Title:	'R' Bindings for the 'Boost' Math Functions
Version:	1.0.0
Description:	'R' bindings for the various functions and statistical distributions provided by the 'Boost' Math library https://www.boost.org/doc/libs/boost_1_88_0/libs/math/doc/html/index.html.
License:	MIT + file LICENSE
URL:	https://github.com/andrjohns/boostmath, https://www.boost.org/doc/libs/boost_1_88_0/libs/math/doc/html/index.html
BugReports:	https://github.com/andrjohns/boostmath/issues
Encoding:	UTF-8
RoxygenNote:	7.3.2
Depends:	R (≥ 3.0.2)
LinkingTo:	cpp11, BH (≥ 1.87.0)
NeedsCompilation:	yes
Suggests:	knitr, rmarkdown, tinytest
VignetteBuilder:	knitr
Packaged:	2025-07-24 12:52:37 UTC; andrew
Author:	Andrew R. Johnson [aut, cre]
Maintainer:	Andrew R. Johnson <andrew.johnson@arjohnsonau.com>
Repository:	CRAN
Date/Publication:	2025-07-25 16:20:07 UTC

boostmath: 'R' Bindings for the 'Boost' Math Functions

Description

'R' bindings for the various functions and statistical distributions provided by the 'Boost' Math library https://www.boost.org/doc/libs/boost_1_88_0/libs/math/doc/html/index.html.

Author(s)

Maintainer: Andrew R. Johnson andrew.johnson@arjohnsonau.com (ORCID)

See Also

Useful links:

• https://github.com/andrjohns/boostmath

• https://www.boost.org/doc/libs/boost_1_88_0/libs/math/doc/html/index.html

• Report bugs at https://github.com/andrjohns/boostmath/issues

Airy Functions

Description

Functions to compute the Airy functions Ai and Bi, their derivatives, and their zeros.

Usage

airy_ai(x)

airy_bi(x)

airy_ai_prime(x)

airy_bi_prime(x)

airy_ai_zero(m = NULL, start_index = NULL, number_of_zeros = NULL)

airy_bi_zero(m = NULL, start_index = NULL, number_of_zeros = NULL)

Arguments

Value

Single numeric value for the Airy functions and their derivatives, or a vector of length number_of_zeros for the multiple zero functions.

See Also

Boost Documentation for more details on the mathematical background.

Examples

airy_ai(2)
airy_bi(2)
airy_ai_prime(2)
airy_bi_prime(2)
airy_ai_zero(1)
airy_bi_zero(1)
airy_ai_zero(start_index = 1, number_of_zeros = 5)
airy_bi_zero(start_index = 1, number_of_zeros = 5)

Arcsine Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the arcsine distribution.

Usage

arcsine_pdf(x, x_min = 0, x_max = 1)

arcsine_lpdf(x, x_min = 0, x_max = 1)

arcsine_cdf(x, x_min = 0, x_max = 1)

arcsine_lcdf(x, x_min = 0, x_max = 1)

arcsine_quantile(p, x_min = 0, x_max = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

arcsine_pdf(0.5)
arcsine_lpdf(0.5)
arcsine_cdf(0.5)
arcsine_lcdf(0.5)
arcsine_quantile(0.5)

Basic Mathematical Functions

Description

Functions to compute sine, cosine, logarithm, exponential, cube root, square root, power, hypotenuse, and inverse square root.

Usage

sin_pi(x)

cos_pi(x)

log1p_boost(x)

expm1_boost(x)

cbrt(x)

sqrt1pm1(x)

powm1(x, y)

hypot(x, y)

rsqrt(x)

Arguments

Value

A single numeric value with the computed result of the function.

See Also

Boost Documentation) for more details on the mathematical background.

Examples

# sin(pi * 0.5)
sin_pi(0.5)
# cos(pi * 0.5)
cos_pi(0.5)
# log(1 + 0.5)
log1p_boost(0.5)
# exp(0.5) - 1
expm1_boost(0.5)
cbrt(8)
# sqrt(1 + 0.5) - 1
sqrt1pm1(0.5)
# 2^3 - 1
powm1(2, 3)
hypot(3, 4)
rsqrt(4)

Bernoulli Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Bernoulli distribution.

Usage

bernoulli_pdf(x, p_success)

bernoulli_lpdf(x, p_success)

bernoulli_cdf(x, p_success)

bernoulli_lcdf(x, p_success)

bernoulli_quantile(p, p_success)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

bernoulli_pdf(1, 0.5)
bernoulli_lpdf(1, 0.5)
bernoulli_cdf(1, 0.5)
bernoulli_lcdf(1, 0.5)
bernoulli_quantile(0.5, 0.5)

Bessel Functions, Their Derivatives, and Zeros

Description

Functions to compute Bessel functions of the first and second kind, their modified versions, spherical Bessel functions, and their derivatives and zeros.

Usage

cyl_bessel_j(v, x)

cyl_neumann(v, x)

cyl_bessel_j_zero(v, m = NULL, start_index = NULL, number_of_zeros = NULL)

cyl_neumann_zero(v, m = NULL, start_index = NULL, number_of_zeros = NULL)

cyl_bessel_i(v, x)

cyl_bessel_k(v, x)

sph_bessel(v, x)

sph_neumann(v, x)

cyl_bessel_j_prime(v, x)

cyl_neumann_prime(v, x)

cyl_bessel_i_prime(v, x)

cyl_bessel_k_prime(v, x)

sph_bessel_prime(v, x)

sph_neumann_prime(v, x)

Arguments

Value

Single numeric value for the Bessel functions and their derivatives, or a vector of length number_of_zeros for the multiple zero functions.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Bessel function of the first kind J_0(1)
cyl_bessel_j(0, 1)
# Bessel function of the second kind Y_0(1)
cyl_neumann(0, 1)
# Modified Bessel function of the first kind I_0(1)
cyl_bessel_i(0, 1)
# Modified Bessel function of the second kind K_0(1)
cyl_bessel_k(0, 1)
# Spherical Bessel function of the first kind j_0(1)
sph_bessel(0, 1)
# Spherical Bessel function of the second kind y_0(1)
sph_neumann(0, 1)
# Derivative of the Bessel function of the first kind J_0(1)
cyl_bessel_j_prime(0, 1)
# Derivative of the Bessel function of the second kind Y_0(1)
cyl_neumann_prime(0, 1)
# Derivative of the modified Bessel function of the first kind I_0(1)
cyl_bessel_i_prime(0, 1)
# Derivative of the modified Bessel function of the second kind K_0(1)
cyl_bessel_k_prime(0, 1)
# Derivative of the spherical Bessel function of the first kind j_0(1)
sph_bessel_prime(0, 1)
# Derivative of the spherical Bessel function of the second kind y_0(1)
sph_neumann_prime(0, 1)
# Finding the first zero of the Bessel function of the first kind J_0
cyl_bessel_j_zero(0, 1)
# Finding the first zero of the Bessel function of the second kind Y_0
cyl_neumann_zero(0, 1)
# Finding multiple zeros of the Bessel function of the first kind J_0 starting from index 1
cyl_bessel_j_zero(0, start_index = 1, number_of_zeros = 5)
# Finding multiple zeros of the Bessel function of the second kind Y_0 starting from index 1
cyl_neumann_zero(0, start_index = 1, number_of_zeros = 5)

Beta Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Beta distribution.

