalized singular
values of mat. Singular values with absolute value less than a default
tolerance of 5E-13 will be considered to be
zero. If you store a value such as 1E-10 to the variable ‘tol’, then that will be used as the tolerance
instead.
§
Circulnt(col)
returns the circulant matrix defined by the column col
Example: Circulnt({1,2,3,4}) Þ [[1,4,3,2][2,1,4,3][3,2,1,4][4,3,2,1]]
Note: A circulant matrix is a special case of the Toeplitz matrix.
§
Frank(n)
returns the Frank matrix of order n
Example: Frank(4) Þ [[4,3,2,1][3,3,2,1][0,2,2,1][0,0,1,1]]
§
Hadamard(n)
returns a Hadamard matrix of order n
Needs: KProduct
Example: Hadamard(4) Þ [[1,1,1,1][1,-1,1,-1][1,1,-1,-1][1,-1,-1,1]]
Notes: Hadamard matrices exist only for n = 2 or n a multiple of 4. For an n´n Hadamard matrix H, HT×H = n×I, where I is the identity
matrix, and det(H) = nn/2. Hadamard matrices are used in various
combinatorics problems such as the Reed-Muller error-correcting codes.
§
Hankel(list)
returns the corresponding Hankel matrix
Example: Hankel({1,1,1,6,21}) Þ [[1,1,1][1,1,6][1,6,21]]
Note: Hankel matrices are symmetric.
§
Hilbert(n)
returns the Hilbert matrix of order n
Example: Hilbert(3) Þ [[1,1/2,1/3][1/2,1/3,1/4][1/3,1/4,1/5]]
Note: The inverse of a Hilbert matrix has integer elements. A Hilbert matrix is
a special case of the Hankel matrix.
§
IHilbert(n)
returns the inverse of the order-n Hilbert matrix
Needs: Hilbert
Example: IHilbert(3) Þ [[9,-36,30][-36,192,-180][30,-180,180]]
Note: IHilbert(n) is usually faster than Hilbert(n)-1. For example,
Hilbert(10)-1 takes about 17.2 seconds while IHilbert(10) takes
about 4.0 seconds.
§
Pascal(n)
returns the Pascal matrix of order n
Example: Pascal(4) Þ [[1,1,1,1][1,2,3,4][1,3,6,10][1,4,10,20]]
§
Toeplitz(row,col)
returns the corresponding Toeplitz matrix
Needs: Reverse
Example: Toeplitz({a,b,c},{a,d,e}) Þ [[a,b,c][d,a,b][e,d,a]]
Note: row[1] must equal col[1].
§
Vanderm(list)
returns the corresponding Vandermonde matrix
Example: Vanderm({v1,v2,v3}) Þ [[1,1,1][v1,v2,v3][v12,v22,v32]]
Note: Vandermonde matrices appear in association with fitting polynomials to
data. In particular, you can create a Vandermonde matrix from the x-values,
augment the y-values to the bottom, perform an rref of the transpose, and
obtain an interpolating polynomial by doing the sum xk-1×mat[k,colDim(mat)], k=1, …, rowDim(mat),
where mat is the row-reduced matrix.
§
cCondNum(matrix)
returns the column-norm condition number of matrix
Example: cCondNum(Hilbert(4)) Þ 28375
Note: Condition numbers can be useful for studying the propagation of errors. Higher values for the
condition number indicate that the matrix is close to singular, which means
that approximate numerical operations like inversion or linear solution will
return inaccurate or unreliable results. The base-b logarithm of the matrix
condition number is a worst-case estimate of how many base-b digits are lost in
solving a linear system with that matrix (see MathWorld entry).
§
Cofactor(mat)
returns the matrix of cofactors of mat
Needs: Minor
Example: Cofactor([[a,b][c,d]]) Þ [[d,-c][-b,a]]
Notes: The adjoint of the matrix is simply the transpose of the matrix of
cofactors. The determinant of the matrix mati j can be computed from
its matrix of cofactors Ci j by multiplying one of the rows or
columns of mat by the corresponding row or column of C and summing the
elements.
§
CompMat(poly,x)
returns the companion matrix of poly
Needs: Degree, Reverse
Example: CompMat(x^2+1,x) Þ [[0,-1][1,0]], eigVl([[0,-1][1,0]]) Þ {i, -i}
Note: The eigenvalues of the companion matrix are the roots of poly, as
demonstrated in the example above.
§
CondNum(matrix)
returns the Turing condition number estimate
Example: CondNum(Hilbert(4)) Þ 25920
Note: Condition numbers can be useful for studying the propagation of errors. Higher values for the
condition number indicate that the matrix is close to singular, which means
that approximate numerical operations like inversion or linear solution will
return inaccurate or unreliable results. The base-b logarithm of the matrix
condition number is a worst-case estimate of how many base-b digits are lost in
solving a linear system with that matrix (see MathWorld entry).
§
FastDet(mat)
returns the determinant of the matrix mat
Example: det([[a,b,c][d,sin(e),f][g,Ö(h),i]]) Þ a×(sin(e)×i-f×Ö(h))-b×(d×i-f×g)+
c×(d×Ö(h)-sin(e)×g) [takes about 5.5
seconds],
FastDet([[a,b,c][d,sin(e),f][g,Ö(h),i]]) Þ a×(sin(e)×i-f×Ö(h))-
b×(d×i-f×g)+c×(d×Ö(h)-sin(e)×g) [takes about 0.6 seconds]
Note: FastDet is usually faster than the built-in det() function, especially
for symbolic and exact numeric matrices. The determinant of an orthogonal
matrix is ±1. One can compute the
absolute value of the determinant of a matrix M from its QR factorization as
product(diag(R)), and from its SVD as product(SingVals(M)).
§
Givens(matrix,i,j)
does a Givens rotation to zero out the [i,j] element of matrix
Example: Givens([[1][1][1][1]],2,1) Þ [[Ö(2)][0][1][1]] (zero out 2nd
element),
Givens([[Ö(2)][0][1][1]],3,1) Þ [[Ö(3)][0][0][1]] (zero out 3rd
elem),
Givens([[Ö(3)][0][0][1]],4,1) Þ [[2][0][0][0]] (zero out 4th
element).
Note: A Givens rotation is similar to a Householder reflection, but a Givens
rotation is useful when you want to zero out a particular element (or small set
of elements), whereas a Householder reflection is more effective when you want
to zero out many elements. Givens rotations are skew-symmetric and orthogonal.
§
KProduct(a,b)
returns the Kronecker/matrix/tensor product of matrices a and b
Example: KProduct([[a11,a12][a21,a22]],[[b11,b12]]) Þ [[a11×b11,a11×b12][a12×b11,a12×b12][a21×b11,a21×b12][a22×b11,a22×b12]]
Note: KProduct can be used to create Dirac matrices from quantum field theory.
The Kronecker product is distributive with respect to addition as in (A+B)ÄC = AÄC+BÄC, and is associative as in AÄ(BÄC) = (AÄB)ÄC. For the transpose and
inverse of a Kronecker product, (AÄB)T = ATÄBT and (AÄB)-1 = A-1ÄB-1. For the determinant and
trace, if A is n´n and B is m´m, det(AÄB) = det(A)m×det(B)n, and trace(AÄB) = trace(A)×trace(B).
§
KSum(a,b)
returns the Kronecker sum of square matrices a and b
Needs: KProduct
Example: KSum([[a,b][c,d]],[[e,f][g,h]]) Þ [[a+e,f,b,0][g,a+h,0,b]
[c,0,d+e,f][0,c,g,d+h]]
§
MatFunc(str,mat)
applies the function given in the string str to the matrix mat
Needs: Jordan
Example: MatFunc(“e^#”,[[1,2][3,4]]) Þ [[(1/2-Ö(33)/22)×e^(…)+…][…]]
Notes: Use “#” in the string to denote the input matrix. MatFunc can handle
functions of matrices (e^A, sin(A), Ö(A), etc.) on a
diagonalizable square matrix.
§
MatRank(mat)
returns the matrix rank of mat
Needs: NullVecs
Examples: MatRank([a,b;c,d;e,f]) Þ 2, MatRank([[a,0,b][c,d,0][0,0,e]]) Þ 3,
MatRank([[a,0,b][c,d,0][0,0,0]]) Þ 2
§
Minor(mat)
returns the matrix of minors for mat
Needs: DelCol, DelRow
Example: Minor([[a,b][c,d]]) Þ [[d,c][b,a]]
§
mMinPoly(mat,var)
returns the minimal polynomial of mat in terms of var
Needs: NullVecs
Examples: mMinPoly([[0,0][0,0]],x) Þ x, mMinPoly([[0,1][0,0]],x) Þ x2,
mMinPoly([[1,0][0,1]],x) Þ x-1
Note: The minimal polynomial of a matrix M is the lowest degree polynomial such
that 
, where n is the polynomial degree. It divides the
characteristic polynomial of M.
§
NullVecs(mat)
returns a matrix whose columns form a basis for the nullspace of mat
Examples: NullVecs([[4,2,-4][2,1,-2][-4,-2,4]]) Þ [[-1/2,1][1,0][0,1]],
NullVecs({{1,1},{1,0}}) Þ "nullspace={0}"
Note: As shown in the second example above, the null space of a nonsingular
matrix mat is trivial, so that there is no nonzero vector v such that mat×v = 0.
§
SparsRnd({m,n},nz)
returns an m´n random sparse array with
nz nonzero values
Needs: ToSparse
Example: RandSeed 0:SparsRnd({2,3},2) Þ
sparse(2,{2,3},{0,1,2},{2,3},{0.733812311248,0.405809641769},0)
§
ToDense(arr)
converts a sparse matrix arr to a matrix in the usual form
Examples: ToDense(ToSparse([[4,-2][0,-7][8,0]],a)) Þ [[4,-2][0,-7][8,0]],
ToDense(sparse(4,{3,2},{0,2,3,4},{1,2,2,1},{4,-2,-7,8},a))
Þ [[4,-2][a,-7][8,a]]
§
ToSparse(mat,pat)
converts a matrix in the usual form to a sparse matrix with default element pat
Needs: MatchQ
Example: ToSparse([[4,-2][0,-7][8,0]],0) Þ
sparse(4,{3,2},{0,2,3,4},{1,2,2,1},{4,-2,-7,8},0)
Note: The format used is the CSR (compressed sparse row) format.
§
VecNorm(vec,p)
returns the p-norm of the vector vec
Examples: VecNorm({a,b},2) Þ Ö(a2+b2),
VecNorm({a,b},3) Þ (|a3|+|b3|)^(1/3),
VecNorm({a,b},¥) Þ max(|b|,|a|)
§
Polynomial algebra
§
Coef(poly,x)
returns the coefficients of poly in the variable x
Example: Coef(x3-6×x2+11×x-6,x) Þ {1,-6,11,-6}
Note: The exponents must be numeric (for example, x2 is valid but xa is not).
§
CoefList(poly,x)
is an alternative function for returning the coefficients of poly
Needs: FreeQ, IsCmplxN, Pad, PolyDeg
Examples: CoefList((a+b×i)×x3+(c+d×i)×x+(e+f×i),x) Þ {e+f×i,c+d×i,0,a+b×i}
Note: This function is experimental and slow, and may return incorrect results.
§
Degree(poly,x)
returns the degree of poly in x
Needs: Coef
Example: Degree(x2-x+1,x) Þ 2
§
LeadCoef(poly,x)
returns the leading coefficient of poly in x
Needs: Coef
Example: LeadCoef((x+x)(x+2×x2)(x+3×x3),x) Þ 12
§
LeadTerm(poly)
returns the leading term in poly
Example: LeadTerm(4×x×y×z2+4×x3-5×y4+7×x×y2×z) Þ 4×x3
§
NthCoeff(poly,x,n)
returns the coefficient of xn in poly
Examples: seq(NthCoeff(x3-x+1,x,i),i,4,0,-1) Þ {0,1,0,-1,1},
seq(NthCoeff(-@n12+Ö(a)×@n1+eb,@n1,i),i,3,0,-1) Þ {0,-1,Ö(a),eb},
NthCoeff(-3×x100+2×x4-7,x,100) Þ -3
Note: NthCoeff can handle polynomials with degrees upto 449.
§
PolyDeg(poly,x)
is an alternative function that returns the degree of poly in x
Needs: MatchQ
Example: PolyDeg(-5×x4-x2+4×x-5,x) Þ 4
§
sFactors(xpr)
returns a list of all symbolic factors in xpr
Needs: nChars
Example: sFactors((x+1)×(x-2)) Þ {x-2, x+1}
Note: sFactors is very similar to Terms, except that it deals with
multiplication instead of addition. If xpr is provided as a string, the factors
are found without any evaluation.
§
Terms(xpr)
returns a list of terms in xpr
Needs: nChars
Example: Terms(cos(a+b)×x2+e2×c×x) Þ {cos(a+b)×x2, (ec)2×x}
Note: If xpr is provided as a string, the terms are found without any
evaluation.
§
ExtGCD(poly1,poly2)
returns polynomials {d,{s,t}} such that
d = s×poly1+t×poly2 = GCD(poly1,poly2)
Needs: IsCmplxN, Quotient, VarList
Example: ExtGCD(5×x2+x-4, x2-1) Þ {d = x+1, st = {1, -5}}
Note: ExtGCD also works for integers. The extended gcd is useful for solving
linear Diophantine equations.
§
MPolyDiv(poly1,poly2,x)
returns polynomial quotient and remainder for multivariate polynomials
Needs: LeadTerm, Remaindr
Example: MPolyDiv(x2-y,x+y,{x,y}) Þ {x-y,y2-y}
§
PolyDiv(poly1,poly2,x)
returns the univariate polynomial quotient and remainder
Needs: Coef, Degree, LeadCoef
Example: PolyDiv(3×x4+2×x3+x+5,x2+2×x+3,x) Þ {3×x2-4×x-1,15×x+8}
§
PolyGCD(poly1,poly2,x)
returns the polynomial GCD of poly1 and poly2
Needs: Coef, PrimPoly, Remaindr
Example: PolyGCD(12×x3-28×x2+20×x-4,-12×x2+10×x-2,x) Þ 3×x-1
Note: Intermediate expression swell is encountered in polynomial gcd
computation, especially with the classical Euclidean algorithm.
§
PolyLCM(poly1,poly2,x)
returns the polynomial LCM of poly1 and poly2
Needs: PolyGCD
Example: PolyLCM((1-x)2×(1+x)2,(1-x)×(2+x),x) Þ (x-1)2×(x+1)2×(x+2)
§
Quotient(poly1,poly2,x)
returns PolyDiv[1] (polynomial quotient)
Needs: PolyDiv
Example: Quotient(3×x4+2×x3+x+5,x2+2×x+3,x) Þ 3×x2-4×x-1
§
Remaindr(poly1,poly2,x)
returns PolyDiv[2] (polynomial remainder)
Needs: PolyDiv
Example: Remaindr(3×x4+2×x3+x+5,x2+2×x+3,x) Þ 15×x+8
§
Resultnt(poly1,poly2,x)
returns the polynomial resultant
Needs: Sylvestr
Example: Resultnt(x2-2,x3-1,x) Þ -7
§
Subres(poly1,poly2,x)
returns a list of polynomial subresultants
Needs: Coef
Example: Subres(2×x4+x2-4,3×x2+2,x) Þ {1156,102}
§
Sylvestr(poly1,poly2,x)
returns the Sylvester matrix
Needs: Coef
Example: Sylvestr(x2-3×x,x3-1,x)
Þ [[1,0,0,1,0][-3,1,0,0,1][0,-3,1,0,0][0,0,-3,-1,0][0,0,0,0,-1]]
Note: Sylvestr cannot handle constant (degree 0) polynomials.
§
BerlQMat(poly,x,m)
returns the Berlekamp Q-matrix modulo m
Needs: Coef, Reverse, sMod
Example: BerlQMat(x2+3×x,x,2) Þ [[1,0,0][0,1,0][0,1,0]]
Note: Finding the null-space of the Q-matrix modulo a prime is a first step in
the Berlekamp factorization algorithm.
§
SqFree(poly,x)
finds the squarefree factorization for a primitive polynomial poly
Needs: PolyGCD
Example: SqFree(x3+5×x2+8×x+4,x) Þ (x+1)×(x+2)2
Note: Squarefree
factorization is a first step in many factoring algorithms. There is a Flash
app extension function MathTool.SqrFree that does the same thing as
SqFree, but faster.
Note: SqFree finds the derivative of poly and then iteratively computes GCDs.
§
SqFreeQ(poly,x)
returns true if poly is squarefree, else false
Needs: PolyGCD
Examples: SqFreeQ(x5+x3+2×x2+x+2,x) Þ true
SqFreeQ(x8-2×x6+2×x2-1,x) Þ false
§
Groebner(polylist,vars)
computes a symbolic Gröbner basis for the set of multivariate polynomials in
polylist
Needs: ListPair, NormalF, Spoly
Example: Groebner({x-y-z,x+y-z2,x2+y2-1},{x,y,z}) [will take »30 minutes!]
