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	mathematica	sympy	sage	maxima
symbolic expressions
	mathematica	sympy	sage	maxima
literal	expr = 1 + x + x^2	x = symbols('x') expr = 1 + x + x^2	expr = 1 + x + x^2	expr = 1 + x + x^2;
prevent simplification	HoldForm[x + x] x + x // HoldForm
variable update	expr = 1 + x x = 3 (* 4: *) expr	x = symbols('x') expr = 1 + x x = 3 # 1 + x: expr	expr = 1 + x x = 7 # 1 + x: expr	expr: 1 + x; x: 3; /* 1 + x: */ expr;
substitute	(* {3, 3}: *) ReplaceAll[{x, x}, x -> 3] (* {3, 3}: *) {x, x} /. x -> 3 (* {3, 4}: *) {x, y} /. {x -> 3, y -> 4}	Matrix([x, x]).subs(x, 3)	vector([x, x]).subs({x: 3})	/* [3, 3]: */ subst(3, x, [x, x]);
piecewise-defined expression	Piecewise[{{x, x >= 0}, {-x, x < 0}}] (* otherwise case: *) Piecewise[{{-x, x < 0}}, x]	Piecewise((-x, x < 0), (x, x >= 0)) # otherwise case: Piecewise((-x, x < 0), (x, True))	piecewise([   ((-infinity,0), -x),   ((0,infinity), x)])	if x < 0 then -x else x; /* integrating over piecewise-defined expression fails */
simplify	Simplify[Cos[x]^2 + Sin[x]^2] (* perform more simplications: *) FullSimplify[-(1/2) I E^(-I x) (-1 + E^(2 I x))]	simplify(cos(x)**2 + sin(x)**2)
assumption	Simplify[Sqrt[x^2], Assumptions -> x >= 0] Simplify[(-1)^(n * (n + 1)),   Assumptions -> Element[n, Integers]] (* perform fewer simplications: *) Refine[Sqrt[x^2], Assumptions -> x >= 0] Refine[(-1)^(n * (n + 1)), Element[n, Integers]]	x = symbols('x', positive=True) sqrt(x ** 2) n = symbols('n', integer=True) (-1)**((n) * (n + 1))	assume(x > 0) sqrt(x^2)	assume(x > 0); sqrt(x^2); /* There is no assumption predicate for    integer variables. */
assumption predicates	Element[x, Complexes] Element[x, Reals] Element[x, Algebraics] Element[x, Rationals] Element[x, Integers] Element[x, Primes] Element[x, Integers] && Mod[x, 5] == 0 Element[x, Booleans] (* assumptions can use inequalities and logical operators: *) x > 0 || x < 0	# a partial list: complex real algebraic rational integer positive nonpositive negative nonnegative nonzero prime odd even	assume(x, 'complex') assume(x, 'real') assume(x, 'rational') assume(x, 'integer') assume(x, 'odd') assume(x, 'even')	Assumptions can only be created using relational operators.
list assumptions	None. Assumptions are always local.	x.assumptions0	assumptions()	facts(x); # assumptions on all symbols: facts();
remove assumption	None. Assumptions are always local.	# removes all assumptions about x: x = symbols('x')	forget(x > 0) # rm all assumptions: forget()	forget(x > 0);
calculus
	mathematica	sympy	sage	maxima
limit	Limit[Sin[x]/x, x -> 0]	limit(sin(x)/x, x, 0)	limit(sin(x)/x, x=0)	limit(sin(x)/x, x, 0);
limit at infinity	Limit[1/x, x -> Infinity]	limit(1/x, x, oo)	limit(1/x, x=infinity)	limit(1/x, x, inf);
one-sided limit from left, from right	Limit[1/x, x -> 0, Direction -> 1] Limit[1/x, x -> 0, Direction -> -1]	limit(1/x, x, 0, '-') limit(1/x, x, 0, '+')	limit(1/x, x=0, dir='-') limit(1/x, x=0, dir='+')	limit(1/x, x, 0, minus); limit(1/x, x, 0, plus);
derivative	D[x^3 + x + 3, x] D[x^3 + x + 3, x] /. x -> 2	diff(x**3 + x + 3, x) diff(x**3 + x + 3, x).subs(x, 2)	diff(x^3 + x + 3, x) diff(x^3 + x + 3, x).subs({x: 2}) # derivative is synonym of diff	diff(x^3 + x + 3, x); at(diff(x^3 + x + 3, x), [x=2]);
derivative of a function	f[x_] = x^3 + x + 3 (* returns expression: *) D[f[x, x]] (* return functions: *) f' Derivative[1][f] (* evaluating derivative at a point: *) f'[2] Derivative[1][f][2]		f(x) = x^3 + x + 3 diff(f) diff(f)(2)
constants	(* a depends on x; b does not: *) D[a x + b, x, NonConstants -> {a}] Dt[a x + b, x, Constants -> {b}]			/* symbols constant unless declared with depends: */ depends(a, x); diff(a * x + b, x); /* makes a constant again: */ remove(a, dependency);
higher order derivative	D[Log[x], {x, 3}] Log'''[x] Derivative[3][Log][x]	diff(log(x), x, 3)	diff(log(x), x, 3)	diff(log(x), x, 3);
mixed partial derivative	D[x^9 * y^8, x, y, y] D[x^9 * y^8, x, {y, 2}]	diff(x**9 * y**8, x, y, y)	diff(x^9 * y^8, x, 1).diff(y, 2)	diff(x^9 * y^8, x, 1, y, 2);
div, grad, and curl	Div[{x^2, x * y, x * y * z}, {x, y, z}] Grad[2 * x * y * z^2, {x, y, z}] Curl[{x * y * z, y^2, 0}, {x, y, z}]
antiderivative	Integrate[x^3 + x + 3, x]	integrate(x**3 + x + 3, x)	integral(x^3 + x + 3, x)	integrate(x^3 + x + 3, x);
definite integral	Integrate[x^3 + x + 3, {x, 0, 1}]	integrate(x**3 + x + 3, [x, 0, 1])	integral(x^3 + x + 3, x, 0, 1)	integrate(x^3 + x + 3, x, 0, 1);
improper integral	Integrate[Exp[-x], {x, 0, Infinity}]	integrate(exp(-x), (x, 0, oo))	integral(exp(-x), x, 0, infinity)	integrate(exp(-x), x, 0, inf);
double integral	(* integrates over y first: *) Integrate[x^2 + y^2, {x, 0, 1}, {y, 0, x}]	f = integrate(x**2 + y**2, (y, 0, x)) integrate(f, (x, 0, 1))	integral(integral(x^2+y^2, y, 0, x), x, 0, 1)	integrate(   integrate(x^2+y^2, y, 0, x), x, 0, 1);
find poles
residue	Residue[1/(z - I), {z, I}]	residue(1/(z-I), z, I)	f(z) = 1/(z - I) f.maxima_methods().residue(z, I)	residue(1 / (z - %i), z, %i);
sum	Sum[2^i, {i, 1, 10}]	Sum(2**i, (i, 1, 10)).doit()	sum(2^i for i in (1..10))	sum(2^i, i, 1, 10);
series sum	Sum[2^-n, {n, 1, Infinity}]	Sum(2**(-n), (n, 1, oo)).doit()	sum(2^-n, n, 1, infinity)	sum(2^-n, n, 1, inf), simpsum;
series expansion of function	Series[Cos[x], {x, 0, 10}]	series(cos(x), x, n=11)	taylor(cos(x), x, 0, 10)	taylor(cos(x), [x, 0, 10]);
omitted order term	expr = 1 + x + x/2 + x^2/6 + O[x]^3 (* remove omitted order term: *) Normal[expr]
product	Product[2*i + 1, {i, 0, 9}]	Product(2*i + 1, (i, 0, 9)).doit()	prod(2*i + 1 for i in (0..9))	product(2*i + 1, i, 0, 9);
equations and unknowns
	mathematica	sympy	sage	maxima
solve equation	Solve[x^3 + x + 3 == 0, x]	solve(x**3 + x + 3, x)	solve(x^3 + x + 3 == 0, x)	solve(x^3 + x + 3 = 0, x);
solve equations	Solve[x + y == 3 && x == 2 * y,   {x, y}] (* or: *) Solve[{x + y == 3, x == 2 * y}, {x, y}]	solve([x + y - 3, 3*x - 2*y], [x, y])	solve([x + y == 3, x == 2*y], x, y)	solve([x + y = 3, x = 2*y], [x, y]);
differential equation	DSolve[y'[x] == y[x], y[x], x]	y = Function('y') dsolve(Derivative(y(x), x) - y(x), y(x))	y = function('y')(x) desolve(diff(y, x) == y, y)	desolve([diff(y(x), x) = y(x)], [y(x)]);
differential equation with boundary condition	DSolve[{y'[x] == y[x], y[0] == 1}, y[x], x] DSolve[{y''[x] == y[x], y[0] == 1, y'[0] == 2},   y[x], x]	support for boundary conditions is limited	y = function('y')(x) # y(0) = 1: desolve(diff(y, x) == y, y, [0, 1]) # y(0) = 1 and y'(0) = 2: desolve(diff(y, x, x) == y, y, [0, 1, 2])	atvalue(y(x), x=0, 1); desolve([diff(y(x), x) = y(x)], [y(x)]);
differential equations	eqn1 = x'[t] == x[t] - x[t] * y[t] eqn2 = y'[t] == x[t] * y[t] - y[t] DSolve[{eqn1, eqn2}, {x[t], y[t]}, t]			eqn1: diff(x(t), t) = x(t) - x(t) * y(t); eqn2: diff(y(t), t) = x(t) * y(t) - y(t); desolve([eqn1, eqn2], [x(t), y(t)]);
recurrence equation	eqns = {a[n + 2] == a[n + 1] + a[n],   a[0] == 0,   a[1] == 1} RSolve[eqns, a[n], n] (* remove Fibonacci[] from solution: *) FunctionExpand[RSolve[eqns, a[n], n]]	n = symbols('n') a = Function('a') eqn = a(n+2) - a(n+1) - a(n) rsolve(eqn, a(n), {a(0): 0, a(1): 1})		solve_rec(a[n]=a[n-1]+a[n-2], a[n], a[0] = 0, a[1] = 1);
optimization
	mathematica	sympy	sage	maxima
minimize	(* returns list of two items: min value and rule    transforming x to argmin *) Minimize[x^2 + 1, x] (* 2 ways to get min value: *) Minimize[x^2 + 1, x][[1]] MinValue(x^2 + 1, x] (* 2 ways to get argmin: *) x /. Minimize[x^2 + 1, x][[2]] ArgMin[x^2 + 1, x]
maximize	Maximize[-x^4 + 3 x^3, x] Maxvalue[-x^4 + 3 x^3, x] ArgMax[-x^4 + 3 x^3, x]
objective with unknown parameter	(* minval and argmin are expressions    containing a: *) Minimize[(x - a)^2 + x, x]
unbounded behavior	(* MaxValue will be Infinity; MinValue will be    -Infinity *)
multiple variables	(* returns one solution: *) Minimize[x^4 - 2 x^2 + 2 y^4 - 3 y^2, {x, y}]
constraints	Minimize[{-x - 2 y^2, y^2 <= 17, 2 x + y <= 5},   {x, y}]
infeasible behavior	(* MaxValue will be -Infinity; MinValue will be    Infinity; ArgMax or ArgMin will be    Indeterminate *)
integer variables	Maximize[{x^2 + 2*y,     x >= 0, y >= 0,     2 x + Pi * y <= 4},   {x, y}, Integers]
vectors
	mathematica	sympy	sage	maxima
vector literal	(* row vector is same as array: *) {1, 2, 3}	# column vector: Matrix([1, 2, 3])	vector([1, 2, 3])	/* row vector is same as array: */ [1, 2, 3];
constant vector all zeros, all ones	Table[0, {i, 1, 100}] Table[1, {i, 1, 100}]	Matrix([0] * 100) Matrix([1] * 100)	vector([0] * 100) vector([1] * 100)	makelist(0, 100); makelist(1, 100);