Usage

beta_pdf(x, alpha, beta)

beta_lpdf(x, alpha, beta)

beta_cdf(x, alpha, beta)

beta_lcdf(x, alpha, beta)

beta_quantile(p, alpha, beta)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Beta distribution with shape parameters alpha = 2, beta = 5
beta_pdf(0.5, 2, 5)
beta_lpdf(0.5, 2, 5)
beta_cdf(0.5, 2, 5)
beta_lcdf(0.5, 2, 5)
beta_quantile(0.5, 2, 5)

Beta Functions

Description

Functions to compute the Euler beta function, normalised incomplete beta function, and their complements, as well as their inverses and derivatives.

Usage

beta_boost(a, b, x = NULL)

ibeta(a, b, x)

ibetac(a, b, x)

betac(a, b, x)

ibeta_inv(a, b, p)

ibetac_inv(a, b, q)

ibeta_inva(b, x, p)

ibetac_inva(b, x, q)

ibeta_invb(a, x, p)

ibetac_invb(a, x, q)

ibeta_derivative(a, b, x)

Arguments

Value

A single numeric value with the computed beta function, normalised incomplete beta function, or their complements, depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

## Not run: 
# Euler beta function B(2, 3)
beta_boost(2, 3)
# Normalised incomplete beta function I_x(2, 3) for x = 0.5
ibeta(2, 3, 0.5)
# Normalised complement of the incomplete beta function 1 - I_x(2, 3) for x = 0.5
ibetac(2, 3, 0.5)
# Full incomplete beta function B_x(2, 3) for x = 0.5
beta_boost(2, 3, 0.5)
# Full complement of the incomplete beta function 1 - B_x(2, 3) for x = 0.5
betac(2, 3, 0.5)
# Inverse of the normalised incomplete beta function I_x(2, 3) = 0.5
ibeta_inv(2, 3, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(2, 3) = 0.5
ibetac_inv(2, 3, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(a, b)
# with respect to a for x = 0.5 and q = 0.5
ibetac_inva(3, 0.5, 0.5)
# Inverse of the normalised incomplete beta function I_x(a, b)
# with respect to b for x = 0.5 and p = 0.5
ibeta_invb(0.8, 0.5, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(a, b)
# with respect to b for x = 0.5 and q = 0.5
ibetac_invb(2, 0.5, 0.5)
# Derivative of the incomplete beta function with respect to x for a = 2, b = 3, x = 0.5
ibeta_derivative(2, 3, 0.5)

## End(Not run)

Binomial Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Binomial distribution.

Usage

binomial_pdf(k, n, prob)

binomial_lpdf(k, n, prob)

binomial_cdf(k, n, prob)

binomial_lcdf(k, n, prob)

binomial_quantile(p, n, prob)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Binomial dist ribution with n = 10, prob = 0.5
binomial_pdf(3, 10, 0.5)
binomial_lpdf(3, 10, 0.5)
binomial_cdf(3, 10, 0.5)
binomial_lcdf(3, 10, 0.5)
binomial_quantile(0.5, 10, 0.5)

Cauchy Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Cauchy distribution.

Usage

cauchy_pdf(x, location = 0, scale = 1)

cauchy_lpdf(x, location = 0, scale = 1)

cauchy_cdf(x, location = 0, scale = 1)

cauchy_lcdf(x, location = 0, scale = 1)

cauchy_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Cauchy distribution with location = 0, scale = 1
cauchy_pdf(0)
cauchy_lpdf(0)
cauchy_cdf(0)
cauchy_lcdf(0)
cauchy_quantile(0.5)

Chebyshev Polynomials and Related Functions

Description

Functions to compute Chebyshev polynomials of the first and second kind.

Usage

chebyshev_next(x, Tn, Tn_1)

chebyshev_t(n, x)

chebyshev_u(n, x)

chebyshev_t_prime(n, x)

chebyshev_clenshaw_recurrence(c, x)

chebyshev_clenshaw_recurrence_ab(c, a, b, x)

Arguments

Value

A single numeric value with the computed Chebyshev polynomial, its derivative, or related functions.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Chebyshev polynomial of the first kind T_2(0.5)
chebyshev_t(2, 0.5)
# Chebyshev polynomial of the second kind U_2(0.5)
chebyshev_u(2, 0.5)
# Derivative of the Chebyshev polynomial of the first kind T_2'(0.5)
chebyshev_t_prime(2, 0.5)
# Next Chebyshev polynomial of the first kind T_3(0.5) using T_2(0.5) and T_1(0.5)
chebyshev_next(0.5, chebyshev_t(2, 0.5), chebyshev_t(1, 0.5))
# Chebyshev polynomial of the first kind using Clenshaw's recurrence with coefficients
# c = c(1, 0, -1) at x = 0.5
chebyshev_clenshaw_recurrence(c(1, 0, -1), 0.5)
# Chebyshev polynomial of the first kind using Clenshaw's recurrence with interval [0, 1]
chebyshev_clenshaw_recurrence_ab(c(1, 0, -1), 0, 1, 0.5)

Chi-Squared Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Chi-Squared distribution.

Usage

chi_squared_pdf(x, df)

chi_squared_lpdf(x, df)

chi_squared_cdf(x, df)

chi_squared_lcdf(x, df)

chi_squared_quantile(p, df)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Chi-Squared distribution with 3 degrees of freedom
chi_squared_pdf(2, 3)
chi_squared_lpdf(2, 3)
chi_squared_cdf(2, 3)
chi_squared_lcdf(2, 3)
chi_squared_quantile(0.5, 3)

Double Exponential Quadrature

Description

Functions for numerical integration using double exponential quadrature methods such as tanh-sinh, sinh-sinh, and exp-sinh quadrature.

Usage

tanh_sinh(f, a, b, tol = sqrt(.Machine$double.eps), max_refinements = 15)

sinh_sinh(f, tol = sqrt(.Machine$double.eps), max_refinements = 9)

exp_sinh(f, a, b, tol = sqrt(.Machine$double.eps), max_refinements = 9)

Arguments

Value

A single numeric value with the computed integral.

Examples

# Tanh-sinh quadrature of log(x) from 0 to 1
tanh_sinh(function(x) { log(x) * log1p(-x) }, a = 0, b = 1)
# Sinh-sinh quadrature of exp(-x^2)
sinh_sinh(function(x) { exp(-x * x) })
# Exp-sinh quadrature of exp(-3*x) from 0 to Inf
exp_sinh(function(x) { exp(-3 * x) }, a = 0, b = Inf)

Elliptic Integrals

Description

Functions to compute various elliptic integrals, including Carlson's elliptic integrals and incomplete elliptic integrals.