Notes: Groebner is a program, unlike most other components of this
package. It is an experimental program
and is usually very slow. Groebner computes the (unreduced) Gröbner basis via
the Buchberger algorithm. At each step, it shows the combination of polynomials
selected ({ pi , pj }) and the normal form of Spoly(pi
, pj). Computing a symbolic Gröbner basis with a lexicographic
ordering can be computationally intensive. When Groebner ends, it displays the
Gröbner basis. Gröbner bases can be used to solve systems of polynomial
equations, and are also useful for other applications such as integer
programming and converting parametric curves and surfaces to implicit form. The
reduced Gröbner basis has the same set of roots as the original polynomials. If
the Gröbner basis reduces to 1, the nonlinear system polylist has no solution.
§
NormalF(poly,polylist,vars)
returns the normal form of poly modulo polylist
Needs: LeadTerm, MPolyDiv, Terms
Example: NormalF(x+y-z2,x-y-z,{x,y,z}) Þ 2×y-z2+z
Note: The “normal form” is guaranteed to be unique only if the polynomials in
polylist form a Gröbner basis.
§
Spoly(poly1,poly2,vars)
returns the S-polynomial of poly1 and poly2
Needs: LeadTerm, Multideg
Example: Spoly(x3-2×x×y,x2×y-2×y2+x,{x,y}) Þ -x2
§
CompMat(poly,x)
returns the companion matrix of poly
Needs: Degree, Reverse
Example: CompMat(x2+1,x) Þ [[0,-1][1,0]], eigVl([[0,-1][1,0]]) Þ {i, -i}
Note: The eigenvalues of the companion matrix are the roots of poly, as
demonstrated in the example above.
§
Content(poly,x)
returns the “content” of poly
Needs: Coef, LGcd
Example: Content(8×x4-2×x2-2×x+12,x) Þ 2
Note: The “content” of a polynomial is the gcd of its coefficients.
§
Cyclotom(n,x)
returns the nth cyclotomic polynomial
Needs: RelPrime
Example: Cyclotom(6,x) Þ x2-x+1
Note: The nth cyclotomic polynomial is the minimal polynomial of the
primitive nth root of unity and is irreducible over 
with degree j(n), where j(n) is the Euler totient function.
§
mMinPoly(mat,var)
returns the minimal polynomial of mat in terms of var
Needs: NullVecs
Examples: mMinPoly([[0,0][0,0]],x) Þ x, mMinPoly([[0,1][0,0]],x) Þ x2,
mMinPoly([[1,0][0,1]],x) Þ x-1
Note: The minimal polynomial of a matrix M is the lowest degree polynomial such
that 
, where n is the polynomial degree. It divides the
characteristic polynomial of M.
§
Multideg(poly,vars)
returns the multidegree of poly
Needs: Degree, LeadTerm
Example: Multideg(4×x×y×z2+4×x3-5×y4+7×x×y2×z,{x,y,z}) Þ {3,0,0}
§
Normal(poly,x)
returns the normal form of poly
Needs: LeadCoef
Example: expand(Normal(8×x4-2×x2-2×x+12,x)) Þ x4-x2/4-x/4+3/2
Note: This is not the same normal form as that given by NormalF.
§
PolyMod(poly,m,x)
returns poly mod m
Needs: Coef, IsCmplxN, Quotient
Example: PolyMod(x6+6×x5+15×x4+20×x3+15×x2+6×x+1,2,x) Þ x6+x4+x2+1
§
PolyQ(xpr,x)
returns true if xpr is a polynomial, else false
Example: PolyQ((x+1)2, x) Þ true
Note: If you have variables raised to unknown powers, PolyQ will return false.
For example, PolyQ(x2×yb,x) Þ false
§
PrimPoly(poly,x)
returns the corresponding primitive polynomial
Needs: Content
Example: PrimPoly(6×x3+26×x2+26×x+6,x) Þ 3×x3+13×x2+13×x+3
§
SymPoly(list,k)
returns the kth symmetric polynomial of the variables in list
Needs: KSubsets
Example: seq(SymPoly({x,y,z},i),i,1,3) Þ {x+y+z, x×y+x×z+y×z, x×y×z}
w
Arbitrary-precision arithmetic (GMP)
§
GMPFMul(numstr1,numstr2)
does arbitrary-precision multiplication of the floating point real numbers
given in numstr1 and numstr2 and returns the result as a list
{string,intdigits}
Needs: GMPMulDt
Example: GMPFMul("32.868852111489014254382668691",
"4.882915734918426996824267") Þ
{"160495835163896471004691233980227653746755621409924496999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998",3},
which means 160.4958…
Note: GMPFMul is experimental and currently has a small memory leak.
§
CPVInt(f(x),x,a,b)
returns the Cauchy principal value of the integral ò(f(x),x,a,b)
Examples: CPVInt(1/x,x,-1,2) Þ ln(2), CPVInt(1/(x+x2),x,-2,1) Þ -2×ln(2),
CPVInt(tan(x),x,p/4,3×p/4) Þ 0,
CPVInt(x/(x3-3),x,0,¥) Þ 0.302299894039
Note: CPVInt may not be able to handle some improper integrals (i.e. integrals
with one or both integration limits infinite) because it does not compute
residues.
§
ExactDif(flist,vars)
returns true if flist is an exact differential, else returns false or the
conditions that must be satisfied
Needs: FuncEval, list2eqn, Permut
Examples: ExactDif({f(x),g(y),h(z)},{x,y,z}) Þ true,
ExactDif({f(x,y),g(y),h(z)},{x,y,z}) Þ 
§
FracDer(xpr,{x,ordr[,a]})
returns the fractional derivative (from Riemann-Liouville calculus) of xpr with
respect to variable x to order ordr
Needs: FreeQ, Gamma, IsCmplxN, MatchQ
Examples: FracDer(1,{x,-1/2}) Þ 2×Ö(x)/Ö(p),
FracDer(d(f(x),x),{x,-2/3})
Þ 0.664639300459×x5/3
Notes: The default for a is zero (a is what is called the ‘lower limit’). The
fractional derivative of a constant is not zero, as the first example shows.
Because the gamma function cannot be evaluated symbolically for many fractions,
you may get a numeric result, as shown in the second example. Weyl calculus is
a subset of Riemann-Liouville calculus.
§
GetExpLn(xpr)
is a subroutine for Risch that returns exponential or logarithmic
subexpressions
Needs: mand
§
Horowitz(numerator,denominator,var)
reduces an integral using the Horowitz method
Needs: Coef, Degree, list2eqn, Pad, PolyDiv, PolyGCD, Quotient, RTInt
Example: Horowitz(x,x4-1,x) Þ ln(x2-1)/4+(-ln(x2+1)/4)-p/4×i
Note: This is an experimental function.
§
ImpDifN(eq,x,y,n)
returns the nth implicit derivative of the equation eq
Example: ImpDifN(x4+y4=0,x,y,2) Þ -3×x6/y7-3×x2/y3
§
mTaylor(f(x,y,…),{x,y,…},{x0,y0,…},k)
returns the multivariate Taylor expansion of f(x,y,…) to order k
Needs: list2eqn
Examples: mTaylor(x2+x×y2, x,x0,5) Þ (x-x0)2 + (x-x0)×(y2+2×x0) + x0×y2 + x02
mTaylor(x2+y2,
{x,y},{x0,y0},2)
Þ x0×x2-2×x0×(x0-1)×x+y0×y2-2×y0×(y0-1)×y+x03-x02+y02×(y0-1)
§
nIntSub(f,x,a,b,n)
numerically integrates f from x=a to x=b by subdividing the interval into n
subintervals
Example: nIntSub(tan(x),x,0,1.5707,5) Þ 9.24776403537
§
Pade(f,x,x0,n,d)
returns the Padé approximant of f(x) about x=x0 with degrees n for
the numerator and d for the denominator
Examples: Pade(ex,x,0,1,2) Þ 2×(x+3)/(x2-4×x+6), Pade(xx,x,1,0,2) Þ -1/(x-2)
Note: Padé approximants are a kind of generalization of Taylor series and are a
good way of approximating functions that have poles.
§
Risch(xpr,var)
does indefinite integration
Needs: Coef, Degree, FreeQ, GetExpLn, list2eqn, mand, PolyGCD, Resultnt, Terms,
VarList
Examples: Risch(ln(x),x) Þ (ln(x)-1)×x,
Risch(1/ln(x),x) Þ "Not integrable in
terms of elementary functions"
Notes: Risch uses a partial implementation of the Risch algorithm, for
logarithmic and exponential extensions. The Risch algorithm builds a tower of
logarithmic, exponential, and algebraic extensions. Liouville’s principle,
which dates back to the 19th century, is an important part of the
Risch algorithm. Hermite reduction is applicable to arbitrary elementary
functions (functions that can be obtained from the rational functions in x
using a finite number of nested logarithms, exponentials, and algebraic numbers
or functions) in the algorithm. At this time, this (Risch) function is
experimental and should be seen as a framework for future functionality rather
than a currently useful tool.
§
RTInt(numerator,denominator,var)
computes ò(numerator/denominator,x)
using the Rothstein-Trager method
Needs: Degree, LeadCoef, PolyGCD, Resultnt, SqFreeQ
Example: RTInt(1,x3+x,x) Þ ln(x)-ln(x2+1)/2
Note: numerator and denominator should be polynomials. RTInt is an experimental
function.
§
TanPlane(f,vars,pt)
returns the equation of the tangent plane to the curve/surface f at the point
pt
Needs: list2eqn
Example: TanPlane(x2+y2,{x,y},{1,1}) Þ 2×x+2×y-4=0
§
Cyc2Perm(clist)
returns the permutation that has the cycle decomposition clist
Needs: Sort
Example: Cyc2Perm({cycle_={1},cycle_={3,2}}) Þ {1,3,2}
Note: Perm2Cyc and Cyc2Perm use the cycle_ = list structure because the AMS
does not allow arbitrary nesting of lists.
§
InvPerm(perm)
returns the inverse permutation of the permutation perm
Example: InvPerm({2,4,3,1}) Þ {4,1,3,2}
Note: InvPerm does not check that the permutation is valid.
§
isPermut(list)
returns true if list is a valid permutation, else false
Needs: Sort
Examples: isPermut({1,3,2}) Þ true, isPermut({2,3,2}) Þ false
Note: For valid permutations, the elements of the list must be numbers.
§
KSubsets(plist,k)
returns subsets of plist with k elements
Example: KSubsets({a,b,c},2) Þ [[a,b][a,c][b,c]]
§
LevCivit(indexlist,g)
returns the Levi-Civita symbol for the given metric
Needs: PermSig
Example: LevCivit({1,3,2},[[1,0,0][0,r2,0][0,0,r2×sin(q)2]]) Þ -|sin(q)|×r2
Note: LevCivit could be used for computing cross products.
§
Ordering(list)
returns the positions at which the elements of list occur when the list is
sorted
Needs: Sort
Example: Ordering({4,1,7,2}) Þ {2,4,1,3}
Note: For the example above, element #2 (1) occurs at position 1 in the sorted
list, element #4 (2) occurs at position 2, element #1 (4) occurs at position 3,
and element #3 (7) occurs at position 4.
§
Perm2Cyc(perm)
returns the cyclic decomposition for the permutation perm
Example: Perm2Cyc({1,3,2}) Þ {cycle_={1}, cycle_={3,2}}
Notes: Perm2Cyc does not check that the permutation is valid. Perm2Cyc and
Cyc2Perm use the cycle_ = list structure because the AMS does not allow
arbitrary nesting of lists.
§
PermSig(list)
returns the signature of the permutation (even = 1, odd = -1, invalid = 0)
given by list
Needs: isPermut, Position
Example: PermSig({3,2,1}) Þ -1 (this is an odd
permutation)
Notes: For valid permutations, the elements of the corresponding list must be
numbers. The permutation is even(odd) if an even(odd) number of element
transpositions will change the permutation to {1,2,3,…}; in the example, 321 ® 231 ® 213 ® 123 Þ 3 transpositions Þ odd permutation.
§
Permut(list)
returns all possible permutations of list
Needs: ListSwap
Example: Permut({a,b,c}) Þ
[[a,b,c][b,a,c][c,a,b][a,c,b][b,c,a][c,b,a]]
§
RandPerm(n)
returns a random permutation of {1,2,…,n}
Needs: Sort
Example: RandSeed 0:RandPerm(4) Þ {3,4,2,1}
§
CFracExp(num,ordr)
returns the continued fraction expansion of the number num to order ordr
Examples: CFracExp(p,5) Þ {3,7,15,1,292}, CFracExp(Ö(2),4) Þ {1,2,2,2},
CFracExp(5,11) Þ {5}, CFracExp(5,¥) Þ {5},
CFracExp(5/2,¥) Þ {2,2},
CFracExp(Ö(28),¥) Þ {5, rep={3,2,3,10}}
Note: Rational numbers have terminating continued fraction expansions, while
quadratic irrational numbers have eventually repeating continued fraction
representations. The continued fraction of Ö(n) is represented as {a,
rep={b1,b2, b3,…}}, where the bi
are cyclically repeated. Continued fractions are used in calendars and musical
theory, among other applications.
§
ContFrac({num(k),denom(k)},k,n)
returns the continued fraction num(k)/(denom(k)+num(k+1)/(denom(k+1)+…))
Example: ContFrac({-(k+a)×z/((k+1)×(k+b)), 1+(k+a)×z/((k+1)×(k+b))},k,2) Þ
§
ContFrc1(list) returns the continued fraction
list[1]+1/(list[2]+1/(list[3]+…
Example: approx(ContFrc1({3,7,15,1,292})) Þ 3.14159265301
§
DaubFilt(k)
returns the FIR coefficients for the Daubechies wavelets
Needs: Map, Reverse, Select
Example: factor(DaubFilt(4)) Þ {(Ö(3)+1)×Ö(2)/8, (Ö(3)+3)×Ö(2)/8,
-(Ö(3)-3)×Ö(2)/8, -(Ö(3)-1)×Ö(2)/8}
§
DCT(list/mat)
returns the discrete cosine transform of list/mat
Example: DCT({2+3×i, 2-3×i}) Þ {2×Ö(2), 3×Ö(2)×i}
§
DST(list/mat)
returns the discrete sine transform of list/mat
Example: DCT({2+3×i, 2-3×i}) Þ {0, 2×Ö(2)}
§
Fit(mat,f(par,vars),par,vars)
returns a least-squares fit of the model f with parameter par to the data in
mat
Needs: FuncEval, VarList
Example: {{0.3,1.28},{-0.61,1.19},{1.2,-0.5},{0.89,-0.98},{-1,-0.945}}®mat
Fit(mat,x2+y2-a2,a,{x,y}) Þ a = 1.33057 or a = -1.33057
Note: The data in mat is given in the form {{x1,y1,…},{x2,y2,…},...}. The number of
columns of mat should be equal to the number of variables vars.
§
Lagrange(mat,x)
returns the Lagrange interpolating polynomial for the data in mat
Example: Lagrange({{2,9},{4,833},{6,7129},{8,31233},{10,97001},
{12,243649}},x) Þ x5-3×x3+1
Note: The data matrix mat is in the form {{x1,y1},{x2,y2},…}.
§
LinFilt(list,q) uses the linear filter given by q to filter the data list
Examples: LinFilt({a,b,c},{d,e}) Þ {a×e+b×d, b×e+c×d},
LinFilt({a,b,c},[[d,e][f,g]]) Þ [[a×f+b×d, b×f+c×d][a×g+b×e, b×g+c×e]]
§
Residual(mat,poly,x)
returns the residual of a polynomial fit of the data in mat
Examples: [[1,3][2,6][3,11]]®mat:Residual(mat,Lagrange(mat,x),x)
Þ 0,
[[0.9][3.82][3,11.1]]®mat:Residual(mat,Lagrange(mat,x),x) Þ 2.E-26
Note: ‘poly’ is the polynomial from the polynomial fit.
w Differential and Recurrence Equations
§
Casorati({f1(t),f2(t),…,fn(t)},t)
returns the Casorati matrix for the functions fi(t)
Example: det(Casorati({2t,3t,1},t)) Þ 2×6t
Note: The Casorati matrix is useful in the study of linear difference
equations, just as the Wronskian is useful with linear differential equations.
§
ChebyPts(x0,L,n)
returns n points with the Chebyshev-Gauss-Lobatto spacing with leftmost point x0
and interval length L
Example: ChebyPts(x0,len,4) Þ {x0, len/4+x0, 3×len/4+x0, len+x0}
§
FDApprox(m,n,s,x,i)
returns an nth order finite-difference approximation of 
on a uniform grid
Needs: Weights
Example: FDApprox(1,2,1,x,i) Þ (f(x[i+1])-f(x[i-1]))/(2×h),
which corresponds to
= nDeriv(f(x),x,h)
Note: Please see the notes for Weights for information on the arguments n and
s.
