Specialized Mathematical Functions (specfun)

Bessel functions, Airy functions, Gamma, Beta, Error functions, and Number Theory primitives. More...

Files
file	specfun.hpp
	Special mathematical functions (gamma, beta, erf, Bessel, Airy, combinatorics).

Classes
struct	nk::builtin::EllipKE
	Complete elliptic integrals of the first and second kinds ([K, E] = ellipke(M)). More...
struct	nk::builtin::EllipJ
	Jacobi elliptic functions ([sn, cn, dn] = ellipj(u, M)). More...

Functions
Value	nk::builtin::gamma (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Complete gamma function (Γ(x)).
Value	nk::builtin::gammaln (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Logarithm of the absolute value of the gamma function (ln|Γ(x)|).
Value	nk::builtin::gammainc (const Value &x, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Incomplete gamma function (P(a, x)).
Value	nk::builtin::gammaincinv (const Value &y, const Value &a, std::pmr::memory_resource *mr=nullptr)
	Inverse incomplete gamma function.
Value	nk::builtin::psi (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Digamma / polygamma function (ψ(x) = d/dx ln Γ(x)).
Value	nk::builtin::beta (const Value &z, const Value &w, std::pmr::memory_resource *mr=nullptr)
	Beta function (B(z, w) = Γ(z)Γ(w) / Γ(z+w)).
Value	nk::builtin::betaln (const Value &z, const Value &w, std::pmr::memory_resource *mr=nullptr)
	Natural logarithm of the beta function (ln B(z, w)).
Value	nk::builtin::betainc (const Value &x, const Value &z, const Value &w, std::pmr::memory_resource *mr=nullptr)
	Incomplete beta function (I_x(z, w)).
Value	nk::builtin::betaincinv (const Value &y, const Value &z, const Value &w, std::pmr::memory_resource *mr=nullptr)
	Inverse incomplete beta function.
Value	nk::builtin::erf (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Error function (erf(x) = 2/√π ∫_0^x e^(-t^2) dt).
Value	nk::builtin::erfc (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Complementary error function (erfc(x) = 1 - erf(x)).
Value	nk::builtin::erfcx (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Scaled complementary error function (erfcx(x) = exp(x^2) * erfc(x)).
Value	nk::builtin::erfinv (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Inverse error function (erfinv(y)).
Value	nk::builtin::erfcinv (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Inverse complementary error function (erfcinv(y)).
Value	nk::builtin::besselj (const Value &nu, const Value &z, std::pmr::memory_resource *mr=nullptr)
	Bessel function of the first kind (J_ν(z)).
Value	nk::builtin::bessely (const Value &nu, const Value &z, std::pmr::memory_resource *mr=nullptr)
	Bessel function of the second kind (Neumann function Y_ν(z)).
Value	nk::builtin::besseli (const Value &nu, const Value &z, std::pmr::memory_resource *mr=nullptr)
	Modified Bessel function of the first kind (I_ν(z)).
Value	nk::builtin::besselk (const Value &nu, const Value &z, std::pmr::memory_resource *mr=nullptr)
	Modified Bessel function of the second kind (K_ν(z)).
Value	nk::builtin::besselh (const Value &nu, int k, const Value &z, std::pmr::memory_resource *mr=nullptr)
