Mathematics

Module math

Elementary mathematical functions.

e

Euler's number: 2.71828...

pi

Half length of the unit circle: 3.14159...

tau

Length of the unit circle: 2*pi.

nan

Not a number.

inf

Infinity.

floor(x)

Round down. Returns floating point numbers.

ceil(x)

Round up. Returns floating point numbers.

sqrt(x)

Square root of x.

exp(x)

Exponential function.

log2(x)

Logarithm to base 2.

ln(x)

Logarithm to base e.

lg(x)

Logarithm to base 10.

lg(x,b)

Logarithm to base b.

sin(x), cos(x), tan(x)

Sine, cosine and tangent.

asin(x), acos(x), atan(x)

Arc sine, arc cosine and arc tangent.

sinh(x), cosh(x), tanh(x)

Hyperbolic sine, hyperbolic cosine and hyperbolic tangent.

asinh(x), acosh(x), atanh(x)

Inverse functions of the hyperbolic functions.

fac(x)

Factorial function. Returns floating point numbers.

gamma(x)

Gamma function.

lgamma(x)

Return ln(abs(gamma(x))).

sgngamma(x)

Return sgn(gamma(x)).

erf(x)

Error function.

hypot(x1,...,xn)

Return sqrt(x1^2+...+xn^2).

atan2(y,x)

Return the phase angle of the coordinate vector [x,y].

expm1(x)

Return exp(x)-1.

ln1p(x)

Return ln(1+x).

isfinite(x)

Return true if x is not infinite and not a NaN.

isnan(x)

Return true if x is a NaN.

isinf(x)

Return true if x is infinite.

frexp(x)

Take x==m*2^n and return [m,n]. The type of m is float, the type of n is int.

ldexp(m,n)

Return m*2^n.

Module cmath

Elementary mathematical functions that can take or return complex numbers.

re(z)

Real part of z.

im(z)

Imaginary part of z.

conj(z)

Complex conjugate.

sqrt(z)

Square root.

exp(z)

Exponential function.

ln(z)

Natural logarithm.

sin(z), cos(z), tan(z)

Sine, cosine and tangent.

sinh(z), cosh(z), tanh(z)

Hyperbolic sine, hyperbolic cosine and hyperbolic tangent.

asinh(z), acosh(z), atanh(z)

Inverse functions of the hyperbolic functions.

gamma(z)

Gamma function.

Module math.rational

Rat

Rational numbers data type.

rat(n,d=1)

Rational number n/d.

r.n

Numerator.

r.d

Denominator.

Module math.nt

Number theory.

base(n,b)

Transform the number n into positional notation by base b. The result is in little endian (least significant digit first).

base(n,b).rev()

Big endian (least significant digit last) of the positional notation above.

isprime(n)

Deterministic primality test.

isprime(n,e)

Probalistic primality test with false positive probability of less than 1/(4^e).

gcd(a,b)

Greatest common divisor of a and b.

lcm(a,b)

Least common multiple of a and b.

lcm(a)

Least common multiple of the numbers in the iterable a.

factor(n)

Prime factorization of n.

divisors(n)

The list of divisors of n.

phi(n)

Euler's totient function.

lambda(n)

Carmichael function.

Module math.cf

Combinatorical functions.

fac(n)

Factorial function.

rf(n,k)

Raising factorial.

ff(n,k)

Falling factorial.

bc(n,k)

Binomial coefficient.

mc([k1,...,kn])

Multinomial coefficient.

stirling1(n,k)

Stirling number of the first kind.

stirling2(n,k)

Stirling number of the second kind.

euler1(n,k)

Eulerian number.

euler2(n,k)

Eulerian number of the second order.

bell(n)

Bell numbers.

pf(n)

Partition function.

pf(n,k)

Number of partitions of n into exactly k parts.

permutations(a)

Permutations of the list a.

combinations(k,s)

Combinations of set/list/string s into sets of k elements.

partitions(n,k)

Partitions of n into k parts.

Module math.sf

Special functions.

PP(n,m,x)

Associated Legendre function P_n,m(x).

PH(n,x)

Hermite polynomial H_n(x).

PT(n,x)

Chebyshev polynomial of the first kind: T_n(x).

PU(n,x)

Chebyshev polynomial of the second kind: U_n(x).

gamma(s,x)

Lower incomplete gamma function γ(s,x).