§
Lin1ODEs(mat,x)
returns solutions of the system of ODEs represented by the matrix mat
Needs: FreeQ, MatchQ, MatFunc, NthCoeff, Reverse
Examples: Lin1ODEs([[0,t][-t,0]],t) Þ [[cos(t2/2),sin(t2/2)][-sin(t2/2),cos(t2/2)]],
Lin1ODEs([[0,3][5,x]],x) Þ {},
factor(Lin1ODEs([[7,3][5,1]],x))|x³0 Þ
[[¼,Ö(6)×(e4×Ö(6)×x-1)×(e(4-2×Ö(6))×x)/8][5×Ö(6)×(e4×Ö(6)×x-1)×(e(4-2×Ö(6))×x)/24,¼]]
Note: The matrix represents the coefficients of the right-hand sides of the
ODEs. If no solution can be found, Lin1ODEs returns {}.
§
ODE2Sys(ode,x,y)
breaks up the ordinary differential equation ode into a system of first-order
ODEs
Needs: RepChars
Example: ODE2Sys(y'' = -y,x,y) Þ {ÿ1'(x) = ÿ2, ÿ2'(x) = -ÿ1}
§
ODEExact(plist,
r) solves linear nth-order exact ordinary differential equations (ODEs) of the
form: p[1](x)×y + p[2](x)×y' + p[3](x)×y'' + … = r(x)
Needs: RepChars
Example: Solve the ODE (1+x+x2)×y'''(x) + (3+6×x)×y''(x) + 6×y'(x) = 6×x
ODEExact({0,6,3+6×x,1+x+x2},6×x)
Þ (x2+x+1)×y = x4/4 + @3×x2/2 + @4×x + @5
Notes: The independent and dependent variables must be x and y, respectively.
If the ODE is not exact, the function will say so. The input syntax for this
function may be changed in the future.
§
ODEOrder(ode)
returns the order of the ordinary differential equation ode
Needs: nChars, Terms
Examples: ODEOrder(y'+y) Þ 1, ODEOrder(-y''''+2×x×y''-x2 = 0) Þ 4
Note: ‘ode’ must be given in the prime notation (e.g. y''+y = 0).
§
PDESolve(pde,
u, x, y, u(var=0) = u0) solves the PDE in u(x,y)
F(x,y,u,
,
) = 0
Needs: FreeQ, ReplcStr, Terms
Example: PDESolve(d(u(x,y),x)2×d(u(x,y),y)-1=0,u,x,y,u(y=0)=x) Þ u(x,y)=x+y
For more examples, see
the included calculator text file PDEDemo.
Notes: Please do not use the variables p, q, s and t, as they are used
internally. PDESolve uses the Lagrange-Charpit and Exact Equation methods.
Sometimes, integral transforms can also be used to solve linear differential
equations. Use Laplace transforms for initial value problems and Fourier/Hankel
transforms for boundary value problems. The Fourier/Hankel transform removes
the spatial dependence, while the Laplace transform removes the temporal
dependence. PDEs usually have the following types of boundary conditions:
Dirichlet condition: u(x,y)
specified on the boundary
Neumann condition: 
specified on the
boundary
Cauchy condition: u and 
specified on the
boundary.
§
PDEType(pde,f(x,y))
returns the type of a 2nd-order partial differential equation (hyperbolic, elliptic,
parabolic, or mixed), with the function given as e.g. u(x,y)
Needs: FreeQ, Terms
Examples: PDEType(d(u(x,y),x,2)+y2, u(x,y)) Þ "parabolic"
PDEType(d(u(x,y),x,2)+d(d(u(x,y),x),y)+y2,
u(x,y)) Þ "hyperbolic"
PDEType(d(u(x,y),x,2)+d(u(x,y),y,2)+f(x,y),
u(x,y)) Þ "elliptic"
PDEType(x×d(u(x,y),x,2)-d(u(x,y),y,2), u(x,y)) Þ "mixed"
§
RSolve(list,u,t)
solves a linear recurrence/difference equation list[1]×u(t+k)+list[2]×u(t+k-1)+…+list[dim(list)-1]×u(t)+list[dim(list)]=0 for
u(t)
Needs: FreeQ, Freq, mZeros
Examples: RSolve({1,-p,r},u,t) Þ u(t) = ç[1,1]×pt-r×(pt-1)/(p-1),
RSolve({1,-7,6,0},y,n)
Þ y(n)
= 6n×ç[2,1]+ç[1,1]
Note: At this time, initial conditions must be substituted manually.
§
Weights(d,n,s)
returns the weights for an nth order finite-difference approximation
for the dth-order derivative on a uniform grid
Needs: Coef, Reverse
Examples: Weights(1,2,1) Þ {-1/(2×h), 0, 1/(2×h)},
Weights(1,3,3/2) Þ {1/(24×h), -9/(8×h), 9/(8×h), -1/(24×h)}
Notes: n is the total number of grid intervals enclosed in the finite
difference stencil and s is the number of grid intervals between the left edge
of the stencil and the point at which the derivative is approximated. s = n/2
for centered differences. When s is not an integer, the grid is called a
staggered grid, because the derivative is requested between grid points.
§
Wronsk({f1(x),f2(x),…,fn(x)},x)
returns the Wronskian of the functions fi(x)
Example: Wronsk({x2,x+5,ln(x)},x) Þ [[x2,x+5,ln(x)][2×x,1,1/x][2,0,-1/x2]]
Note: The functions are linearly independent if det(Wronskian)¹0.
§
Cubic({a,b,c,d})
returns the exact roots of the cubic a×x3+b×x2+c×x+d = 0
Example: Cubic({35,-41,77,3})
Þ {(315×Ö(4948777)-477919)1/3/105-…+41/105,…,…}
§
Diophant({a,b,c,d,e,f},{x,y})
solves the quadratic bivariate diophantine equation
a×x2+b×x×y+c×y2+d×x+e×y+f = 0, {a,b,c,d,e,f}Î
,
for integers {x,y}
Needs: CFracExp, ContFrc1, Divisors, ExtGCD, ListSwap, mor
Examples: Diophant({42,8,15,23,17,-4915},{x,y}) Þ x=-11 and y=-1,
exp►list(Diophant({0,2,0,5,56,7},{x,y}),{x,y})
Þ
[[105,-2][-9,1][-21,7][-27,64][-29,-69][-35,-12][-47,-6][-161,-3]],
Diophant({0,0,0,5,22,18},{x,y})
Þ x=22×@n1+8 and y=-5×@n1-1,
Diophant({8,-24,18,5,7,16},{x,y})
Þ x=41×@n2-174×@n22-4 and y=37×@n2-116×@n22-4 or
x=17×@n2-174×@n22-2 and y=21×@n2-116×@n22-2,
Diophant({1,0,4,0,0,-1},{x,y}) Þ x=1 and y=0 or x=-1 and
y=0,
Diophant({1,0,3,0,0,1},{x,y}) Þ false,
Diophant({1,0,-2,0,0,-1},{x,y}) Þ x=cosh(@n3×ln(…)) and
y=sinh(@n3×ln(…))×Ö(2)/2 and @n3³0 or …
Note: Diophant returns false if there are no solutions for {x,y} in the
integers. Diophant can currently solve any solvable linear, elliptic, or
parabolic equation, as well as many hyperbolic (including some Pell-type) equations.
§
iSolve(xpr,var)
solves an inequality given by xpr for the variable var
Needs: IsCmplxN, mor, Sort, VarList
Examples: iSolve(x3-2×x2+x<0,x) Þ x<0, iSolve(x2³2×x+3,x) Þ x³3 or x£-1,
iSolve(ex+x£1,x) Þ x£0., iSolve(abs(x-3)³0,x) Þ true,
iSolve(x2×ex³1/2,x) Þ x£-1.48796 and x³-2.61787 or x³0.539835
Note: Wrapping approx() around the iSolve() or evaluating in Approx mode gives
faster results, especially for complicated results.
§
ModRoots(poly,p)
returns roots of poly mod prime p
Needs: Degree, PolyMod, Select, sFactors
Examples: ModRoots(x2=16,x,7) Þ {4,3}, ModRoots(x2=16,x,11)
Þ
{4,7},
ModRoots((x2+1)2-25,x,97) Þ {2,95},
ModRoots(x3+1,x,137)
Þ
{136},
ModRoots(x3+11×x2-25×x+13,x,5) Þ {2}
Note: ModRoots often does not return all the roots of poly mod p.
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mZeros(poly,x)
returns the zeros of the polynomial poly, with multiplicities
Needs: Degree, Sort
Example: mZeros(x2+2×x+1,x) Þ {-1,-1}
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NewtRaph(f,vars,start,tol,intermresults)
returns a solution for f = 0
Needs: Jacobian, list2eqn
Example: nSolve(x×ex = 2, x=0) Þ 0.852605502014,
NewtRaph(x×ex = 2, x, 0, 1E-10, false) Þ {x=0.852605502014},
NewtRaph(x×ex = 2, x, 0, 1E-10, true) Þ {solution={x=…}, …}
Notes: NewtRaph uses Newton-Raphson iteration. The ‘intermresults’ argument
should be true or false; it specifies whether or not intermediate results are
given.
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NRoots(coeflist)
finds approximate roots of the polynomial with coefficients in coeflist in
descending order
Needs: Reverse
Example: NRoots({1,0,0,0,-1,-1}) Þ {1.1673, 0.18123+1.084×i,
0.18123-1.084×i, -0.76488+0.35247×i,
-0.76488-0.35247×i}
Note: NRoots uses the eigenvalue method, which can be computationally expensive
but is quite robust at finding close and multiple roots. The example polynomial
above is x5-x-1.
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NumRoots(poly,var,istart,iend)
returns the number of real roots of the polynomial poly in the open interval
(istart,iend)
Needs: Sturm
Examples: NumRoots(x2+1,x,-2,2) Þ 0, NumRoots(P(x-i,i,-3,2),x,-2,0) Þ 2
Note: The computation may take some time due to a slowness in Sturm.
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Quartic({a,b,c,d,e})
gives the exact roots of the quartic a×x4+b×x3+c×x2+d×x+e = 0
Needs: Cubic
Example: Quartic({1,0,0,1,4})
Þ {-Ö(2×cos(tan-1(Ö(49071)/9)/3))×33/4/3-…,…,…,…}
Note: Abel and Galois proved that general equations of fifth and higher order
cannot be solved in terms of nth roots of numbers (see MathWorld
entry).
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Sturm(poly,var)
returns the Sturm sequence for the polynomial poly
Needs: Degree, Remaindr
Example: Sturm(x3+2×x+1,x) Þ {x3+2×x+1, 3×x2+2, -4×x-3, -59}
Note: Sturm may be slow.
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DelCol(mat,c)
returns the matrix mat with column c deleted
Example: DelCol([[a,b][c,d]],2) Þ [[a][c]]
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DelElem(list,i)
returns list with the ith element removed
Example: DelElem({a,b,c,d},3) Þ {a,b,d}
§
DelRow(mat,r)
returns the matrix mat with row r deleted
Example: DelRow([[a,b][c,d]],1) Þ [[c,d]]
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LGcd(list)
returns the GCD of the elements of list
Example: LGcd({2,16,4,0,2}) Þ 2
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list2eqn(list)
inserts ‘and’ between the elements of list
Example: list2eqn({a¹b,c=d}) Þ a-b¹0 and c=d
Note: list2eqn is now deprecated, since the MathTools Flash app provides a
faster function MathTool.ListAnd.
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ListPair(list)
returns possible two-element combinations from list as rows in a matrix
Example: ListPair({a,b,c,d}) Þ [[a,b][a,c][b,c][a,d][b,d][c,d]]
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ListPlot(ylist,joined)
plots the points given in the list ylist versus x, with ‘joined’ specifying
whether the points should be connected by lines
Example: ListPlot(seq(2i,i,1,6),true)
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ListRpt(list,n)
returns the list repeated n times
Example: ListRpt({a,b,c},3) Þ {a,b,c,a,b,c,a,b,c}
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ListSubt(list1,list2)
returns elements from list1 that are not in list2
Needs: FreeQ
Example: ListSubt({a,b,c,d,e},{d,b,e}) Þ {a,c}
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ListSwap(list,i,j)
returns list with the ith and jth elements swapped
Example: ListSwap({a,b,c,d,e,f},2,5) Þ {a,e,c,d,b,f}
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LLcm(list)
returns the LCM of the elements of list
Example: LLcm({3,11,9,1,15}) Þ 495
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mand(list)
is an auxiliary function that returns true if all elements of list are true,
else returns false
Examples: mand({a=a,2>3,1<2}) Þ false, mand({x+x=2×x,2×2=4}) Þ true
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MemberQ(list/matrix,e)
returns true if e is an element of list/matrix, else false
Needs: MatchQ
Examples: MemberQ({a, b, c},b) Þ true, MemberQ({x, y2, z},y) Þ false,
MemberQ({2, ex, x2}, ez) Þ false, MemberQ({2, ex, x2}, ez_) Þ true
Note: MemberQ works with patterns, as shown in the last example above.
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mor(list)
is an auxiliary function that inserts an “or” between the elements of list
Needs: ReplcStr
Example: mor({a>b,a>c}) Þ a-b>0 or a-c>0
Note: mor is now deprecated, since the MathTools Flash app provides a faster
function MathTool.ListOr.
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Pad(list,n)
simply pads list with zeros so that the total number of elements is |n|
Examples: Pad({a,b,c},5) Þ {a,b,c,0,0}, Pad({a,b,c},-4) Þ {0,a,b,c}
Note: If n is negative, the zeros are prepended to the list; if n is positive,
they are appended.
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Position(list/mat,elem/list)
returns the position of element/list in the list/matrix
Needs: mand
Examples: Position({a,b,c,d},c) Þ 3, Position({a,b,c,d},e) Þ 0,
Position([[a,b][c,d]],c) Þ {2,1} (row 2, column 1),
Position([[1,2][3,4]],{3,4}) Þ 2
Note: Searching for a symbolic list is not currently recommended, because it
can give incorrect answers (e.g. Position([[a,b][c,d]],{c,d}) Þ 1 while it is 2).
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Reverse(list)
simply reverses the elements of the list or matrix
Examples: Reverse({a,b,c,d}) Þ {d,c,b,a}
Reverse([[a,b][c,d]]) Þ [[c,d][a,b]]
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RmDup(list)
returns list with duplicate elements removed
Needs: DelElem, MemberQ
Example: RmDup({a,b,c,d2,c}) Þ {a,b,c,d2}
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Select(list,str)
returns elements of list that satisfy the criteria given in str
Needs: ReplcStr
Example: Select({-4.5,-3,2,0,7.2,¥},"not isApprox(#) and #³0") Þ {2,0, ¥}
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Sequency(list)
returns the number of sign changes in list
Examples: Sequency({3,-2,1}) Þ 2, Sequency({-3,-2,1}) Þ 1
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Sort(list)
sorts the elements of list in ascending order
Needs: ListSwap
Example: Sort({4,1,9,3,7,1}) Þ {1,1,3,4,7,9}
Note: Sort works only for real numeric elements. If a matrix is provided, it
sorts the rows of the matrix, treating columns past the first column as
dependent.
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Union(list1,list2)
returns the union of list1 and list2
Needs: RmDup
Example: Union({c,a,b},{d,c,a}) Þ {c,a,b,d}
Note: The union is a joined list with duplicate elements removed.
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IfThen(boolxpr1,boolxpr2)
gives the conditional boolxpr1Þboolxpr2, which means ‘if
boolxpr1 then boolxpr2’
Example: IfThen(a, b or c) Þ ~a or b or c
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IfOnlyIf(boolxpr1,boolxpr2)
gives the biconditional boolxpr1Ûboolxpr2, which means
‘boolxpr1 iff boolxpr2’
Example: IfOnlyIf(a and b, c) Þ ~a and ~c or a and b and c or ~b and ~c
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Nand(xpr1,xpr2)
returns the logical nand function
Example: Nand(x or y, z) Þ ~x and ~y or ~z
§
Nor(xpr1,xpr2)
returns the logical nor function
Example: Nor(a, b xor c) Þ ~a and ~b and ~c or ~a and b and c
§
TruthTbl(boolxpr)
returns a truth table for the expression/list boolxpr
Needs: BaseConv, Str2Chrs, VarList
Example: TruthTbl(a xor b) Þ 
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ANumNorm(a)
returns the norm of the algebraic number a
Needs: Coef, MinPoly, RatNumQ
Example: ANumNorm(AlgNum(x4-10×x2+1,{1,-9/2,0,1/2})) Þ -1
Note: The norm of a is the product of all roots of its minimal polynomial. An
algebraic number field is a finite extension field of the field of rational
numbers. Within an algebraic number field is a ring of algebraic integers,
which plays a role similar to the usual integers in the rational numbers.
Algebraic integers and the usual “rational” integers have their similarities
and differences. For example, one may lose the unique factorization of elements
into products of prime powers (the Fundamental Theorem of Arithmetic).
Algebraic numbers are roots of polynomial equations. A non-algebraic number is
called transcendental. A good reference on algebraic number theory is Henri
Cohen’s book “A Course in Computational Algebraic Number Theory.”
