Package 'boostmath'

Description

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

The Airy functions are the two linearly independent solutions to the differential equation:

$y'' - xy = 0$

Airy $Ai$ Function: The first solution to the Airy differential equation. For negative $x$ values, $Ai(x)$ exhibits oscillatory behavior. For positive $x$ values, $Ai(x)$ monotonically decreases toward zero.

Airy $Bi$ Function: The second solution to the Airy differential equation. For negative $x$ values, $Bi(x)$ exhibits cyclic oscillation. For positive $x$ values, $Bi(x)$ tends toward infinity.

Airy $Ai'$ Function: The derivative of the first solution to the Airy differential equation. For negative $x$ values, $Ai'(x)$ displays cyclic oscillation. For positive $x$ values, $Ai'(x)$ approaches zero asymptotically.

Airy $Bi'$ Function: The derivative of the second solution to the Airy differential equation. For negative $x$ values, $Bi'(x)$ oscillates cyclically. For positive $x$ values, $Bi'(x)$ increases toward infinity.

Zeros of Airy Functions: The zeros are the values where $Ai(x) = 0$ or $Bi(x) = 0$. The zeros are indexed starting from 1. The first few zeros are approximately:

• $Ai$: -2.33811, -4.08795, -5.52056, ...

• $Bi$: -1.17371, -3.27109, -4.83074, ...

All functions are implemented using relationships to Bessel functions for numerical accuracy.

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)

Description

Performs the Anderson-Darling test for normality by computing the $A^2$ test statistic:

$A^2 = n\int_{-\infty}^{\infty}\frac{(F_n(x)-F(x))^2F'(x)}{F(x)(1-F(x))}dx$

The Anderson-Darling test evaluates whether a sample comes from a normal distribution by computing an integral over the empirical cumulative distribution function (ECDF) and comparing it against the normal distribution's CDF.

Interpretation:

• When $A^2/n$ approaches zero as sample size increases, the normality hypothesis is supported

• When $A^2/n$ converges to a positive finite value, the normality hypothesis lacks support

Important: The input data vector x must be sorted in ascending order. Unsorted data will trigger an error.

Usage

anderson_darling_normality_statistic(x, mu = 0, sd = 1)

Arguments

Value

The Anderson-Darling $A^2$ test statistic.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Anderson-Darling test for normality with sorted data
x <- sort(rnorm(100))
anderson_darling_normality_statistic(x, 0, 1)

Description

Functions to compute the probability density function, cumulative distribution function, and quantile function for the arcsine distribution on the interval $[x_{min}, x_{max}]$.

The arcsine distribution is a U-shaped distribution with infinite density at the endpoints.

For $x_{min} < x < x_{max}$:

The PDF is:

$f(x; x_{min}, x_{max}) = \frac{1}{\pi\sqrt{(x - x_{min})(x_{max} - x)}}$

The CDF is:

$F(x; x_{min}, x_{max}) = \frac{2}{\pi}\arcsin\left(\sqrt{\frac{x - x_{min}}{x_{max} - x_{min}}}\right)$

The quantile for $0 < p < 1$ is

$F^{-1}(p; x_{min}, x_{max}) = x_{min} + (x_{max} - x_{min})\sin^2\left(\frac{\pi p}{2}\right)$

For the standard distribution on $[0, 1]$, these reduce to

$f(x) = 1/(\pi\sqrt{x(1-x)})$

$F(x) = \frac{2}{\pi}\arcsin(\sqrt{x})$

Usage

arcsine_distribution(x_min = 0, x_max = 1)

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 distribution with default parameters
dist <- arcsine_distribution()
# Apply generic functions
cdf(dist, 0.5)
logcdf(dist, 0.5)
pdf(dist, 0.5)
logpdf(dist, 0.5)
hazard(dist, 0.5)
chf(dist, 0.5)
mean(dist)
median(dist)
range(dist)
quantile(dist, 0.2)
standard_deviation(dist)
support(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
kurtosis_excess(dist)

# Convenience functions
arcsine_pdf(0.5)
arcsine_lpdf(0.5)
arcsine_cdf(0.5)
arcsine_lcdf(0.5)
arcsine_quantile(0.5)

Description

Barycentric rational interpolation is a high-accuracy interpolation method for non-uniformly spaced samples.

Performance and Accuracy:

It requires O(N) time for construction and O(N) time for each evaluation. If the approximation order is d, the error is O(h^(d+1)).

Caveats:

This method is robust but can behave unexpectedly if the sample spacing at the endpoints is much larger than in the center.

Usage

barycentric_rational(x, y, order = 3)

Arguments

Value

An object of class barycentric_rational_interpolator with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

See Also

Boost Documentation

Examples

x <- c(0, 1, 2, 3)
y <- c(1, 2, 0, 2)
order <- 3
interpolator <- barycentric_rational(x, y, order)
xi <- 1.5
interpolator$interpolate(xi)
interpolator$prime(xi)

Description

High-precision implementations of basic mathematical functions with enhanced numerical stability for special cases.