vector coordinate	(* indices start at one: *) {6, 7, 8}[[1]]	Matrix([6, 7, 8])[0]	vector([6, 7, 8])[0]	[6, 7, 8][1];
vector dimension	Length[{1, 2, 3}]	len(Matrix([6, 7, 8])) Matrix([6, 7, 8]).shape[0]	len(vector([1, 2, 3]))	length([1, 2, 3]);
element-wise arithmetic operators	+ - * / adjacent lists are multiplied element-wise	+ - # element-wise multiplication: A = Matrix([1, 2, 3]) B = Matrix([2, 3, 4]) A.multiply_elementwise(B)	+ -	+ - * /
vector length mismatch	error	raises ShapeError	raises TypeError	error
scalar multiplication	3 {1, 2, 3} {1, 2, 3} 3 * may also be used	3 * Matrix([1, 2, 3]) Matrix([1, 2, 3]) * 3	3 * vector([1, 2, 3]) vector([1, 2, 3]) * 3	3 * [1, 2, 3]; [1, 2, 3] * 3;
dot product	{1, 1, 1} . {2, 2, 2} Dot[{1, 1, 1}, {2, 2, 2}]	v1 = Matrix([1, 1, 1]) v2 = Matrix([2, 2, 2]) v1.dot(v2)	vector([1, 1, 1]) * vector([2, 2, 2]) vector([1,1,1]).dot_product(vector([2,2,2]))	[1, 1, 1] . [2, 2, 2];
cross product	Cross[{1, 0, 0}, {0, 1, 0}]	e1 = Matrix([1, 0, 0]) e2 = Matrix([0, 1, 0]) e1.cross(e2)	e1 = vector([1, 0, 0]) e2 = vector([0, 1, 0]) e1.cross_product(e2)
norms	Norm[{1, 2, 3}, 1] Norm[{1, 2, 3}] Norm[{1, 2, 3}, Infinity]	vec = Matrix([1, 2, 3]) vec.norm(1) vec.norm() vec.norm(inf)	vector([1, 2, 3]).norm(1) vector([1, 2, 3]).norm() vector([1, 2, 3]).norm(infinity)
orthonormal basis	Orthogonalize[{{1, 0, 1}, {1, 1, 1}}]		A = matrix([[1, 0, 1], [1, 1, 1]] # Rows of B are orthogonal and span same # space as rows of A. 2nd return value # expresses rows of A as linear combos # of rows of B. B, _ = A.gram_schmidt()	load(eigen); gramschmidt([[1, 0, 1], [1, 1, 1]]);
matrices
	mathematica	sympy	sage	maxima
literal or constructor	(* used a nested array for each row: *) {{1, 2}, {3, 4}} (* display as grid with aligned columns: *) MatrixForm[{{1, 2}, {3, 4}}]	Matrix([[1, 2], [3, 4]])	matrix([[1, 2], [3, 4]])	matrix([1, 2], [3, 4]);
construct from sequence	ArrayReshape[{1, 2, 3, 4, 5, 6}, {2, 3}]	Matrix(2, 3, [1, 2, 3, 4, 5, 6])	matrix([1, 2, 3, 4, 5, 6], nrows=2)
constant matrices all zeros, all ones	Table[0, {i, 3}, {j, 3}] Table[1, {i, 3}, {j, 3}]	zeros(3, 3) ones(3, 3)	matrix([0] * 9, nrows=3) matrix([1] * 9, nrows=3)	zeromatrix(3, 3); f[i, j] := 1; genmatrix(f, 3, 3);
diagonal matrices and identity	DiagonalMatrix[{1, 2, 3}] IdentityMatrix[3]	diag(*[1, 2, 3]) eye(3)	diag = [1, 2, 3] d = {(i, i): v for (i, v) in enumerate(diag)} Matrix(3, 3, d) matrix.identity(3)	ident(3) * [1, 2, 3]; ident(3);
matrix by formula	Table[1/(i + j - 1), {i, 1, 3}, {j, 1, 3}]		Matrix(3, 3, lambda i, j: 1/(i + j + 1))	h2[i, j] := 1/(i + j -1); genmatrix(h2, 3, 3);
dimensions	(* returns {3, 2}: *) Dimensions[{{1, 2}, {3, 4}, {5, 6}}]	A = matrix([[1, 2], [3, 4], [5, 6]]) # returns (3, 2): A.shape	A = matrix([[1, 2], [3, 4], [5, 6]]) A.nrows() A.ncols()	A: matrix([1, 2, 3], [4, 5, 6]); matrix_size(A);
element lookup	(* top left corner: *) {{1, 2}, {3, 4}}[[1, 1]]	A = Matrix([[1, 2], [3, 4]]) # top left corner: A[0, 0]	A = matrix([[1, 2], [3, 4]]) A[0, 0] A[0][0]	A: matrix([1, 2], [3, 4]); A[1, 1]; A[1][1];
extract row	(* first row: *) {{1, 2}, {3, 4}}[[1]]	# first row: A[0, :]	# first row as vector: A[0] A.rows()[0]	row(matrix([1, 2], [3, 4]), 1); matrix([1, 2], [3, 4])[1];
extract column	(* first column as array: *) {{1, 2}, {3, 4}}[[All, 1]]	# first column as 1x2 matrix: A[:, 0]	# first column as vector: A.columns()[0]	col(matrix([1, 2], [3, 4]), 1);
extract submatrix	A = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} A[[1;;2, 1;;2]]	rows = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] A = Matrix(rows) A[0:2, 0:2]	A = matrix(range(1, 10), nrows=3) # takes two lists of indices: A.matrix_from_rows_and_columns([0, 1], [0, 1])
scalar multiplication	3 * {{1, 2}, {3, 4}} {{1, 2}, {3, 4}} * 3	3 * Matrix([[1, 2], [3, 4]]) Matrix([[1, 2], [3, 4]]) * 3	3 * matrix([[1, 2], [3, 4]]) matrix([[1, 2], [3, 4]]) * 3	3 * matrix([1, 2], [3, 4]); matrix([1, 2], [3, 4]) * 3;
element-wise operators	+ - * / adjacent matrices are multiplied element-wise	+ - # for Hadamard product: A.multiply_elementwise(B)	+ -	+ - * /
product	A = {{1, 2}, {3, 4}} B = {{4, 3}, {2, 1}} Dot[A, B] (* or use period: *) A . B	A = matrix([[1, 2], [3, 4]]) B = matrix([[4, 3], [2, 1]]) A * B	A = matrix([[1, 2], [3, 4]]) B = matrix([[4, 3], [2, 1]]) A * B	A: matrix([1, 2], [3, 4]); B: matrix([4, 3], [2, 1]); A . B;
multiply by vector	{{1, 2}, {3, 4}} . {7, 8} Dot[{{1, 2}, {3, 4}}, {5, 6}]		matrix([[1, 2], [3, 4]]) * vector([5, 6])	matrix([1, 2], [3, 4]) . transpose([5, 6]);
power	MatrixPower[{{1, 2}, {3, 4}}, 3] (* element-wise operator: *) A ^ 3	A ** 3	A ^ 3 A ** 3	matrix([1, 2], [3, 4]) ^^ 3;
exponential	MatrixExp[{{1, 2}, {3, 4}}]		exp(matrix([[1, 2], [3, 4]]))
log	MatrixLog[{{1, 2}, {3, 4}}]
kronecker product	A = {{1, 2}, {3, 4}} B = {{4, 3}, {2, 1}} KroneckerProduct[A, B]		A = matrix([[1, 2], [3, 4]]) B = matrix([[4, 3], [2, 1]]) A.tensor_product(B)	A: matrix([1, 2], [3, 4]); B: matrix([4, 3], [2, 1]); kronecker_product(A, B);
norms	A = {{1, 2}, {3, 4}} Norm[A, 1] Norm[A, 2] Norm[A, Infinity] Norm[A, "Frobenius"]		A = matrix([[1, 2], [3, 4]]) # floating point values: A.norm(1) A.norm() A.norm(infinity) A.norm('frob')	A: matrix([1, 2], [3, 4]); mat_norm(A, 1); /* none */ mat_norm(A, inf); mat_norm(A, frobenius);
transpose	Transpose[{{1, 2}, {3, 4}}] (* or ESC tr ESC for T exponent notation *)	A.T	A.transpose()	transpose(A);
conjugate transpose	A = {{1, I}, {2, -I}} ConjugateTranspose[A] (* or ESC ct ESC for dagger exponent notation *)	M = Matrix([[1, I], [2, -I]]) M.adjoint()	M = matrix([[1, I], [2, -I]]) M.conjugate_transpose()	ctranspose(matrix([1, %i], [2, -%i]));
inverse	Inverse[{{1, 2}, {3, 4}}] (* expression left unevaluated: *) Inverse[{{1, 0}, {0, 0}}]	A.inv() # raises ValueError: Matrix([[1, 0], [0, 0]]).inv()	A.inverse() A ^ -1 A ** -1	invert(A); A ^^ -1; /* error: */ invert(matrix([1, 0], [0, 0]));
row echelon form	RowReduce[{{1, 1}, {1, 1}}]		matrix([[1, 1], [1, 1]]).echelon_form()	echelon(matrix([1, 1], [1, 1]));
pseudoinverse	PseudoInverse[{{1, 0}, {3, 0}}]
determinant	Det[{{1, 2}, {3, 4}}]	A.det()	A.determinant()	determinant(A);
trace	Tr[{{1, 2}, {3, 4}}]		A.trace()	load("nchrpl"); mattrace(matrix([1, 2], [3, 4]));
characteristic polynomial	CharacteristicPolynomial[{{1, 2}, {3, 4}}, x]		matrix([[1, 2], [3, 4]]).charpoly('x')	A: matrix([1, 2], [3, 4]); charpoly(A, x);
minimal polynomial			matrix.identity(3).minpoly('x')	load(diag); minimalPoly(jordan(ident(3)));
rank	MatrixRank[{{1, 1}, {0, 0}}]		matrix([[1, 1], [0, 0]]).rank()	rank(matrix([1, 1], [0, 0]));
nullspace basis	NullSpace[{{1, 1}, {0, 0}}]			nullspace(matrix([1, 1], [0, 0]));
eigenvalues	Eigenvalues[{{1, 2}, {3, 4}}]	A.eigenvals()	matrix([[1, 2], [3, 4]]).eigenvalues()	/* returns list of two lists:    first is the eigenvalues,    second is their multiplicities */ eigenvalues(A);
eigenvectors	Eigenvectors[{{1, 2}, {3, 4}}]	A.eigenvects()	A = matrix([[1, 2], [3, 4]]) # returns list of triples: # (eigenval, eigenvec, multiplicity) A.eigenvectors_right()	/* returns list of two lists. The first item is the return value of eigenvalues(). The second item is a list containing a list of eigenvectors for each eigenvalue. */ eigenvectors(A);
LU decomposition	{lu, p, c} = LUDecomposition[{{1, 2}, {3, 4}}] L = LowerTriangularize[lu] U = UpperTriangularize[lu] P = Permute[IdentityMatrix[2], p]		P, L, U = matrix([[1, 2], [3, 4]]).LU()	A: matrix([1, 2], [3, 4]); [P, L, U]: get_lu_factors(lu_factor(A));
QR decomposition	A := {{1, 2}, {3, 4}} {Q, R} = QRDecomposition[A] A == Q . R		# numerical result: Q, R = matrix(CDF, [[1, 2], [3, 4]]).QR()
spectral decomposition	A = {{1, 2}, {2, 1}} z := Eigensystem[A] d := DiagonalMatrix[z[[1]]] P := Transpose[z[[2]]] P . d . Inverse[P] == A
singular value decomposition	A := {{1, 2}, {3, 4}} z := SingularValueDecomposition[A] U := z[[1]] S := z[[2]] V := z[[3]] N[A] == N[U . S . ConjugateTranspose[V]]		A = matrix(CDF, [[1, 2], [3, 4]]) # numerical result: U, D, V = A.SVD() norm(A - U * D * V.conjugate_transpose())
jordan decomposition	A := {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} z := JordanDecomposition[A] P := z[[1]] J := z[[2]] A . P == P . J		A = matrix([[0, 1], [1, 0]]) # eigenvalues must be rational: J, P = A.jordan_form(   subdivide=False, transformation=True)
polar decomposition	A := {{1, 2}, {3, 4}} {u, s, v} := SingularValueDecomposition[A] vt := ConjugateTranspose[v] U := u * vt P = v * s * vt
combinatorics
	mathematica	sympy	sage	maxima
factorial and permutations	5! Factorial[5] Permutations[Range[1, 5]]	factorial(5)	factorial(5) 5.factorial()	5! factorial(5);