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ANumTrac(a)
returns the trace of the algebraic number a
Needs: Coef, MinPoly, RatNumQ
Example: ANumTrac(AlgNum(x4-10×x2+1,{1,-9/2,0,1/2})) Þ 2
Note: The trace of a is the sum of all roots of its minimal polynomial.
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CarLamb(n)
returns the Carmichael lambda l(n), the least universal
exponent
Needs: FactList, LLcm, Totient
Example: CarLamb(5040) Þ 12
Note: A universal exponent for an integer n is an integer m such that for all k
relatively prime to n, km º 1 (mod n).
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CarNumQ(n)
returns true if n is a Carmichael number, else false
Needs: FactList, mand
Examples: CarNumQ(561) Þ true, CarNumQ(563) Þ false
Note: A Carmichael number n is a composite integer such that for every integer
b relatively prime to n, bn-1 º 1 (mod n). For any Carmichael number n, n-1 is divisible by p-1, where p represents any of the prime
factors of n.
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CentrRem(i,j,x)
returns the central remainder
Example: CentrRem(5,8,11) Þ 2
Note: The central remainder, as its name suggests, is the middle remainder in
the remainder sequence (from the Euclidean algorithm) for finding a
representation of a prime p as a sum of two squares.
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cGCD(a,b)
returns the GCD of complex numbers a and b
Example: cGCD(70+6×i, 150+10×i) Þ -2+2×i
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CharPoly(a,x)
gives the characteristic polynomial of the algebraic number a relative to its
number field
Needs: Degree, LLcm, Resultnt, VarList
Example: CharPoly(AlgNum(x3-x+1,{2,7/3,-1}),x) Þ x3-4×x2-67×x/9+925/27
Note: CharPoly currently works only for algebraic numbers. It will be extended
to work for matrices in the future.
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ChinRem({{a1,
m1},{a2, m2},…}) solves the system of
congruences
k º ai (mod mi) for the integer k
Needs: ExtGCD, LLcm
Example: ChinRem({{1,3},{3,5},{5,8}}) Þ 120×@n1 + 13
Note: ChinRem returns the general solution. For a particular solution, set the
constant @n1 to zero.
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Divisors(n)
returns the divisors of n
Needs: FactList, ListPair
Example: Divisors(147) Þ {1,2,19,38}, Divisors(541) Þ {1,541},
Divisors(37) Þ {1,3,9,27,81,243,729,2187}
Note: The divisor list returned by Divisors occasionally has duplicates, and
may not be sorted. A perfect number equals the sum of its proper divisors, i.e.
divisors excluding itself.
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ExtGCD(a,b)
returns integers {d,{s,t}} such that d=s×a+t×b=GCD(a,b)
Needs: IsCmplxN, Quotient, VarList
Example: ExtGCD(541,547) Þ {d = 1, st = {91,-90}}
Notes: ExtGCD can be used to find the multiplicative inverse of a (mod b). For
example, to solve the congruence 3×k º 4 (mod 5), multiply both sides by the mod-5
inverse (the s returned from ExtGCD(3,5)), so k = 2 × 4 = 8 is a solution. ExtGCD also works for
univariate polynomials.
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FactList(n)
returns a matrix of the factors of n, with multiplicities
Needs: ReplcStr
Example: FactList(123456) Þ [[2,6][3,1][643,1]],
FactList(x2+1)
Þ [[x-i,1][x+i,1]],
FactList((x+i)2)
Þ [[x+i, 2]],
FactList((x+i)2/2)
Þ "Error: cannot handle division at present"
Note: If xpr is an integer, FactList returns its prime factors with
multiplicities. If xpr is provided as a string, the factors are found without
any evaluation.
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FundDisc(n)
returns true if n is a fundamental discriminant of a quadratic number field,
else returns false
Needs: Moebius
Examples: seq(FundDisc(i),i,-5,5) Þ
{false,true,true,false,false,false,false,false,false,false,true}
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gFactor(n)
returns the Gaussian factors of n
Needs: FactList, IsGPrime, Sum2Sqr
Examples: gFactor(2) Þ [[1+i,1][1-i,1]], gFactor(3) Þ [[3, 1]],
gFactor(2) Þ [[1+i,2][1-i,2]], gFactor(6) Þ [[1+i,1][1-i,1][3,1]]
Note: The returned factors may not be coprime. The result of this function may
be changed in the future.
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IsGPrime(z)
returns true if z is a Gaussian prime, else false
Examples: IsGPrime(2) Þ false, (1+i)×(1-i) Þ 2, IsGPrime(1+i) Þ true
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IsMPrime(n)
returns true if n is a Mersenne prime, else returns false
Needs: LogBase, Nest
Example: seq(IsMPrime(i),i,1,8) Þ {false,false,true,false,false,false,true,false}
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JacSymb(k,n)
returns the Jacobi symbol
Needs: FactList, LegSymb
Examples: JacSymb(2,8) Þ 0, JacSymb(3,8) Þ -1
Note: JacSymb(k,n) equals LegSymb(k,n) for prime n.
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KronSymb(m,n)
returns the Kronecker symbol (m/n)
Needs: JacSymb
Example: KronSymb(2,3) Þ 1
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LegSymb(k,n)
returns the Legendre symbol
Needs: QuadResQ
Examples: LegSymb(2,3) Þ -1, LegSymb(2,7) Þ 1
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LucasNum(n)
returns the nth Lucas number
Needs: LucasSeq
Example: LucasNum(21) Þ 24476
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LucasSeq(p,q,L0,L1,n)
returns the nth term Ln of the generalized Lucas sequence
with Ln+2 = p×Ln+1-q×Ln
Example: LucasSeq(1,-1,0,1,30) Þ 832040
Note: L0 and L1 are the initial values for the
recurrence.
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MinPoly(a,x)
returns the minimal polynomial of the algebraic number a
Needs: CharPoly, ConstQ, Degree, FactList, NthCoeff, Select, SqrFree, VarList
Example: MinPoly(AlgNum(x3-x+1,{2,7/3,-1}),x) Þ 27×x3-108×x2-201×x+925
Note: The minimal polynomial of an algebraic number a is the unique irreducible
monic polynomial p of smallest degree such that p(a) = 0. MinPoly currently
works only for algebraic numbers. It will be extended to work for matrices in
the future.
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Moebius(n)
returns the Möbius function m(n)
Needs: ReplcStr
Examples: Moebius(12) Þ 0, Moebius(12345) Þ -1, Moebius(1234567) Þ 1
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MultMod(a,b,p)
returns (a×b) (mod p) for large
positive integers a, b, and p
Example: MultMod(123456789,987654321,7) Þ 3, MultMod(25,34,5) Þ 2
Note: MultMod is part of the MathTools Flash app; for installation instructions
see the “Installation” section. Of course, you can use this function for small
integers too. The full path for the MultMod function is MathTool.MultMod.
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NFMahler(a)
returns the Mahler measure of the algebraic number a
Needs: Coef, MinPoly
Example: NFMahler(AlgNum(x4-10×x2+1,{1,-9/2,0,1/2})) Þ Ö(2)+1
Note: The Mahler measure is widely used in the study of polynomials.
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NFNorm(a)
gives the norm of the algebraic number a relative to its number field
Needs: Degree, LLcm, Resultnt, VarList
Example: NFNorm(AlgNum(x3-x+1,{2,7/3,-1})) Þ -925/27
Note: The relative norm of a is the product of all roots of its characteristic
polynomial.
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NFTrace(a)
gives the trace of the algebraic number a relative to its number field
Needs: CharPoly, Degree, VarList
Example: NFTrace(AlgNum(x3-x+1,{2,7/3,-1})) Þ 4
Note: The relative trace of a is the sum of all roots of its characteristic
polynomial.
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NxtPrime(n[,dir])
returns the next/previous prime
Examples: NxtPrime(10621) Þ 10627, NxtPrime(10621,-1) Þ 10613
Note: NxtPrime returns the next prime if dir is positive, and the previous
prime if dir is negative.
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OrderMod(a,m)
returns the multiplicative (mod-m) order ordm(a)
Needs: PowerMod, RelPrime
Examples: OrderMod(2,3) Þ 2, OrderMod(2,5) Þ 4, OrderMod(2,137) Þ 68
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PowerMod(a,b,p)
returns (ab) (mod p) for large positive integers a, b, and p
Example: PowerMod(129140163,488281255,7) Þ 5, PowerMod(2,47,5) Þ 1
Note: The modular power is perhaps the most important function in elementary
number theory. PowerMod is part of the MathTools Flash app; for installation
instructions see the “Installation” section. Of course, you can use this
function for small integers as well. The full path for the PowerMod function is
MathTool.PowerMod.
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PowTower(x)
returns the value of the infinite power tower x^x^x^… = x¥
Needs: LambertW
Example: PowTower(1.1) Þ 1.11178201104,
1.1^1.1^1.1^1.1^1.1 Þ 1.11178052653
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Prime(n)
returns the nth prime number
Examples: Prime(10001) Þ 104743, Prime(0) Þ undef, Prime(-7) Þ undef
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PrimePi(n)
returns the number of primes less than or equal to n
Needs: LogInt, Prime
Example: PrimePi(179) Þ 41
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Primes(n)
returns the first n prime numbers
Example: Primes(11) Þ
{2,3,5,7,11,13,17,19,23,29,31}
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QuadResQ(k,p)
returns true if k is a quadratic residue mod p, else false
Needs: PowerMod
Examples: QuadResQ(7,11) Þ false, QuadResQ(7,31) Þ true
Note: A number k is a quadratic residue mod p if it is relatively prime to p
and there is an integer j such that k º j2 (mod p).
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RelPrime(list)
returns true when all elements of list are relatively prime, else returns false
Examples: RelPrime({5,7}) Þ true, RelPrime({2,4}) Þ false,
RelPrime({x-2,x3+2×x2-x-2}) Þ true
Notes: RelPrime works for numbers and univariate polynomials. The probability
that n integers chosen at random are relatively prime is 1/z(n), where z(n) is the zeta function.
The Fermat numbers 2^2^n+1 are relatively prime.
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RndPrime(n)
returns a random prime number in [1,n]
Example: RandSeed 0:RndPrime(109) Þ 951898333
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SqrtModP(a,p)
returns the smallest nonnegative square-root of a (mod p)
Needs: JacSymb, PowerMod
Examples: SqrtModP(123456,987) Þ 360, SqrtModP(1234567,987) Þ {}
Note: SqrtModP uses Shanks’ method. It returns {} if there is no square-root.
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SqrtNOMP(p)
returns a square-root of -1 mod p
Needs: JacSymb, PowerMod, Sort
Example: SqrtNOMP(1741) Þ 59
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Sum2Sqr(p)
returns {a,b} where p = a2+b2
Needs: CentrRem, SqrtNOMP
Examples: Sum2Sqr(5) Þ {1,2}, Sum2Sqr(103841) Þ {104,305}
Note: Sum2Sqr uses Smith’s algorithm to find {a,b}.
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SumOfSqN(d,n)
returns the number of representations rd(n) of n as a sum of d
squares of integers
Needs: Divisors, DivSigma, FactList, IntExp, JacSymb, Moebius
Examples: SumOfSqN(2,5) Þ 8, SumOfSqN(3,4) Þ 6,
SumOfSqN(7,3) Þ 280, SumOfSqN(5,217) Þ 27840
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Totient(n)
returns the Euler j(n) function, which is the
number of positive integers less than n that are relatively prime to n
Needs: FactList
Examples: Totient(143055667) Þ 142782138, Totient(17100)
Þ 1043669997…
Note: The Euler totient function is a multiplicative function, which is a
function f: 
® 
such that whenever gcd(i,j) = 1, f(i×j) = f(i)×f(j), so that one can obtain
f(n) by multiplying f(pk), where pk is a prime power from
the factorization of n. For all n greater than 1, the value of j(n) lies between 1 and n-1, inclusive. j(n) = n-1 iff n is prime. Also, for n > 2, j(n) is even.
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LagrMult(f(x,y,…),{cond1,cond2,…},{x,y,…})
finds extrema for f(x,y,…) subject to the conditions {cond1=0, cond2=0, …}
Needs: list2eqn
Example: LagrMult(x2+y2,{x2+y-1},{x,y}) Þ x = Ö(2)/2 and y = 1/2 and …
Note: If you want LagrMult to return the maximum/minimum values of f(x,y,…) in
addition to the values of x,y,…, open the function in the Program Editor and
delete the “Return hh” line. The extrema returned may be local or global
extrema. You can use the Hessian to determine the type of extremum.
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Minimize({f,
constraints},{vars, start},{{opt1,val1},…}) finds a minimum of the objective
function f subject to the given constraints
Needs: FuncEval, mand
Examples: RandSeed 0: Minimize({x^2+(y-1/2)^2, y>=0, y>=x+1},
{{x,y},{5,5}},{{“MaxIterations”,2}})
Note: Minimize is a program (instead of a function) so that the user can get an
overview of the progress. It currently uses the method of simulated annealing.
Minimize is experimental and is generally slow, especially with constraints.
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Simplex(obj,cmat)
returns the minimum for the linear programming problem with the objective
function given by obj and the constraints given by cmat
Needs: ListSubt, Select, SimpStep, Sort
Example: Minimize f = 5×x-3×y-8×z subject to the constraints
{2×x+5×y-z£1, -2×x-12×y+3×z£9, -3×x-8×y+2×z£4, x³0, y³0, z³0}
Simplex({5,-3,-8,0,0,0,-1,0},
[[2,5,-1,1,0,0,0,1][-2,-12,3,0,1,0,0,9][-3,-8,2,0,0,1,0,4]]) Þ
{-121,
vars = {0,3,14,0,3,0}}
Note: There are two widely used solution techniques for linear programming:
simplex methods and interior point methods. While simplex methods move from
vertex to vertex along the edges of the boundary of the feasible region,
interior point methods visit points within the interior of the feasible region.
The Simplex function currently uses the tableau form of the simplex method. The
input list is obtained in the above example by setting 5×x-3×y-8×z-f = 0, so that f has a
coefficient of -1. You can also enter a shortened form of the list as
{5,-3,-8}. The rows in cmat give the coefficients of the original variables and
the “slack” variables in the constraints. There is a slack variable for each
inequality constraint. A slack variable allows you to turn an inequality
constraint into an equality constraint. So, for example, the first constraint
in the above example would become 2×x+5×y-z+s1 = 1. The
final result displays the minimum of f and the values of the original
and slack variables at the minimum. So, for the above example, f has a
minimum at -121, where {x=0, y=3, z=14, s1=0, s2=3, s3=0}.
To find the maximum rather than the minimum, simply find the minimum of –f
and flip the sign of f in the result.
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SimpStep(obj,mat,bvars)
does pivoting for a single step of the simplex method and gives the tableau for
the next basic feasible solution with the specified basic variables bvars
Needs: ListSubt
Example: SimpStep({5,-3,-8,0,0,0,-1,0}, [[2,5,-1,1,0,0,0,1][-2,-12,3,0,1,0,0,9]
[-3,-8,2,0,0,1,0,4]], {3,4,5}) Þ
[[-3/2,-4,1,0,0,1/2,0,2][1/2,1,0,1,0,1/2,0,3]
[5/2,0,0,0,1,-3/2,0,3][-7,-35,0,0,0,4,1,16]]
Note: The rows represent the results for {bvars[1],bvars[2],…,bvars[n],f},
where f is the objective function. Non-basic variables (the variables
not in bvars, in this case {1,2,6}) are assumed to be zero, so that one can
read off the result for each basic variable by looking at the number in the
last column.
§
FreeQ(expression1,expression2)
returns true if expression1 is free of expression2, else returns false
Needs: mand
Examples: FreeQ(y^(a×x2+b),x) Þ false, FreeQ(ln(cos(x)),y) Þ true,
FreeQ({a,b2×x,c,d2},2×x) Þ false, FreeQ(x2,"cos") Þ true,
FreeQ(cos(x)2,"cos") Þ false
Note: Please see MatchQ entry for a note on auto-simplification.
§
MatchQ(expression,form)
returns true if the structure of expression matches that of form, else returns
false
Needs: mand
Examples: MatchQ(2,x) Þ false, MatchQ(2,x_) Þ true, MatchQ(a2,x_) Þ true,
MatchQ(a2,x_+y_)
Þ
false, MatchQ(ax×f(x),2x×f(x)) Þ false,
MatchQ(2x×f(x), ax×f(x)) Þ false, MatchQ(2x×f(x), a_x×f(x)) Þ true,
MatchQ(a(b,c),d(e,f,g))
Þ
false, MatchQ(a(b,c),d(e,f)) Þ false,
MatchQ(a_(b_,c_),d_(e_,f_)) Þ true
Note: The AMS performs some auto-simplification before MatchQ can use the
expression, so for example MatchQ(x+x,a_+b_) Þ false, because x+x is
simplified to 2×x. This can affect the form
as well. Here is another example: MatchQ(x2×y(n),a_×f_(n)) Þ false, while MatchQ(x2×c(n),a_×f_(n)) Þ true. This auto-simplification
can be good or bad, depending on the context. In general, I would suggest
trying expressions and forms out on the home screen to see how they are
auto-simplified before using them in MatchQ. Also, until I implement default
values for MatchQ, x will not match a_n_ ; this is because the
structure does not match exactly unless you use a default value of 1 for
powers.