These functions provide numerically stable alternatives to standard operations, particularly useful when working with values near zero or when high precision is required.

Trigonometric Functions with $\pi$:

• sin_pi(x): Computes $\sin(\pi x)$

• cos_pi(x): Computes $\cos(\pi x)$

Logarithmic and Exponential Functions:

• log1p_boost(x): Computes $\log(1 + x)$ accurately for small $|x|$

• expm1_boost(x): Computes $\exp(x) - 1$ accurately for small $|x|$

Root Functions:

• cbrt(x): Computes the cube root $x^{1/3}$

• sqrt1pm1(x): Computes $\sqrt{1 + x} - 1$ accurately for small $|x|$

• rsqrt(x): Computes the reciprocal square root $\tfrac{1}{\sqrt{x}}$

Power Functions:

• powm1(x, y): Computes $x^y - 1$ accurately

Geometric Functions:

• hypot(x, y): Computes $\sqrt{x^2 + y^2}$ without overflow/underflow

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/2) = 1 (exact)
sin_pi(0.5)
# cos(pi/2) = 0 (exact)
cos_pi(0.5)
# log(1 + x) for small x
log1p_boost(0.001)
# exp(x) - 1 for small x
expm1_boost(0.001)
# Cube root
cbrt(8)  # Returns 2
# sqrt(1 + x) - 1 for small x
sqrt1pm1(0.001)
# x^y - 1 accurately
powm1(2, 3)  # Returns 7 (2^3 - 1)
# Euclidean distance
hypot(3, 4)  # Returns 5
# Reciprocal square root
rsqrt(4)  # Returns 0.5

Description

Functions to compute the probability mass function (pmf), cumulative distribution function, and quantile function for the Bernoulli distribution.

The Bernoulli distribution models a single trial with outcomes $k \in \{0, 1\}$ and success probability $p$:

$P(X = 0) = 1 - p, \quad P(X = 1) = p$

$F(0)=1-p, \quad F(1)=1$

Usage

bernoulli_distribution(p_success)

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 distribution with p_success = 0.5
dist <- bernoulli_distribution(0.5)
# Apply generic functions
cdf(dist, 0.5)
logcdf(dist, 0.5)
pdf(dist, 0.5)
logpdf(dist, 0.5)
hazard(dist, 0.5)
chf(dist, 0.5)
mean(dist)
median(dist)
mode(dist)
range(dist)
quantile(dist, 0.2)
standard_deviation(dist)
support(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
kurtosis_excess(dist)

# Convenience functions
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)

Description

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

Bessel functions are solutions to Bessel's ordinary differential equation and appear in many problems with cylindrical or spherical symmetry, such as wave propagation, heat conduction, and quantum mechanics.

Bessel Functions of the First and Second Kinds

• cyl_bessel_j(v, x): Computes the Bessel function of the first kind $J_v(x)$:

$J_v(x) = \left(\frac{1}{2}x\right)^v\sum_{k=0}^\infty{\frac{\left(-\frac{1}{4}x^2\right)^k}{k!\Gamma(v+k+1)}}$

• cyl_neumann(v, x): Computes the Bessel function of the second kind $Y_v(x) = N_v(x)$:

$Y_v(x) = \frac{J_v(x)\cos(v\pi) - J_{-v}(x)}{\sin(v\pi)}$

Modified Bessel Functions of the First and Second Kinds

• cyl_bessel_i(v, x): Computes the modified Bessel function of the first kind $I_v(x)$:

$I_v(x) = \left(\frac{1}{2}x\right)^v\sum_{k=0}^\infty{\frac{\left(\frac{1}{4}x^2\right)^k}{k!\Gamma(v+k+1)}}$

• cyl_bessel_k(v, x): Computes the modified Bessel function of the second kind $K_v(x)$:

$K_v(x) = \frac{\pi}{2} \frac{I_{-v}(x) - I_{v}(x)}{\sin(v\pi)}$

Spherical Bessel Functions of the First and Second Kinds

• sph_bessel(v, x): Computes the Spherical Bessel function of the first kind $j_v(x)$:

$j_v(x) = \sqrt{\frac{\pi}{2x}}J_{v+\frac{1}{2}}(x)$

• sph_neumann(v, x): Computes the Spherical Bessel function of the second kind $y_v(x) = n_v(x)$:

$y_v(x) = \sqrt{\frac{\pi}{2x}}Y_{v+\frac{1}{2}}(x)$

Derivatives:

The _prime functions compute the derivatives with respect to x of the corresponding Bessel functions:

$J'_v(x) = \frac{J_{v-1}(x)-J_{v+1}(x)}{2}$

$Y'_v(x) = \frac{Y_{v-1}(x)-Y_{v+1}(x)}{2}$

$I'_v(x) = \frac{I_{v-1}(x)-I_{v+1}(x)}{2}$

$K'_v(x) = \frac{K_{v-1}(x)-K_{v+1}(x)}{-2}$

$j'_v(x) = \left(\frac{v}{x}\right)j_v(x)-j_{v+1}(x)$

$y'_v(x) = \left(\frac{v}{x}\right)y_v(x)-y_{v+1}(x)$

Zeros:

The zero functions find the zeros of J_v and Y_v (where the function equals zero), indexed starting from 1.