Usage

ellint_rf(x, y, z)

ellint_rd(x, y, z)

ellint_rj(x, y, z, p)

ellint_rc(x, y)

ellint_rg(x, y, z)

ellint_1(k, phi = NULL)

ellint_2(k, phi = NULL)

ellint_3(k, n, phi = NULL)

ellint_d(k, phi = NULL)

jacobi_zeta(k, phi)

heuman_lambda(k, phi)

Arguments

Value

A single numeric value with the computed elliptic integral.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Carlson's elliptic integral Rf with parameters x = 1, y = 2, z = 3
ellint_rf(1, 2, 3)
#' # Carlson's elliptic integral Rd with parameters x = 1, y = 2, z = 3
ellint_rd(1, 2, 3)
# Carlson's elliptic integral Rj with parameters x = 1, y = 2, z = 3, p = 4
ellint_rj(1, 2, 3, 4)
# Carlson's elliptic integral Rc with parameters x = 1, y = 2
ellint_rc(1, 2)
# Carlson's elliptic integral Rg with parameters x = 1, y = 2, z = 3
ellint_rg(1, 2, 3)
# Incomplete elliptic integral of the first kind with k = 0.5, phi = pi/4
ellint_1(0.5, pi / 4)
# Complete elliptic integral of the first kind
ellint_1(0.5)
# Incomplete elliptic integral of the second kind with k = 0.5, phi = pi/4
ellint_2(0.5, pi / 4)
# Complete elliptic integral of the second kind
ellint_2(0.5)
# Incomplete elliptic integral of the third kind with k = 0.5, n = 0.5, phi = pi/4
ellint_3(0.5, 0.5, pi / 4)
# Complete elliptic integral of the third kind with k = 0.5, n = 0.5
ellint_3(0.5, 0.5)
# Incomplete elliptic integral D with k = 0.5, phi = pi/4
ellint_d(0.5, pi / 4)
# Complete elliptic integral D
ellint_d(0.5)
# Jacobi zeta function with k = 0.5, phi = pi/4
jacobi_zeta(0.5, pi / 4)
# Heuman's lambda function with k = 0.5, phi = pi/4
heuman_lambda(0.5, pi / 4)

Error Functions and Inverses

Description

Functions to compute the error function, complementary error function, and their inverses.

Usage

erf(x)

erfc(x)

erf_inv(p)

erfc_inv(p)

Arguments

Value

A single numeric value with the computed error function, complementary error function, or their inverses.

See Also

Boost Documentation for more details

Examples

# Error function
erf(0.5)
# Complementary error function
erfc(0.5)
# Inverse error function
erf_inv(0.5)
# Inverse complementary error function
erfc_inv(0.5)

Exponential Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Exponential distribution.

Usage

exponential_pdf(x, lambda)

exponential_lpdf(x, lambda)

exponential_cdf(x, lambda)

exponential_lcdf(x, lambda)

exponential_quantile(p, lambda)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Exponential distribution with rate parameter lambda = 2
exponential_pdf(1, 2)
exponential_lpdf(1, 2)
exponential_cdf(1, 2)
exponential_lcdf(1, 2)
exponential_quantile(0.5, 2)

Exponential Integrals

Description

Functions to compute various exponential integrals, including En and Ei.

Usage

expint_en(n, z)

expint_ei(z)

Arguments

Value

A single numeric value with the computed exponential integral.

See Also

Boost Documentation for

Examples

# Exponential integral En with n = 1 and z = 2
expint_en(1, 2)
# Exponential integral Ei with z = 2
expint_ei(2)

Extreme Value Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Extreme Value distribution.

Usage

extreme_value_pdf(x, location = 0, scale = 1)

extreme_value_lpdf(x, location = 0, scale = 1)

extreme_value_cdf(x, location = 0, scale = 1)

extreme_value_lcdf(x, location = 0, scale = 1)

extreme_value_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Extreme Value distribution with location = 0, scale = 1
extreme_value_pdf(0)
extreme_value_lpdf(0)
extreme_value_cdf(0)
extreme_value_lcdf(0)
extreme_value_quantile(0.5)

Factorials and Binomial Coefficients

Description

Functions to compute factorials, double factorials, rising and falling factorials, and binomial coefficients.

Usage

factorial_boost(i)

unchecked_factorial(i)

max_factorial()

double_factorial(i)

rising_factorial(x, i)

falling_factorial(x, i)

binomial_coefficient(n, k)

Arguments

Value

A single numeric value with the computed factorial, double factorial, rising factorial, falling factorial, or binomial coefficient.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Factorial of 5
factorial_boost(5)
# Unchecked factorial of 5 (using table lookup)
unchecked_factorial(5)
# Maximum factorial value that can be computed
max_factorial()
# Double factorial of 6
double_factorial(6)
# Rising factorial of 3 with exponent 2
rising_factorial(3, 2)
# Falling factorial of 3 with exponent 2
falling_factorial(3, 2)
# Binomial coefficient "5 choose 2"
binomial_coefficient(5, 2)

Fisher F Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Fisher F distribution.

Usage

fisher_f_pdf(x, df1, df2)

fisher_f_lpdf(x, df1, df2)

fisher_f_cdf(x, df1, df2)

fisher_f_lcdf(x, df1, df2)

fisher_f_quantile(p, df1, df2)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Fisher F distribution with df1 = 5, df2 = 2
fisher_f_pdf(1, 5, 2)
fisher_f_lpdf(1, 5, 2)
fisher_f_cdf(1, 5, 2)
fisher_f_lcdf(1, 5, 2)
fisher_f_quantile(0.5, 5, 2)

Gamma Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Gamma distribution.

Usage

gamma_pdf(x, shape, scale)

gamma_lpdf(x, shape, scale)

gamma_cdf(x, shape, scale)

gamma_lcdf(x, shape, scale)

gamma_quantile(p, shape, scale)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Gamma distribution with shape = 3, scale = 4
gamma_pdf(2, 3, 4)
gamma_lpdf(2, 3, 4)
gamma_cdf(2, 3, 4)
gamma_lcdf(2, 3, 4)
gamma_quantile(0.5, 3, 4)

Gamma Functions

Description

Functions to compute the gamma function, its logarithm, digamma, trigamma, polygamma, and various incomplete gamma functions.

Usage

tgamma(z)

tgamma1pm1(z)

lgamma_boost(z)

digamma_boost(z)

trigamma_boost(z)

polygamma(n, z)

tgamma_ratio(a, b)

tgamma_delta_ratio(a, delta)

gamma_p(a, z)

gamma_q(a, z)

tgamma_lower(a, z)

tgamma_upper(a, z)

gamma_q_inv(a, q)

gamma_p_inv(a, p)

gamma_q_inva(z, q)

gamma_p_inva(z, p)

gamma_p_derivative(a, z)

Arguments

Value

A single numeric value with the computed gamma function, logarithm, digamma, trigamma, polygamma, or incomplete gamma functions.

See Also

Boost Documentation for more details on the mathematical background.

Examples

## Not run: 
# Gamma function for z = 5
tgamma(5)
# Gamma function for (1 + z) - 1, where z = 5
tgamma1pm1(5)
# Logarithm of the gamma function for z = 5
lgamma_boost(5)
# Digamma function for z = 5
digamma_boost(5)
# Trigamma function for z = 5
trigamma_boost(5)
# Polygamma function of order 1 for z = 5
polygamma(1, 5)
# Ratio of gamma functions for a = 5, b = 3
tgamma_ratio(5, 3)
# Ratio of gamma functions with delta for a = 5, delta = 2
tgamma_delta_ratio(5, 2)
# Normalised lower incomplete gamma function P(a, z) for a = 5, z = 2
gamma_p(5, 2)
# Normalised upper incomplete gamma function Q(a, z) for a = 5, z = 2
gamma_q(5, 2)
# Full lower incomplete gamma function for a = 5, z = 2
tgamma_lower(5, 2)
# Full upper incomplete gamma function for a = 5, z = 2
tgamma_upper(5, 2)
# Inverse of the normalised upper incomplete gamma function for a = 5, q = 0.5
gamma_q_inv(5, 0.5)
# Inverse of the normalised lower incomplete gamma function for a = 5, p = 0.5
gamma_p_inv(5, 0.5)
# Inverse of the normalised upper incomplete gamma function with respect to a for z = 2, q = 0.5
gamma_q_inva(2, 0.5)
# Inverse of the normalised lower incomplete gamma function with respect to a for z = 2, p = 0.5
gamma_p_inva(2, 0.5)
# Derivative of the normalised upper incomplete gamma function for a = 5, z = 2
gamma_p_derivative(5, 2)

## End(Not run)

Gegenbauer Polynomials and Related Functions

Description

Functions to compute Gegenbauer polynomials, their derivatives, and related functions.