§
AiryAi(x)
returns the Airy function Ai(x) for real x
Needs: Gamma
Examples: AiryAi(0.1) Þ 0.329203129944, AiryAi(7.3)
Þ 0.000000332514
§
AiryBi(x)
returns the Airy function Bi(x) for real x
Needs: Gamma
Examples: AiryBi(-5.2) Þ -0.275027044184,
AiryBi(9.5) Þ 96892265.5804
§
BellNum(n)
returns the nth Bell number
Needs: BellPoly
Example: seq(BellNum(i),i,0,6) Þ {1,1,2,5,15,52,203}
§
BellPoly(n,x)
returns the nth Bell polynomial
Needs: BellNum
Example: BellPoly(5,x) Þ x5+10×x4+25×x3+15×x2+x
Note: BellPoly(n,1) returns the nth Bell number.
§
BernNum(n)
returns the nth Bernoulli number Bn
Needs: Zeta
Example: seq(BernNum(i),i,1,6) Þ {-1/2,1/6,0,-1/30,0,1/42}
§
BernPoly(n,x)
returns the nth Bernoulli polynomial Bn(x)
Needs: IsCmplxN, Zeta
Example: BernPoly(3,x) Þ x3-3×x2/2+x/2
§
BesselI(n,z) returns the nth modified Bessel function of
the first kind In(z)
Needs: BesselJ
Examples: BesselI(3/2,x) Þ Ö(2)×((x-1)×(ex)2+x+1)/(2×Ö(p)×x3/2×ex),
BesselI(0.92,0.35) Þ 0.210999194087-2.2083340901E-16×i
Note: Please see the note in the help entry for BesselJ.
§
BesselJ(n,z) returns the nth Bessel function of the
first kind Jn(z)
Needs: Gamma, IsCmplxN
Examples: BesselJ(3/2,x) Þ Ö(2)×sin(x)/(Ö(p)×x3/2)-Ö(2)×cos(x)/(Ö(p)×Ö(x)),
BesselJ(2,1/2) Þ 0.030604023459,
BesselJ(51/2,1.×π) Þ 0.
Note: Symbolic results from BesselJ for half-integer n may give incorrect results for subsequent
floating point evaluation because of catastrophic cancellation. Do
comDenom(symbresult|x=«value», q) and then approximate it, or calculate the
numeric result directly (see the last example above).
§
BesselK(n,z) returns the nth modified Bessel function of
the second kind Kn(z)
Needs: BesselI
Example: BesselK(5/2,x) Þ
Ö(p)×Ö(2)/(2×Ö(x)×ex)+3×Ö(p)×Ö(2)/(2×x3/2×ex)+3×Ö(p)×Ö(2)/(2×x5/2×ex)
Note: Please see the note in the help entry for BesselJ.
§
BesselY(n,z) returns the nth Bessel function of the
second kind Yn(z)
Needs: BesselJ
Example: BesselY(3/2,x) Þ -Ö(2)×cos(x)/(Ö(p)×x3/2)-Ö(2)×sin(x)/(Ö(p)×Ö(x)),
BesselY(1,10.3) Þ 0.24706994
Note: Please see the note in the help entry for BesselJ.
§
Beta(p,q)
returns the beta function B(p,q)
Needs: Gamma
Example: Beta(1/2,2) Þ 4/3
Note: The (complete) beta function is symmetric in its arguments, that is,
B(p,q) = B(q,p).
§
Catalan(n)
returns the nth Catalan number
Example: seq(Catalan(i),i,1,6) Þ {1,2,5,14,42,132}
§
ChebyT(n,x)
returns the nth Chebyshev polynomial of the first kind Tn(x)
Needs: Gamma
Example: ChebyT(5,x) Þ 16×x5-20×x3+5×x, ChebyT(n,0) Þ cos(n×p)/2
§
ChebyU(n,x)
returns the nth Chebyshev polynomial of the second kind Un(x)
Needs: Gamma
Example: ChebyU(6,x) Þ 64×x6-80×x4+24×x2-1, ChebyU(n,1) Þ n+1
§
Clebsch(j1,m1,j2,m2,j,m)
returns the Clebsch-Gordan coefficient
Needs: ReplcFac
Examples: Clebsch(1/2,1/2,1/2,-1/2,1,0) Þ Ö(2)/2,
Clebsch(1,1,1,1/2,2,3/2) Þ Ö(105)/12
§
Cyclotom(n,x)
returns the nth cyclotomic polynomial
Needs: RelPrime
Example: Cyclotom(6,x) Þ x2-x+1
Note: The nth cyclotomic polynomial is the minimal polynomial of the
primitive nth root of unity and is irreducible over 
with degree j(n), where j(n) is the Euler totient function.
§
DblFact(n)
returns the double factorial n!!
Needs: IsCmplxN
Example: DblFact(14) Þ 645120, DblFact(2.1) Þ 2.10479089216
§
DivSigma(n,k)
returns the sum sk(n) of kth powers
of the divisors of n
Needs: FactList
Examples: DivSigma(111,0) Þ 4, DivSigma(111,1) Þ 152,
DivSigma(111,2) Þ 13700
Note: As special cases, DivSigma(n,0) returns the number of divisors and
DivSigma(n,1) returns the sum of the divisors. The divisors in the example
above are {1,3,37,111}. A perfect number equals the sum of its proper divisors,
i.e. divisors excluding itself.
§
Erf(x)
returns the error function
Needs: Erfi
Examples: Erf(0.5) Þ 0.520499877813, Erf(3.7) Þ 0.999999832845,
Erf(i) Þ 1.6504257588×i
Note: The nth derivative of erf(x) involves the n-1th Hermite polynomial.
§
Erfc(x)
returns the complementary error function
Needs: Erf
Example: Erfc(0.5) Þ 0.479500122187
§
Erfi(z)
returns the imaginary error function erf(i×z)/i
Needs: Erf, Hypg1F1
Examples: Erfi(1) Þ 1.6504257588, Erfi(-i) Þ -0.84270079295×i
§
Euler(n,x)
returns the nth Euler polynomial
Needs: BernPoly, IsCmplxN
Example: Euler(4,x) Þ x4-2×x3+x
Note: The nth Euler number is given by 2n×Euler(n,1/2).
§
ExpIntE(n,x)
returns the exponential integral En(x)
Needs: IncGamma
Examples: ExpIntE(-1,z) Þ ((z+1)×e-z)/z2,
ExpIntE(8,0) Þ 1/7
§
ExpIntEi(x)
returns the exponential integral Ei(x)
Examples: ExpIntEi(2.3) Þ 6.15438079133,
ExpIntEi(98.1) Þ 4.14110745294E40,
ExpIntEi(-5) Þ -0.001148295591,
ExpIntEi(-6) Þ -0.000360082452
§
FibNum(n)
returns the nth Fibonacci number Fn
Examples: seq(FibNum(i),i,1,6) Þ {1,1,2,3,5,8},
FibNum(2000) Þ <418-digit integer> (takes about 29
seconds),
FibNum(2000.) Þ 4.22469633342E417 (takes about 0.21
seconds)
Note: The ratios of alternate Fibonacci numbers are said to measure the
fraction of a turn between successive leaves on the stalk of a plant. Also, the
number of petals on flowers are said to be Fibonacci numbers. (See MathWorld entry.)
FibNum uses a combination of the Binet formula and the Q-matrix method.
§
FibPoly(n,x)
returns the nth Fibonacci polynomial Fn(x)
Example: FibPoly(7,x) Þ x6+5×x4+6×x2+1
Note: Fibonacci polynomials are used in a variety of fields, such as
cryptography and number theory.
§
FresnelC(x)
returns the Fresnel cosine integral C(x) for real x
Examples: FresnelC(0.78) Þ 0.748357308436,
FresnelC(6.4) Þ 0.549604555704, FresnelC(0)
Þ 0
§
FresnelS(x)
returns the Fresnel sine integral S(x) for real x
Needs: FresnelC
Examples: FresnelS(1.1) Þ 0.625062823934,
FresnelS(13.7) Þ 0.479484940352, FresnelS(0)
Þ 0
§
Gamma(z)
returns the gamma function G(z)
Needs: DblFact, IsCmplxN
Examples: Gamma(11/2) Þ 945×Ö(p)/32, Gamma(1/3) Þ 2.67893853471,
Gamma(1+5×i) Þ -0.001699664494-0.001358519418×i
Note: The gamma function generalizes the factorial function from the natural
numbers to the complex plane. However, it is not a unique solution to the
problem of extending x! into complex values of x. Other solutions are of the
form f(x) = G(x)×g(x), where g(x) is an arbitrary periodic
function with period 1.
§
Gegenbau(n,m,x)
returns the nth Gegenbauer polynomial for the parameter m
Needs: Gamma, IsCmplxN
Example: Gegenbau(4,2,x) Þ 80×x4-48×x2+3
§
Hermite(n,x)
returns the nth Hermite polynomial Hn(x)
Example: Hermite(4,x) Þ 16×x4-48×x2+12,
Hermite(n,0) Þ 2n×Ö(p)/(-n/2-1/2)!,
Hermite(100,1) Þ -1448706729…,
Hermite(10,-3) Þ -3093984,
Hermite(-1,3) Þ 0.15863565
Note: For any given interval (a,b) there is always a Hermite polynomial that
has a zero in the interval.
§
Hypg1F1(a,c,z)
returns Kummer’s confluent hypergeometric function 1F1(a;c;z)
Needs: Gamma
Examples: 1/Ö(p)×Hypg1F1(1/2,3/2,-(1/2)2)
Þ
0.520499877813,
Hypg1F1(1,2,1) Þ e-1, Hypg1F1(3,1,x) Þ (x2/2+2×x+1)×ex,
Hypg1F1(-2,3/2,x) Þ ((4×x2-20×x+15)/15), Hypg1F1(a,a,x) Þ ex,
expand(Hypg1F1(-2,b,x),x) Þ x2/(b×(b+1))-2×x/b+1,
Note: 1/Ö(p)×Hypg1F1(1/2,3/2,-x2)
gives the error function erf(x).
§
Hypg2F1(a,b,c,z)
returns Gauss’ hypergeometric function 2F1(a,b;c;z)
Needs: ChebyT, Gamma, Gegenbau, Jacobi, Legendre
Examples: Hypg2F1(1/2,1/2,3/2,x2) Þ sin-1(x)/x, Hypg2F1(1,1,1,-x) Þ 1/(x+1),
Hypg2F1(1,1,2,1-x) Þ ln(x)/(x-1),
Hypg2F1(-3,3,-5,z) Þ z3+9×z2/5+9×z/5+1,
Hypg2F1(1/3,1/3,4/3,1/2) Þ 1.05285157425,
§
HypgPFQ({a1,a2,…,ap},{b1,b2,…,bq},z) returns the generalized
hypergeometric function pFq(a;b;z)
Needs: Gamma, Hypg1F1, Hypg2F1
Examples: HypgPFQ({},{},x) Þ ex, HypgPFQ({},{1/2},-x2/4) Þ cos(x)
§
HZeta(n,z)
returns the Hurwitz zeta function 
Needs: Psi
Examples: HZeta(4,1/2) Þ p4/6, HZeta(0,-x/2) Þ (x+1)/2,
HZeta(2,1/4) Þ 8×çatalan+p2, HZeta(n,3) Þ -2-n+z(n)-1,
HZeta(-3,1/4) Þ -7/15360, HZeta(4.2,5) Þ 0.002471293458
§
IncBeta(z,a,b)
returns the incomplete beta function Bz(a,b)
Needs: Beta, Pochhamm
Examples: IncBeta(1,-3/2,2) Þ 4/3, IncBeta(5,4,9) Þ 1142738125/396
§
IncGamma(n,x)
returns the incomplete gamma function G(n,x)
Needs: ExpIntEi, Gamma, IsCmplxN, Pochhamm
Examples: IncGamma(2,z) Þ (z+1)×e-z, IncGamma(11,1) Þ 9864101×e-1,
IncGamma(1.2,5.8) Þ 0.004435405791,
IncGamma(n,1) Þ G(n,1)
§
InvErf(x)
returns the inverse erf-1(x) of the error function
Example: InvErf(0.013) Þ 0.011521459812, Erf(0.011521459812) Þ 0.013
§
InvErfc(x)
returns the inverse erfc-1(x) of the complementary
error function
Needs: InvErf
Example: InvErfc(1/2) Þ 0.476936276204, Erfc(0.476936276204) Þ 0.5
§
Jacobi(n,a,b,x) returns the Jacobi
polynomial 
Needs: Gamma, IsCmplxN
Examples: Jacobi(3,2,2,1) Þ 10, Jacobi(3,2,1,x) Þ (21×x3+7×x2-7×x-1)/2
Note: Jacobi polynomials are the most general orthogonal polynomials for x in
the interval [-1, 1].
§
KDelta({n1,n2,…})
returns 1 if n1=n2=…, else returns 0
Examples: KDelta({0,0,1,0}) Þ 0, KDelta({0,0,0,0}) Þ 1,
KDelta({1,1,1}) Þ 1, KDelta({a,1}) Þ 0, KDelta({a,a}) Þ 1
§
Laguerre({n,k},x)
returns the associated Laguerre polynomial 
Examples: Laguerre(3,x) Þ (-x3+9×x2-18×x+6)/6, Laguerre(n,0) Þ 1,
Laguerre({3,1},x) Þ (-x3+12×x2-36×x+24)/6
Note: If you give an integer n as the first argument, you get the ordinary
Laguerre polynomial Ln(x). Laguerre polynomials appear
as eigenfunctions of the hydrogen atom in quantum mechanics.
§
LambertW(x)
returns the principal branch of the Lambert W function
Example: LambertW(2/3×e2/3) Þ 2/3,
LambertW(0.12) Þ 0.10774299816228
[approximately 0.68 seconds]
(solving the equation
without an initial guess takes about 8 seconds)
Note: The Lambert W function is defined by x = W(x)×eW(x)
§
Legendre({n,m},x)
returns the associated Legendre function of the first kind 
Needs: Gamma, ReplcStr
Examples: Legendre(3,x) Þ (5×x3)/2-(3×x)/2, Legendre(10,0) Þ -63/256,
Legendre({3,1},x) Þ (15×x2×Ö(1-x2))/2-(3×Ö(1-x2))/2
Note: The returned function may have an extra factor of (-1)m. If you give an integer n
as the first argument, you get the ordinary Legendre polynomial Pn(x).
Because of the rather low value of the maximum representable number on these
devices, a bug can crop up when trying to evaluate the Legendre polynomials for
large n, e.g. Legendre(1000,0) incorrectly returns zero.
§
Legendr2({n,m},x)
returns the associated Legendre function of the second kind 
Needs: Gamma, Legendre
Examples: Legendr2(70,0) Þ 0, Legendr2(1/2,0) Þ -0.847213084794,
Legendr2(0,x) Þ ln(x+1)/2-ln(-(x-1))/2
Note: If you give an integer n as the first argument, you get the ordinary
Legendre polynomial Qn(x). Please also see the notes
for Legendre.
§
LogInt(x)
returns the logarithmic integral li(x)
Needs: ExpIntEi
Example: LogInt(0.1) Þ -0.032389789593
§
LucasNum(n)
returns the nth Lucas number
Needs: LucasSeq
Example: LucasNum(21) Þ 24476
§
LucasSeq(p,q,L0,L1,n)
returns the nth term Ln of the generalized Lucas sequence
with Ln+2 = p×Ln+1-q×Ln
Example: LucasSeq(1,-1,0,1,30) Þ 832040
Note: L0 and L1 are the initial values for the
recurrence.
§
Moebius(n)
returns the Möbius function m(n)
Needs: ReplcStr
Examples: Moebius(12) Þ 0, Moebius(12345) Þ -1, Moebius(1234567) Þ 1
§
Multinom(list)
returns the number of ways of partitioning N=sum(list) objects into m=dim(list)
sets of sizes list[i]
Example: Multinom({a,b,c}) Þ (a+b+c)!/(a!×b!×c!)
§
PellNum(n)
returns the nth Pell number
Example: seq(PellNum(i),i,1,6) Þ {1,2,5,12,29,70}
Note: The Pell numbers are the Un’s in the Lucas sequence.