	Bessel function of the third kind (Hankel function H_ν^(k)(z)).
Value	nk::builtin::airy (int k, const Value &z, std::pmr::memory_resource *mr=nullptr)
	Airy function (Ai(z) or k-th derivative / Bi).
Value	nk::builtin::airy (const Value &z, std::pmr::memory_resource *mr=nullptr)
	Airy function of the first kind (Ai(z)).
Value	nk::builtin::expint (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Exponential integral function (E_1(x)).
EllipKE	nk::builtin::ellipke (const Value &m, std::pmr::memory_resource *mr=nullptr)
	Complete elliptic integrals of the first and second kind ([K, E] = ellipke(m)).
EllipJ	nk::builtin::ellipj (const Value &u, const Value &m, std::pmr::memory_resource *mr=nullptr)
	Jacobi elliptic functions ([sn, cn, dn] = ellipj(u, m)).
Value	nk::builtin::legendre (int n, const Value &x, std::pmr::memory_resource *mr=nullptr)
	Associated Legendre functions (P_n^m(x)).
Value	nk::builtin::factorial (const Value &n, std::pmr::memory_resource *mr=nullptr)
	Factorial of non-negative integer elements (n!).
Value	nk::builtin::primes (double n, std::pmr::memory_resource *mr=nullptr)
	Generates row vector of all prime numbers less than or equal to n (primes(n)).
Value	nk::builtin::primes (const Value &n, std::pmr::memory_resource *mr=nullptr)
	Generates row vector of all prime numbers less than or equal to n (primes(n)).
Value	nk::builtin::isprime (const Value &x, std::pmr::memory_resource *mr=nullptr)
	Tests if array elements are prime numbers (isprime(x)).
Value	nk::builtin::factor (double n, std::pmr::memory_resource *mr=nullptr)
	Computes prime factors of positive integer (factor(n)).
Value	nk::builtin::factor (const Value &n, std::pmr::memory_resource *mr=nullptr)
	Computes prime factors of positive integer value (factor(n)).
Value	nk::builtin::nchoosek (double n, double k, std::pmr::memory_resource *mr=nullptr)
	Computes binomial coefficient n choose k (nchoosek(n, k)).
Value	nk::builtin::nchoosekCombinations (const Value &v, double kd, std::pmr::memory_resource *mr=nullptr)
	Generates all combinations of vector elements taken k at a time (nchoosek(v, k)).
Value	nk::builtin::nchoosek (const Value &v, int k, std::pmr::memory_resource *mr=nullptr)
	Binomial coefficient or all combinations (nchoosek(v, k)).
Value	nk::builtin::nchoosek (const Value &v, const Value &k, std::pmr::memory_resource *mr=nullptr)
	Binomial coefficient or all combinations (nchoosek(v, k)).
Value	nk::builtin::perms (const Value &v, std::pmr::memory_resource *mr=nullptr)
	All permutations of a vector elements.
Value	nk::builtin::gcd (const Value &a, const Value &b, std::pmr::memory_resource *mr=nullptr)
	Greatest common divisor of two integers or arrays (gcd(a, b)).
Value	nk::builtin::lcm (const Value &a, const Value &b, std::pmr::memory_resource *mr=nullptr)
	Least common multiple of two integers or arrays (lcm(a, b)).
Value	nk::builtin::colperm (const Value &s, std::pmr::memory_resource *mr=nullptr)
	Column permutation vector for sparse matrix bandwidth/fill-in reduction (colperm(s)).
Value	nk::builtin::symrcm (const Value &s, std::pmr::memory_resource *mr=nullptr)
	Symmetric reverse Cuthill-McKee ordering permutation vector (symrcm(s)).