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)

Description

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

The Beta distribution is defined on $x \in [0, 1]$ with shape parameters $\alpha > 0$ and $\beta > 0$.

The PDF is:

$f(x;\alpha, \beta) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{B(\alpha, \beta)}$

Where $B(\alpha, \beta)$ is the beta function.

The CDF is:

$F(x; \alpha, \beta) = I_x(\alpha, \beta)$

Where $I_x$ is the regularised incomplete beta function.

The quantile is:

$F^{-1}(p; \alpha, \beta) = I_{p}^{-1}(\alpha, \beta)$

Where $I_{p}^{-1}$ is the inverse of the regularised incomplete beta function.

Usage

beta_distribution(alpha, beta)

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)

beta_find_alpha(mean = NULL, variance = NULL, beta = NULL, x = NULL, p = NULL)

beta_find_beta(mean = NULL, variance = NULL, alpha = NULL, x = NULL, p = NULL)

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
dist <- beta_distribution(2, 5)
# Apply generic functions
cdf(dist, 0.5)
logcdf(dist, 0.5)
pdf(dist, 0.5)
logpdf(dist, 0.5)
hazard(dist, 0.5)
chf(dist, 0.5)
mean(dist)
median(dist)
mode(dist)
range(dist)
quantile(dist, 0.2)
standard_deviation(dist)
support(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
kurtosis_excess(dist)

# Convenience functions
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)

## Not run: 
# Find alpha given mean and variance
beta_find_alpha(mean = 0.3, variance = 0.02)
# Find alpha given beta, x, and probability
beta_find_alpha(beta = 5, x = 0.4, p = 0.6)
# Find beta given mean and variance
beta_find_beta(mean = 0.3, variance = 0.02)
# Find beta given alpha, x, and probability
beta_find_beta(alpha = 2, x = 0.4, p = 0.6)

## End(Not run)

Description

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

Beta Function $B(a, b)$:

• beta_boost(a, b)

$B(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

Incomplete Beta Functions:

• Normalised (Regularised) Functions:

  • ibeta(a, b, x): Normalised incomplete beta function $I_x(a, b)$

$I_x(a,b) = \frac{1}{B(a, b)}\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$

• ibetac(a, b, x): Normalised complement, $1 - I_x(a, b) = I_{1-x}(b, a)$

• Non-normalised Functions:

  • beta_boost(a, b, x): Full incomplete beta function $B_x(a, b)$

$\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$

• betac(a, b, x): Full complement , $1 - B_x(a, b) = B_{1-x}(b, a)$

Inverse Functions:

• Primary inverses (solving for x):

  • ibeta_inv(a, b, p): Returns $x$ such that $p = I_x(a, b)$

  • ibetac_inv(a, b, q): Returns $x$ such that $q = 1 - I_x(a, b)$

• Parameter inverses (solving for a or b):

  • ibeta_inva(b, x, p): Returns a such that $p = I_x(a, b)$

  • ibetac_inva(b, x, q): Returns a such that $q = 1 - I_x(a, b)$

  • ibeta_invb(a, x, p): Returns b such that $p = I_x(a, b)$

  • ibetac_invb(a, x, q): Returns b such that $q = 1 - I_x(a, b)$s

Derivatives:

ibeta_derivative(a, b, x): Computes the partial derivative with respect to x of the incomplete beta function

$\frac{\partial}{\partial x}I_x(a,b) = \frac{(1-x)^{b-1}x^{a-1}}{B(a,b)}$

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)

Description

Bezier polynomials are smooth curves which approximate a set of control points. They are commonly used in computer-aided geometric design.

Properties:

The curve is approximating, meaning it does not necessarily pass through the control points. Passing n control points creates a polynomial of degree n-1. Evaluation is O(N^2) via de Casteljau's algorithm.

Usage

bezier_polynomial(control_points)

Arguments

Value

An object of class bezier_polynomial with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

• edit_control_point(new_control_point, index): Insert a new control point at the specified index.

See Also

Boost Documentation

Examples

control_points <- list(c(0, 0, 0), c(1, 2, 0), c(2, 0, 0), c(3, 3, 0))
interpolator <- bezier_polynomial(control_points)
xi <- 1.5
interpolator$interpolate(xi)
interpolator$prime(xi)
new_control_point <- c(1.5, 1, 0)
interpolator$edit_control_point(new_control_point, 2)

Description

The bilinear uniform interpolator takes a grid of data points specified by a linear index and interpolates between each segment using a bilinear function.

Details:

"Bilinear" means it is the product of two linear functions. The interpolant is continuous and its evaluation is constant time. The interpolator is point-centered.