§
Pochhamm(a,n)
returns the Pochhammer symbol (a)n
Needs: Gamma
Example: Pochhamm(x,3) Þ x×(x+1)×(x+2)
§
Polylog(n,z)
returns the polylogarithm Lin(z)
Needs: FuncEval, Gamma, HZeta, IsCmplxN, Psi, Zeta
Examples: Polylog(-1,x) Þ x/(x-1)2,
Polylog(2,-1) Þ -p2/12,
Polylog(2,-2) Þ
-1.43674636688
§
Psi(n,z)
returns the polygamma function
Needs: BernNum, Zeta
Examples: Psi(3,1/2) Þ p4,
Psi(0,i)
Þ 0.094650320623+2.07667404747×i,
Psi(2,6.38) Þ -0.028717418078,
Psi(0,11/3) Þ -g - (3×ln(3))/2 + (p×Ö(3))/6 + 99/40,
Psi(13,2) Þ (2048×p14)/3 - 6227020800
Note: Psi(0,z) returns the digamma function 
. Psi(n,z) is the (n+1)th
logarithmic derivative of the gamma function G(z). The g returned in results is the Euler gamma
constant g » 0.57721566490153, sometimes also called the
Euler-Mascheroni constant.
§
RegGamma(a,x)
returns the regularized incomplete gamma function Q(a,x)
Needs: Erfc, Gamma, IncGamma, Pochhamm
Examples: RegGamma(0,x) Þ 0, RegGamma(-91,x) Þ 0,
RegGamma(3,x) Þ (x2/2+x+1)×e-x
§
SBesselH(k,n,x)
returns the spherical Bessel function 
for k=1 or k=2
Examples: SBesselH(1,1,x) Þ -((x+i)×ei×x)/x2,
SBesselH(2,1,x) Þ -((x+-i)×e-i×x)/x2
§
SBesselI(n,x)
returns the modified spherical Bessel function 
Example: SBesselI(n,x) Þ (e-x×((x-1)×e2×x+x+1))/(2×x2)
§
SBesselJ(n,x)
returns the spherical Bessel function 
Example: SBesselJ(1,x) Þ -(x×cos(x)-sin(x))/x2
§
SBesselK(n,x)
returns the modified spherical Bessel function 
Example: SBesselK(1,x) Þ ((x+1)×e-x)/x2
§
SBesselN(n,x)
returns the spherical Bessel function 
Example: SBesselN(1,x) Þ -cos(x)/x2-sin(x)/x
§
SinInt(x)
returns the sine integral function Si(x) for real x
Examples: SinInt(0.007) Þ 0.00699998, SinInt(0.1) Þ 0.0999445,
SinInt(-0.1) Þ -0.0999445, SinInt(18.2) Þ 1.529091
§
SphrHarm(l,m,q,j) returns the spherical
harmonic Ylm(q,j)
Needs: Legendre
Example: SphrHarm(2,0,q,f) Þ ((3×(cos(q))2-1)×Ö(5))/(4×Ö(p))
§
StirNum1(n,k)
returns the Stirling number of the first kind
Needs: StirNum2
Example: StirNum1(5,3) Þ 35
Note: (-1)n-k×StirNum1(n,k) gives the number of
permutations of n items that have exactly k cycles.
§
StirNum2(n,k)
returns the Stirling cycle number
Example: StirNum2(11,4) Þ 145750
Note: The Stirling cycle number gives the number of ways one can partition an
n-element set into k non-empty subsets. It also occurs in summing powers of
integers.
§
StruveH(n,z)
returns the Struve function Hn(z)
Needs: BesselJ, BesselY, Pochhamm
Examples: StruveH(1/2,0) Þ 0,
StruveH(1/2,x) Þ Ö(2)×(Ö(p)×Ö(x))-Ö(2)×cos(x)/(Ö(p)×Ö(x))
Note: This function currently works mainly only for half-integer n.
§
Subfact(n)
returns the number of permutations of n objects which leave none of them in the
same place
Needs: IsCmplxN
Example: Subfact(20) Þ 895014631192902121
§
Totient(n)
returns the Euler j(n) function, which is the
number of positive integers less than n that are relatively prime to n
Needs: FactList
Examples: Totient(143055667) Þ 142782138, Totient(17100)
Þ 1043669997…
§
Wigner3j(j1,m1,j2,m2,j3,m3)
returns the Wigner 3-j symbol 
Needs: Clebsch
Example: Wigner3j(2,1,5/2,-3/2,1/2,1/2) Þ Ö(30)/15
§
Zeta(z)
returns the Riemann zeta function z(z)
Examples: Zeta(2) Þ p2/6, Zeta(1-i) Þ 0.00330022-0.418155×i,
Zeta(1/2+7×i) Þ 1.02143 + 0.396189×i, Zeta(k) Þ z(k)
Notes: The Riemann zeta function has a central role in number theory. It also
appears in integration and summation. For small approximate values, Zeta seems
to have an error of 10-9 or less. For large negative values, the error may
be significant due to the limited precision on these calculators, even though
most of the digits returned are correct. For example, the error for Zeta(-98.4)
is approximately 4.17×1062, and the
error for Zeta(-251.189) is approximately 3.99×10281. Zeta seems
quite accurate for large positive values, where the zeta function approaches 1.
Two other commonly used sums of reciprocal powers of integers can be related to
the zeta function: 
, 
§
ZetaPrim(z)
returns the first derivative of the zeta function z'(z)
Needs: BernNum
Examples: ZetaPrim(2) Þ -0.937548, ZetaPrim(i) Þ 0.0834062-0.506847×i,
ZetaPrim(-1) Þ 1/12-ln(glaisher),
ZetaPrim(-1.0) Þ -0.165421143696,
ZetaPrim(k) Þ z'(k)
Note: The symbol ‘glaisher’ returned in results represents the
Glaisher-Kinkelin constant A » 1.2824271291006, and
satisfies the equation ln(A) = 1/12 - z'(-1). It appears in sums and integrals of some
special functions.
§
CDF(StatDist("DistributionName",parameters),
x) returns the distribution probability between -¥ and x for the given distribution and
parameters
Needs: Erf, Erfc, Gamma, IncGamma
Examples: CDF(statdist("Cauchy",{1,2}),0.7) Þ 0.452606857723,
CDF(statdist("Cauchy",{1,2}),x) Þ tan-1((x-1)/2)/p+1/2,
CDF(statdist("Chi^2",{0.3}),0.11) Þ 0.688767224365,
CDF(statdist("Exponential",{0.83}),0.04) Þ 0.032654928773,
CDF(statdist("Gamma",{1/2,1/4}),2) Þ 0.999936657516,
CDF(statdist("Gamma",{1/2,1/4}),x) Þ 1-G(1/2,4×x)/Ö(p),
CDF(statdist("Hypergeometric",{2,3,7}),1/3) Þ 2/7,
CDF(statdist("Laplace",{0.96,0.32}),0.5) Þ 0.118760409548,
CDF(statdist("Logistic",{2.5,1}),4) Þ 0.817574476194,
CDF(statdist("Normal",{0,1}),0.2) Þ 0.579259709439,
CDF(statdist("Poisson",{4/3}),9/4) Þ 29×e-4/3/9
§
CentrMom(list,r)
returns the rth central moment of list
Example: CentrMom({a,b},2) Þ (a-b)2/4
§
Freq(list)
returns the distinct elements of list with their frequencies
Needs: Sort
Example: Freq({a,b,b,c,a}) Þ [[a,b,c][2,2,1]]
§
GeomMean(list)
returns the geometric mean of list
Example: GeomMean({a,b}) Þ Ö(a×b)
§
HarmMean(list)
returns the harmonic mean of list
Example: HarmMean({a,b}) Þ 2×a×b/(a+b)
§
IntQuant(list,q)
returns the qth interpolated quantile of a probability distribution
Needs: Sort
Example: IntQuant({a,b,c,d},0.75) Þ c
Note: The quantile is obtained from a linear interpolation of list. IntQuant
can be used to find quartiles, deciles, etc.
§
Kurtosis(list)
returns the Pearson kurtosis coefficient for list
Needs: CentrMom, PStdDev
Example: Kurtosis({a,b,c}) Þ 3/2
§
MeanDev(list)
returns the mean absolute deviation about the mean for list
Example: MeanDev({a,b}) Þ |a-b|/2
§
MedDev(list)
returns the median absolute deviation about the median for list
Example: MedDev({2,3}) Þ 1/2
§
Mode(list)
returns the most frequent element(s) of list
Needs: Freq
Example: Mode({a,b,b,c,a}) Þ {a,b}
§
MovAvg(list,n)
smooths list using an n-point moving average
Example: MovAvg({a,b,c},2) Þ {(a+b)/2,(b+c)/2}
§
MovMed(list,n)
smooths list using a span-n moving median
Example: MovMed({3,7,1,0,5},3) Þ {3,1,1}
§
PDF(StatDist("DistributionName",parameters),
x) returns the probability density function at x for the given distribution and
parameters
Needs: Gamma
Examples: PDF(statdist("Cauchy",{0.22,0.81}),0.75) Þ 0.275166497128,
PDF(statdist("Chi^2",{1/2}),0.4) Þ 0.377535275007,
PDF(statdist("Exponential",{5}),1/4) Þ 5×e-5/4,
PDF(statdist("Exponential",{a}),x) Þ a×e-a×x,
PDF(statdist("FRatio",{0.03,0.14}),0.9) Þ 0.013213046749,
PDF(statdist("Gamma",{0.3,0.72}),0.45)
Þ
0.178995053492,
PDF(statdist("Hypergeometric",{4,5,8}),3) Þ 3/7,
PDF(statdist("Laplace",{0.61,0.05}),0.37) Þ 0.08229747049,
PDF(statdist("Logistic",{0.14,0.8}),5.7) Þ 0.001195999788,
PDF(statdist("Normal",{0,1}),x) Þ Ö(2)×
/(2×Ö(p)),
PDF(statdist("Normal",{0.2,3}),1/3) Þ 0.132849485949,
PDF(statdist("Poisson",{9}),2) Þ 81×e-9/2,
PDF(statdist("StudentT",{0.66}),0.5)
Þ
0.222461859264,
PDF(statdist("StudentT",{3}),x) Þ 6×Ö(3)/(p×(x2+3)2)
§
PStdDev(list)
returns the most likely estimate for the population standard deviation of list
Example: stdDev({a,b}) Þ |a-b|×Ö(2)/2, PStdDev({a,b}) Þ |a-b|/2
§
Quantile(list,q)
returns the qth quantile
Needs: Sort
Example: Quantile({a,b},0.6) Þ b
§
Random(StatDist("DistributionName",parameters))
returns a pseudorandom number using the given statistical distribution
Examples: RandSeed 0:Random(statdist("Bernoulli",{14})) Þ 1,
RandSeed
0:Random(statdist("Beta",{2.4,-0.11})) Þ 0.304489,
RandSeed
0:Random(statdist("Binomial",{4,2/3})) Þ 2,
RandSeed
0:Random(statdist("Cauchy",{-3/2,4})) Þ 20.8374,
RandSeed
0:Random(statdist("Chi^2",{1})) Þ 0.0603222,
RandSeed
0:Random(statdist("DiscreteUniform",{-21,37})) Þ 34,
RandSeed
0:Random(statdist("Gamma",{0.5,0.5})) Þ 0.0150806,
RandSeed
0:Random(statdist("Geometric",{0.027})) Þ 2,
RandSeed
0:Random(statdist("Hypergeometric",{5,3,10})) Þ 3,
RandSeed
0:Random(statdist("Logistic",{0.5,0.5})) Þ 1.90859,
RandSeed
0:Random(statdist("LogNormal",{0.5,0.5})) Þ 1.90195,
RandSeed
0:Random(statdist("Normal",{1,0.4})) Þ 2.01958,
RandSeed
0:Random(statdist("Poisson",{3.2})) Þ 6,
RandSeed
0:Random(statdist("Uniform",{})) Þ 0.943597,
RandSeed 0:Random(statdist("Uniform",{-13,21.2})) Þ 19.2710
§
RMS(list)
returns the root-mean-square of list
Example: RMS({a,b,c}) Þ Ö(a2+b2+c2)/3
§
Skewness(list)
returns the skewness coefficient for list
Needs: CentrMom, PStdDev
Examples: Skewness({a,b}) Þ 0, Skewness({1,2,2.1}) Þ -0.685667537489
§
TrimMean(list,a) returns the a-trimmed mean
Needs: Sort
Examples: TrimMean({a,b,c},1/10) Þ (7×a+10×b+7×c)/24,
TrimMean({2,8,4},1/10)
Þ 55/12
Note: The trimmed mean is equal to the arithmetic mean when a=0.
§
nChars(str,chr)
returns the number of occurrences of chr in the string str
Examples: nChars("mathtools","o") Þ 2,
nChars("mathematics","ma") Þ 2
§
RepChars(chars,n)
returns a string with chars repeated n times
Example: RepChars("math",3) Þ "mathmathmath"
§
ReplcStr(str1,str2,str3)
replaces occurrences of the string str2 with str3 in str1
Example: ReplcStr("aububu","ub","qq") Þ "aqqqqu"
§
Str2Chrs(str)
returns a list of the characters in the string str
Example: Str2Chrs("a bc") Þ {"a"," ","b","c"}
§
StrCount(str,substr)
is an alternative function that returns the number of occurrences of substr in
str
Example: StrCount("more & more","or") Þ 2
§
Aitken(list)
accelerates series with Aitken’s d2 method and returns {sn,
delta}
Example: Aitken(seq((-1)k/(2×k+1),k,0,10)) Þ {0.785459904732,
0.00014619883}
Notes: The first few terms of the series are given in ‘list’. For example, to
accelerate the series å(x(i),i,a,b), one could use
seq(x(i),i,a,a+10). ‘sn’ is the last accelerated sum, and ‘delta’ is the
difference between the last two sums.
§
EMSum(x(i),i,a,expstart,b,exporder)
returns an asymptotic (Euler-Maclaurin) expansion for the sum å(x(i),i,a,b)
Examples: å(1/(2^k),k,0,¥) Þ 2,
EMSum(1/(2^k),k,0,1,¥,5.) Þ 2.0000000000047,
å(e1/x^2-1,x,1,¥) Þ å(e1/x^2-1,x,1,¥) [unevaluated],
EMSum(e1/x^2-1,x,1,5,¥,4.) Þ 2.40744653864
Note: expstart is the start of the expansion and exporder is the expansion
order. The formula was taken from the “Handbook of Mathematical Functions”, by
Abramowitz and Stegun.
§
Gosper(xpr,var,low,up)
does hypergeometric summation å(xpr,var,low,up)
Examples: Gosper(k×k!,k,0,n-1) Þ n!-1,
Gosper(4k/nCr(2×k,k),k,0,n-1) Þ (2×n-1)×(n!)2×4n/(3×(2×n)!)+1/3
Notes: Gosper returns "No solution found" if the requested sum cannot
be summed as a hypergeometric series, and gives an "Argument error"
if the series is not hypergeometric or xpr(n+1)/xpr(n) contains variable powers
of n. It will display a "Questionable accuracy" warning if
computations are done using approximations, because due to rounding errors, you
can end up with a solution where there is none. Gosper returns an
"Overflow" if one of the polynomials used in the computations has an
excessive degree (256 or higher), and gives an error "Too many undefined
variables" if the difference between a root of the numerator and a root of
the denominator is not constant.
§
PsiSum(xpr,x)
does the sum å(xpr,x,1,¥) in terms of polygamma
functions
Needs: Coef, Degree, FreeQ, IsApprox, IsCmplxN, Psi, Terms
Example: PsiSum(1/(n2×(8n+1)2),n) Þ y(1,9/8)+y(1,1)-16×y(0,9/8)+16×y(0,1),
Psi(1,9/8)+Psi(1,1)-16×Psi(0,9/8)+16×Psi(0,1) Þ 0.013499486145
Note: PsiSum(xpr,{x,lowlimit}) does the sum å(xpr,x,lowlimit,¥).
§
Sum2Rec(xpr,var)
does indefinite summation (from var=1 to var=N)
Needs: Coef, Degree, PolyGCD, Resultnt
Examples: Sum2Rec(k×k!,k) Þ
{n=f(n)×(n+1)-f(n-1), f(1)=1/2,
s(n)=n!×f(n)×(n+1)},
Sum2Rec(2k/k!,k)
Þ false (the result involves
special functions)
Note: Sum2Rec implements the Gosper algorithm and returns a recurrence equation
to be solved. The result is given in terms of the variable n. Many recurrence
equations can be solved using z-transforms (using Glenn Fisher’s z-transform
routines, for example). The solution for f(n) needs to be substituted into the
last element of the list to obtain the final result. If the indefinite sum
cannot be computed in terms of elementary functions, Sum2Rec returns false.
§
Wynn(x[i],i,k,n)
returns an nth-order Wynn convergence acceleration formula
Example: factor(Wynn(x[i],i,k,2)) Þ
(x[k]×x[k+2]-x[k+1]2)/(x[k+2]-2×x[k+1]+x[k])
§
Curl({E1,E2,E3},{x1,x2,x3},coordstr|{g11,g22,g33})
returns the curl of the vector field E for arbitrary 3-coordinates xa with the coordinate system
given as a string or by the metric gab
Needs: Metric
Example: Curl({x2,x+y,x-y+z},{x,y,z},"Rectangular")
Þ {-1,-1,1}
Notes: The third argument specifies the coordinate system. Any orthogonal
coordinate system can be used. The name of a common coordinate system can be
used ("Rectangular",
"Cylindrical"|"CircularCylinder",
"EllipticCylinder", "ParabolicCylinder",
"Spherical", "ProlateSpheroidal",
"OblateSpheroidal", "Parabolic", "Conical"|"Conic",
"Ellipsoidal", "Paraboloidal") or the diagonal elements of
the metric can be explicitly provided.