binomial coefficient and combinations	Binomial[10, 3]	binomial(10, 3)	binomial(10, 3)	binomial(10, 3);
multinomial coefficient	Multinomial[3, 4, 5]		multinomial([3, 4, 5])	multinomial(12, [3, 4, 5]);
rising and falling factorial	Pochhammer[1/2, 3] FactorialPower[1/2, 3]		rising_factorial(1/2, 3) falling_factorial(1/2, 3)	pochhammer(1/2, 3); pochhammer(1/2 - 2, 3);
subfactorial and derangments	Needs["Combinatorica`"] NumberOfDerangements[10]	subfactorial(10)	subfactorial(10)
integer partitions	(* number of partitions: *) PartitionsP[10] (* the partitions as an array: *) IntegerPartitions[10]	from sympy.utilities.iterables \   import partitions len(list(partitions(10))) [p.copy() for p in partitions(10)]	Partitions(10).cardinality() Partitions(10).list()	length(integer_partitions(10)); /* the partitions as an array: */ integer_partitions(10);
compositions	Needs["Combinatorica`"] (* weak compositions of size 3 is 66: *) NumberOfCompositions[10, 3] Compositions[10, 3]		# compositions of all lengths: Compositions(10).cardinality() Compositions(10).list() # of length 3: Compositions(10, min_length=3,   max_length=3).list()
set partitions	StirlingS2[10, 3] Needs["Combinatorica`"] KSetPartitions[10, 3] SetPartititions[10]		stirling_number2(10, 3)	stirling2(10, 3);
bell number	BellB[10]	bell(10)	bell_number(10)	belln(10);
permutations with k disjoint cycles	Abs[StirlingS1[10, 3]]		stirling_number1(10, 3)	abs(stirling1(10, 3));
fibonacci number and lucas number	Fibonacci[10] LucasL[10]	fibonacci(10) lucas(10)	fibonacci(10) lucas_number2(10, 1, -1)	fib(10); lucas(10);
bernoulli number	BernoulliB[100]	bernoulli(100)	bernoulli(100)	bern(100);
harmonic number	HarmonicNumber[100]	harmonic(100)
catalan number	CatalanNumber[10]	catalan(10)	catalan_number(10)
number theory
	mathematica	sympy	sage	maxima
divisible test	Divisible[1001, 7]	1001 % 7 == 0	7.divides(1001)	is(mod(1001, 7) = 0);
divisors	(* {1, 2, 4, 5, 10, 20, 25, 50, 100}: *) Divisors[100]	ntheory.divisors(100)	divisors(100)	divisors(100);
pseudoprime test	PrimeQ[7]	ntheory.primetest.isprime(7) ntheory.primetest.mr(7, [2, 3])	is_prime(7) is_pseudoprime(7)	primep(7);
prime factors	(* returns {{2, 2}, {3, 1}, {7, 1}} *) FactorInteger[84]	# {2: 2, 3: 1, 7: 1}: ntheory.factorint(84)	# 2^2 * 3 * 7: factor(84)	/* 2^2 3 7: */ factor(84); /* [[2,2],[3,1],[7,1]]: */ ifactors(84);
next prime and preceding	NextPrime[1000] NextPrime[1000, -1]	ntheory.generate.nextprime(1000) ntheory.generate.prevprime(1000)	next_prime(1000) previous_prime(1000)	next_prime(1000); prev_prime(1000);
nth prime	(* 541: *) Prime[100]	ntheory.generate.prime(100)	primes_first_n(100)[-1]
prime counting function	(* 25: *) PrimePi[100]	ntheory.generate.primepi(100)	prime_pi(100)
divmod	QuotientRemainder[7, 3]	divmod(7, 3)	divmod(7, 3)	divide(7, 3);
greatest common divisor and relatively prime test	GCD[14, 21] GCD[14, 21, 777] CoprimeQ[14, 21]	gcd(14, 21) gcd(gcd(14, 21), 777)	gcd(14, 21) gcd(gcd(14, 21), 777)	gcd(14, 21); gcd(gcd(14, 21), 777);
extended euclidean algorithm	(* {1, {2, -1}}: *) ExtendedGCD[3, 5]	from sympy.ntheory.modular import igcdex # (2, -1, 1): igcdex(3, 5)	# (1, 2, -1): xgcd(3, 5)	/* [2,-1,1]: */ gcdex(3, 5);
least common multiple	LCM[14, 21]	lcm(14, 21)	lcm(14, 21)	lcm(14, 21);
power modulus	PowerMod[3, 212, 7]		power_mod(3, 212, 7)
multiplicative inverse	(* inverse of 2 mod 7: *) PowerMod[2, -1, 7] (* left unevaluated: *) PowerMod[2, -1, 4]		r = Integers(7) r(2)^-1 r2 = Integers(4) # raises ZeroDivisionError: r2(4)^-1
chinese remainder theorem	(* returns 173, which is equal to 3 mod 17 and 8 mod 11: *) ChineseRemainder[{3, 8}, {17, 11}]		crt(3, 8, 17, 11)	/* 173: */ chinese([3, 8], [17, 11]);
euler totient	EulerPhi[256]	ntheory.totient(256)	euler_phi(256)	totient(256);
carmichael function	CarmichaelLambda[561]		from sage.crypto.util import carmichael_lambda carmichael_lambda(561)
multiplicative order	MultiplicativeOrder[7, 108]		Mod(7, 108).multiplicative_order()
primitive roots	PrimitiveRoot[11] (* all primitive roots: *) PrimitiveRootList[11]		primitive_root(11) # raises ValueError if none
discrete logarithm	(* solves 10 = 2^x (mod 11): *) MultiplicativeOrder[2, 11, 10]		log(Mod(10, 11), Mod(2, 11))
quadratic residues	Select[Range[0, 4], KroneckerSymbol[#, 5] == 1 &]		quadratic_residues(5)
discrete square root	PowerMod[4, 1/2, 5]		Mod(4, 5).sqrt()
kronecker symbol and jacobi symbol	KroneckerSymbol[3, 5] JacobiSymbol[3, 5]		kronecker_symbol(3, 5)	?? jacobi(3, 5);
moebius function	MoebiusMu[11]		moebius(11)	moebius(11);
riemann zeta function	Zeta[2]	mpmath.zeta(2)	zeta(2)	zeta(2);
continued fraction	(* {0, 1, 1, 1, 5}: *) ContinuedFraction[11/17] (* arrray of first 100 digits for for pi: *) ca= ContinuedFraction[Pi, 100] (* rational approximation of pi: *) FromContinuedFraction[a]		continued_fraction(11/17) continued_fraction(pi, 100)	/* [0,1,1,1,5]: */ cf(11/17); float_pi: %pi, numer; a = cf(float_pi); /* as continued fraction: */ as_cf: cfdisrep(a); /* as simple fraction: */ ratsimp(as_cf);
convergents	Convergents[11/17] (* for continued fraction: *) Convergents[{0, 1, 1, 1, 5}] (* first 100 rational approximations: *) Convergents[Pi, 100]		# [0, 1, 1/2, 2/3, 11/17]: continued_fraction(11/17).convergents() # iterable infinite list: continued_fraction(pi, 100).convergents()
polynomials
	mathematica	sympy	sage	maxima
literal	p = 2 -3 * x + 2* x^2 p2 = (1 + x)^10		p = x^2 - 3*x + 2 p2 = (x + 1)^10	p: x^2 - 3*x + 2; p2: (x + 1)^10;
extract coefficient	Coefficient[(1 + x)^10, x, 3]		p = (1 + x)^10 # coefficients() returns (power, coeff) pairs: [pair[0] for pair in p.coefficients()  if pair[1] == 3][0]	coeff(expand((x + 1)^10), x^3); coeff(expand((x + 1)^10), x, 3);
extract coefficients	CoefficientList[(x + 1)^10, x]			p: expand((x+1)^10); makelist(coeff(p, x^i), i, 0, 10);
from array of coefficients	a = {2, -3, 1} Sum[a[[i]] * x^i, {i, 1, 3}]			a: [2, -3, 1]; sum(x^i * a[i + 1], i, 0, 2);
degree	Exponent[(x + 1)^10, x]			hipow(expand((1 + x)^10), x);
expand	Expand[(1 + x)^5]	expand((1 + x)**5)	expand((1 + x)^5)	expand((1 + x)^5);
factor	Factor[3 + 10 x + 9 x^2 + 2 x^3] Factor[x^10 - y^10]	factor(3 + 10*x + 9*x**2 + 2*x**3)	factor(2*x^3 + 9*x^2 + 10*x + 3)	factor(2*x^3 + 9*x^2 + 10*x + 3);
roots	Solve[x^3 + 3 x^2 + 2 x - 1 == 0, x] (* just the 2nd root: *) Root[x^3 + 3 x^2 + 2 x - 1, 2]			solve(x^3 + 3*x^2 + 2*x - 1 = 0, x);
quotient and remainder	PolynomialReduce[x^10 - 1, x - 1, {x}]			[q, r]: divide(x^10-1, x - 1);
greatest common divisor	p1 = -2 - x + 2 x^2 + x^3 p2 = 6 - 7 x + x^3 PolynomialGCD[p1, p2]			p1: -2 - x + 2 * x^2 + x^3; p2: 6 - 7*x + x^3; gcd(p1, p2);
extended euclidean algorithm	p1 = -2 - x + 2 x^2 + x^3 p2 = 6 - 7 x + x^3 (* returns list; first element is GCD; 2nd element is list of two polynomials *) PolynomialExtendedGCD[p1, p2, x]
resultant	Resultant[(x-1) * (x-2), (x-3) * (x-3), x]			resultant((x - 1)*(x - 2), (x - 3)*(x - 3), x);
discriminant	Discriminant[(x + 1) * (x - 2), x]
collect terms	(* write as polynomial in x: *) Collect[(1 + x + y)^3, x]	collect(expand((x+y+1)**3), x)		load(facexp); facsum(expand((1 + x + y)^5), x);
multivariate quotient and remainder	PolynomialReduce[x^10 - y^10, x - y, {x, y}]			[q, r]: divide(x^10 - y^10, x - y);
groebner basis	p1 = x^2 + y + z - 1 p2 = x + y^2 + z - 1 p3 = x + y + z^2 - 1 (* uses lexographic order by default: *) GroebnerBasis[{p1, p2, p3}, {x, y, z}]
specify ordering	GroebnerBasis[{p1, p2, p3}, {x, y, z},   MonomialOrder -> DegreeReverseLexicographic] (* possible values for MonomialOrder:    Lexicographic    DegreeLexicographic    EliminationOrder    {1, 2, 3} *)
elementary symmetric polynomial	SymmetricPolynomial[3, {x1, x2, x3, x4}]
symmetric reduction	(* returns list of two elements; 2nd element is remainder if polynomial not symmetric: *) SymmetricReduction[x^3 + y^3 + z^3, {x, y, z}]
cyclotomic polynomial	Cyclotomic[10, x]
hermite polynomial	HermiteH[4, x]
chebyshev polynomial first and second kind	ChebyshevT[4, x] ChebyshevU[4, x]
interpolation polynomial	pts = Inner[List, {1, 2, 3}, {2, 4, 7}, List] InterpolatingPolynomial[pts, x]
spline
partial fraction decomposition	Apart[(3*x + 2)/ (x^2 + x)] (* can handle multiple vars in denominator: *) Apart[(b * c + a * d)/(b * d)]	apart((3*x+2) / (x*(x+1)))		partfrac((3*x + 2) / (x^2 + x), x);
add fractions	Together[a/b + c/d]	together(x/y + z/w)		ratsimp(a/b + c/d);
pade approximant	PadeApproximant[Log[x], {x, 1, {2, 3}}]			p: taylor(log(x + 1), [x, 0, 5]); pade(p, 3, 2);
trigonometry
	mathematica	sympy	sage	maxima