Usage

gegenbauer(n, lambda, x)

gegenbauer_prime(n, lambda, x)

gegenbauer_derivative(n, lambda, x, k)

Arguments

Value

A single numeric value with the computed Gegenbauer polynomial, its derivative, or k-th derivative.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Gegenbauer polynomial C_2^(1)(0.5)
gegenbauer(2, 1, 0.5)
# Derivative of the Gegenbauer polynomial C_2^(1)'(0.5)
gegenbauer_prime(2, 1, 0.5)
# k-th derivative of the Gegenbauer polynomial C_2^(1)''(0.5)
gegenbauer_derivative(2, 1, 0.5, 2)

Geometric Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Geometric distribution.

Usage

geometric_pdf(x, prob)

geometric_lpdf(x, prob)

geometric_cdf(x, prob)

geometric_lcdf(x, prob)

geometric_quantile(p, prob)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Geometric distribution with probability of success prob = 0.5
geometric_pdf(3, 0.5)
geometric_lpdf(3, 0.5)
geometric_cdf(3, 0.5)
geometric_lcdf(3, 0.5)
geometric_quantile(0.5, 0.5)

Hankel Functions

Description

Functions to compute cyclic and spherical Hankel functions of the first and second kinds.

Usage

cyl_hankel_1(v, x)

cyl_hankel_2(v, x)

sph_hankel_1(v, x)

sph_hankel_2(v, x)

Arguments

Value

A single complex value with the computed Hankel function.

See Also

Boost Documentation for more details on the mathematical background.

Examples

cyl_hankel_1(2, 0.5)
cyl_hankel_2(2, 0.5)
sph_hankel_1(2, 0.5)
sph_hankel_2(2, 0.5)

Hermite Polynomials and Related Functions

Description

Functions to compute Hermite polynomials.

Usage

hermite(n, x)

hermite_next(n, x, Hn, Hnm1)

Arguments

Value

A single numeric value with the computed Hermite polynomial or its next value.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Hermite polynomial H_2(0.5)
hermite(2, 0.5)
# Next Hermite polynomial H_3(0.5) using H_2(0.5) and H_1(0.5)
hermite_next(2, 0.5, hermite(2, 0.5), hermite(1, 0.5))

Holtsmark Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Holtsmark distribution.

Usage

holtsmark_pdf(x, location = 0, scale = 1)

holtsmark_lpdf(x, location = 0, scale = 1)

holtsmark_cdf(x, location = 0, scale = 1)

holtsmark_lcdf(x, location = 0, scale = 1)

holtsmark_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Distribution only available with Boost version 1.87.0 or later.
## Not run: 
# Holtsmark distribution with location 0 and scale 1
  holtsmark_pdf(3)
  holtsmark_lpdf(3)
  holtsmark_cdf(3)
  holtsmark_lcdf(3)
  holtsmark_quantile(0.5)

## End(Not run)

Hyperexponential Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Hyperexponential distribution.

Usage

hyperexponential_pdf(x, probabilities, rates)

hyperexponential_lpdf(x, probabilities, rates)

hyperexponential_cdf(x, probabilities, rates)

hyperexponential_lcdf(x, probabilities, rates)

hyperexponential_quantile(p, probabilities, rates)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Hyperexponential distribution with probabilities = c(0.5, 0.5) and rates = c(1, 2)
hyperexponential_pdf(2, c(0.5, 0.5), c(1, 2))
hyperexponential_lpdf(2, c(0.5, 0.5), c(1, 2))
hyperexponential_cdf(2, c(0.5, 0.5), c(1, 2))
hyperexponential_lcdf(2, c(0.5, 0.5), c(1, 2))
hyperexponential_quantile(0.5, c(0.5, 0.5), c(1, 2))

Hypergeometric Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Hypergeometric distribution.

Usage

hypergeometric_pdf(x, r, n, N)

hypergeometric_lpdf(x, r, n, N)

hypergeometric_cdf(x, r, n, N)

hypergeometric_lcdf(x, r, n, N)

hypergeometric_quantile(p, r, n, N)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Hypergeometric distribution with r = 5, n = 10, N = 20
hypergeometric_pdf(3, 5, 10, 20)
hypergeometric_lpdf(3, 5, 10, 20)
hypergeometric_cdf(3, 5, 10, 20)
hypergeometric_lcdf(3, 5, 10, 20)
hypergeometric_quantile(0.5, 5, 10, 20)

Hypergeometric Functions

Description

Functions to compute various hypergeometric functions.

Usage

hypergeometric_1F0(a, z)

hypergeometric_0F1(b, z)

hypergeometric_2F0(a1, a2, z)

hypergeometric_1F1(a, b, z)

hypergeometric_1F1_regularized(a, b, z)

log_hypergeometric_1F1(a, b, z)

hypergeometric_pFq(a, b, z)

Arguments

Value

A single numeric value with the computed hypergeometric function.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Hypergeometric Function 1F0
hypergeometric_1F0(2, 0.2)
# Hypergeometric Function 0F1
hypergeometric_0F1(1, 0.8)
# Hypergeometric Function 2F0
hypergeometric_2F0(0.1, -1, 0.1)
# Hypergeometric Function 1F1
hypergeometric_1F1(2, 3, 1)
# Regularised Hypergeometric Function 1F1
hypergeometric_1F1_regularized(2, 3, 1)
# Logarithm of the Hypergeometric Function 1F1
log_hypergeometric_1F1(2, 3, 1)
# Hypergeometric Function pFq
hypergeometric_pFq(c(2, 3), c(4, 5), 6)

Inverse Chi-Squared Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Inverse Chi-Squared distribution.

Usage

inverse_chi_squared_pdf(x, df, scale = 1)

inverse_chi_squared_lpdf(x, df, scale = 1)

inverse_chi_squared_cdf(x, df, scale = 1)

inverse_chi_squared_lcdf(x, df, scale = 1)

inverse_chi_squared_quantile(p, df, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Inverse Chi-Squared distribution with 3 degrees of freedom, scale = 1
inverse_chi_squared_pdf(2, 3, 1)
inverse_chi_squared_lpdf(2, 3, 1)
inverse_chi_squared_cdf(2, 3, 1)
inverse_chi_squared_lcdf(2, 3, 1)
inverse_chi_squared_quantile(0.5, 3, 1)

Inverse Gamma Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Inverse Gamma distribution.

Usage

inverse_gamma_pdf(x, shape, scale)

inverse_gamma_lpdf(x, shape, scale)

inverse_gamma_cdf(x, shape, scale)

inverse_gamma_lcdf(x, shape, scale)

inverse_gamma_quantile(p, shape, scale)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Inverse Gamma distribution with shape = 3, scale = 4
inverse_gamma_pdf(2, 3, 4)
inverse_gamma_lpdf(2, 3, 4)
inverse_gamma_cdf(2, 3, 4)
inverse_gamma_lcdf(2, 3, 4)
inverse_gamma_quantile(0.5, 3, 4)

Inverse Gaussian Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Inverse Gaussian distribution.