Detailed Description

Bessel functions, Airy functions, Gamma, Beta, Error functions, and Number Theory primitives.

Function Documentation

airy() [1/2]

Value nk::builtin::airy	(	const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Airy function of the first kind (Ai(z)).

Parameters

z	Argument.
mr	Memory resource.

Returns

Airy function values.

airy() [2/2]

Value nk::builtin::airy	(	int	k,
		const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Airy function (Ai(z) or k-th derivative / Bi).

Parameters

k	Function kind / derivative index (0: Ai, 1: Ai', 2: Bi, 3: Bi').
z	Argument.
mr	Memory resource.

Returns

Airy function values.

besselh()

Value nk::builtin::besselh	(	const Value &	nu,
		int	k,
		const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Bessel function of the third kind (Hankel function H_ν^(k)(z)).

Parameters

nu	Order.
k	Kind index (1 or 2).
z	Argument.
mr	Memory resource.

Returns

Hankel function values.

See also

besselj, bessely

besseli()

Value nk::builtin::besseli	(	const Value &	nu,
		const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Modified Bessel function of the first kind (I_ν(z)).

Parameters

nu	Order.
z	Argument.
mr	Memory resource.

Returns

Modified Bessel function values.

See also

besselk

besselj()

Value nk::builtin::besselj	(	const Value &	nu,
		const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Bessel function of the first kind (J_ν(z)).

Parameters

nu	Order of the Bessel function.
z	Argument.
mr	Memory resource.

Returns

Bessel function values.

See also

bessely, besseli, besselk

besselk()

Value nk::builtin::besselk	(	const Value &	nu,
		const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Modified Bessel function of the second kind (K_ν(z)).

Parameters

nu	Order.
z	Argument.
mr	Memory resource.

Returns

Modified Bessel function values.

See also

besseli

bessely()

Value nk::builtin::bessely	(	const Value &	nu,
		const Value &	z,
		std::pmr::memory_resource *	mr = nullptr )

Bessel function of the second kind (Neumann function Y_ν(z)).

Parameters

nu	Order of the Bessel function.
z	Argument.
mr	Memory resource.

Returns

Bessel function values.

See also

besselj

beta()

Value nk::builtin::beta	(	const Value &	z,
		const Value &	w,
		std::pmr::memory_resource *	mr = nullptr )

Beta function (B(z, w) = Γ(z)Γ(w) / Γ(z+w)).

Parameters

z	First parameter.
w	Second parameter.
mr	Memory resource.

Returns

Beta function values.

See also

betaln, betainc

betainc()

Value nk::builtin::betainc	(	const Value &	x,
		const Value &	z,
		const Value &	w,
		std::pmr::memory_resource *	mr = nullptr )

Incomplete beta function (I_x(z, w)).

Parameters

x	Upper limit in [0, 1].
z	First parameter.
w	Second parameter.
mr	Memory resource.

Returns

Regularized incomplete beta values.

See also

beta, betaincinv

betaincinv()

Value nk::builtin::betaincinv	(	const Value &	y,
		const Value &	z,
		const Value &	w,
		std::pmr::memory_resource *	mr = nullptr )

Inverse incomplete beta function.

Parameters

y	Value in [0, 1].
z	First parameter.
w	Second parameter.
mr	Memory resource.

Returns

Value x such that betainc(x, z, w) == y.

See also

betainc

betaln()

Value nk::builtin::betaln	(	const Value &	z,
		const Value &	w,
		std::pmr::memory_resource *	mr = nullptr )

Natural logarithm of the beta function (ln B(z, w)).

Parameters

z	First parameter.
w	Second parameter.
mr	Memory resource.

Returns

Log-beta values.

See also

beta

colperm()

Value nk::builtin::colperm	(	const Value &	s,
		std::pmr::memory_resource *	mr = nullptr )

Column permutation vector for sparse matrix bandwidth/fill-in reduction (colperm(s)).

Parameters

s	Sparse or dense matrix.
mr	Memory resource.

Returns

Column permutation vector.

See also

symrcm

ellipj()

EllipJ nk::builtin::ellipj	(	const Value &	u,
		const Value &	m,
		std::pmr::memory_resource *	mr = nullptr )

Jacobi elliptic functions ([sn, cn, dn] = ellipj(u, m)).

Parameters

u	Argument.
m	Parameter.
mr	Memory resource.

Returns

Jacobi elliptic function values.

ellipke()

EllipKE nk::builtin::ellipke	(	const Value &	m,
		std::pmr::memory_resource *	mr = nullptr )

Complete elliptic integrals of the first and second kind ([K, E] = ellipke(m)).

Parameters

m	Parameter m = k^2.
mr	Memory resource.

Returns

Complete elliptic integral values.

erf()

Value nk::builtin::erf	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Error function (erf(x) = 2/√π ∫_0^x e^(-t^2) dt).