Usage

bilinear_uniform(x, rows, cols, dx = 1, dy = 1, x0 = 0, y0 = 0)

Arguments

Value

An object of class bilinear_uniform with methods:

• interpolate(xi, yi): Evaluate the interpolator at point ⁠(xi, yi)⁠.

See Also

Boost Documentation

Examples

x <- seq(0, 1, length.out = 10)
interpolator <- bilinear_uniform(x, rows = 2, cols = 5)
xi <- 0.5
yi <- 0.5
interpolator$interpolate(xi, yi)

Description

Functions to compute the probability mass function (pmf), cumulative distribution function, quantile function, and confidence bounds for the Binomial distribution.

The Binomial distribution models the number of successes $k$ in $n$ independent trials with success probability $p$. The pmf is

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$

Accuracy and Implementation Notes: CDF and related functions are implemented using incomplete beta functions (ibeta, ibetac). The pmf is evaluated via ibeta_derivative for stability. Quantiles are obtained numerically (TOMS 748), since no closed-form inverse exists for discrete $k$. As a discrete distribution, quantiles are rounded outward to ensure at least the requested coverage; use complements for improved tail accuracy.

Confidence Bounds: binomial_find_lower_bound_on_p and binomial_find_upper_bound_on_p implement Clopper-Pearson exact intervals (default) or Jeffreys prior intervals, as described in the Boost documentation.

Usage

binomial_distribution(n, prob)

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)

binomial_find_lower_bound_on_p(n, k, alpha, method = "clopper_pearson_exact")

binomial_find_upper_bound_on_p(n, k, alpha, method = "clopper_pearson_exact")

binomial_find_minimum_number_of_trials(k, prob, alpha)

binomial_find_maximum_number_of_trials(k, prob, alpha)

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 distribution with n = 10, prob = 0.5
dist <- binomial_distribution(10, 0.5)
# Apply generic functions
cdf(dist, 2)
logcdf(dist, 2)
pdf(dist, 2)
logpdf(dist, 2)
hazard(dist, 2)
chf(dist, 2)
mean(dist)
median(dist)
mode(dist)
range(dist)
quantile(dist, 0.2)
standard_deviation(dist)
support(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
kurtosis_excess(dist)

# Convenience functions
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)

## Not run: 
# Find lower bound on p given k = 3 successes in n = 10 trials with 95% confidence
binomial_find_lower_bound_on_p(10, 3, 0.05)
# Find upper bound on p given k = 3 successes in n = 10 trials with 95% confidence
binomial_find_upper_bound_on_p(10, 3, 0.05)
# Find minimum number of trials n to observe k = 3 successes with p = 0.5 at 95% confidence
binomial_find_minimum_number_of_trials(3, 0.5, 0.05)
# Find maximum number of trials n to observe k = 3 successes with p = 0.5 at 95% confidence
binomial_find_maximum_number_of_trials(3, 0.5, 0.05)

## End(Not run)

Description

Functions to compute bivariate statistics including covariance and the Pearson correlation coefficient.

Covariance: The population covariance is

$\operatorname{cov}(x, y) = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$

Correlation Coefficient: The Pearson correlation coefficient is

$\rho_{x,y} = \frac{\operatorname{cov}(x, y)}{\sigma_x \sigma_y}$

Usage

covariance(x, y)

means_and_covariance(x, y)

correlation_coefficient(x, y)

Arguments

Value

A numeric value (or tuple for means_and_covariance) with the computed statistic.

See Also

Boost Documentation for more details on the mathematical background.

Examples

# Covariance
covariance(c(1, 2, 3), c(4, 5, 6))
# Means and Covariance
means_and_covariance(c(1, 2, 3), c(4, 5, 6))
# Correlation Coefficient
correlation_coefficient(c(1, 2, 3), c(4, 5, 6))

Description

The cardinal cubic B-spline interpolator allows for fast and accurate interpolation of a function which is known at equally spaced points.

Mathematical Properties:

It uses compactly supported basis functions constructed via iterative convolution, ensuring numerical stability. The interpolant is O(h^4) accurate for values and O(h^3) accurate for derivatives, where h is the step size.

Conditions:

Ideally, the function being interpolated should be four-times continuously differentiable. If the derivatives at the endpoints are not provided, they are estimated using one-sided finite-difference formulas.

Usage

cardinal_cubic_b_spline(
  y,
  t0,
  h,
  left_endpoint_derivative = NULL,
  right_endpoint_derivative = NULL
)

Arguments

Value

An object of class cardinal_cubic_b_spline with methods:

• interpolate(x): Evaluate the spline at point x.

• prime(x): Evaluate the first derivative of the spline at point x.

• double_prime(x): Evaluate the second derivative of the spline at point x.

See Also

Boost Documentation

Examples

y <- c(1, 2, 0, 2, 1)
t0 <- 0
h <- 1
spline_obj <- cardinal_cubic_b_spline(y, t0, h)
x <- 0.5
spline_obj$interpolate(x)
spline_obj$prime(x)
spline_obj$double_prime(x)

Description

The cardinal cubic Hermite interpolator is similar to the cubic Hermite interpolator but optimised for equispaced data.