Usage

inverse_gaussian_pdf(x, mu, lambda)

inverse_gaussian_lpdf(x, mu, lambda)

inverse_gaussian_cdf(x, mu, lambda)

inverse_gaussian_lcdf(x, mu, lambda)

inverse_gaussian_quantile(p, mu, lambda)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Inverse Gaussian distribution with mu = 3, lambda = 4
inverse_gaussian_pdf(2, 3, 4)
inverse_gaussian_lpdf(2, 3, 4)
inverse_gaussian_cdf(2, 3, 4)
inverse_gaussian_lcdf(2, 3, 4)
inverse_gaussian_quantile(0.5, 3, 4)

Inverse Hyperbolic Functions

Description

Functions to compute the inverse hyperbolic functions: acosh, asinh, and atanh.

Usage

acosh_boost(x)

asinh_boost(x)

atanh_boost(x)

Arguments

Value

A single numeric value with the computed inverse hyperbolic function.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Inverse Hyperbolic Cosine
acosh_boost(2)
# Inverse Hyperbolic Sine
asinh_boost(1)
# Inverse Hyperbolic Tangent
atanh_boost(0.5)

Jacobi Elliptic Functions

Description

Functions to compute the Jacobi elliptic functions: sn, cn, dn, and others.

Usage

jacobi_elliptic(k, u)

jacobi_cd(k, u)

jacobi_cn(k, u)

jacobi_cs(k, u)

jacobi_dc(k, u)

jacobi_dn(k, u)

jacobi_ds(k, u)

jacobi_nc(k, u)

jacobi_nd(k, u)

jacobi_ns(k, u)

jacobi_sc(k, u)

jacobi_sd(k, u)

jacobi_sn(k, u)

Arguments

Value

For jacobi_elliptic, a list containing the values of the Jacobi elliptic functions: sn, cn, dn. For individual functions, a single numeric value is returned.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Jacobi Elliptic Functions
k <- 0.5
u <- 2
jacobi_elliptic(k, u)
# Individual Jacobi Elliptic Functions
jacobi_cd(k, u)
jacobi_cn(k, u)
jacobi_cs(k, u)
jacobi_dc(k, u)
jacobi_dn(k, u)
jacobi_ds(k, u)
jacobi_nc(k, u)
jacobi_nd(k, u)
jacobi_ns(k, u)
jacobi_sc(k, u)
jacobi_sd(k, u)
jacobi_sn(k, u)

Jacobi Polynomials and Related Functions

Description

Functions to compute Jacobi polynomials, their derivatives, and related functions.

Usage

jacobi(n, alpha, beta, x)

jacobi_prime(n, alpha, beta, x)

jacobi_double_prime(n, alpha, beta, x)

jacobi_derivative(n, alpha, beta, x, k)

Arguments

Value

A single numeric value with the computed Jacobi polynomial, its derivative, or k-th derivative.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Jacobi polynomial P_2^(1, 2)(0.5)
jacobi(2, 1, 2, 0.5)
# Derivative of the Jacobi polynomial P_2^(1, 2)'(0.5)
jacobi_prime(2, 1, 2, 0.5)
# Second derivative of the Jacobi polynomial P_2^(1, 2)''(0.5)
jacobi_double_prime(2, 1, 2, 0.5)
# 3rd derivative of the Jacobi polynomial P_2^(1, 2)^(k)(0.5)
jacobi_derivative(2, 1, 2, 0.5, 3)

Jacobi Theta Functions

Description

Functions to compute the Jacobi theta functions (\theta_1, \theta_2, \theta_3, \theta_4) parameterised by either (q) or (\tau).

Usage

jacobi_theta1(x, q)

jacobi_theta1tau(x, tau)

jacobi_theta2(x, q)

jacobi_theta2tau(x, tau)

jacobi_theta3(x, q)

jacobi_theta3tau(x, tau)

jacobi_theta3m1(x, q)

jacobi_theta3m1tau(x, tau)

jacobi_theta4(x, q)

jacobi_theta4tau(x, tau)

jacobi_theta4m1(x, q)

jacobi_theta4m1tau(x, tau)

Arguments

Value

A single numeric value with the computed Jacobi theta function.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Jacobi Theta Functions
x <- 0.5
q <- 0.9
tau <- 1.5
jacobi_theta1(x, q)
jacobi_theta1tau(x, tau)
jacobi_theta2(x, q)
jacobi_theta2tau(x, tau)
jacobi_theta3(x, q)
jacobi_theta3tau(x, tau)
jacobi_theta3m1(x, q)
jacobi_theta3m1tau(x, tau)
jacobi_theta4(x, q)
jacobi_theta4tau(x, tau)
jacobi_theta4m1(x, q)
jacobi_theta4m1tau(x, tau)

Kolmogorov-Smirnov Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Kolmogorov-Smirnov distribution.

Usage

kolmogorov_smirnov_pdf(x, n)

kolmogorov_smirnov_lpdf(x, n)

kolmogorov_smirnov_cdf(x, n)

kolmogorov_smirnov_lcdf(x, n)

kolmogorov_smirnov_quantile(p, n)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Kolmogorov-Smirnov distribution with sample size n = 10
kolmogorov_smirnov_pdf(0.5, 10)
kolmogorov_smirnov_lpdf(0.5, 10)
kolmogorov_smirnov_cdf(0.5, 10)
kolmogorov_smirnov_lcdf(0.5, 10)
kolmogorov_smirnov_quantile(0.5, 10)

Laguerre Polynomials and Related Functions

Description

Functions to compute Laguerre polynomials of the first kind.

Usage

laguerre(n, x)

laguerre_m(n, m, x)

laguerre_next(n, x, Ln, Lnm1)

laguerre_next_m(n, m, x, Ln, Lnm1)

Arguments

Value

A single numeric value with the computed Laguerre polynomial, its derivative, or related functions.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Laguerre polynomial of the first kind L_2(0.5)
laguerre(2, 0.5)
# Laguerre polynomial of the first kind with order 1 L_2^1(0.5)
laguerre_m(2, 1, 0.5)
# Next Laguerre polynomial of the first kind L_3(0.5) using L_2(0.5) and L_1(0.5)
laguerre_next(2, 0.5, laguerre(2, 0.5), laguerre(1, 0.5))
# Next Laguerre polynomial of the first kind with order 1 L_3^1(0.5) using L_2^1(0.5) and L_1^1(0.5)
laguerre_next_m(2, 1, 0.5, laguerre_m(2, 1, 0.5), laguerre_m(1, 1, 0.5))

Lambert W Function and Its Derivatives

Description

Functions to compute the Lambert W function and its derivatives for the principal branch (W_0) and the branch -1 (W_{-1}).

Usage

lambert_w0(z)

lambert_wm1(z)

lambert_w0_prime(z)

lambert_wm1_prime(z)

Arguments

Value

A single numeric value with the computed Lambert W function or its derivative.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Lambert W Function (Principal Branch)
lambert_w0(0.3)
# Lambert W Function (Branch -1)
lambert_wm1(-0.3)
# Derivative of the Lambert W Function (Principal Branch)
lambert_w0_prime(0.3)
# Derivative of the Lambert W Function (Branch -1)
lambert_wm1_prime(-0.3)

Landau Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Landau distribution.