eliminate powers and products of trigonometric functions	TrigReduce[Sin[x]^2 + Cos[x] Sin[x]]			trigreduce(sin(x)^2 + cos(x) * sin(x));
eliminate sums and multiples inside trigonometric functions	TrigExpand[Sin[2 * x + 1]]			trigexpand(sin(2*x + 1));
trigonometric to exponential	TrigToExp[Cos[x]]	cos(x).rewrite(cos, exp)		exponentialize(cos(x));
exponential to trigonometric	ExpToTrig[Exp[I x]]	from sympy import exp, sin, I exp(I * x).rewrite(exp, sin)		demoivre(exp(%i * x));
fourier expansion	(* in sin and cos: *) FourierTrigSeries[SquareWave[x / (2*Pi)],   x, 10] (* in complex exponentials: *) FourierSeries[SquareWave[x / (2*Pi)], x, 10]
periodic functions on unit interval	(*1: [0, 0.5); -1: [0.5, 1.0) *) SquareWave[x] (* 0 at 0 and 0.5; 1 at 0.25; -1 at 0.75 *) TriangleWave[x] (* x on [0, 1) *) SawtoothWave[x]
fourier transform	f[w_] = FourierTransform[ Sin[t], t, w] InverseFourierTransform[f[w], w, t]
heaviside step function
dirac delta
special functions
	mathematica	sympy	sage	maxima
gamma function	Gamma[1/2]	gamma(Rational(1, 2))	gamma(1/2)	gamma(1/2);
error function	Erf[1/2] // N Erfc Erfi InverseErf InverseErfc	N(erf(Rational(1, 2))) erfc erfi	n(erf(1/2))	erf(1/2), numer; erfc erfi
hyperbolic functions	Sinh Cosh Tanh ArcSinh ArcCosh ArcTanh	sinh cosh tanh asinh acosh atanh	sinh cosh tanh asinh acosh atanh	sinh cosh tanh asinh acosh atanh
elliptic integerals	EllipticK EllipticF EllipticE EllipticPi			elliptic_f elliptic_e elliptic_pi
bessel functions	BesselJ BesselY BesselI BesselK			bessel_j bessel_y bessel_i bessel_k
permutations
	mathematica	sympy	sage	maxima
from disjoint cycles	p = Cycles[{{1, 2}, {3, 4}}]	import sympy.combinatorics as comb p = combinatorics.Permutation(0, 1)(2, 3)	Permutation([(1, 2), (3, 4)])
to disjoint cycles
from array	p = PermutationCycles[{2, 1, 4, 3}]	import sympy.combinatorics as comb p = combinatorics.Permutation([1, 0, 3, 2])	Permutation((2, 1, 4, 3))
from two arrays with same elements	FindPermutation[{a, b, c}, {b, c, a}]
size
support and fixed points	PermutationSupport[Cycles[{{1, 3, 5}, {7, 8}}]]	import sympy.combinatorics as comb p = comb.Permutation(0, 2, 4)(6, 7) p.support()
act on element	p = Cycles[{{1, 2}, {3, 4}}] PermutationReplace[1, p]	p(0)	Permutation((2, 1, 4, 3))(1)
act on list	(* if list is too big, extra elements retain    their positions; if list is too small,    expression is left unevaluated. *) Permute[{a, b, c, d}, p12n34]	import sympy.combinatorics as comb p = comb.Permutation(0, 1)(2, 3) p([a, b, c, d])	a, b, c, d = var('a b c d') p = Permutation([(1, 2), (3, 4)]) p.action([a, b, c, d])
compose	p1 = Cycles[{{1, 2}, {3, 4}}] p2 = Cycles[{{1, 3}}] PermutationProduct[p1, p2]	import sympy.combinatorics as comb p1 = comb.Permutation(0, 1)(2, 3) p2 = comb.Permutation(0, 2) p1 * p2	p1 = Permutation([(1, 2), (3, 4)]) p2 = Permutation((1, 3)) p1 * p2
inverse	InversePermutation[Cycles[{{1, 2, 3}}]]	import sympy.combinatorics as comb comb.Permutation(0, 1, 2) ** -1	Permutation((1, 2, 3)).inverse()
power	PermutationPower[Cycles[{{1, 2, 3, 4, 5}}], 3]	import sympy.combinatorics as comb comb.Permutation(0, 1, 2, 3, 4) ** 3	Permutation((1, 2, 3, 4, 5))^3
order	PermutationOrder[Cycles[{{1, 2, 3, 4, 5}}]]	import sympy.combinatorics as comb comb.Permutation(0, 1, 2, 3, 4).order()	p = Permutation((1,2,3,4,5)) p.to_permutation_group_element().order()
number of inversions		import sympy.combinatorics as comb comb.Permutation(0, 2, 1).inversions()	Permutation((1, 3, 2)).length()
parity		import sympy.combinatorics as comb comb.Permutation(0, 2, 1).parity()	Permutation((1, 3, 2)).is_even()
to inversion vector		import sympy.combinatorics as comb comb.Permutation(0, 2, 1).inversion_vector()	Permutation((1, 3, 2)).to_inversion_vector()
from inversion vector		import sympy.combinatorics as comb comb.Permutation.from_inversion_vector([2, 0])
list permutations	GroupElements[SymmetricGroup[4]] (* of a list: *) Permutations[{a, b, c, d}]		list(SymmetricGroup(4))
random permutation	RandomPermutation[10]	Permutation.random(10)
descriptive statistics
	mathematica	sympy	sage	maxima
first moment statistics	vals = {1, 2, 3, 8, 12, 19} X = NormalDistribution[0, 1] Mean[vals] Total[vals] Mean[X]			load(distrib); /* Other distributions have similar functions: */ mean_normal(0, 1);
second moment statistics	Variance[X] StandardDeviation[X]			load(distrib); /* Other distributions have similar functions: */ var_normal(0, 1); std_normal(0, 1);
second moment statistics for samples	Variance[vals] StandardDeviation[vals]
skewness	Skewness[vals] Skewness[X]			load(distrib); /* Other distributions have similar functions: */ skewness_normal(0, 1);
kurtosis	Kurtosis[vals] Kurtosis[X]			load(distrib); /* Other distributions have similar functions: */ kurtosis_normal(0, 1);
nth moment and nth central moment	Moment[vals, 5] CentralMoment[vals, 5] Moment[X, 5] CentralMoment[X, 5] MomentGeneratingFunction[X, t]
cumulant	Cumulant[vals, 1] Cumulant[X, 1] CumulantGeneratingFunction[X, t]
entropy	Entropy[vals]
mode	Commonest[{1, 2, 2, 2, 3, 3, 8, 12}]
quantile statistics	Min[vals] Median[vals] Max[vals] InterquartileRange[vals] Quantile[vals, 9/10]			load(distrib); /* Other distributions have similar functions: */ quantile_normal(9/10, 0, 1);
bivariate statistiscs correlation, covariance, Spearman's rank	Correlation[{1, 2, 3}, {2, 4, 7}] Covariance[{1, 2, 3}, {2, 4, 7}] SpearmanRho[{1, 2, 3}, {2, 4, 7}]
data set to frequency table	data = {1, 2, 2, 2, 3, 3, 8, 12} (* list of pairs: *) tab = Tally[data] (* dictionary: *) dict = Counts[data]
frequency table to data set	f = Function[a, Table[a[[1]], {i, 1, a[[2]]}]] data = Flatten[Map[f, tab]]
bin	data = {1.1, 3.7, 8.9, 1.2, 1.9, 4.1} (* bins are [0, 3), [3, 6), and [6, 9): *) bins = BinCounts[data, 0, 3, 6, 9]
distributions
	mathematica	sympy	sage	maxima
binomial density, cumulative distribution, sample	X = BinomialDistribution[100, 1/2] PDF[X][50] CDF[X][50] RandomVariate[X]	from sympy.stats import * X = Binomial('X', 100, Rational(1, 2)) density(Y).dict[Integer(50)] P(X < 50) sample(X)		load(distrib); pdf_binomial(x, 50, 1/2); cdf_binomial(x, 50, 1/2); random_binomial(50, 1/2);
poisson	X = PoissonDistribution[1]	# P(X < 4) raises NotImplementedError: X = Poisson('X', 1)		load(distrib); pdf_poisson(x, 1); cdf_poisson(x, 1); random_poisson(1);
discrete uniform	X = DiscreteUniformDistribution[{0, 99}]	X = DiscreteUniform('X', list(range(0, 100)))		load(distrib); /* {1, 2, …, 100}: */ pdf_discrete_uniform(x, 100); cdf_discrete_uniform(x, 100); random_discrete_uniform(100);
normal density, cumulative distribution, quantile, sample	X = NormalDistribution[0, 1] PDF[X][0] CDF[X][0] InverseFunction[CDF[X]][1/2] RandomVariate[X, 10]	from sympy.stats import * X = Normal('X', 0, 1) density(X)(0) P(X < 0) ?? sample(X)	X = RealDistribution('gaussian', 1) X.distribution_function(0) X.cum_distribution_function(0) X.cum_distribution_function_inv(0.5) X.get_random_element()	with(distrib); pdf_normal(x, 0, 1); cdf_normal(x, 0, 1); /* no inverse cdf */ random_normal(0, 1);
gamma	X = GammaDistribution[1, 1]	X = Gamma('X', 1, 1)		with(distrib); pdf_gamma(x, 1, 1); cdf_gamma(x, 1, 1); random_gamma(1, 1);
exponential	X = ExponentialDistribution[1]	X = Exponential('X', 1)		with(distrib); pdf_exponential(x, ); cdf_exponential(x, 1); random_exponential(1);
chi-squared	X = ChiSquareDistribution[2]	X = ChiSquared('X', 2)	X = RealDistribution('chisquared', 2)	with(distrib); pdf_chi2(x, 2); cdf_chi2(x, 2); random_chi2(2);
beta	X = BetaDistribution[10, 90]	X = Beta('X', 10, 90)	X = RealDistribution('beta', [10, 90])	with(distrib); pdf_beta(x, 10, 90); cdf_beta(x, 10, 90); random_beta(10, 90);
uniform	X = UniformDistribution[{0, 1}]	X = Uniform('X', 0, 1)	X = RealDistribution('uniform', [0, 1]) X.distribution_function(0.5) X.cum_distribution_function(0.5) X.cum_distribution_function_inv(0.5) X.get_random_element()	with(distrib); pdf_continuous_uniform(x, 0, 1); cdf_continuous_uniform(x, 0, 1); random_continuous_uniform(0, 1);
student's t	X = StudentTDistribution[2]	X = StudentT('X', 2)	X = RealDistribution('t', 2)	with(distrib); pdf_student_t(x, 2); cdf_student_t(x, 2); random_student_t(2);
snedecor's F	X = FRatioDistribution[1, 1]	X = FDistribution('X', 1, 1)	X = RealDistribution('F', [1, 1])	with(distrib); pdf_f(x, 1, 1); cdf_f(x, 1, 1); random_f(1, 1);
empirical density function	X = NormalDistribution[0, 1] data = Table[RandomVariate[X], {i, 1, 30}] Y = EmpiricalDistribution[data] PDF[Y]
empirical cumulative distribution	X = NormalDistribution[0, 1] data = Table[RandomVariate[X], {i, 1, 30}] Y = EmpiricalDistribution[data] Plot[CDF[Y][x], {x, -4, 4}]