Performance:

This allows for constant-time evaluation.

Usage

cardinal_cubic_hermite(y, dydx, x0, dx)

Arguments

Value

An object of class cardinal_cubic_hermite with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

• domain(): Get the domain of the interpolator.

See Also

Boost Documentation

Examples

y <- c(0, 1, 0)
dydx <- c(1, 0, -1)
interpolator <- cardinal_cubic_hermite(y, dydx, 0, 1)
xi <- 0.5
interpolator$interpolate(xi)
interpolator$prime(xi)
interpolator$domain()

Description

The cardinal quadratic B-spline interpolator is very nearly the same as the cubic B-spline interpolator, but uses quadratic basis functions.

Use Cases:

Basis functions are constructed by convolving a box function with itself twice. Since the basis functions are less smooth than the cubic B-spline, this is primarily useful for approximating functions of reduced smoothness. It is appropriate for functions which are two or three times continuously differentiable.

Usage

cardinal_quadratic_b_spline(
  y,
  t0,
  h,
  left_endpoint_derivative = NULL,
  right_endpoint_derivative = NULL
)

Arguments

Value

An object of class cardinal_quadratic_b_spline with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

See Also

Boost Documentation

Examples

y <- c(0, 1, 0, 1)
t0 <- 0
h <- 1
interpolator <- cardinal_quadratic_b_spline(y, t0, h)
xi <- 0.5
interpolator$interpolate(xi)
interpolator$prime(xi)

Description

The cardinal quintic B-spline interpolator is similar to the cubic B-spline but uses basis functions constructed by convolving a box function with itself five times.

Properties:

The basis functions are more smooth (C4) than the cubic B-spline (C2), making this useful for computing second derivatives. The second derivative of the quintic B-spline is a cubic spline.

Usage

cardinal_quintic_b_spline(
  y,
  t0,
  h,
  left_endpoint_derivatives = NULL,
  right_endpoint_derivatives = NULL
)

Arguments

Value

An object of class cardinal_quintic_b_spline with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

• double_prime(xi): Evaluate the second derivative of the interpolator at point xi.

See Also

Boost Documentation

Examples

y <- seq(0, 1, length.out = 20)
t0 <- 0
h <- 1
interpolator <- cardinal_quintic_b_spline(y, t0, h)
xi <- 0.5
interpolator$interpolate(xi)
interpolator$prime(xi)
interpolator$double_prime(xi)

Description

The cardinal quintic Hermite interpolator is similar to the quintic Hermite interpolator but optimised for equispaced data.

Performance:

This allows for constant-time evaluation.

Usage

cardinal_quintic_hermite(y, dydx, d2ydx2, x0, dx)

Arguments

Value

An object of class cardinal_quintic_hermite with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

• double_prime(xi): Evaluate the second derivative of the interpolator at point xi.

• domain(): Get the domain of the interpolator.

See Also

Boost Documentation

Examples

y <- c(0, 1, 0)
dydx <- c(1, 0, -1)
d2ydx2 <- c(0, -1, 0)
x0 <- 0
dx <- 1
interpolator <- cardinal_quintic_hermite(y, dydx, d2ydx2, x0, dx)
xi <- 0.5
interpolator$interpolate(xi)
interpolator$prime(xi)
interpolator$double_prime(xi)
interpolator$domain()

Description

Catmull-Rom splines are a family of interpolating curves which are commonly used in computer graphics and animation.

Properties:

They enjoy affine invariance, local support, C2-smoothness, and interpolation of control points. The curve is internally closed, however the user specifies if it should be treated as open or closed via the parameters. Evaluation is O(log N).

Usage

catmull_rom(control_points, closed = FALSE, alpha = 0.5)

Arguments

Value

An object of class catmull_rom with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

• max_parameter(): Get the maximum parameter value of the spline.

• parameter_at_point(i): Get the parameter value at index i.

See Also

Boost Documentation

Examples

control_points <- list(c(0, 0, 0), c(1, 1, 0), c(2, 0, 0), c(3, 1, 0))
interpolator <- catmull_rom(control_points)
xi <- 1.5
interpolator$interpolate(xi)
interpolator$prime(xi)
interpolator$max_parameter()
interpolator$parameter_at_point(2)

Description

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

The PDF is:

$f(x; x_0, \gamma) = \frac{1}{\pi}\left(\frac{\gamma}{(x-x_0)^2 + \gamma^2}\right)$

The CDF:

$F(x; x_0, \gamma) = \frac{1}{\pi}\text{arctan}\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}$

Usage

cauchy_distribution(location = 0, scale = 1)

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
dist <- cauchy_distribution(0, 1)
# Apply generic functions
cdf(dist, 0.5)
logcdf(dist, 0.5)
pdf(dist, 0.5)
logpdf(dist, 0.5)
hazard(dist, 0.5)
chf(dist, 0.5)
median(dist)
mode(dist)
range(dist)
quantile(dist, 0.2)
support(dist)

# Convenience functions
cauchy_pdf(0)
cauchy_lpdf(0)
cauchy_cdf(0)
cauchy_lcdf(0)
cauchy_quantile(0.5)

Description

Computes the Chatterjee correlation coefficient, a robust measure of dependence. Unlike classical correlation coefficients (Pearson, Spearman), Chatterjee's coefficient measures the degree to which y is a function of x (functional dependence), capturing non-linear relationships.