Usage

landau_pdf(x, location = 0, scale = 1)

landau_lpdf(x, location = 0, scale = 1)

landau_cdf(x, location = 0, scale = 1)

landau_lcdf(x, location = 0, scale = 1)

landau_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Distribution only available with Boost version 1.87.0 or later.
## Not run: 
# Landau distribution with location 0 and scale 1
  landau_pdf(3)
  landau_lpdf(3)
  landau_cdf(3)
  landau_lcdf(3)
  landau_quantile(0.5)

## End(Not run)

Laplace Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Laplace distribution.

Usage

laplace_pdf(x, location = 0, scale = 1)

laplace_lpdf(x, location = 0, scale = 1)

laplace_cdf(x, location = 0, scale = 1)

laplace_lcdf(x, location = 0, scale = 1)

laplace_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Laplace distribution with location = 0, scale = 1
laplace_pdf(0)
laplace_lpdf(0)
laplace_cdf(0)
laplace_lcdf(0)
laplace_quantile(0.5)

Legendre Polynomials and Related Functions

Description

Functions to compute Legendre polynomials of the first and second kind, their derivatives, zeros, and related functions.

Usage

legendre_p(n, x)

legendre_p_prime(n, x)

legendre_p_zeros(n)

legendre_p_m(n, m, x)

legendre_q(n, x)

legendre_next(n, x, Pl, Plm1)

legendre_next_m(n, m, x, Pl, Plm1)

Arguments

Value

A single numeric value with the computed Legendre polynomial, its derivative, zeros, or related functions.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Legendre polynomial of the first kind P_2(0.5)
legendre_p(2, 0.5)
# Derivative of the Legendre polynomial of the first kind P_2'(0.5)
legendre_p_prime(2, 0.5)
# Zeros of the Legendre polynomial of the first kind P_2
legendre_p_zeros(2)
# Legendre polynomial of the first kind with order 1 P_2^1(0.5)
legendre_p_m(2, 1, 0.5)
# Legendre polynomial of the second kind Q_2(0.5)
legendre_q(2, 0.5)
# Next Legendre polynomial of the first kind P_3(0.5) using P_2(0.5) and P_1(0.5)
legendre_next(2, 0.5, legendre_p(2, 0.5), legendre_p(1, 0.5))
# Next Legendre polynomial of the first kind with order 1 P_3^1(0.5) using P_2^1(0.5) and P_1^1(0.5)
legendre_next_m(2, 1, 0.5, legendre_p_m(2, 1, 0.5), legendre_p_m(1, 1, 0.5))

Logistic Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Logistic distribution.

Usage

logistic_pdf(x, location = 0, scale = 1)

logistic_lpdf(x, location = 0, scale = 1)

logistic_cdf(x, location = 0, scale = 1)

logistic_lcdf(x, location = 0, scale = 1)

logistic_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Logistic distribution with location = 0, scale = 1
logistic_pdf(0)
logistic_lpdf(0)
logistic_cdf(0)
logistic_lcdf(0)
logistic_quantile(0.5)

Log Normal Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Log Normal distribution.

Usage

lognormal_pdf(x, location = 0, scale = 1)

lognormal_lpdf(x, location = 0, scale = 1)

lognormal_cdf(x, location = 0, scale = 1)

lognormal_lcdf(x, location = 0, scale = 1)

lognormal_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Log Normal distribution with location = 0, scale = 1
lognormal_pdf(0)
lognormal_lpdf(0)
lognormal_cdf(0)
lognormal_lcdf(0)
lognormal_quantile(0.5)

Map-Airy Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Map-Airy distribution.

Usage

mapairy_pdf(x, location = 0, scale = 1)

mapairy_lpdf(x, location = 0, scale = 1)

mapairy_cdf(x, location = 0, scale = 1)

mapairy_lcdf(x, location = 0, scale = 1)

mapairy_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Distribution only available with Boost version 1.87.0 or later.
## Not run: 
# Map-Airy distribution with location 0 and scale 1
  mapairy_pdf(3)
  mapairy_lpdf(3)
  mapairy_cdf(3)
  mapairy_lcdf(3)
  mapairy_quantile(0.5)

## End(Not run)

Negative Binomial Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Negative Binomial distribution.

Usage

negative_binomial_pdf(x, successes, success_fraction)

negative_binomial_lpdf(x, successes, success_fraction)

negative_binomial_cdf(x, successes, success_fraction)

negative_binomial_lcdf(x, successes, success_fraction)

negative_binomial_quantile(p, successes, success_fraction)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

negative_binomial_pdf(3, 5, 0.5)
negative_binomial_lpdf(3, 5, 0.5)
negative_binomial_cdf(3, 5, 0.5)
negative_binomial_lcdf(3, 5, 0.5)
negative_binomial_quantile(0.5, 5, 0.5)

Noncentral Beta Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Noncentral Beta distribution.

Usage

non_central_beta_pdf(x, alpha, beta, lambda)

non_central_beta_lpdf(x, alpha, beta, lambda)

non_central_beta_cdf(x, alpha, beta, lambda)

non_central_beta_lcdf(x, alpha, beta, lambda)

non_central_beta_quantile(p, alpha, beta, lambda)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Noncentral Beta distribution with shape parameters alpha = 2, beta = 3
# and noncentrality parameter lambda = 1
non_central_beta_pdf(0.5, 2, 3, 1)
non_central_beta_lpdf(0.5, 2, 3, 1)
non_central_beta_cdf(0.5, 2, 3, 1)
non_central_beta_lcdf(0.5, 2, 3, 1)
non_central_beta_quantile(0.5, 2, 3, 1)

Noncentral Chi-Squared Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Noncentral Chi-Squared distribution.

Usage

non_central_chi_squared_pdf(x, df, lambda)

non_central_chi_squared_lpdf(x, df, lambda)

non_central_chi_squared_cdf(x, df, lambda)

non_central_chi_squared_lcdf(x, df, lambda)

non_central_chi_squared_quantile(p, df, lambda)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

## Not run: 
# Noncentral Chi-Squared distribution with 3 degrees of freedom and noncentrality
# parameter 1
non_central_chi_squared_pdf(2, 3, 1)
non_central_chi_squared_lpdf(2, 3, 1)
non_central_chi_squared_cdf(2, 3, 1)
non_central_chi_squared_lcdf(2, 3, 1)
non_central_chi_squared_quantile(0.5, 3, 1)

## End(Not run)

Noncentral T Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Noncentral T distribution.

Usage

non_central_t_pdf(x, df, delta)

non_central_t_lpdf(x, df, delta)

non_central_t_cdf(x, df, delta)

non_central_t_lcdf(x, df, delta)

non_central_t_quantile(p, df, delta)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Noncentral T distribution with 3 degrees of freedom and noncentrality parameter 1
non_central_t_pdf(0, 3, 1)
non_central_t_lpdf(0, 3, 1)
non_central_t_cdf(0, 3, 1)
non_central_t_lcdf(0, 3, 1)
non_central_t_quantile(0.5, 3, 1)

Normal Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Normal distribution.

Usage

normal_pdf(x, mean = 0, sd = 1)

normal_lpdf(x, mean = 0, sd = 1)

normal_cdf(x, mean = 0, sd = 1)

normal_lcdf(x, mean = 0, sd = 1)

normal_quantile(p, mean = 0, sd = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Normal distribution with mean = 0, sd = 1
normal_pdf(0)
normal_lpdf(0)
normal_cdf(0)
normal_lcdf(0)
normal_quantile(0.5)

Number Series

Description

Functions to compute Bernoulli numbers, tangent numbers, fibonacci numbers, and prime numbers.