§
DirDeriv(f,{x1,x2,x3},coordstr|{g11,g22,g33},u)
returns the directional derivative of f in the direction of the vector u
Needs: Gradient
Example: factor(DirDeriv(x2×ey×sin(z),{x,y,z},"Rectangular",{2,3,-4}),z)
Þ Ö(29)/29×(sin(z)×(3×x+4)-4×cos(z)×x)×x×ey
Note: The vector u is
given as a list {u1,u2,u3}. The third argument
specifies the coordinate system. Any orthogonal coordinate system can be used.
The name of a common coordinate system can be used (see Curl) or the diagonal
elements of the metric can be explicitly provided.
§
Div({E1,E2,E3},{x1,x2,x3},coordstr|{g11,g22,g33})
returns the divergence of the vector field E for arbitrary 3-coordinates xa with the coordinate system
given as a string or by the metric gab
Needs: Metric
Example: Div({x2,x+y,x-y+z},{x,y,z},"Rectangular")
Þ 2×x+2
Note: The third argument specifies the coordinate system. Any orthogonal
coordinate system can be used. The name of a common coordinate system can be
used (see Curl) or the diagonal elements of the metric can be explicitly
provided.
§
Gradient(f,{x1,x2,x3},coordstr|{g11,g22,g33})
returns the gradient of the scalar field f for arbitrary 3-coordinates
xa
with the coordinate system given as a string or by the metric gab
Needs: Metric
Example: Gradient(ex×y+z,{x,y,z},"Rectangular")
Þ {ex×y+z×y, x×ex×y+z, ex×y+z}
Note: The third argument specifies the coordinate system. Any orthogonal
coordinate system can be used. The name of a common coordinate system can be
used (see Curl) or the diagonal elements of the metric can be explicitly
provided.
§
Hessian(f(a[i]),a)
returns the Hessian matrix
Example: Hessian(x×y2 + x3,{x,y})
Þ [[6×x,2×y][2×y,2×x]]
Note: ‘a’ must be a list of variables.
§
Jacobian(funcs,vars)
returns the Jacobian matrix of functions funcs w.r.t. vars
Example: Jacobian({x+y2,x-z,y×ez},{x,y,z}) Þ [[1,2×y,0][1,0,-1][0,ez,y×ez]]
§
Laplacn(f,{x1,x2,x3},coordstr|{g11,g22,g33})
returns the Laplacian of the scalar or vector field f for arbitrary 3-coordinates xa with the coordinate system
given as a string or by the metric gab
Needs: Metric
Example: factor(Laplacn(ex×y+z,{x,y,z},"Rectangular"))
Þ (x2+y2+1)×ex×y+z
Notes: For a vector
field, input f as a list. The third argument
specifies the coordinate system. Any orthogonal coordinate system can be used.
The name of a common coordinate system can be used (see Curl) or the diagonal
elements of the metric can be explicitly provided.
§
Metric(coordstr,vars)
returns the diagonal elements of the metric for the coordinate system specified
by coordstr, with the variable list given in vars
Example: Metric("Spherical",{r,q,f}) Þ {1, r2, sin(q)2×r2}
Note: The coordinate system names are given in the help entry for Curl. They
are case-insensitive, so that you can use "Cylindrical" or
"cyLinDRical".
§
AGM(a,b)
returns the arithmetic-geometric mean of a and b, alongwith the number of
iterations
Needs: IsCmplxN
Example: AGM(1,1.1) Þ {1.0494043395823,3}
Note: The AGM can be used to compute some special functions. It can also be
used to compute p via the Brent-Salamin
algorithm.
§
Apply(funcstr,xpr)
applies expr(funcstr) to xpr
Needs: ReplcStr
Example: Apply("ln(#)+1",2×a) Þ 1+ln(2×a)
§
BaseConv(numstr,b1,b2)
converts a number string from base b1 to base b2
Examples: BaseConv("11",10,2) Þ "1011",
BaseConv("1011",2,10) Þ "11",
BaseConv("0.1",10,2) Þ
"0.00011001100110011001100110011…"
§
BSOptVal(sprice,eprice,vol,intrate,time)
computes the Black-Scholes value of a call option
Needs: Erf
Example: BSOptVal(20,25,3.1,0.027,3/4) Þ 16.0335379358
Note: ‘sprice’ is the stock trading price, ‘eprice’ is the exercise price,
‘vol’ is the stock volatility (the standard deviation of the annualized
continuously compounded rate of return), ‘intrate’ is the risk-free interest
rate (the annualized continuously compounded rate on a safe asset with the same
maturity as the stock option), and ‘time’ is the time to maturity of the call
option (in years). A stock option is a contract that gives you the right to
buy/sell a stock at the pre-specified exercise price (this is called
“exercising the option”) until the option expires. There are two kinds of stock
options: American (you can exercise the option any time until the option
expires) and European (you can exercise the option only at the expiration
date). Instead of exercising the option, you can sell it to someone else before
the option expires. Note that the Black-Scholes model makes several assumptions
about the underlying securities and their behavior.
§
Conic(f(x,y))
returns a description of the conic given by f(x,y)
Needs: FreeQ
Examples: Conic(x2-y2=0) Þ “Intersecting lines”, Conic(x2+y2=4)
Þ “Ellipse”
Note: The function must be in terms of x and y.
§
Curvatur(f,x)
returns the extrinsic curvature k of the function f(x)
Examples: Curvatur(x2-y2=1,{x,y}) Þ -(x2-y2)/(x2+y2)3/2,
Curvatur(y=x2,{x,y})
Þ -2/(4×x2+1)3/2,
Curvatur({t,t2},t)
Þ 2/(4×t2+1)3/2,
Curvatur({t,t2,t},t)
Þ 1/(2×t2+1)3/2
Note: f can be a 2D scalar function or a 2D/3D vector-valued function.
§
Date()
returns the current date and time on Hardware 2 units
Example: Date() Þ "Thu 03/06/03 10:45
PM"
Note: The date and time are shown using the formats specified in the Clock app.
§
FactList(xpr)
returns a matrix of the factors of xpr, with multiplicities
Needs: ReplcStr
Examples: FactList(123456) Þ [[2,6][3,1][643,1]],
FactList(x2+1)
Þ [[x-i,1][x+i,1]],
FactList((x+i)2)
Þ [[x+i, 2]],
FactList((x+i)2/2)
Þ "Error: cannot handle division at
present"
Note: If xpr is an integer, FactList returns its prime factors with
multiplicities. If xpr is provided as a string, the factors are found without
any evaluation.
§
FixPoint(funcstr,xpr)
applies expr(funcstr) repeatedly to xpr until the result doesn’t change
Needs: Apply
Example: FixPoint("cos(#)",0.) Þ 0.739085133215
§
FnPeriod(f(x),x)
returns the period of a univariate function f(x)
Needs: IsCmplxN, LLcm, RmDup, Select
Examples: FnPeriod(sin(3×x+2),x) Þ 2×p/3, FnPeriod(sin(a×x+b),x) Þ ¥,
FnPeriod(x2,x) Þ ¥, FnPeriod(sin(sin(x)),x) Þ 2×p,
FnPeriod(sin(x2),x) Þ ¥, FnPeriod(ln(x),x) Þ ¥, FnPeriod(a,x) Þ 0
§
FuncEval(xpr,vals)
evaluates xpr with constraints given by the list vals
Needs: list2eqn
Examples: FuncEval(x,{x=y,y=z}) Þ z, FuncEval(x2,
x=3) Þ 9
Note: FuncEval(xpr,{constr1,constr2,…}) is equivalent to ‘xpr|constr1 and
constr2 and …’
§
GetArg(xpr,head)
returns the argument of the function with the given head
Examples: GetArg(ln(a×tan(x+b)),“tan(”) Þ b+x,
GetArg(sin(x+a),“sin(”) Þ a+x, GetArg(cos(x),“ln(”) Þ cos(x)
Note: head = part(expression,0). If head is not present, xpr is returned.
§
GrayCode(m)
returns the reflected Gray code ordering of the integers {1,…,2m}
Needs: Nest, Reverse
Examples: GrayCode(2) Þ {0,1,3,2}, GrayCode(2) Þ {0,1,3,2,6,7,5,4}
Note: Two integers are considered adjacent if they differ by a power of two.
§
IntExp(n,b)
returns the largest integer k such that bk divides n
Examples: IntExp(2,2) Þ 1, IntExp(2,7) Þ 0, IntExp(5,2) Þ 0,
IntExp(100,2) Þ 2, IntExp(71!,3) Þ 32
Note: IntExp(n,b) gives the number of trailing zeros in the base-b
representation of n.
§
IsApprox(x)
returns true if x is an approximate number, else returns false
Examples: IsApprox(p) Þ false, IsApprox(approx(p)) Þ true,
IsApprox(1-Ö(2)×i) Þ false, IsApprox(1-Ö(2.0)×i) Þ true
§
IsCmplxN(xpr)
returns true if xpr is real or complex, else returns false
Examples: IsCmplxN(p) Þ true, IsCmplxN(1-i) Þ true, IsCmplxN(x) Þ false,
IsCmplxN({1,i,7.3})
Þ true,
IsCmplxN({e,2k}) Þ false
Note: xpr can be a list or matrix, besides a scalar expression.
§
IsMarkov(mat)
returns true if mat is a valid Markov matrix, else false
Examples: IsMarkov([[a,b][c,d]]) Þ false, IsMarkov([[1,2][3,4]])
Þ true
§
Laplace(xpr,var1,var2)
returns the Laplace transform of xpr with the result given in terms of var2
Needs: FreeQ, RepChars
Examples: Laplace(et,t,s) Þ 1/(s-1), Laplace(sin(t),t,s) Þ 1/(s2+1),
Laplace(d(f(t),t,2),t,s) Þ f(s)×s2-f(0)×s-f'(0), which means
F(s)×s2-F(0)×s-F'(0)
§
LeafCnt(xpr)
returns the number of leaves in xpr
Examples: LeafCnt(x2) Þ 2, LeafCnt(ln(a×x+b)) Þ 3, LeafCnt(1/4) Þ 1
Note: The leaves of an expression are the lowest nodes in its tree structure.
§
LnFrcLim(f1,f2,x)
is a subroutine for the MRV limit algorithm
Needs: MatchQ
Note: This function is experimental.
§
LogBase(b,xpr)
returns the base-b logarithm of xpr
Example: LogBase(2,1.1) Þ 0.137504
Note: ‘b’ and ‘xpr’ need not be numbers.
§
Map(funcstr,xpr)
maps expr(funcstr) over the head of the expression xpr
Needs: ReplcStr
Example: Map("sin(#)",f(1,2,3)) Þ {sin(1),sin(2),sin(3)}
Note: Use “#” in the string to denote the parts of xpr.
§
MarkovEP(mat)
returns the stationary probabilities from a Markov matrix
Example: MarkovEP([[1/2,1/4,1/4][1/4,1/2,1/4][1/4,1/4,1/2]]) Þ {1/3,1/3,1/3}
Note: MarkovEP returns a list representing a vector p such that mat×p = p.
§
MRVList(xpr,x)
returns a list of the most rapidly varying subexpressions of xpr
Needs: FreeQ, LnFrcLim, MemberQ, Union
Note: MRVList is an experimental subroutine for the MRV limit algorithm.
§
NaturalQ(x)
returns true if x is a natural number, else returns false
Examples: NaturalQ(72581) Þ true, NaturalQ(0) Þ true, NaturalQ(-1) Þ false,
NaturalQ(z) Þ false
§
Nest(funcstr,xpr,n)
applies expr(funcstr) to xpr n times
Needs: Apply
Example: Nest("#^#",x,2) Þ x^x^x^x
§
NestList(funcstr,xpr,n)
applies expr(funcstr) to xpr n times and returns a list showing each
application
Needs: Apply
Example: NestList("mod(2×#,1)",approx(1/3),10) Þ {0.33333333333333,
0.666666666666,0.333333333332,0.666666666664,0.333333333328,
0.666666666656,0.333333333312,0.666666666624,0.333333333248,
0.666666666496,.333333332992}
§
NIrrPoly(q,d)
returns the number of degree d irreducible polynomials over the finite field
GF(q)
Needs: Moebius
Example: seq(NIrrPoly(2,i),i,1,10) Þ {2,1,2,3,6,9,18,30,56,99}
§
nResidue(f(x),x,x0)
returns a numeric value for the residue of f(x) at x = x0
Needs: Chop
Examples: nResidue(1/x,x,0) Þ 1., nResidue(1/sin(x),x,0) Þ 1.000008333
Note: nResidue assumes that there are no poles within 0.01 of x0.
§
ParCurve(f,x,y)
returns a parametrization (x(t),y(t)) of a curve f(x,y)=0
Example: ParCurve(x3+x2-y2,x,y) Þ x = t2-1 and y = (t2-1)×t
Note: The parametrization may not be unique. For parametric graphing, you may
need to modify window variables such as tmin and tmax.
§
PolyArea(vertxmat)
returns the area of the simple polygon
Example: PolyArea([[a,b][c,d][e,f]]) Þ (a×(d-f)-b×(c-e)+c×f-d×e)/2
This is the area of a
triangle (on a plane) with the given vertices.
Note: The vertices are given in rectangular coordinates. The vertex matrix is given
as {{x1,y1},{x2,y2},...}.
§
PowerExp(xpr)
expands products and powers of products in xpr
Needs: VarList
Example: expand(ln(xm×yn)) Þ ln(xm×yn) [not completely expanded],
PowerExp(ln(xm×yn)) Þ m×ln(x)+n×ln(y)
§
PtInPgon(vertxmat,{x,y})
returns true if point {x,y} is in the polygon, else false
Examples: PtInPgon({{0,0},{0,1},{2,1},{2,0},{0,0}},{1/2,1/2}) Þ true,
PtInPgon({{0,0},{0,1},{2,1},{2,0},{0,0}},{0,0}) Þ true,
PtInPgon({{0,0},{0,1},{2,1},{2,0},{0,0}},{1,2}) Þ false
Note: The vertices are given in rectangular coordinates. The vertex matrix is
given as {{x1,y1},{x2,y2},...}. The last vertex in vertxmat must be the same as the first
(closed polygon).
§
RatNumQ(xpr)
returns true if xpr is a rational number, else returns false
Examples: RatNumQ(13) Þ true, RatNumQ(1/7) Þ true,
RatNumQ(1/e) Þ false, RatNumQ(Ö(2)) Þ false,
RatNumQ(x) Þ false
§
ReplcFac(xpr)
returns xpr with all occurrences of _! replaced by gamma(_+1)
Needs: Gamma
Example: ReplcFac(Ö(3×(3/2)!)/Ö((5/2)!)) Þ Ö(30)/5
Note: xpr should be a number.
§
RotMat(axis,q) returns the rotation matrix for a rotation
by angle q around the given axis
Example: RotMat({1,0,1},p/3) Þ
[[3/4,Ö(6)/4,1/4][-Ö(6)/4,1/2,Ö(6)/4][1/4,-Ö(6)/4,3/4]]
§
sMod(num,n)
returns mod(num,n) where num is a signed number
Example: mod(-7,3) Þ 2, sMod(-7,3) Þ -1
§
Torsion(v,t)
returns the torsion of a parametrized vector function v(t)
Example: Torsion({sin(s),ln(s),s},s) Þ -s×(s×cos(s)+2×sin(s))/(s4×(sin(s))2+1)
§
UNormVec(v,t)
returns the unit normal vector of a parametrized curve v(t)
Needs: Curvatur, UTanVec
Examples: UNormVec({t, t2, t3},t) Þ {-t×(9×t2+2)/Ö(9×t4+9×t2+1),
-(9×t4-1)/Ö(9×t4+9×t2+1), 3×t×(2×t2+1)/Ö(9×t4+9×t2+1)}
§
UTanVec(v,t)
returns the unit tangent vector for a parametrized curve v(t)
Example: UTanVec({t, t2, et},t) Þ
{1/Ö(e2×t+4×t2+1), 2×t/Ö(e2×t+4×t2+1), et/Ö(e2×t+4×t2+1)}
§
VarList(xpr)
returns a list of all variables in the expression xpr
Needs: IsCmplxN, RmDup
Example: VarList(sin(a×tan(b^f(x))+c×_m)) Þ {b,x,c,a}
(f is considered to be
an operator, and units are not returned)
§ FindHead(expr|exprstring,headexpr) returns a list of subexpressions with the specified head
§ HelpMsg(str) displays the message given in the string str in the status bar
§ IntDig(integer[,base]) returns the digits of the integer in the given base
§ RealDig(realnum[,base]) returns the digits of the real number in the given base
§ VarList(xpr) returns a list of variables in the expression xpr, just like the TI-Basic function VarList under Miscellaneous®OtherFunctions
w MathTools Flash app
§
AllApprx(xpr)
returns true if every elements of xpr is an approximate number (real or
complex), else returns false
Examples: AllApprx(2/3) Þ false, AllApprx(2/3.) Þ true,
AllApprx({3.4,1}) Þ false, AllApprx({3.4,1+i}) Þ true,
AllApprx([[2.74,8.613][x,1.1]]) Þ false
§
AppsInfo()
returns information about installed applications
Example: AppsInfo() Þ 
Note: AppsInfo returns a matrix with one row per application. Of course, the
result depends on what apps you have installed and your AMS version (e.g. the
DeskTop app was introduced in AMS 2.07). Some apps may have blank token names
because they are not intended to be called from TI-Basic. The size of built-in
apps is currently shown as 4 bytes.