statistical tests
	mathematica	sympy	sage	maxima
wilcoxon signed-rank test variable is symmetric around zero	X = UniformDistribution[{-1/2, 1/2}] data = RandomVariate[X, 100] (* null hypothesis is true: *) SignedRankTest[data] (* alternative hypothesis is true: *) SignedRankTest[data + 1/2]			load(distrib); load(stats); data: makelist(   random_continuous_uniform(-1/2, 1/2),   i, 1, 100); /* null hypothesis is true: */ test_signed_rank(data); /* alternative hypothesis is true: */ test_signed_rank(data + 1/2);
kruskal-wallis rank sum test variables have same location parameter	X = NormalDistribution[0, 1] Y = UniformDistribution[{0, 1}] (* null hypothesis is true: *) LocationEquivalenceTest[   {RandomVariate[X, 100],    RandomVariate[X, 200]}] (* alternative hypothesis is true: *) LocationEquivalenceTest[   {RandomVariate[X, 100],    RandomVariate[Y, 200]}]			load(distrib); load(stats); x1: makelist(   random_normal(0, 1),   i, 1, 100); x2: makelist(   random_normal(0, 1),   i, 1, 100); y: makelist(   random_continuous_uniform(-1/2, 1/2),   i, 1, 100); /* null hypothesis is true: */ test_rank_sum(x1, x2); /* alternative hypothesis is true: */ test_rank_sum(x1, y);
kolmogorov-smirnov test variables have same distribution	X = NormalDistribution[0, 1] Y = UniformDistribution[{-1/2, 1/2}] (* null hypothesis is true: *) KolmogorovSmirnovTest[RandomVariate[X, 200], X] (* alternative hypothesis is true: *) KolmogorovSmirnovTest[RandomVariate[X, 200], Y]
one-sample t-test mean of normal variable with unknown variance is zero	X = NormalDistribution[0, 1] (* null hypothesis is true: *) TTest[RandomVariate[X, 200]] (* alternative hypothesis is true: *) TTest[RandomVariate[X, 200] + 1]
independent two-sample t-test two normal variables have same mean	X = NormalDistribution[0, 1] (* null hypothesis is true: *) TTest[   {RandomVariate[X, 100],    RandomVariate[X, 200]}] (* alternative hypothesis is true: *) TTest[   {RandomVariate[X, 100],    RandomVariate[X, 100] + 1}]
paired sample t-test population has same mean across measurements
one-sample binomial test binomial variable parameter are as given
two-sample binomial test parameters of two binomial variables are equal
chi-squared test parameters of multinomial variable are all equal
poisson test parameter of poisson variable is as given
F test ratio of variance of normal variables are the same	X = NormalDistribution[0, 1] Y = NormalDistribution[0, 2] (* null hypothesis is true: *) FisherRatioTest[   {RandomVariate[X, 100],    RandomVariate[X, 200]}] (* alternative hypothesis is true: *) FisherRatioTest[   {RandomVariate[X, 100],    RandomVariate[Y, 100]}]
pearson product moment test normal variables are not correlated	X = NormalDistrubtion[0, 1] x = RandomVariate[X, 100] y = RandomVariate[X, 100] x2 = x + RandomVariate[X, 100] data1 = Inner[List, x, y, List] data2 = Inner[List, x, x2, List] (* null hypothesis is true: *) CorrelationTest[data1, 0, "PearsonCorrelation"] (* alternative hypothesis is true: *) CorrelationTest[data2, 0, "PearsonCorrelation"]
spearman rank test variables are not correlated	X = UniformDistribution[{0, 1}] x = RandomVariate[X, 100] y = RandomVariate[X, 100] x2 = x + RandomVariate[X, 100] data1 = Inner[List, x, y, List] data2 = Inner[List, x, x2, List] (* null hypothesis is true: *) CorrelationTest[data1, 0, "SpearmanRank"] (* alternative hypothesis is true: *) CorrelationTest[data2, 0, "SpearmanRank"]
shapiro-wilk test variable has normal distribution	X = NormalDistribution[0, 1] Y = UniformDistribution[{0, 1}] (* null hypothesis is true: *) ShapiroWilkTest[RandomVariate[X, 100]] (* alternative hypothesis is true: *) ShapiroWilkTest[RandomVariate[Y, 100]]			load(distrib); load(stats); x: makelist(   random_normal(0, 1),   i, 1, 100); y: makelist(   random_continuous_uniform(-1/2, 1/2),   i, 1, 100); /* null hypothesis is true: */ test_normality(x); /* alternative hypothesis is true: */ test_normality(y);
bartlett's test two or more normal variables have same variance
levene's test two or more variables have same variance	X = NormalDistribution[0, 1] Y = NormalDistribution[0, 2] (* null hypothesis is true: *) LeveneTest[   {RandomVariate[X, 100],    RandomVariate[X, 200]}] (* alternative hypothesis is true: *) LeveneTest[   {RandomVariate[X, 100],    RandomVariate[Y, 100]}]
one-way anova two or more normal variables have same mean	Needs["ANOVA‘"] X = NormalDistribution[0, 1] ones = Table[1, {i, 1, 100}] x1 = Inner[   List, ones, RandomVariate[X, 100], List] x2 = Inner[   List, 2 * ones, RandomVariate[X, 100], List] x3 = Inner[   List, 3 * ones, RandomVariate[X, 100], List] y = Inner[   List,   3 * ones,   RandomVariate[X, 100] + 0.5,   List] (* null hypothesis is true: *) ANOVA[Join[x1, x2, x3]] (* alternative hypothesis is true: *) ANOVA[Join[x1, x2, y]]
two-way anova
bar charts
	mathematica	sympy	sage	maxima
vertical bar chart	BarChart[{7, 3, 8, 5, 5},   ChartLegends->     {"a","b","c","d","e"}]			x: [7, 3, 8, 5, 5]; labs: [a, b, c, d, e]; data: makelist(makelist(labs[i], j, x[i]), i, 5); wxbarsplot(flatten(data));
horizontal bar chart	BarChart[{7, 3, 8, 5, 5}, BarOrigin -> Left]			none
grouped bar chart	data = {{7, 1}, {3, 2}, {8, 1}, {5, 3}, {5, 1}} BarChart[data]			x: [7, 3, 8, 5, 5]; y: [1,2,1,3,1]; labs: [a, b, c, d, e]; d1: makelist(makelist(labs[i], j, x[i]), i, 5); d2: makelist(makelist(labs[i], j, y[i]), i, 5); wxbarsplot(flatten(d1), flatten(d2));
stacked bar chart	data = {{7, 1}, {3, 2}, {8, 1}, {5, 3}, {5, 1}} BarChart[data, ChartLayout -> "Stacked"]			x: [7, 3, 8, 5, 5]; y: [1,2,1,3,1]; labs: [a, b, c, d, e]; d1: makelist(makelist(labs[i], j, x[i]), i, 5); d2: makelist(makelist(labs[i], j, y[i]), i, 5); wxbarsplot(flatten(d1), flatten(d2),   grouping=stacked);
pie chart	PieChart[{7, 3, 8, 5, 5}]			x: [7, 3, 8, 5, 5]; labs: [a, b, c, d, e]; data: makelist(makelist(labs[i], j, x[i]), i, 5); wxpiechart(flatten(data));
histogram	X = NormalDistribution[0, 1] (* 2nd arg is approx number of bins: *) Histogram[RandomReal[X, 100], 10]			load(distrib); data: makelist(random_normal(0, 1), i, 1, 100); wxhistogram(data);
box plot	X = NormalDistribution[0, 1] n100 = RandomVariate[X, 100] BoxWhiskerChart[n100] Y = ExponentialDistribution[1] e100 = RandomVariate[Y, 100] u100 = RandomReal[1, 100] data = {n100, e100, u100} BoxWhiskerChart[data]			load(distrib); data: makelist(random_normal(0, 1), i, 1, 100); wxboxplot(data);
scatter plots
	mathematica	sympy	sage	maxima
strip chart	X = NormalDistribution[0, 1] data = {RandomReal[X], 0} & /@ Range[1, 50] ListPlot[data]
strip chart with jitter	X = NormalDistribution[0, 1] Y = UniformDistribution[{-0.05, 0.05}] data = {RandomReal[X], RandomReal[Y]} & /@   Range[1, 50] ListPlot[data,   PlotRange -> {Automatic, {-1, 1}}]
scatter plot	X = NormalDistribution[0, 1] rand = Function[RandomReal[X]] data = {rand[], rand[]} & /@ Range[1, 50] ListPlot[data]			load(distrib); x: makelist(random_normal(0, 1), i, 1, 50); y: makelist(random_normal(0, 1), i, 1, 50); wxplot2d([discrete, x, y], [style, points]);
additional point set	X = NormalDistribution[0, 1] rand = Function[RandomReal[X]] data1 = {rand[], rand[]} & /@ Range[1, 50] data2 = {rand[]+1, rand[]+1} & /@ Range[1, 50] Show[ListPlot[data1],   ListPlot[data2, PlotStyle -> Red]]			load(distrib); x1: makelist(random_normal(0, 1), i, 1, 50); y1: makelist(random_normal(0, 1), i, 1, 50); x2: makelist(random_normal(0, 1), i, 1, 50) + 1; y2: makelist(random_normal(0, 1), i, 1, 50); + 1; wxplot2d([[discrete, x1, y1],     [discrete, x2, y2]],   [style, points], [color, black, red]);
point types	ListPlot[data, PlotMarkers -> {"*"}] (* shows standard sequence of point types: *) Graphics`PlotMarkers[] (* The elements of the PlotMarkers array can be strings, symbols, expressions, or images. *)			wxplot2d([discrete, x, y],   [style, points],   [point_type, asterisk]); /* possible point_type values:   asterisk   box   bullet   circle   diamond   plus   square   times   triangle The bullet and box are filled versions of circle and square. */
point size	X = NormalDistribution[0, 1] rand = Function[RandomReal[X]] data = {rand[], rand[]} & /@ Range[1, 50] (* point size is fraction of plot width: *) ListPlot[data, PlotStyle -> {PointSize[0.03]}]
scatter plot matrix	Needs["StatisticalPlots`"] X = NormalDistribution[0, 1] x = RandomReal[X, 50] y = RandomReal[X, 50] z = x + 3 * y w = y + RandomReal[X, 50] PairwiseScatterPlot[Transpose[{x, y, z, w}]]			load(distrib); x: makelist(random_normal(0, 1), i, 1, 50); y: makelist(random_normal(0, 1), i, 1, 50); z: x + 3 * y; w: y + makelist(random_normal(0, 1), i, 1, 50); wxscatterplot(transpose(matrix(x, y, z, w)));