Characteristics:

• Functional Dependence: Value is 1 if and only if y is a measurable function of x.

• Independence: Value is 0 if x and y are independent.

• Range: The coefficient is theoretically in $[0, 1]$.

• Asymmetry: The measure is asymmetric; $C(X, Y) \neq C(Y, X)$. It specifically tests if $Y = f(X)$.

Usage

chatterjee_correlation(x, y)

Arguments

Details

The coefficient is calculated using the ranks of y when sorted by x. This implementation computes the sample version of the coefficient as described by Chatterjee (2021).

Formula: Given pairs $(X_i, Y_i)$, sort them such that $X_{(1)} \le \dots \le X_{(n)}$. Let $r_i$ be the rank of $Y_{(i)}$. The coefficient is:

$\xi_n(X, Y) = 1 - \frac{3 \sum_{i=1}^{n-1} |r_{i+1} - r_i|}{n^2 - 1}$

Value

A numeric value representing the Chatterjee correlation coefficient.

A numeric vector containing:

1. Correlation Coefficient: The Chatterjee correlation estimate.

References

Chatterjee, S. (2021). A new coefficient of correlation. Journal of the American Statistical Association, 116(536), 2009-2022.

See Also

Boost Documentation

Examples

# Functional dependence (Y = X^2)
x <- runif(50, -1, 1)
y <- x^2
chatterjee_correlation(x, y) # Should be high (near 1)

# Independence
x <- runif(50)
y <- runif(50)
chatterjee_correlation(x, y) # Should be low (near 0)

# Asymmetry check
chatterjee_correlation(x, y)
chatterjee_correlation(y, x)

Description

Functions to compute Chebyshev polynomials of the first and second kind, and efficiently evaluate Chebyshev series using Clenshaw's recurrence algorithm.

Chebyshev polynomials are orthogonal polynomials used extensively in approximation theory, numerical analysis, and spectral methods. They minimize the Runge phenomenon in polynomial interpolation.

Chebyshev Polynomials of the First Kind T_n(x):

• chebyshev_t(n, x): Evaluates $T_n(x)$

• Recurrence relation: $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ for n > 0

• Initial conditions: T_0(x) = 1, T_1(x) = x

• chebyshev_t_prime(n, x): Derivative of T_n(x)

• Stable evaluation for x in $[-1, 1]$ (mixed forward-backward stable)

Chebyshev Polynomials of the Second Kind U_n(x):

• chebyshev_u(n, x): Evaluates U_n(x)

• Related to T_n by differentiation

Recurrence Relation:

• chebyshev_next(x, Tn, Tn_1): Computes $T_{n+1}(x)$ from T_n and $T_{n-1}$

Clenshaw's Recurrence Algorithm:

Efficient O(n) evaluation of Chebyshev series (alternative to O(n^2) direct computation):

• chebyshev_clenshaw_recurrence(c, x): Evaluates Chebyshev series with coefficients c at point x on standard interval $[-1, 1]$

• chebyshev_clenshaw_recurrence_ab(c, a, b, x): Evaluates Chebyshev series on arbitrary interval $[a, b]$ using Reinsch's modification

Stability:

Evaluation by three-term recurrence is known to be mixed forward-backward stable for x in $[-1, 1]$. Stability outside this interval is not established.

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 or series evaluation.

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))
# Evaluate Chebyshev series with Clenshaw's algorithm
chebyshev_clenshaw_recurrence(c(1, 0, -1), 0.5)
# Evaluate Chebyshev series on interval [0, 1]
chebyshev_clenshaw_recurrence_ab(c(1, 0, -1), 0, 1, 0.5)

Description

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

With degrees of freedom

$\nu > 0$

, the PDF is:

$f(x; \nu) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)} x^{\nu/2 - 1} e^{-x/2}, \quad x \ge 0$

and the CDF is given by the regularised incomplete gamma function

$F(x;\nu) = P(\nu/2, x/2)$

Accuracy and Implementation Notes: The CDF and quantiles are implemented via incomplete gamma functions. Specifically, the PDF uses ⁠gamma_p_derivative(\nu/2, x/2)/2⁠, the CDF uses gamma_p, the complement uses gamma_q, and quantiles use gamma_p_inv/gamma_q_inv. Accuracy therefore follows that of the incomplete gamma functions.

Sample Size Estimation: chi_squared_find_degrees_of_freedom estimates the sample size required to detect a difference from a nominal variance. The sign of difference_from_variance determines whether the test is for higher or lower variance.

