| Numeric Functions | |
| abs(x) | absolute value |
| ceil(x) | integer above x |
| floor(x) | integer below x |
| round(x) | round to nearest integer |
| trunc(x) | truncate towards zero |
| ln(x) | natural logarithm |
| log(x) | logarithm base 10 |
| exp(x) | exponential |
| sin(x) | sine |
| cos(x) | cosine |
| tan(x) | tangent |
| sinh(x) | hyperbolic sine |
| cosh(x) | hyperbolic cosine |
| tanh(x) | hyperbolic tangent |
| arcsin(x) | inverse sine |
| arccos(x) | inverse cosine |
| arctan(x) | inverse tangent |
| arcsinh(x) | inverse hyperbolic sine |
| arccosh(x) | inverse hyperbolic cosine |
| arctanh(x) | inverse hyperbolic tangent |
| pi() | approximate pi |
| Number Theory | |
| binomial(n,k) | binomial coefficient |
| continuedfraction(n) | continued fraction of n |
| factorial(n) | factorial of n, i.e. n! |
| gcd(a,b) | greatest common divisor |
| gcdext(a,b) | extended Euclidean algorithm |
| invmod(a,n) | inverse of a modulo n |
| jacobi(a,n) | Jacobi symbol (a | n) |
| latticereduce(A) | lattice basis reduction |
| lcm(a,b) | least common multiple |
| logint(a,b) | floor of log(a) base b |
| logmod(a,b,n) | solve a = b^x mod n |
| mod(a,n) | reduce a modulo n |
| mods(a,n) | reduce a modulo n (symmetric) |
| mulmod(a,b,n) | compute a*b mod n |
| order(a,n) | order of a modulo n |
| powmod(a,b,n) | compute a^b mod n |
| prime(n) | primality test |
| primecount(n) | count primes <= n |
| primefactors(n) | integer factorization |
| primesieve(n) | list primes <= n |
| quotient(a,b) | quotient of a/b |
| randominteger(n) | random integer mod n |
| remainder(a,b) | remainder of a/b |
| rootint(a,b) | floor of a^(1/b) |
| rootmod(a,b,n) | solve a = x^b mod n |
| totient(n) | Euler's totient |
| Formulae | |
| approximate(a) | convert to floating point |
| assign(x,i,a) | assign to variable xi |
| coefficient(a,x,d) | coefficient of x^d in a |
| concatenate(x,i) | create new variable xi |
| content(a) | rational content |
| degree(a,x) | degree of a in x |
| depend(a,x) | does a depend on x? |
| expand(a) | multiply out an expression |
| functions(a) | functions in expression |
| Im(a) | imaginary part of a |
| length(a) | length of a |
| numden(a) | numerator and denominator |
| normal(a) | simplify rational function |
| rationalize(a) | convert to rational numbers |
| Re(a) | real part of a |
| recover(a,p) | rational reconstruction |
| replace(a,u,v) | replace u by v in a |
| symbols(a) | symbols in expression |
| type(a,...) | test for a type |
| variables(a) | variables in expression |
| Numerical Methods | |
| fft(A) | fast Fourier transform |
| fftinv(A) | inverse fft |
| kaczmarz(A,B,n) | solve A*X=B iteratively |
| montecarlo(f,x,y,x0,x1,y0,y1) | integrate where f(x,y) is true |
| montecarlo3d(f,x,y,z,x0,x1,y0,y1,z0,z1) | integrate where f(x,y,z) is true |
| plot(f,x,x0,x1,y0,y1) | plot y = f(x) |
| plot3d(f,x,y,x0,x1,y0,y1,z0,z1) | plot z = f(x,y) |
| plotimplicit(f,x,y,x0,x1,y0,y1) | plot f(x,y) = 0 |
| plotimplicit3d(f,x,y,z,x0,x1,y0,y1,z0,z1) | plot f(x,y,z) = 0 |
| plotinequality(f,x,y,x0,x1,y0,y1) | plot f(x,y) is true |
| plotinequality3d(f,x,y,z,x0,x1,y0,y1,z0,z1) | plot f(x,y,z) is true |
| plotsize(x,y) | set plot dimensions |
| quadrature(f,x,x0,x1) | numerical integration |
| realroot(f,x) | numerical root finding |
| Polynomials | |
| collect(a,x) | group terms by powers of x |
| discriminant(a,x) | discriminant of a w.r.t. x |
| divide(a,b) | test if a is divisible by b |
| factor(a) | polynomial factorization |
| gcd(a,b) | greatest common divisor |
| gcdext(a,b,x) | extended Euclidean algorithm |
| homogenize(a,x) | homogenize polynomial with x |
| hornerform(a,x) | convert to Horner form |
| invmod(a,b,x) | invert a mod b w.r.t. x |
| lcm(a,b) | least common multiple |
| powmod(a,n,b,x) | expand a^n mod b w.r.t. x |
| quotient(a,b,x) | quotient of a/b w.r.t. x |
| randompoly(x,d,t,c) | random polynomial in x |
| remainder(a,b,x) | remainder of a/b w.r.t. x |
| resultant(a,b,x) | resultant of a,b w.r.t. x |
| Calculus | |
| differentiate(a,x) | derivative of a w.r.t. x |
| integrate(a,x) | integral of a w.r.t. x |
| jacobian(a,x) | Jacobian matrix |
| series(a,x,b,n) | series of a about x=b to degree n |
| Linear Algebra | |
| adjoint(A) | matrix adjoint |
| append(A,B) | augment A with rows of B |
| augment(A,B) | augment A with columns of B |
| characteristicpoly(A) | characteristic polynomial |
| determinant(A) | determinant |
| gausselimination(A) | row echelon form |
| gaussjordan(A) | reduced row echelon form |
| gramschmidt(A) | make rows orthogonal |
| hessenbergform(A) | upper Hessenberg form |
| identity(n) | n x n identity matrix |
| interpolate(a,b) | polynomial interpolation |
| leastsquares(A,b) | least squares solution |
| linearsolve(A,b) | solve matrix equations |
| matrix(m,n) | create m x n matrix |
| minimalpoly(A) | minimal polynomial |
| nullspace(A) | nullspace basis |
| polynomial(A,X) | create polynomial from vector |
| randommatrix(n,i,j,p) | create a random matrix |
| randomtensor(n,i,j,k,p) | create a random tensor |
| randomvector(n,i,p) | create a random vector |
| rank(A) | rank of a matrix |
| submatrix(A,i,j,k,l) | extract a submatrix A[i..j,k..l] |
| symmetric(A) | test if symmetric |
| trace(A) | trace of a matrix |
| transpose(A) | transpose matrix |
| vandermonde(a,b) | transposed vandermonde solver |
| vector(n) | create vector |
| Graph Theory | |
| indegree(A,u) | in-degree |
| outdegree(A,u) | out-degree |
| pathdistance(A,P) | path distance |
| shortestpath(A,u,v) | shortest path |
| spanningtree(A) | minimal spanning tree |
| subgraph(A,P) | induced subgraph |
| Algebraic Geometry | |
| groebnerbasis(A,X) | Groebner basis |
| hilbertdimension(G,X) | Hilbert dimension |
| leadingterm(A,X) | leading terms |
| normalform(A,B,X) | normal forms |
| System | |
| digits(n) | number of digits |
| info(n) | print info |
| load(s) | load output |
| save(x) | save output |
| time() | elapsed time |