3d scatter plot	X = NormalDistribution[0, 1] data = RandomReal[X, {50, 3}] ListPointPlot3D[data]			none
bubble chart	X = NormalDistribution[0, 1] data = RandomReal[X, {50, 3}] BubbleChart[data]
linear regression line	data = Table[{i, 2 * i + RandomReal[{-5, 5}]},   {i, 0, 20}] model = LinearModelFit[data, x, x] Show[ListPlot[data],   Plot[model["BestFit"],     {x, 0, 20}]]			load(distrib); load(lsquares); X: makelist(i, i, 50); Y: makelist(X[i] + random_normal(0, 1), i, 50); M: transpose(matrix(X, Y)); fit: lsquares_estimates(M, [x, y], y = A*x + B,   [A, B]); A: second(fit[1][1]), numer; B: second(fit[1][2]), numer; Xhat: makelist(A*X[i] + B, i, 50); wxplot2d([[discrete, X, Y], [discrete, X, Xhat]],   [style, points, lines], [color, black, red]);
quantile-quantile plot	X = NormalDistribution[0, 1] data1 = RandomReal[1, 50] data2 = RandomReal[X, 50] QuantilePlot[data1, data2]			load(distrib); x: makelist(random_continuous_uniform(0, 1),   i, 200); y: makelist(random_normal(0, 1), i, 200); wxplot2d([discrete, sort(x), sort(y)],   [style, points]);
line charts
	mathematica	sympy	sage	maxima
polygonal line plot	X = NormalDistribution[0, 1] rand = Function[RandomReal[X]] f = Function[i, {i, rand[]}] data = f /@ Range[1, 20] ListLinePlot[data]			load(distrib); x: makelist(random_normal(0, 1), i, 1, 20); wxplot2d([discrete, makelist(i, i, 20), x]);
additional line	X = NormalDistribution[0, 1] data1 = RandomReal[X, 20] data2 = RandomReal[X, 20] ListLinePlot[{data1, data2}   PlotStyle->{Black, Red}]			load(distrib); x: makelist(random_normal(0, 1), i, 1, 20); y: makelist(random_normal(0, 1), i, 1, 20); wxplot2d([[discrete, makelist(i, i, 20), x],     [discrete, makelist(i, i, 20), y]],   [color, black, red]);
line types	ListLinePlot[data, PlotStyle -> Dashed] (* PlotStyle values:   Dashed   DotDashed   Dotted *)			none
line thickness	X = NormalDistribution[0, 1] data1 = RandomReal[X, 20] data2 = RandomReal[X, 20] (* thickness is fraction of plot width: *) ListLinePlot[{data1, data2},   PlotStyle -> {Thickness[0.01], Thickness[0.02]}]
function plot	Plot[Sin[x], {x, -4, 4}]			wxplot2d(sin(x), [x, -4, 4]);
parametric plot	ParametricPlot[{Sin[u], Sin[2 * u]},   {u, 0, 2 * Pi}]			wxplot2d([parametric, sin(t), sin(2*t),   [t, 0, 2*%pi]]);
implicit plot	ContourPlot[x^2 + y^2 == 1, {x, -1, 1},   {y, -1, 1}]			load(implicit_plot); wximplicit_plot(x^2 + y^2 = 1, [x, -1, 1],   [y, -1, 1]);
polar plot	PolarPlot[Sin[3 * t], {t, 0, Pi}]			f(x) := sin(3 * x); wxplot2d([parametric, cos(t)*f(t), sin(t)*f(t),   [t, 0, %pi]]);
cubic spline	X = NormalDistribution[0, 1] data = Table[{i, RandomReal[X]},   {i, 0, 20}] f = Interpolation[data,   InterpolationOrder -> 3] Show[ListPlot[data],   Plot[f[x], {x, 0, 20}]]			load(interpol); load(distrib); load(draw); data: makelist([i, random_normal(0, 1)], i, 20); cspline(data); f(x):=''%; wxdraw2d(explicit(f(x),x,0,20));
area chart	data = {{7, 1, 3, 2, 8}, {1, 5, 3, 5, 1}} stacked = {data[[1]], data[[1]] + data[[2]]} ListLinePlot[stacked, Filling ->   {1 -> {Axis, LightBlue},    2 -> {{1}, LightRed}}]
surface charts
	mathematica	sympy	sage	maxima
contour plot	(* of function: *) ContourPlot[x * (y - 1), {x, 0, 10},   {y, 0, 10}] (* of data: *) X = NormalDistribution[0, 1] rand = Function[RandomReal[X]] data = Table[x * (y - 1) + 5 * rand[],   {x, 0, 10}, {y, 0, 10}] ListContourPlot[data]			wxcontour_plot(x * (y-1), [x, 0, 10],   [y, 0, 10]);
heat map	(* of function: *) DensityPlot[Sin[x] * Sin[y],   {x, -4, 4},   {y, -4, 4}] (* of data: *) X = NormalDistribution[0, 1] rand = Function[RandomReal[X]] data = Table[x * y + 10 * rand[],   {x, 1, 10},   {y, 1, 10}] ListDensityPlot[data]			wxplot3d (sin(x) * sin(y), [x,-4,4], [y,-4,4],   [elevation, 0], [azimuth, 0],   [grid, 100, 100], [mesh_lines_color, false]);
shaded surface plot	Plot3D[Exp[-(x^2 + y^2)], {x, -2, 2},   {y, -2, 2}, MeshStyle -> None]
light source	lot3D[Exp[-(x^2 + y^2)],   {x, -2, 2}, {y, -2, 2},   MeshStyle -> None,   Lighting -> {{"Point", White, {5, -5, 5}}}]
mesh surface plot	Plot3D[Exp[-(x^2 + y^2)], {x, -2, 2},   {y, -2, 2}, Lighting -> {White},   PlotStyle -> White]			wxplot3d(exp(-(x^2 + y^2)),   [x, -2, 2], [y, -2, 2],   [palette, false], [color, black]);
view point	(* (x, y, z) coordinates;    (0, 0, 3) is from above: *) Plot3D[Exp[-(x^2 + y^2)],   {x, -2, 2}, {y, -2, 2},   MeshStyle -> None,   ViewPoint -> {0, 0, 3}]
vector field plot	StreamPlot[{x^2 + y, 1 + x - y^2}, {x, -4, 4}, {y, -4, 4}]			plotdf([x^2 + y, 1 + x - y^2], [x, -4, 4],   [y, -4, 4]);
chart options
	mathematica	sympy	sage	maxima
chart title	(* title on top by default *) Plot[Sin[x], {x, -4, 4},   PlotLabel -> "title example"]			wxplot2d(sin(x), [x, -4, 4],   [title, "title example"]); data: 1, 1, 2, 2, 3, 3, 3, 3, 4]; wxboxplot(data, title="title example"); /* title on top by default */
axis label	data = Table[{i, i^2}, {i, 1, 20}] ListLinePlot[data, AxesLabel -> {x, x^2}]			x: makelist(i, i, 20); y: makelist(i^2, i, 20); wxplot2d([discrete, x, y],   [xlabel, "x"], [ylabel, "x^2"]);
legend	X = NormalDistribution[0, 1] data1 = RandomReal[X, 20] data2 = RandomReal[X, 20] ListLinePlot[{data1, data2},     PlotLegends -> {"first", "second"}]			/* wxplot2d includes a legend by default. Provide [legend, false] as argument to suppress it. */
data label	data = {{313, 3.7}, {62, .094}, {138, 6.6},   {113, 0.76}, {126, 0.15}} (* The {0, -1} argument of Text[] centers the label above the data point. *) Show[ListPlot[data,     AxesLabel -> {"pop", "area"}],   Graphics[Text["USA", data[[1]], {0, -1}]],   Graphics[Text["UK", data[[2]], {0, -1}]],   Graphics[Text["Russia", data[[3]], {0, -1}]],   Graphics[Text["Mexico", data[[4]], {0, -1}]],   Graphics[Text["Japan", data[[5]], {0, -1}]]]
named colors	White Gray Black Transparent Blue Brown Cyan Green Magenta Orange Pink Purple Red Yellow LightBlue LightBrown LightCyan LightGray LightGreen LightMagenta LightOrange LightPink LightPurple LightRed LightYellow			white gray black gray0 gray10 gray100 gray20 gray30 gray40 gray50 gray60 gray70 gray80 gray90 gray100 grey grey0 grey10 grey20 grey30 grey40 grey50 grey60 grey70 grey80 grey90 grey100 aquamarine beige blue brown coral cyan forest_green gold goldenrod green khaki magenta medium_blue midnight_blue navy orange orange_red pink plum purple red royalblue salmon sea_green skyblue spring_green turquoise violet yellow dark_blue dark_cyan dark_goldenrod dark_gray dark_green dark_grey dark_khaki dark_magenta dark_orange dark_pink dark_red dark_salmon dark_turquoise dark_violet dark_yellow light_blue light_coral light_cyan light_goldenrod light_gray light_green light_grey light_magenta light_pink light_red light_salmon light_turquoise light_yellow
rgb color	RGBColor[1, 0, 0] (* with opacity: *) RGBColor[1, 0, 0, 0.5]			[color, "#FF0000"]
background color	Plot[Sin[x], {x, 0, 2 Pi},   Background -> Black,   PlotStyle -> White,   AxesStyle -> White,   TicksStyle -> White,   GridLines -> Automatic,   GridLinesStyle -> White]
axis limits	Plot[x^2, {x, 0, 20},   PlotRange -> {{0, 20}, {-200, 500}}]
logarithmic y-axis	LogPlot[{x^2, x^3, x^4, x^5},   {x, 0, 20}]			x: makelist(i, i, 20); wxplot2d([     [discrete, x, makelist(i^2, i, 20)],     [discrete, x, makelist(i^3, i, 20)]],   [logy, true]);
aspect ratio	(* aspect ratio is height divided by width: *) Plot[Sin[x], {x, 0, 2 Pi}, AspectRatio -> 0.25] (* In the notebook, dragging the corner of an image increases or decreases the size, but aspect ratio is preserved. *)			wxplot2d(sin(x), [x, -4, 4],   [yx_ratio, 0.25]); /* Image size can’t be changed in notebook. */
ticks	Plot[Sin[x], {x, 0, 2 Pi}, Ticks -> None] Plot[Sin[x], {x, 0, 2 Pi},   Ticks -> {{0, Pi, 2*Pi}, {-1, 0, 1}}]			wxplot2d(sin(x), [x, -4, 4],   [xtics, -4, 2, 4],   [ytics, -1, 0.5, 1]);
grid lines	Plot[Sin[x], {x, 0, 2 Pi},   GridLines -> Automatic] Plot[Sin[x], {x, 0, 2 Pi},   GridLines -> {{0, 1, 2, 3, 4, 5, 6},     {-1, -0.5, 0, 0.5, 1}}]
grid of subplots	GraphicsGrid[Table[Table[       Histogram[RandomReal[X, 100], 10],       {i, 1, 2}],     {j, 1, 2}]]			load(distrib); x: makelist(makelist(random_normal(0, 1), i, 50),   j, 4); p: makelist(histogram_description(x[i]), i, 4); wxdraw(gr2d(p[1]), gr2d(p[2]), gr2d(p[3]),   gr2d(p[4]), columns=2);
save plot as png	Export["hist.png",   Histogram[RandomReal[X, 100], 10]]			After creating a plot, run gnuplot on the .gnuplot file generated in the home directory: $ gnuplot maxout.gnuplot
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Symbolic Expressions