Usage

chi_squared_distribution(df)

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)

chi_squared_find_degrees_of_freedom(
  difference_from_variance,
  alpha,
  beta,
  variance,
  hint = 100
)

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
dist <- chi_squared_distribution(3)
# Apply generic functions
cdf(dist, 0.5)
logcdf(dist, 0.5)
pdf(dist, 0.5)
logpdf(dist, 0.5)
hazard(dist, 0.5)
chf(dist, 0.5)
mean(dist)
median(dist)
mode(dist)
range(dist)
quantile(dist, 0.2)
standard_deviation(dist)
support(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
kurtosis_excess(dist)

# Convenience functions
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)

# Find degrees of freedom needed to detect a difference from variance of 2.0
# with alpha = 0.05 and beta = 0.2 when the nominal variance is 5.0
chi_squared_find_degrees_of_freedom(2.0, 0.05, 0.2, 5.0)

Description

Functions to compute condition numbers for summation operations.

Usage

summation_condition_number(x = 0, kahan = TRUE)

evaluation_condition_number(f, x)

Arguments

Value

For summation_condition_number, an object with methods to compute the condition number, sum, L1 norm, and to add or subtract values. For evaluation_condition_number, a numeric value representing the condition number of the function evaluation at x.

Examples

# Create a summation condition number object
scn <- summation_condition_number(kahan = TRUE)
# Add some values
scn$add(1.0)
scn$add(2.0)
scn$add(3.0)
# Compute sum, condition number, and L1 norm
print(scn$sum())
print(scn$condition_number())
print(scn$l1_norm())
# Compute evaluation condition number for a function
f <- function(x) { x^2 + 3*x + 2 }
print(evaluation_condition_number(f, 1.0))

Description

Provides access to mathematical constants used in the Boost Math library.

The available constants include rational fractions,

$\pi$

-related values,

$e$

-related values, and assorted special constants (e.g., Euler-Mascheroni, Catalan). Integer values are intentionally omitted since they can be constructed exactly from literals.

Accuracy and Implementation Notes: The constants are provided at high precision by Boost.Math; refer to the Boost constants table for names and values.

Usage

constants(constant = NULL)

Arguments

Value

Requested constant value if constant is specified, otherwise a list of all available constants.

See Also

Boost Documentation for more details on the constants.

Examples

constants()

Description

The cubic Hermite interpolant takes non-equispaced data and interpolates between them via cubic Hermite polynomials whose slopes must be provided.

Applications:

The interpolant is C1 and evaluation has O(log N) complexity. This interpolator is useful for solution skeletons of ODE steppers.

Usage

cubic_hermite(x, y, dydx)

Arguments

Value

An object of class cubic_hermite with methods:

• interpolate(xi): Evaluate the interpolator at point xi.

• prime(xi): Evaluate the derivative of the interpolator at point xi.

• push_back(x, y, dydx): Add a new control point to the interpolator.

• domain(): Get the domain of the interpolator.

See Also

Boost Documentation

Examples

x <- c(0, 1, 2)
y <- c(0, 1, 0)
dydx <- c(1, 0, -1)
interpolator <- cubic_hermite(x, y, dydx)
xi <- 0.5
interpolator$interpolate(xi)
interpolator$prime(xi)
interpolator$push_back(3, 0, 1)
interpolator$domain()

Description

Numerical integration using double exponential quadrature methods (Tanh-Sinh, Sinh-Sinh, Exp-Sinh). These methods use variable transformations to achieve high-order convergence, often optimal for functions in the Hardy space (holomorphic in the unit disk).

Tanh-Sinh Quadrature: Best for integration over a finite interval $(a, b)$. Can handle singularities at the endpoints of the integration domain. Converges rapidly for holomorphic integrands.

Sinh-Sinh Quadrature: Designed for integration over the entire real line $(-\infty, \infty)$. Handles integrands with large features or decay properties.

Exp-Sinh Quadrature: Designed for integration over a semi-infinite interval, typically $(0, \infty)$, or ranges like $(a, \infty)$ or $(-\infty, b)$. Supports endpoint singularities.

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.

See Also

numerical_integration

Examples

# Tanh-sinh quadrature of log(x) from 0 to 1 (Endpoint singularity)
tanh_sinh(function(x) { log(x) * log1p(-x) }, a = 0, b = 1)

# Sinh-sinh quadrature of exp(-x^2) over (-Inf, Inf)
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)

Description

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

Elliptic Integrals - Carlson Form

• ellint_rf(x, y, z): Carlson's Elliptic Integral $R_F$

$R_F(x, y, z) = \frac{1}{2}\int_{0}^\infty[(t+x)(t+y)(t+z)]^{-\frac{1}{2}}dt$

• ellint_rd(x, y, z): Carlson's Elliptic Integral $R_D$

$R_D(x, y, z) = \frac{3}{2}\int_{0}^\infty[(t+x)(t+y)]^{-\frac{1}{2}}(t+z)^{-\frac{3}{2}}dt$

• ellint_rj(x, y, z, p): Carlson's Elliptic Integral $R_J$

$R_J(x, y, z) = \frac{3}{2}\int_{0}^\infty(t+p)^{-1}[(t+x)(t+y)(t+z)]^{-\frac{1}{2}}dt$