§
ByteCnt(xpr)
returns the number of bytes needed for xpr on the estack
Examples: ByteCnt(2+2) Þ 7, ByteCnt(4) Þ 3, ByteCnt(x+x) Þ 3,
ByteCnt(2×x) Þ 5, ByteCnt(x2) Þ 5, ByteCnt(a×x2) Þ 7,
ByteCnt(sin(x)) Þ 2
Note: As shown above, ByteCnt does not evaluate xpr.
§
ConstQ(xpr)
returns true if xpr is a constant expression, free of any variables, else
returns false
Examples: ConstQ(p) Þ true, ConstQ(x3) Þ false, ConstQ(@n1) Þ false,
ConstQ("str") Þ true
Note: Arbitrary real (@k) and integer (@nk) constants are
considered variables.
§
FFT(list[,direction[,convention]])
computes the radix-2 FFT (or inverse FFT) of a list or matrix, using the
signal-processing convention for the FFT by default
Examples: FFT({1+i,1–i}) Þ {2, 2×i}, FFT({2, 2×i},-1) Þ {1+i,1–i}
Notes: The list can have complex numbers, but it must be numeric, and it also
cannot have numbers like e. Its length (or, if it is a matrix, both
dimensions) must be a power of 2. The convention can be:
"s" or "signal" – signal processing (default)
"p" or "physics" – physical sciences
"d" or "data" – data analysis
§
IsUnivar(xpr,var)
returns true if xpr has no variables other than var, else false
Examples: IsUnivar(ln(x),x) Þ true, IsUnivar(1,x) Þ true,
IsUnivar(x,x) Þ true, IsUnivar(y,x) Þ false,
IsUnivar(@5,x) Þ false
Note: Arbitrary real (@k) and integer (@nk) constants are
considered variables.
§
ListAnd(list)
“ands” together the elements of list
Examples: ListAnd({a, b or c}) Þ a and b or a and c,
ListAnd({x>0,x<0}) Þ false
§
ListOr(list)
“ors” together the elements of list
Example: ListOr({a,b}) Þ a or b
§
MultMod(a,b,p)
returns (a×b) (mod p) for large
positive integers a, b, and p
Example: MultMod(123456789,987654321,7) Þ 3, MultMod(25,34,5) Þ 2
Note: MultMod is part of the MathTools Flash app; for installation instructions
see the “Installation” section. Of course, you can use this function for small
integers too. The full path for the MultMod function is MathTool.MultMod.
§
PowerMod(a,b,p)
returns (ab) (mod p) for large positive integers a, b, and p
Example: PowerMod(129140163,488281255,7) Þ 5, PowerMod(2,47,5) Þ 1
Note: The modular power is perhaps the most important function in elementary
number theory. PowerMod is part of the MathTools Flash app; for installation
instructions see the “Installation” section. Of course, you can use this
function for small integers as well. The full path for the PowerMod function is
MathTool.PowerMod.
§
ResetArb()
resets the CAS’ arbitrary real (@k)
and integer (@nk) constants
Example: ResetArb() Þ "Arbitrary constant counters set to 1",
zeros(sin(x),x) Þ {@n1×p}, deSolve(y'=0,x,y) Þ y=@1
§
SqrFree(xpr
[,var]) returns the squarefree factorization of xpr
Examples: SqrFree(x3+5×x2+8×x+4,x) Þ (x+1)×(x+2)2,
SqrFree(x6×z+2×x3×y3×z+2×x5×z2+x2×y3×z2) Þ
x2×(x4+2×x3×z+2×x×y3+y3×z)×z
Note: The factorization is done with respect to variable ‘var’ if it is
specified, or with respect to all variables otherwise.
§
Timing(cmd
[,returntiming]) gives the execution time of a command cmd
Examples: Timing(expand((x+y)15)) Þ <Degree-15
polynomial>
[Status bar shows 0.32 sec],
Timing(expand((x+y)15),true)
Þ 0.315
Note: If ‘returntiming’ is specified, the execution time is returned instead of
the result of the calculation.
Important notes
Known issues
§ GMPFMul has a small memory leak (2)
§ The Position function returns incorrect results when searching for a symbolic list (2)
§ Cubic(Coef((x-i)3,x)) returns a list of undef’s (3)
§ IncGamma(n,x) often returns incorrect results for complex x
§
NumRoots((x+1)n,x,-2,0)
incorrectly returns 1 for n Î 
(3)
§ Cannot use the variables p, q, s and t in PDESolve, as they are reserved for use by PDESolve. Please use other variable names until this is fixed. (3)
§ Sylvestr cannot handle constant (degree 0) polynomials (3)
§ Laguerre({2,-3},x) returns undef (it should return x2/2+x+1) (3)
§ SubRes currently does not return the last subresultant (4)
§ Legendre gives incorrect results for large n, e.g. Legendre(1000,0) returns 0. This is because the calculation hits the maximum representable number (note the warning). (4)
§ Hermite(n,num) and Laguerre(n,num) give Domain Error for most values of num
§ FibNum(32768) gives a Domain Error because of an issue with matrix powers in the AMS
§ PsiSum gives an incorrect result with a non-default lower limit for the sum
§ mZeros returns incorrect results for multiple complex roots when the variable is not specified to be complex, e.g. mZeros((x+i)2,x)
§ Groebner is currently in beta and is very slow. Groebner seems to spend most of its time in NormalF, so if you can optimize NormalF and/or Groebner, please let me know.
§ RTInt and Horowitz are also in beta, and may give Domain Errors. They currently work best for polynomials and rational functions.
§
Risch sometimes cannot solve transformed integrands
when the resultant is constant in q.
Risch is a “framework” for future functionality and may not be particularly
useful (in the sense of being able to handle integrals that the built-in
symbolic integrator cannot do). Risch also sometimes incorrectly claims that
the integral cannot be done in terms of elementary functions (e.g. for ln(x)2).
Thanks to…
§
My
dear cockatiel Pi, who doesn’t hurt me the way humans do
§
My
friends, especially Heidi, Doug, E.W., Ray, Brett, Dave, Craig, Roberto, Chris,
Patrick, and Matt, for being great friends
§
My
family as well as my friends at Wolfram Research, for their support during a
personal crisis
§
Alex
Astashyn, for his PolyQ function
§
Doug
Burkett, for his ListSwap function and for offering to let me reuse some
routines from the tip list (including a couple of random number generation
functions)
§
Glenn
Fisher, for his Coef function and the idea for PowerExp
§
S.L.
Hollis, for his NullVecs and GSchmidt functions
§
Glenn
Fisher, Lars Frederiksen, and Brett Sauerwein, for trying to optimize MathTools
§
Brett
Sauerwein, for his Grab function (renamed to GetArg)
§
Everyone
who has reported bugs (cj, Moshe Mesieles, George Dorner, OAA, Samuel Stearley,
and Steve Zalek)
§
Artraze,
Nevin McChesney, and Brett Sauerwein, for helping to test iSolve
§
David
Stoutemyer, for his base conversion routines (integrated into BaseConv), for
ImpDifN, and for the reference to the method for quickly computing the inverse
Hilbert matrix
§
Kevin
Kofler, for his help in compiling GMP and for the amazing Gosper program
§
Zeljko
Juric, for his PowerMod/MultMod functions
§
Scott
Noveck, for his help in getting PowerMod and MultMod signed as a Flash app
§
Cybernesto
and Doug, who wrote routines similar to NRoots
§
Timité
Hassan, for his FastDet function
§ Gosia Brothers and Brian Dean from TI, for their help with TI Flash Studio and the TI-68k platform in general
History
- Version 2.4.3:
·
Updated:
ParCurve, SingVals
·
Bugfixes:
Gegenbau, MatFunc, mZeros, SVD (SVD was completely rewritten)
·
Removed:
CoefList, Minimize
- Version 2.4.2:
·
Added
around 40 routines
·
Added
MathTools Flash app, with several TI-Basic extensions
·
Updated
around 15 routines
·
Bugfixes:
AiryAi, AiryBi, FactList, FresnelC, FresnelS
·
Renamed:
Interpol ® Lagrange
·
Removed:
Install (and InstData)
- Version 2.4.1:
·
Added
around 10 routines
·
Updated
around 8 routines
·
Renamed:
Gosper ® Sum2Rec
·
Bugfixes:
SVD, demo text files (they were mysteriously corrupted earlier)
- Version 2.4.0:
·
Added
around 50 routines
·
Updated
over 30 routines
·
Renamed:
ContFrac ® ContFrc1, Grab ® GetArg
·
Removed:
Interval (if it is present in your MathTool folder, please delete it)
·
Modified
all routines so that they can be used from any folder
- Versions 2.3 to 2.331:
·
Added
around 165 routines
·
Updated
around 35 routines
·
Renamed:
GenEigVc ® Eigenvec, RepEigVl ® Eigenval, PInv ® PInverse,
PGcd ® PolyGCD, PLcm ® PolyLCM, PMod ® PolyMod
·
Removed:
FNorm (it was equivalent to the built-in norm() function)
·
Bugfixes:
ExtGCD, FreeQ, GenEigVc, Install (and InstData), iSolve, list2eqn, mZeros,
ReplcFac, RTInt, Terms, VarList, ZetaPrim
- Version 2.2:
·
Added
around 15 routines
·
Bugfixes:
Resultnt, Sort, Sylvestr, VarList
- Version 2.1:
·
Added
around 35 routines
·
Updated
a couple of routines
·
Improved
documentation
- Version 2.0:
·
First
major update (many functions added)
Future plans
· Bessel K and Y functions for integer n (high priority)
· Update GMP stuff for AMS 2.07/2.08
· More experimental GMP functions
· Nonlinear multivariate local and global optimization
· Merge nSeries and CmplxMap into MathTools
· Include E.W.’s UniWrite in MathTools, and add feature requested by user
· Fix known bugs and limitations
· Support for trig functions in iSolve
· Hermite, Smith, and Frobenius normal form for matrices
· Matrix tests (HermMatQ, UnitaryQ, PosDefQ, etc.)
· Clement and Wilkinson matrices
· Improved speed for BesselJ for complex arguments
· Return symbolic result for Psi(1,n/4) in terms of çatalan
· IncGamma(n/2,x) and RegGamma(n/2,x), in terms of the error function
· Cosine integral Ci(z), and hyperbolic counterparts for Ci(z) and Si(z)
· Elliptic integrals
· Polylog({n,p},z) should return the Nielsen polylogarithm Sn,p(z)
· More statistical distributions, such as the Erlang and Weibull distributions
· Some inferential and non-parametric statistics
· Goodness-of-fit tests
· Exponential smoothing for data
· Higher-order non-homogeneous constant-coefficient recurrence equations
· Systems of recurrence equations
· Most-rapidly-varying (MRV) algorithm for computing limits
· Asymptotic expansions
· Tensor analysis functions for multidimensional arrays
· Series convergence tests
· Roots of polynomials modulo a composite integer using Hensel lifting and the CRT
·
Improved support for congruence systems and nonlinear
Diophantine equations
(as a solution to Hilbert’s 10th problem (MathWorld entry),
Matiyasevich proved that there is no way to determine solvability of a general
Diophantine equation; however, many special classes of Diophantine equations
can be solved)
· Quantifier elimination using cylindrical algebraic decomposition (CAD)
· Improve RTInt and Horowitz so that they can integrate all rational functions
· Complete Risch and Minimize functions
· Allow ChinRem to take univariate polynomials as arguments
· Primitive roots (from number theory)
· Generalize PolyQ for multivariate polynomials
·
Implement PSLQ with multiple precision to find integer
relations
(I implemented it in TI-Basic, but the 14-digit precision is too low)
· Sugarcube selection strategy and basis reduction for Groebner
· Series solution of ordinary differential equations
· Multivariate limits
· Inflection points of a function
· LQ, Schur (using QR iteration) and Hessenberg matrix decompositions
· Matrix scaling
· More list operations, such as list convolution and correlation
· Linear algebra functions MatDet(), MatInv() and LinSolve() with modulus argument
· Derivatives of special functions
· Multidimensional interpolation
· More filters and discrete transforms for signal processing
· Trigonometric fits for data with regular spacing (using Fourier transform)
· Spline fits
· 3D parametric plots
· 3D vector plots
· 3D matrix plots
· Support for error bars in ListPlot
· Computational geometry
· Lattice reduction to give a reduced basis for a set of vectors (LLL algorithm)
· Improve Hadamard matrix function
· Default values for MatchQ; function Replace for pattern-based substitution
·
Complexity analysis for various types of simplification
(this would work only if the auto-simplification of the AMS were disabled)
· Second-order or higher PDEs and PDE systems, using Lie symmetry analysis, integral transforms, and separation of variables
· Support for integral and integro-differential equations
· Multidimensional finite differences
· More functions for numerical solution of PDEs
· Solvers for stochastic and delay differential equations
· More integral transforms such as the Mellin transform
· Symbolic Laurent series expansions and symbolic residues
·
Closed-form nth derivative, e.g. d(sin(x),x,n)
is
· Graph theory
· Differential geometry
· Coulomb wave functions
· StruveL and Weber functions
· Euler numbers
· Lucas polynomials
· Derivatives of the Airy functions
· Improve speed for PermSig
· PowerSet function (combinatorics)
· Number partitions
· Cartesian product
· Sum-of-products and product-of-sums forms (logic)
· Surface, volume and contour integrals
·
Implicit representations of parametric curves and
surfaces
(transform to a polynomial form, do trig expansion, and use trig identities)
· Wigner 6-j and 9-j symbols for couplings of quantum momentum states
· PolyGCD(poly1,poly2,prime) [multivariate polynomial GCD modulo a prime]
· Heuristic polynomial GCD
· Arithmetic and squarefree factorization over finite fields, such as Galois fields
· Berlekamp factorization (implement nullspace computation mod a prime)
· More finance functions (cash flows, bonds, interest rates, etc.)
· Add some basics on computer algebra to the documentation (ongoing process)
· Optimization of methods used (for speed and size)
· Any suggestions?
Computer Algebra links
· A = B, an online book on hypergeometric identities by Petkovsek/Wilf/Zeilberger (foreword by Knuth)
· ATLAS (automatically tuned linear algebra software)
· Combinatorics electronic journal
· Derivation of Numerical Methods using Computer Algebra (article)
· Factoring Polynomials over Finite Fields of Small Characteristic (article)
· Finite Element Methods universal resource
· GMP, the GNU multiple-precision library
· Gröbner Bases Algorithm with a tutorial
· Gröbner Bases and Elimination of Variables
· Linear Algebra (thanks to E.W. for this link)
· Mathematical Functions from Wolfram Research
· Mathematical Programming Glossary
· Mathematics textbooks online
· MathWorld, the online math encyclopedia
· NEOS optimization server, to solve linear programming problems online
· Numerical Evaluation of Special Functions from NIST
· Numerical Methods for Partial Differential Equations
· On Computing Limits in a Symbolic Manipulation System (article)
· Online Encyclopedia of Integer Sequences
· Optimization Technology Center from Argonne Nat’l Lab and Northwestern Univ.
· Papers by Manuel Bronstein
· Risch algorithm overview (Lecture notes in computer algebra)
· Symbolic integration tutorial (article)
· UMFPACK (sparse linear algebra; uses Level-3 BLAS)
· Weights for finite-difference formulas (article)
· Wilf-Zeilberger theory quick review (slides)
Copyright notice and disclaimer
MathTools may be freely distributed, provided it is not modified, is
distributed as a whole (besides Gosper and MultMod/PowerMod), is not
sold/licensed for monetary profit, and this documentation is provided with it.
If you wish to distribute a modified version, please contact the author.
MathTools is subject to change without notice. MathTools is not intended for
use in a mission/life-critical environment, and although I have tested it, there
may still be bugs and/or inaccuracies. The author and copyright holder will not
be liable for any damages or losses related to the MathTools software or
documentation. If something happens to me, all rights for all of my software
transfer to Doug Burkett.
Thank you for using MathTools! I hope you like it and it is useful to you.
Contact information
Questions/comments/suggestions/bug-reports are welcomed at bbhatt1@towson.edu
For updates and other packages/resources, visit: http://triton.towson.edu/users/bbhatt1/ti/