In many programming languages, attempting to evaluate an expression with an undefined variable results in an error. Some languages assign a default value to variables so that such expressions can be evaluated.

In a CAS, undefined variables are treated as unknowns; expressions which contains them are symbolic expressions. When evaluating them, if the unknowns cannot be eliminated, the expression cannot be reduced to a numeric value. The expression then evaluates to a possibly simplified or normalized version of itself. Symbolic expressions are first class values; they can be stored in variables or passed to functions.

An application of symbolic expressions is a function which solves a system of equations. Without symbolic expressions, it would be awkward for the caller to specify the equations to be solved.

literal

How to create a symbolic expression.

In most CAS systems, any expression an undefined variables is automatically a a symbolic expression.

sympy:

In SymPy, unknowns must be declared. This is a consequence of SymPy being implemented as a library in a language which throws exceptions when undefined variables are encountered.

prevent simplification

variable update

Do symbolic expressions "see" changes to the unknown variables they contain.

substitute

piecewise-defined expression

simplify

assumption

assumption predicates

list assumptions

remove assumption

Calculus

limit

limit at infinity

one-sided limit

derivative

derivative of a function

constants

higher order derivative

mixed partial derivative

div, grad, and curl

antiderivative

definite integral

improper integral

double integral

find poles

residue

sum

series sum

series expansion of function

omitted order term

product

Equations and Unknowns

solve equation

solve equations

differential equation

differential equation with boundary condition

differential equations

recurrence equation

Optimization

An optimization problem consists of a real-valued function called the objective function.

The objective function takes one or more input variables. In the case of a maximization problem, the goal is to find the value for the input variables where the objective function achieves its maximum value. Similarly for a minimization function one looks for the values for which the objective function achieves its minimum value.

minimize

How to solve a minimization problem in one variable.

maximize

How to solve a maximization problem.

We can use a function which solves minimization problems to solve maximization problems by negating the objective function. The downside is we might forget the minimum value returned is the negation of the maximum value we seek.

objective with unknown parameter

How to solve an optimization when the objective function contains unknown parameters.

unbounded behavior

What happens when attempting to solve an unbounded optimization problem.

multiple variables

How to solve an optimization problem with more than one input variable.

constraints

How to solve an optimization with constraints on the input variable. The constrains are represented by inequalities.

infeasible behavior

What happens when attempting to solve an optimization problem when the solution set for the constraints is empty.

integer variables

How to solve an optimization problem when the input variables are constrained to linear values.

Vectors

vector literal

The notation for a vector literal.

constant vector

How to create a vector with components all the same.

vector coordinate

How to get one of the coordinates of a vector.

vector dimension

How to get the number of coordinates of a vector.

element-wise arithmetic operators

How to perform an element-wise arithmetic operation on vectors.

vector length mismatch

What happens when an element-wise arithmetic operation is performed on vectors of different dimension.

scalar multiplication

How to multiply a scalar with a vector.

dot product

How to compute the dot product of two vectors.

cross product

How to compute the cross product of two three-dimensional vectors.

norms

How to compute the norm of a vector.

Matrices

literal or constructor

Literal syntax or constructor for creating a matrix.

mathematica:

Matrices are represented as lists of lists. No error is generated if one of the rows contains too many or two few elements. The MatrixQ predicate can be used to test whether a list of lists is matrix: i.e. all of the sublists contain numbers and are of the same length.

Matrices are displayed by Mathematica using list notation. To see a matrix as it would be displayed in mathematical notation, use the MatrixForm function.

construct from sequence

constant matrices

diagonal matrices

matrix by formula

dimensions

How to get the number of rows and columns of a matrix.

element lookup

How to access an element of a matrix.

The anguages described here follow the mathematical convention of putting the row index before the column index.

extract row

How to access a row.

extract column

How to access a column.

extract submatrix

How to access a submatrix.

scalar multiplication

How to multiply a matrix by a scalar.

element-wise operators

Operators which act on two identically sized matrices element by element. Note that element-wise multiplication of two matrices is used less frequently in mathematics than matrix multiplication.

product

How to multiply matrices.

Matrix multiplication in non-commutative and only requires that the number of columns of the matrix on the left match the number of rows of the matrix. Element-wise multiplication, by contrast, is commutative and requires that the dimensions of the two matrices be equal.

power

How to compute the power of a square matrix.

For non-negative integers, the power of a matrix is defined recursively with A⁰ = I and Aⁿ = A^n-1 ⋅ A.

If the matrix is invertible, the power is defined for negative integers by Aⁿ = (A^-1)^-n.

exponential

log

kronecker product

The Kronecker product is a non-commutative operation defined on any two matrices. If A is m x n and B is p x q, then the Kronecker product is a matrix with dimensions mp x nq.

norms

How to compute the 1-norm, the 2-norm, the infinity norm, and the frobenius norm.

transpose

conjugate transpose

inverse

row echelon form

pseudoinverse

determinant

The determinant of a square matrix is equal to the product of the eigenvalues of a matrix. It is zero if and only if the matrix is nonsingular.

The determinant can be computed by calculating the matrix cofactors along a row or column, multiplying each by the row or column entry, and then summing. This technique, called Laplace's formula, requires n! multiplications, where n is number of rows in matrix, and thus is impractical for large matrices.

trace

The trace of a matrix is the sum of the diagonal entries. It is equal to the sum of the eigenvalues of the matrix.

characteristic polynomial

The charateristic polynomial of a matrix A can be defined by:

The eigenvalues of A are the roots of p(t).

minimal polynomial

rank

nullspace basis

eigenvalues

eigenvectors

LU decomposition

A factorization into a lower triangular matrix L, an upper triangular matrix U, and a permutation matrix P. Typically the identity is PA = LU.

An LU factorization of a square matrix always exists. It can be found using a modified variant of Gaussian elimination. For an n × n matrix it requires about n³ scalar multiplications.

LU factorization is an efficient way (1) to solve a system of equations, (2) to find the inverse of a matrix, and (3) to compute the determinant of a matrix.

To use PA = LU to solve Ax = b, first solve for y in Ly = Pb using forward substitution, then solve for x in Ux = y using backward substitution.

To use PA = LU to find the inverse B of A, first solve for Y in LU = P using forward substitution, then solve for UB = Y using backward substitution.

The determinant of A can be computed from the diagonal entries of L and U. To get the sign correctly, one must count the number of row exchanges S in the permutation matrix P:

QR decomposition

A factorization of a square matrix into an orthogonal matrix Q and an upper triangular matrix R.

The QR factorization is unique when the original matrix A is invertible.

The Gram-Schmidt process can be used to compute a QR factorization, though it is not the most numerically stable method.

If a₁, …, a_n are the column vectors of the original matrix A, then the Gram-Schmidt process yields the column vectors e₁, …, e_n of the orthogonal matrix Q.

The QR algorithm uses QR factorizations to iteratively find eigenvalues. For each iteration we perform a QR factorization on A_k:

Then we multiply Q_k and R_k in reverse to get A_k+1:

Usually the sequence of A_k will converge to a triangular matrix with the eigenvalues on the diagonal. The limit matrix is similar to the original matrix because

and hence has the same eigenvalues.

spectral decomposition

The spectral decomposition of a square matrix A is a factorization P ⋅D ⋅ P^-1 where P is invertible and D is diagonal.

The spectral decomposition is also called the eigendecomposition. The values on the diagonal of D are eigenvalues of the matrix A and the rows of P are eigenvectors.

If a spectral decomposition exists, the matrix A is said to be diagonalizable.

If an invertible matrix P exists such that A = P ⋅ B ⋅ P^-1, then A and B are said to be similar.

According to the spectral theorem, a spectral decomposition exists when the matrix A is normal, which means it commutes with its conjugate transpose.

If a matrix A is symmetric, then a spectral decomposition P ⋅ D ⋅ P^-1 exists, and moreover P and D are real matrices.

singular value decomposition

A singluar value decomposition of a matrix A is a factorization into a diagonal matrix S and unitary matrices U and V such that A = U ⋅ S ⋅ V^*.

Unlike the spectral decomposition, an SVD always exists, even if A is not square. The values on the diagonal of S are called the singular values, and they are the eigenvalues of A ⋅ A^*.

jordan decomposition

The Jordan decomposition of a square matrix A is a factorization A = P ⋅ J ⋅ P^-1 where J is in Jordan canonical form.

polar decomposition

A factorization of a square matrix into a unitary matrix U and a positive definite Hermitian matrix P.

All invertible matrices have a polar decomposition.

A unitary matrix corresponds to a linear transformation representing a rotation, reflection, or a combination of the two. It is distance perserving, in that it maps vectors to vectors of the same length. A real valued unitary matrix is called an orthogonal matrix.

Combinatorics

Enumerative combinatorics is the study of the size of finite sets. The sets are defined by some property, and we seek a formula for the size of the set so defined.

For some simple examples, let A and B be disjoint sets of size n and m respectively. The size of the union A ∪ B is n + m and the size of the Cartesian product A × B is nm. The size of the power set of A is 2ⁿ.

factorial

The factorial function n! is the product of the first n positive integers 1 × 2 × … × n.

It is also the number of permutations or bijective functions on a set of n elements. It is the number of orderings that can be given to n elements.

See the section on permutations below for how to iterate through all n! permutations on a set of n elements.

As the factorial function grows rapidly with n, it is useful to be aware of this approximation:

binomial coefficient

A binomial coefficient can be computed using the factorial function:

The binomial coefficient appears in the binomial theorem:

The binomial cofficient ${ n \choose k }$ is the number of sets of size k which can be drawn from a set of size n without replacement.

multinomial coefficient

The multinomial coefficient generalizes the binomial cofficient:

It appears in the multinomial theorem:

The multinomial cofficient ${ n \choose k_1,ldots,k_m }$ is the number of ways to partition a set of n elements into subsets of size k₁, …, k_m where the k_i sum to n.

rising and falling factorial

subfactorial

A derangement is a permutation on a set of n elements where every element moves to a new location.

The number of derangements is thus less than the number of permutations, n!, and the function for the number of derangements is called the subfactorial function.

Using a exclamation point as a prefix to denote the subfactorial, the following equations hold:

integer partitions

The number of multisets of positive integers which sum to a integer.

There are 5 integer partitions of 4:

    4
    3 + 1
    2 + 2
    2 + 1 + 1
    1 + 1 + 1 + 1

compositions

The number of sequences of positive integers which sum to an integer.

There are 8 compositions of 4:

    4
    3 + 1
    1 + 3
    2 + 2
    2 + 1 + 1
    1 + 2 + 1
    1 + 1 + 2
    1 + 1 + 1 + 1

mathematica:

The NumberOfCompositions and Compositions functions use weak compositions, which include zero as a possible summation.

The number of weak compositions of an integer is infinite, since there is no limit on the number of times zero can appear as a summand. The number of weak compositions of a fixed size is finite, however.

set partitions

bell number

permutations with k disjoint cycles

fibonacci number

bernoulli number

harmonic number

catalan number

Number Theory

divisible test

A test whether an integer a is divisible by another integer b.

Equivalently, does there exists a third integer m such that a = mb.

divisors

The list of divisors for an integer.

pseudoprime test

A fast primality test.

An integer p is prime if for any factorization p = ab, where a and b are integers, either a or b are in the set {-1, 1}.

A number of primality tests exists which give occasional false positives. The simplest of these use Fermat's Little Theorem, in which for prime p and a in $\{1, ..., p - 1\}$:

The test for a candidate prime p is to randomly choose several values for a in $\{1, ..., p - 1\}$ and evaluate

If any of them are not equivalent to 1, then the test shows that p is not prime. Unfortunately, there are composite numbers n, the Carmichael numbers, for which

holds for all a in $\{1, ..., n - 1\}$.

A stronger test is the Miller-Rabin primality test. Given a candidate prime n, we factor n - 1 as 2^r ⋅ d where d is odd. If n is prime, then one of the following must be true:

Thus, one checks the above two equations for a small number of primes. If we use all primes p ≤ 41, then it is known that there are no false positives for n ≤ 3 × 10²⁴.

Since pseuodoprime tests are known which are correct for all integers up to a very large size, and since conclusively showing that a number is prime is a slow operation for larger integers, a true prime test is often not practical.

prime factors

The list of prime factors for an integer, with their multiplicities.

next prime

The smallest prime number greater than an integer. Also the greatest prime number smaller than an integer.

nth prime

The n-th prime number.

prime counting function

The number of primes less than or equal to a value.

According to the prime number theorem:

divmod

The quotient and remainder.