• ellint_rc(x, y): Carlson's Elliptic Integral $R_C$

$R_C(x, y) = \frac{1}{2}\int_{0}^\infty(t+x)^{-\frac{1}{2}}(t+y)^{-1}dt$

• ellint_rg(x, y, z): Carlson's Elliptic Integral $R_G$

$R_G(x, y, z) = \frac{1}{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}\sqrt{\left(x\sin^2\theta\cos^2\phi+y\sin^2\theta\sin^2\phi + z\cos^2\theta\right)}\sin\theta d\theta \phi$

Elliptic Integrals of the First Kind - Legendre Form

• ellint_1(k, phi): Incomplete elliptic integral of the first kind: $F(\phi, k)$:

$F(\phi, k) = \int_0^{\phi}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}d\theta$

Elliptic Integrals of the Second Kind - Legendre Form

• ellint_2(k, phi): Incomplete elliptic integral of the second kind: $E(\phi, k)$:

$E(\phi, k) = \int_0^{\phi}\sqrt{1-k^2\sin^2\theta}d\theta$

Elliptic Integrals of the Third Kind - Legendre Form

• ellint_3(k, n, phi): Incomplete elliptic integral of the third kind: $\Pi(n, \phi, k)$:

$\Pi(n, \phi, k) = \int_0^{\phi}\frac{d\theta}{\left(1-n\sin^2\theta\right)\sqrt{1-k^2\sin^2\theta}}d\theta$

Elliptic Integral D - Legendre Form

• ellint_d(k, phi): Incomplete elliptic integral $D(\phi, k)$:

$D(\phi, k) = \frac{(F(\phi, k) - E(\phi, k))}{k^2}$

Jacobi Zeta Function

• jacobi_zeta(k, phi): Jacobi Zeta function $Z(\phi, k)$:

$Z(\phi, k) = E(\phi, k) - \frac{E(\frac{\pi}{2},k)F(\phi,k)}{F(\frac{\pi}{2}, k)}$

Heuman Lambda Function

• heuman_lambda(k, phi): Heuman Lambda function $\Lambda_0(\phi, k)$:

$\Lambda_0(\phi, k) = \frac{F(\phi,\sqrt{1-k^2})}{F(\frac{\pi}{2}, \sqrt{1-k^2})} + \frac{2}{\pi}F(\frac{\pi}{2},k)Z(\phi, \sqrt{1-k^2})$

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)

Description

Create an empirical cumulative distribution function (ECDF) from a given vector.

Usage

empirical_cumulative_distribution_function(data, sorted = FALSE)

Arguments

Details

The ECDF is a step function constructed from observed data that converges to the true CDF as sample size grows. It is commonly used in goodness-of-fit workflows that compare the empirical CDF to a hypothesised distribution.

Implementation Notes: Data must be sorted; by default the constructor sorts at

$O(n \log n)$

cost. If the data is already sorted, set sorted = TRUE to avoid the sort. Evaluation uses binary search (upper_bound) and runs in

$O(\log n)$

time.

Value

An object representing the ECDF, with member function ⁠$ecdf(x)⁠ to evaluate the ECDF at point(s) x.

Examples

data <- c(1.2, 2.3, 3.1, 4.5, 5.0)
ecdf_obj <- empirical_cumulative_distribution_function(data)
ecdf_obj$ecdf(3.0)  # Evaluate ECDF at x = 3.0

Description

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

Error functions appear frequently in probability, statistics, and partial differential equations, particularly in the study of normal distributions and diffusion processes.

Error Function:

The error function is defined by the integral:

$erf(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt$

The error function is an odd function (erf(-z) = -erf(z)). Implementation uses rational approximations optimised for absolute error, particularly for |z| <= 0.5.

Complementary Error Function:

The complementary error function is defined as:

$erfc(z) = 1 - erf(z) = \frac{2}{\sqrt{\pi}} \int_z^\infty e^{-t^2} dt$

Key reflection formulas:

• erfc(-z) = 2 - erfc(z) (preferred when -z < -0.5)

• erfc(-z) = 1 + erf(z) (preferred when -0.5 <= -z < 0)

For large z, uses exponential scaling to maintain numerical stability.

Inverse Functions:

• erf_inv(p): Returns x such that p = erf(x), where -1 <= p <= 1

• erfc_inv(p): Returns x such that p = erfc(x), where 0 <= p <= 2

Inverse functions use rational approximations with different formulas for different ranges of p, achieving accuracy to less than ~2 epsilon for standard precision types.

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 on the mathematical background.

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)

Description

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

With rate parameter $\lambda > 0$, the PDF and CDF are

$f(x; \lambda) = \lambda e^{-\lambda x}, \quad x \ge 0$

$F(x; \lambda) = 1 - e^{-\lambda x}$

and the quantile is

$F^{-1}(p; \lambda) = -\frac{\log(1-p)}{\lambda}$