Introduction to Maple

Matrices (linalg)

> with(linalg);

Warning, the protected names norm and trace have been redefined and unprotected

> m:=matrix([[1,2,3],[2,3,1],[3,2,1]]);

> det(m);

> im:=inverse(m);

> id3:=evalm(m&*im);

> eigenvalues(m);

> map(evalf,[%]);

> charpoly(m,lambda);

> eig:=[eigenvectors(m)];

> eiv:=eig[2][1];

> eigm2:=op(eig[2][3]);

> evalm((m-eiv*id3)&*eigm2);

> nullspace(m);

Gaussian Elimination

> m:=matrix([[1,2,3,-2],[2,3,1,4],[3,2,1,7]]);

> m:=addrow(addrow(m,1,2,-2),1,3,-3);

> m:=addrow(m,2,3,-4);

> submatrix(m,1..2,1..3);

> minor(m,1,2);

Matrices (LinearAlgebra)

> with(LinearAlgebra);

Warning, the assigned name GramSchmidt now has a global binding

> M:=Matrix(3,[[1,2,3],[-1,-3,-5],[-5,7,9]]);

> sort(CharacteristicPolynomial(M,lambda));

> Eigenvalues(M);

> f:= (i,j) -> x^i+y^j;

> Matrix(3,f);

Differentiation, Integration (diff, int)

> restart;

> diff(x^n,x);

> diff(1/sqrt(x+1),x);

> diff(x^x,x);

> diff(exp(x*ln(x)),x);

> int(x^n,x);

> int(sin(x)^5,x);

> int(exp(-x^2),x);

> int(sqrt(1-x^2),x=-1..1);

> int(1/(1+x^3),x=0..infinity);

> int(1/(1+x^6),x=0..infinity);

> int(1/(1+x^7),x=0..infinity);

> int(1/(1+x^n),x=0..infinity);

Definite integration: Can't determine if the integral is convergent.

Need to know the sign of --> n

Will now try indefinite integration and then take limits.

> int(cos(x)/(1+x^2),x=0..infinity);

> convert(%,exp);

> simplify(%);

> int(log(x)/(1+x)^3,x=0..infinity);

Number Theory (numtheory)

> with(numtheory);

Warning, the protected name order has been redefined and unprotected

$[GIgcd, bigomega, cfrac, cfracpol, cyclotomic, divisors, factorEQ, factorset, fermat, imagunit, index, integral_basis, invcfrac, invphi, issqrfree, jacobi, kronecker, lambda, legendre, mcombine, mersen...$
$[GIgcd, bigomega, cfrac, cfracpol, cyclotomic, divisors, factorEQ, factorset, fermat, imagunit, index, integral_basis, invcfrac, invphi, issqrfree, jacobi, kronecker, lambda, legendre, mcombine, mersen...$
$[GIgcd, bigomega, cfrac, cfracpol, cyclotomic, divisors, factorEQ, factorset, fermat, imagunit, index, integral_basis, invcfrac, invphi, issqrfree, jacobi, kronecker, lambda, legendre, mcombine, mersen...$

> ifactor(2^(2^6)+1);

> divisors(111111);

> n:=2352356;

> phi(n);

> pr:=convert(factorset(n),list);

> n*product('(1-1/pr[i])','i'=1..nops(pr));

> cyclotomic(3,x);

> cyclotomic(6,x);

> cyclotomic(7,x);

> cyclotomic(8,x);

> cyclotomic(100,x);

>    for n from 1 to 500 do
   if {coeffs(cyclotomic(n,x))} minus {1,-1} <> {}
      then lprint(n); fi;
od;

> s:={};
st:=time():
for n from 1 to 2000 do
s:=s union {coeffs(cyclotomic(n,x))}: od:
time()-st;
s;

Count the number of primes less than a given limit

> pi(100);

resultants

> p:=x^6+y^2-1;

> q:=x^3+y^3-5;

> factor(resultant(p,q,x));

> solve({p,q},{x,y});

${y = RootOf(24+_Z^2-10*_Z^3+_Z^6), x = RootOf(_Z^3-5+RootOf(24+_Z^2-10*_Z^3+_Z^6)^3)}$

> with(plots):

Warning, the name changecoords has been redefined

> p:=x^2+y^2-1;q:=x-y-1;

> a:=implicitplot(p,x=-1..1,y=-1..1):

> b:=implicitplot(q,x=-2..2,y=-2..1,color=green):

> display({a,b});

other packages

> with(orthopoly);

> seq(T(n,x),n=1..6);

> map(factor,[%]);

> with(stats);with(stats[statplots]):

> mark:=rand(1..100);

> marks:=[seq(mark(),i=1..30)];

> histogram(marks);