If the divisor is positive, then the remainder is non-negative.

greatest common divisor

The greatest common divisor of a pair of integers. The divisor is always positive.

Two integers are relatively prime if their greatest common divisor is one.

extended euclidean algorithm

How to express a greatest common divisor as a linear combination of the integers it is a GCD of.

The functions described return the GCD in addition to the coefficients.

least common multiple

The least common multiple of a pair of integers.

The LCM can be calculated from the GCD using this formula:

power modulus

Raise an integer to a integer power, modulo a third integer.

Euler's theorem can often be used to reduce the size of the exponent.

integer residues

The integer residues or integers modulo n are the equivalence classes formed by the relation

An element in of these equivalence classes is called a representative. We can extend addition and multiplication to the residues by performing integer addition or multiplication on representatives. This is well-defined in the sense that it does not depend on the representatives chosen. Addition and multiplication defined this way turn the integer residues into commutative rings with identity.

multiplicative inverse

How to get the multiplicative inverse for a residue.

If the representative for a residue is relatively prime to the modulus, then the residue has a multiplicative inverse. For that matter, if the modulus n is a prime, then the ring of residues is a field.

Note that we cannot in general find the inverse using a representative, since the only units in the integers are -1 and 1.

By Euler's theorem, we can find a multiplicative inverse by raising it to the power $\phi(n) - 1$:

When a doesn't have a multiplicative inverse, then we cannot cancel it from both sides of a congruence. The following is true, however:

chinese remainder theorem

A function which finds a solution to a system of congruences.

The Chinese remainder theorem asserts that there is a solution x to the system of k congruences

provided that the n_i are pairwise relatively prime. In this case there are an infinite number of solutions, all which are equal modulo $N = n_1 \cdots n_k$. For this reason the solution is returned as a residue modulo N.

lift integer residue

How to get a representative from the equivalence class of integers modulo n.

Typically an integer in $\{0, ..., n - 1\}$ is chosen. A centered lift chooses a representative x such that $-n/2 < x \leq n/2$.

euler totient

The Euler totient function is defined for any positive integer n as:

Note that the product is over the primes that divide n.

The Euler totient is the number of integers in $\{1, ..., n - 1\}$ which are relatively prime to n. It is thus the size of the multiplicative group of integers modulo n.

The Euler totient appears in Euler's theorem:

carmichael function

The smallest number k such that a^k ≡ 1 (mod n) for all residues a.

By Euler's theorem, the Carmichael function λ(n) is less that or equal to the Euler totient function φ(n). The functions are equal when there are primitive roots modulo n.

multiplicative order

The multiplicative order of a residue a is the smallest exponent k such that

In older literature, it is sometimes said that a belongs to the exponent k modulo n.

primitive roots

A primitive root is a residue module n with multiplicative order φ(n).

The multiplicative group is not necessarily cyclic, though it is when n is prime. If it is not cyclic, then there are no primitive roots.

Any primitive root is a generator for the multiplicative group, so it can be used to find the other primitive roots.

discrete logarithm

For a residue x and a base residue b, find a positive integer such that:

quadratic residues

A quadratic residue is a non-zero residue a which has a square root modulo p. That is, there is x such that

If a is non-zero and doesn't have a square root, then it is a quadratic non-residue.

discrete square root

How to find the square root of a quadratic residue.

kronecker symbol

The Legendre symbol is used to indicate whether a number is a quadratic residue and is defined as follows:

The Legendre symbol is only defined when p is an odd prime, but if n is an odd positive integer with prime factorization

then the Jacobi symbol is defined as

The Kronecker symbol is a generalization of the Jacobi symbol to all integers, but we omit the details.

moebius function

The Möbius function μ(n) is 1, -1, or 0 depending upon when n is a square-free integer with an even number of prime factors, a square-free integer with an odd number of prime factors, or an integer which is divisible by p² for some prime p.

The Möbius function is multiplicative: when a and b are relatively prime, μ(a)μ(b) = μ(ab).

The Möbius function appears in the Möbius inversion formula. If g and f are possibly complex-valued functions defined on the natural numbers such that for all integers n ≥ 1:

then for all integers n ≥ 1:

riemann zeta function

The Riemann zeta function is a complex-valued function defined as the analytic continuation of this series:

The function has zeros (called the trivial zeros) at -2, -4, …. All other zeros must lie in the strip 0 ≤ ℜ(z) ≤ 1. In 1859 Riemann conjectured that all non-trivial zeros are on the line ℜ(z) = 1/2.

continued fraction

Convert a real number to a continued fraction.

A continued fraction is a sequence of integers a₀, a₁, …, a_n representing the fraction:

The sequence can even be infinite, in which case the fraction is the limit of the rational numbers defined by taking the first n digits in the sequence.

A continued fraction for a real number can be computed using the Euclidean algorithm. In the case of a rational number, one starts with the numerator and the denominator. In the case of a rational number, one can start with the number itself and 1.

A continued fraction is finite if and only if the number is a rational.

A continued fraction repeats if and only if it is a quadratic irrational.

convergents

The first n digits of a continued fraction define a sequence of rational numbers called the convergents. The rational numbers converge to the number defined by the continued fraction.

Each convergent r/q is the closest rational number to the continued fraction with denominator of size q or smaller.

Polynomials

literal

extract coefficient

extract coefficients

from array of coefficients

degree

expand

factor

[[ #collect-terms-note]]

collect terms

roots

quotient and remainder

greatest common divisor

extended euclidean algorithm

resultant

discriminant

groebner basis

specify ordering

elementary symmetric polynomial

symmetric reduction

cyclotomic polynomial

hermite polynomial

chebyshev polynomial

interpolation polynomial

spline

add fractions

partial fraction decomposition

pade approximant

Trigonometry

eliminate powers and products of trigonometric functions

eliminate sums and multiples inside trigonometric functions

trigonometric to exponential

exponential to trigonometric

fourier expansion

periodic functions on unit interval

fourier transform

heaviside step function

dirac delta

Special Functions

gamma function

The gamma function is defined for all complex numbers except the non-positive integers.

For positive integers, the following equation holds:

If the real part of t is positive, then

error function

The error function is function from ℝ to [-1, 1] defined by:

The complementary error function is

The cumulative distribution of the standard normal distribution is related to the error function by scaling:

hyperbolic functions

Definitions of the hyperbolic functions:

sinh and cosh are odd and even functions, respectively. Like e^x and e^-x, sinh and cosh span the linear space of solutions to y''(x) = y(x).

elliptic functions

bessel functions

Permutations

A permutation is a bijection on a set of n elements.

The notation that Mathematica uses assumes the set the permutation operates on is indexed by {1, .., n}. The notation that SymPy uses assumes the set is indexed by {0, …, n - 1}.

Cayley two line notation

one line notation

cycle notation

inversions

from disjoint cycles

to disjoint cycles

from array

from two arrays with same elements

size

support

act on element

act on list

compose

inverse

power

order

number of inversions

parity

Permutations are classified as even or odd based on the number of inversions.

The composition of two even permutations is even.

to inversion vector

from inversion vector

list permutations

random permutation

Descriptive Statistics

Distributions

Statistical Tests

A selection of statistical tests. For each test the null hypothesis of the test is stated in the left column.

In a null hypothesis test one considers the p-value, which is the chance of getting data which is as or more extreme than the observed data if the null hypothesis is true. The null hypothesis is usually a supposition that the data is drawn from a distribution with certain parameters.

The extremeness of the data is determined by comparing the expected value of a parameter according to the null hypothesis to the estimated value from the data. Usually the parameter is a mean or variance. In a one-tailed test the p-value is the chance the difference is greater than the observed amount; in a two-tailed test the p-value is the chance the absolute value of the difference is greater than the observed amount.

wilcoxon signed-rank test

A non-parametric est whether a variable is drawn from a distribution that is symmetric about zero.

Often this test is used to test that the mean of the distribution is zero.

kruskal-wallis rank sum test

A non-parametric test whether variables have the same mean.

For two variables, this test is the same as the Mann-Whitney test.

maxima:

The Maxima function only supports testing two variables.

kolmogorov-smirnov test

Test whether two samples are drawn from the same distribution.

one-sample t-test

Student's t-test determines whether a sample drawn from a normal distribution has mean zero.

The test can be used to test for a different mean value; just subtract the value from each value in the sample.

One may know in advance that the sample is drawn from a normal distribution. For example, if the values in the sample are each means of large samples, then the distribution is normal by the central limit theorem.

The Shapiro-Wilk test can be applied to determine if the values come from a normal distribution.

If the distribution is not known to be normal, the Wilcoxon signed-rank test can be used instead.

The Student's t-test used the sample to estimate the variance, and as a result the test statistic has a t-distribution.

By way of contrast, the z-test assumes that the variance is known in advance, and simply scales the data to get a z-score, which has standard normal distribution.

independent two-sample t-test

Test whether two normal variables have same mean.

paired sample t-test

A t-test used when the same individuals are measure twice.

one-sample binomial test

two-sample binomial test

chi-squared test

poisson test

F test

pearson product moment test

pearson spearman rank test

shapiro-wilk test

bartlett's test

A test whether variables are drawn from normal distributions with the same variance.

levene's test

A test whether variables are drawn from distributions with the same variance.

one-way anova

two-way anova

Bar Charts

vertical bar chart

A chart in which the height of bars is used to represent a list of numbers.

maxima:

Maxima plots the frequency of the values, and not the values themselves. Non-positive values cannot be represented.

horizontal bar chart

A bar chart in which zero is the y-axis and the bars extend to the right.

grouped bar chart

stacked bar chart

pie chart

maxima:

Note that Maxima plots the frequency of the values, and not the values themselves.

histogram

A histogram is a bar chart in which each bar represents the frequency of values in a data set within a range. The width of the bars can be used to indicate the ranges.

box plot

Scatter Plots

strip chart

A strip chart represents a list of values by points on a line. The values are converted to pairs by assigning the y-coordinate a constant value of zero. Pairs are then displayed with a scatter plot.

strip chart with jitter

A strip chart in which in which a random variable with small range is used to fill the y-coordinate. Jitter makes it easier to see how many values are in dense regions.

scatter plot

How to plot a list of pairs of numbers by representing the pairs as points in the (x, y) plane.

additional point set

How to add a second list of pairs of numbers to a scatter plot. Color can be used to distinguish the two data sets.

point types

How to select the symbols used to mark data points. Choice of symbols can be use to distinguish data sets.

point size

How to change the size of the symbols used to mark points.

scatter plot matrix

A scatter plot matrix is a way of displaying a multivariate data set by means of a grid of scatter plots. Off-diagonal plots are scatter plots of two of the variables. On-diagonal plots can be used to to display the name or a histogram of one of the variables.

3d scatter plot

How to represent a list of triples of numbers by points in (x, y, z) space.

bubble chart

How to represent a list of triples of numbers by position in the (x, y) plane and size of the point marker.

It is probably better to associate the 3rd component of each triple with the area, and not the diameter of the point marker, but in general bubble charts suffer from ambiguity.

linear regression line

How to add a linear regression line to a scatter plot.

quantile-quantile plot

Line Charts

polygonal line plot

additional line

line types

line thickness

function plot

parametric plot

implicit plot

polar plot

cubic spline

area chart

Surface Charts

contour plot

heat map

shaded surface plot

light source

mesh surface plot

view point

vector field plot

Chart Options

chart title

axis label

legend

data label

named colors

rgb color

background color

axis limits

logarithmic y-axis

aspect ratio

ticks

grid lines

grid of subplots

save plot as png

Mathematica

Mathematica Documentation Center
WolframAlpha

SymPy

Welcome to SymPy’s documentation!

Sage

http://doc.sagemath.org/html/en/index.html

Maxima

http://maxima.sourceforge.net/docs/manual/maxima.html