Parameters

x	Input value or array.
mr	Memory resource.

Returns

Error function values.

See also

erfc, erfinv

erfc()

Value nk::builtin::erfc	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Complementary error function (erfc(x) = 1 - erf(x)).

Parameters

x	Input value or array.
mr	Memory resource.

Returns

Complementary error function values.

See also

erf, erfcx

erfcinv()

Value nk::builtin::erfcinv	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Inverse complementary error function (erfcinv(y)).

Parameters

x	Value in (0, 2).
mr	Memory resource.

Returns

Value y such that erfc(y) == x.

See also

erfc, erfinv

erfcx()

Value nk::builtin::erfcx	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Scaled complementary error function (erfcx(x) = exp(x^2) * erfc(x)).

Parameters

x	Input value or array.
mr	Memory resource.

Returns

Scaled complementary error values.

See also

erfc

erfinv()

Value nk::builtin::erfinv	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Inverse error function (erfinv(y)).

Parameters

x	Value in (-1, 1).
mr	Memory resource.

Returns

Value y such that erf(y) == x.

See also

erf, erfcinv

expint()

Value nk::builtin::expint	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Exponential integral function (E_1(x)).

Parameters

x	Argument.
mr	Memory resource.

Returns

Exponential integral values.

factor() [1/2]

Value nk::builtin::factor	(	const Value &	n,
		std::pmr::memory_resource *	mr = nullptr )

Computes prime factors of positive integer value (factor(n)).

Parameters

n	Positive integer value.
mr	Memory resource.

Returns

Row vector containing prime factors.

factor() [2/2]

Value nk::builtin::factor	(	double	n,
		std::pmr::memory_resource *	mr = nullptr )

Computes prime factors of positive integer (factor(n)).

Parameters

n	Positive integer.
mr	Memory resource.

Returns

Row vector containing prime factors in ascending order.

See also

primes, isprime

factorial()

Value nk::builtin::factorial	(	const Value &	n,
		std::pmr::memory_resource *	mr = nullptr )

Factorial of non-negative integer elements (n!).

Parameters

n	Non-negative integers.
mr	Memory resource.

Returns

Factorials.

See also

gamma, nchoosek

gamma()

Value nk::builtin::gamma	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Complete gamma function (Γ(x)).

Parameters

x	Input value or array.
mr	Memory resource for allocations (nullptr for default).

Returns

Array containing gamma function values.

See also

gammaln, gammainc, psi

gammainc()

Value nk::builtin::gammainc	(	const Value &	x,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Incomplete gamma function (P(a, x)).

Parameters

x	Upper limit of integration.
a	Scaling parameter.
mr	Memory resource.

Returns

Regularized lower incomplete gamma values.

See also

gamma, gammaincinv

gammaincinv()

Value nk::builtin::gammaincinv	(	const Value &	y,
		const Value &	a,
		std::pmr::memory_resource *	mr = nullptr )

Inverse incomplete gamma function.

Parameters

y	Function value in [0, 1].
a	Scaling parameter.
mr	Memory resource.

Returns

Value x such that gammainc(x, a) == y.

See also

gammainc

gammaln()

Value nk::builtin::gammaln	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Logarithm of the absolute value of the gamma function (ln|Γ(x)|).

Parameters

x	Input value or array.
mr	Memory resource.

Returns

Array containing log-gamma values.

See also

gamma, psi

gcd()

Value nk::builtin::gcd	(	const Value &	a,
		const Value &	b,
		std::pmr::memory_resource *	mr = nullptr )

Greatest common divisor of two integers or arrays (gcd(a, b)).

Parameters

a	First operand.
b	Second operand.
mr	Memory resource.

Returns

Greatest common divisor.

See also

lcm

isprime()

Value nk::builtin::isprime	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Tests if array elements are prime numbers (isprime(x)).

Parameters

x	Input array of integers.
mr	Memory resource.