Usage

bernoulli_b2n(n = NULL, start_index = NULL, number_of_bernoullis_b2n = NULL)

max_bernoulli_b2n()

unchecked_bernoulli_b2n(n)

tangent_t2n(n = NULL, start_index = NULL, number_of_tangent_t2n = NULL)

prime(n)

max_prime()

fibonacci(n)

unchecked_fibonacci(n)

Arguments

Details

Efficient computation of Bernoulli numbers, tangent numbers, fibonacci numbers, and prime numbers.

The checked_ functions ensure that the input is within valid bounds, while the unchecked_ functions do not perform such checks, allowing for potentially faster computation at the risk of overflow or invalid input.

The range_ functions allow for computing a sequence of numbers starting from a specified index.

The max_ functions return the maximum index for which the respective numbers can be computed using precomputed lookup tables.

Value

A single numeric value for the Bernoulli numbers, tangent numbers, fibonacci numbers, or prime numbers, or a vector of values for ranges.

See Also

Boost Documentation for more details on the mathematical background.

Examples

bernoulli_b2n(10)
max_bernoulli_b2n()
unchecked_bernoulli_b2n(10)
bernoulli_b2n(start_index = 0, number_of_bernoullis_b2n = 10)
tangent_t2n(10)
tangent_t2n(start_index = 0, number_of_tangent_t2n = 10)
prime(10)
max_prime()
fibonacci(10)
unchecked_fibonacci(10)

Numerical Differentiation

Description

Functions for numerical differentiation using finite difference methods and complex step methods.

Usage

finite_difference_derivative(f, x, order = 1)

complex_step_derivative(f, x)

Arguments

Value

The approximate value of the derivative at the point x.

Examples

# Finite difference derivative of sin(x) at pi/4
finite_difference_derivative(sin, pi / 4)
# Complex step derivative of exp(x) at 1.7
complex_step_derivative(exp, 1.7)

Numerical Integration

Description

Functions for numerical integration using various methods such as trapezoidal rule, Gauss-Legendre quadrature, and Gauss-Kronrod quadrature.

Usage

trapezoidal(f, a, b, tol = sqrt(.Machine$double.eps), max_refinements = 12)

gauss_legendre(f, a, b, points = 7)

gauss_kronrod(
  f,
  a,
  b,
  points = 15,
  max_depth = 15,
  tol = sqrt(.Machine$double.eps)
)

Arguments

Value

A single numeric value with the computed integral.

Examples

# Trapezoidal rule integration of sin(x) from 0 to pi
trapezoidal(sin, 0, pi)
# Gauss-Legendre integration of exp(x) from 0 to 1
gauss_legendre(exp, 0, 1, points = 7)
# Adaptive Gauss-Kronrod integration of log(x) from 1 to 2
gauss_kronrod(log, 1, 2, points = 15, max_depth = 10)

Ooura Fourier Integrals

Description

Computing Fourier sine and cosine integrals using Ooura's method.

Usage

ooura_fourier_sin(
  f,
  omega = 1,
  relative_error_tolerance = sqrt(.Machine$double.eps),
  levels = 8
)

ooura_fourier_cos(
  f,
  omega = 1,
  relative_error_tolerance = sqrt(.Machine$double.eps),
  levels = 8
)

Arguments

Value

A single numeric value with the computed Fourier sine or cosine integral, with attribute 'relative_error' indicating the relative error of the approximation.

Examples

# Fourier sine integral of sin(x) with omega = 1
ooura_fourier_sin(function(x) { 1 / x }, omega = 1)
# Fourier cosine integral of cos(x) with omega = 1
ooura_fourier_cos(function(x) { 1/ (x * x + 1) }, omega = 1)

Owens T Function

Description

Computes the Owens T function of h and a, giving the probability of the event (X > h and 0 < Y < a * X) where X and Y are independent standard normal random variables.

Usage

owens_t(h, a)

Arguments

Value

The value of the Owens T function at (h, a).

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Owens T Function
owens_t(1, 0.5)

Pareto Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Pareto distribution.

Usage

pareto_pdf(x, shape = 1, scale = 1)

pareto_lpdf(x, shape = 1, scale = 1)

pareto_cdf(x, shape = 1, scale = 1)

pareto_lcdf(x, shape = 1, scale = 1)

pareto_quantile(p, shape = 1, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Pareto distribution with shape = 1, scale = 1
pareto_pdf(1)
pareto_lpdf(1)
pareto_cdf(1)
pareto_lcdf(1)
pareto_quantile(0.5)

Poisson Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Poisson distribution.

Usage

poisson_pdf(x, lambda = 1)

poisson_lpdf(x, lambda = 1)

poisson_cdf(x, lambda = 1)

poisson_lcdf(x, lambda = 1)

poisson_quantile(p, lambda = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Poisson distribution with lambda = 1
poisson_pdf(0, 1)
poisson_lpdf(0, 1)
poisson_cdf(0, 1)
poisson_lcdf(0, 1)
poisson_quantile(0.5, 1)

Polynomial Root-Finding

Description

Functions for finding roots of polynomials of various degrees.

Usage

quadratic_roots(a, b, c)

cubic_roots(a, b, c, d)

cubic_root_residual(a, b, c, d, root)

cubic_root_condition_number(a, b, c, d, root)

quartic_roots(a, b, c, d, e)

Arguments

Details

This package provides functions to find roots of quadratic, cubic, and quartic polynomials. The functions return the roots as numeric vectors.

Value

A numeric vector of the polynomial roots, residual, or condition number.

Examples

# Example of finding quadratic roots
quadratic_roots(1, -3, 2)
# Example of finding cubic roots
cubic_roots(1, -6, 11, -6)
# Example of finding quartic roots
quartic_roots(1, -10, 35, -50, 24)
# Example of finding cubic root residual
cubic_root_residual(1, -6, 11, -6, 1)
# Example of finding cubic root condition number
cubic_root_condition_number(1, -6, 11, -6, 1)

Rayleigh Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Rayleigh distribution.

Usage

rayleigh_pdf(x, scale = 1)

rayleigh_lpdf(x, scale = 1)

rayleigh_cdf(x, scale = 1)

rayleigh_lcdf(x, scale = 1)

rayleigh_quantile(p, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Rayleigh distribution with scale = 1
rayleigh_pdf(1)
rayleigh_lpdf(1)
rayleigh_cdf(1)
rayleigh_lcdf(1)
rayleigh_quantile(0.5)

Root-Finding and Minimisation Functions

Description

Functions for root-finding and minimisation using various algorithms.

Usage

bisect(
  f,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

bracket_and_solve_root(
  f,
  guess,
  factor,
  rising,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

toms748_solve(
  f,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

newton_raphson_iterate(
  f,
  guess,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

halley_iterate(
  f,
  guess,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

schroder_iterate(
  f,
  guess,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

brent_find_minima(
  f,
  lower,
  upper,
  digits = .Machine$double.digits,
  max_iter = .Machine$integer.max
)

Arguments

Details

This package provides a set of functions for finding roots of equations and minimising functions using different numerical methods. The methods include bisection, bracket and solve, TOMS 748, Newton-Raphson, Halley's method, Schroder's method, and Brent's method. It also includes functions for finding roots of polynomials (quadratic, cubic, quartic) and computing minima.