Gamma(s,x)

Upper incomplete gamma function Γ(s,x).

zeta(x)

Zeta function.

B(n)

Bernoulli number, B(1) = +1/2.

Module math.sf.ei

Elliptic integrals.

K(m), m=k^2

Complete elliptic integral of the first kind.

E(m), m=k^2

Complete elliptic integral of the second kind.

F(phi,m), m=k^2

Incomplete elliptic integral of the first kind.

E(phi,m), m=k^2

Incomplete elliptic integral of the second kind.

Pi(phi,n,m), m=k^2

Incomplete elliptic integral of the third kind.

RF(x,y,z)

Carlson symmetric form R_F(x, y, z).

RC(x,y)

Carlson symmetric form R_C(x, y) = R_F(x, y, y).

RJ(x,y,z,p)

Carlson symmetric form R_J(x, y, z, p).

RD(x,y,z)

Carlson symmetric form R_D(x, y, z) = R_J(x, y, z, z).

Module math.la

Linear algebra by polymorphic multidimensional arrays.

vector(a1,...,an)

Build a coordinate vector.

matrix([a11,...,a1n],...,[am1,...,amn])

Build a matrix.

matrix(vector(a11,...,am1),...,vector(a1n,...,amn))

Build a matrix from column vectors.

array(N,data)

Return a coordinate tensor of order N. Note that:

array(1,a) = vector(*a),
array(2,a) = matrix(*a).

vec(a1,...,an)

A shorthand for vector.

diag(a1,...,an)

Return a diagonal matrix.

scalar(n,lambda,zero)

Return a scalar n×n matrix.

scalar(n,1,0)

Return the n×n identity matrix.

a.T

Transposed matrix.

a.H

Conjugate transpose.

a.conj

Conjugated complex matrix.

a.tr

Trace.

a.diag

Main diagonal as a coordinate vector.

a.shape

Shape of the array.

a.copy

Shallow copy of the array.

abs(a)

Absolute value of a vector. Frobenius norm of a matrix.

a+b, a-b, -a, r*a, a/r

Pointwise operations.

v*w, v.conj*w

Scalar product, scalar product of complex vectors.

v.T*w, v.H*w

Scalar product of column matrices, scalar product of complex column matrices.

A*v, A*B

Multiplication matrix*vector, multiplication matrix*matrix.

A^n

Matrix power.

a.list()

Convert the array into a list.

a.map(f)

Applies f to every element of a.

a[k]

Component of a vector (k=0 up to k=n-1). Row of a matrix.

a[i,j]

Component of a matrix.

Module math.na

Numerical algorithms.

pli(x0,d,a)

Return a function that computes a piecewise linear interpolation of the given equidistant nodes [x[k],y[k]]=[x0+k*d,a[k]] for k in 0..len(a)-1. Note that a may contain vectors or a matrices. Vectors are interpolated componentwise.

inv(f,y,a,b)

Calculate the preimage x0 by bisection such that y≈f(x0). The function f(x)-y should change its sign on x∈[a,b] only once at x=x0.

diffh({h=0.001,order=false,fast=false})

Create a numerical derivative operation diff=diffh(). Apply diff(f,a) to calculate f'(x) at x=a. If order is true, apply diff(f,a,n) instead to calculate the nth derivative. If fast is true, use a slightly faster algorithm for the first derivative. Note that f may also be a complex function or a vector-valued function. For a given smooth function, a smaller h leads to a better approximation, unless h is too small.

diffvh({h=0.001,order=false,fast=false})

Create a directional derivative operation diffv=diffvh(). Apply diffv(v,f,a) to calculate diff(|t| f(a+t*v),0).

integral(a,b,f,n=1)

Calculate the definite integral of f(x) for x=a to x=b. Uses a Gauss-Legendre quadrature with 32 nodes. To increase accuracy, the interval [a,b] can be subdivided into n parts. Note that f may be a complex-valued or vector-valued function.

Module math.na.quad

Numerical quadrature.

gauss_legendre(N)

Return a new integral operation, using the Gauss-Legendre quadrature with N nodes.

integral = gauss_legendre(64)
ln = |x| integral(1,x,|t| 1/t)
# integral(a,b,f,n=1)

legendre_nodes(N)

The nodes for the Gauss-Legendre quadrature.

gauss(nodes)

Take nodes and return a new integral operation by Gaussian quadrature.

integral = gauss(legendre_nodes(64))