Returns

Logical array where true indicates prime numbers.

See also

primes, factor

lcm()

Value nk::builtin::lcm	(	const Value &	a,
		const Value &	b,
		std::pmr::memory_resource *	mr = nullptr )

Least common multiple of two integers or arrays (lcm(a, b)).

Parameters

a	First operand.
b	Second operand.
mr	Memory resource.

Returns

Least common multiple.

See also

gcd

legendre()

Value nk::builtin::legendre	(	int	n,
		const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Associated Legendre functions (P_n^m(x)).

Parameters

n	Degree of the Legendre function.
x	Argument.
mr	Memory resource.

Returns

Associated Legendre values.

nchoosek() [1/3]

Value nk::builtin::nchoosek	(	const Value &	v,
		const Value &	k,
		std::pmr::memory_resource *	mr = nullptr )

Binomial coefficient or all combinations (nchoosek(v, k)).

Parameters

v	Input vector or scalar n.
k	Subset size value.
mr	Memory resource.

Returns

Binomial coefficient or combinations matrix.

nchoosek() [2/3]

Value nk::builtin::nchoosek	(	const Value &	v,
		int	k,
		std::pmr::memory_resource *	mr = nullptr )

Binomial coefficient or all combinations (nchoosek(v, k)).

Parameters

v	Input vector or scalar n.
k	Subset size.
mr	Memory resource.

Returns

Binomial coefficient or combinations matrix.

nchoosek() [3/3]

Value nk::builtin::nchoosek	(	double	n,
		double	k,
		std::pmr::memory_resource *	mr = nullptr )

Computes binomial coefficient n choose k (nchoosek(n, k)).

Parameters

n	Total number of items.
k	Items chosen.
mr	Memory resource.

Returns

Binomial coefficient value.

See also

perms, factorial

nchoosekCombinations()

Value nk::builtin::nchoosekCombinations	(	const Value &	v,
		double	kd,
		std::pmr::memory_resource *	mr = nullptr )

Generates all combinations of vector elements taken k at a time (nchoosek(v, k)).

Parameters

v	Input vector.
kd	Number of elements to pick.
mr	Memory resource.

Returns

Matrix with nchoosek(length(v), k) rows and k columns.

perms()

Value nk::builtin::perms	(	const Value &	v,
		std::pmr::memory_resource *	mr = nullptr )

All permutations of a vector elements.

Parameters

v	Input vector.
mr	Memory resource.

Returns

Matrix of all n! permutations.

See also

nchoosek, randperm

primes() [1/2]

Value nk::builtin::primes	(	const Value &	n,
		std::pmr::memory_resource *	mr = nullptr )

Generates row vector of all prime numbers less than or equal to n (primes(n)).

Parameters

n	Upper limit value.
mr	Memory resource.

Returns

Row vector of prime numbers.

See also

isprime, factor

primes() [2/2]

Value nk::builtin::primes	(	double	n,
		std::pmr::memory_resource *	mr = nullptr )

Generates row vector of all prime numbers less than or equal to n (primes(n)).

Parameters

n	Upper limit.
mr	Memory resource.

Returns

Row vector of prime numbers.

See also

isprime, factor

psi()

Value nk::builtin::psi	(	const Value &	x,
		std::pmr::memory_resource *	mr = nullptr )

Digamma / polygamma function (ψ(x) = d/dx ln Γ(x)).

Parameters

x	Input value or array.
mr	Memory resource.

Returns

Logarithmic derivative of gamma function.

See also

gamma, gammaln

symrcm()

Value nk::builtin::symrcm	(	const Value &	s,
		std::pmr::memory_resource *	mr = nullptr )

Symmetric reverse Cuthill-McKee ordering permutation vector (symrcm(s)).

Parameters

s	Symmetric sparse or dense adjacency matrix.
mr	Memory resource.

Returns

Permutation vector that reduces matrix bandwidth.

See also

colperm