Value

A list containing the root or minimum value, the value of the function at that point, and the number of iterations performed.

See Also

Boost Documentation for more details on the mathematical background.

Examples

f <- function(x) x^2 - 2
bisect(f, lower = 0, upper = 2)
bracket_and_solve_root(f, guess = 1, factor = 0.1, rising = TRUE)
toms748_solve(f, lower = 0, upper = 2)
f <- function(x) c(x^2 - 2, 2 * x)
newton_raphson_iterate(f, guess = 1, lower = 0, upper = 2)
f <- function(x) c(x^2 - 2, 2 * x, 2)
halley_iterate(f, guess = 1, lower = 0, upper = 2)
schroder_iterate(f, guess = 1, lower = 0, upper = 2)
f <- function(x) (x - 2)^2 + 1
brent_find_minima(f, lower = 0, upper = 4)

S\alphaS Point5 Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the S\alphaS Point5 distribution.

Usage

saspoint5_pdf(x, location = 0, scale = 1)

saspoint5_lpdf(x, location = 0, scale = 1)

saspoint5_cdf(x, location = 0, scale = 1)

saspoint5_lcdf(x, location = 0, scale = 1)

saspoint5_quantile(p, location = 0, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Distribution only available with Boost version 1.87.0 or later.
## Not run: 
# SaS Point5 distribution with location 0 and scale 1
  saspoint5_pdf(3)
  saspoint5_lpdf(3)
  saspoint5_cdf(3)
  saspoint5_lcdf(3)
  saspoint5_quantile(0.5)

## End(Not run)

Sinus Cardinal and Hyperbolic Functions

Description

Functions to compute the sinus cardinal function and hyperbolic sinus cardinal function.

Usage

sinc_pi(x)

sinhc_pi(x)

Arguments

Value

A single numeric value with the computed sinus cardinal or hyperbolic sinus cardinal function.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Sinus cardinal function
sinc_pi(0.5)
# Hyperbolic sinus cardinal function
sinhc_pi(0.5)

Skew Normal Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Skew Normal distribution.

Usage

skew_normal_pdf(x, location = 0, scale = 1, shape = 0)

skew_normal_lpdf(x, location = 0, scale = 1, shape = 0)

skew_normal_cdf(x, location = 0, scale = 1, shape = 0)

skew_normal_lcdf(x, location = 0, scale = 1, shape = 0)

skew_normal_quantile(p, location = 0, scale = 1, shape = 0)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Skew Normal distribution with location = 0, scale = 1, shape = 0
skew_normal_pdf(0)
skew_normal_lpdf(0)
skew_normal_cdf(0)
skew_normal_lcdf(0)
skew_normal_quantile(0.5)

Spherical Harmonics

Description

Functions to compute spherical harmonics and related functions.

Usage

spherical_harmonic(n, m, theta, phi)

spherical_harmonic_r(n, m, theta, phi)

spherical_harmonic_i(n, m, theta, phi)

Arguments

Value

A single complex value with the computed spherical harmonic function, or its real and imaginary parts.

See Also

Boost Documentation

Examples

# Spherical harmonic function Y_2^1(0.5, 0.5)
spherical_harmonic(2, 1, 0.5, 0.5)
# Real part of the spherical harmonic function Y_2^1(0.5, 0.5)
spherical_harmonic_r(2, 1, 0.5, 0.5)
# Imaginary part of the spherical harmonic function Y_2^1(0.5, 0.5)
spherical_harmonic_i(2, 1, 0.5, 0.5)

Student's T Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Student's t distribution.

Usage

students_t_pdf(x, df = 1)

students_t_lpdf(x, df = 1)

students_t_cdf(x, df = 1)

students_t_lcdf(x, df = 1)

students_t_quantile(p, df = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Student's t distribution with 3 degrees of freedom
students_t_pdf(0, 3)
students_t_lpdf(0, 3)
students_t_cdf(0, 3)
students_t_lcdf(0, 3)
students_t_quantile(0.5, 3)

Triangular Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Triangular distribution.

Usage

triangular_pdf(x, lower = 0, mode = 1, upper = 2)

triangular_lpdf(x, lower = 0, mode = 1, upper = 2)

triangular_cdf(x, lower = 0, mode = 1, upper = 2)

triangular_lcdf(x, lower = 0, mode = 1, upper = 2)

triangular_quantile(p, lower = 0, mode = 1, upper = 2)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Triangular distribution with lower = 0, mode = 1, upper = 2
triangular_pdf(1)
triangular_lpdf(1)
triangular_cdf(1)
triangular_lcdf(1)
triangular_quantile(0.5)

Uniform Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Uniform distribution.

Usage

uniform_pdf(x, lower = 0, upper = 1)

uniform_lpdf(x, lower = 0, upper = 1)

uniform_cdf(x, lower = 0, upper = 1)

uniform_lcdf(x, lower = 0, upper = 1)

uniform_quantile(p, lower = 0, upper = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Uniform distribution with lower = 0, upper = 1
uniform_pdf(0.5)
uniform_lpdf(0.5)
uniform_cdf(0.5)
uniform_lcdf(0.5)
uniform_quantile(0.5)

Vector Functionals

Description

Functions to compute various vector norms and distances.

Usage

l0_pseudo_norm(x)

hamming_distance(x, y)

l1_norm(x)

l1_distance(x, y)

l2_norm(x)

l2_distance(x, y)

sup_norm(x)

sup_distance(x, y)

lp_norm(x, p)

lp_distance(x, y, p)

total_variation(x)

Arguments

Value

A single numeric value with the computed norm or distance.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# L0 Pseudo Norm
l0_pseudo_norm(c(1, 0, 2, 0, 3))
# Hamming Distance
hamming_distance(c(1, 0, 1), c(0, 1, 1))
# L1 Norm
l1_norm(c(1, -2, 3))
# L1 Distance
l1_distance(c(1, -2, 3), c(4, -5, 6))
# L2 Norm
l2_norm(c(3, 4))
# L2 Distance
l2_distance(c(3, 4), c(0, 0))
# Supremum Norm
sup_norm(c(1, -2, 3))
# Supremum Distance
sup_distance(c(1, -2, 3), c(4, -5, 6))
# Lp Norm
lp_norm(c(1, -2, 3), 3)
# Lp Distance
lp_distance(c(1, -2, 3), c(4, -5, 6), 3)
# Total Variation
total_variation(c(1, 2, 1, 3))

Weibull Distribution Functions

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Weibull distribution.

Usage

weibull_pdf(x, shape = 1, scale = 1)

weibull_lpdf(x, shape = 1, scale = 1)

weibull_cdf(x, shape = 1, scale = 1)

weibull_lcdf(x, shape = 1, scale = 1)

weibull_quantile(p, shape = 1, scale = 1)

Arguments

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Weibull distribution with shape = 1, scale = 1
weibull_pdf(1)
weibull_lpdf(1)
weibull_cdf(1)
weibull_lcdf(1)
weibull_quantile(0.5)

Riemann Zeta Function

Description

Computes the Riemann zeta function (\zeta(s)) for argument (z).

Usage

zeta(z)

Arguments

Value

The value of the Riemann zeta function (\zeta(z)).

Examples

# Riemann Zeta Function
zeta(2)  # Should return pi^2 / 6