NAG CL Interfaces14cqc (beta_​incomplete_​vector)

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1Purpose

s14cqc computes an array of values for the regularized incomplete beta function ${I}_{x}\left(a,b\right)$ and its complement $1-{I}_{x}\left(a,b\right)$.

2Specification

 #include
 void s14cqc (Integer n, const double a[], const double b[], const double x[], double w[], double w1[], Integer ivalid[], NagError *fail)
The function may be called by the names: s14cqc, nag_specfun_beta_incomplete_vector or nag_incomplete_beta_vector.

3Description

s14cqc evaluates the regularized incomplete beta function ${I}_{x}\left(a,b\right)$ and its complement $1–{I}_{x}\left(a,b\right)$ in the normalized form, for arrays of arguments ${x}_{i}$, ${a}_{i}$ and ${b}_{i}$, for $\mathit{i}=1,2,\dots ,n$. The incomplete beta function and its complement are given by
 $Ix(a,b) = 1 B(a,b) ∫ 0 x ta-1 (1-t) b-1 dt 1–Ix (a,b) = Iy (b,a) , where ​ y=1-x ,$
with
• $0\le x\le 1$,
• $a\ge 0$ and $b\ge 0$,
• and the beta function $B\left(a,b\right)$ is defined as $B\left(a,b\right)=\underset{0}{\overset{1}{\int }}{t}^{a-1}{\left(1-t\right)}^{b-1}dt=\frac{\Gamma \left(a\right)\Gamma \left(b\right)}{\Gamma \left(a+b\right)}$ where $\Gamma \left(y\right)$ is the gamma function.
Several methods are used to evaluate the functions depending on the arguments $a$, $b$ and $x$. The methods include Wise's asymptotic expansion (see Wise (1950)) when $a>b$, continued fraction derived by DiDonato and Morris (1992) when $a$, $b>1$, and power series when $b\le 1$ or $b×x\le 0.7$. When both $a$ and $b$ are large, specifically $a$, $b\ge 15$, the DiDonato and Morris (1992) asymptotic expansion is employed for greater efficiency.
Once either ${I}_{x}\left(a,b\right)$ or ${I}_{y}\left(b,a\right)$ is computed, the other is obtained by subtraction from $1$. In order to avoid loss of relative precision in this subtraction, the smaller of ${I}_{x}\left(a,b\right)$ and ${I}_{y}\left(b,a\right)$ is computed first.
s14cqc is derived from BRATIO in DiDonato and Morris (1992).

4References

DiDonato A R and Morris A H (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios ACM Trans. Math. Software 18 360–373
Wise M E (1950) The incomplete beta function as a contour integral and a quickly converging series for its inverse Biometrika 37 208–218

5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${a}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{a}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{b}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${b}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraints:
• ${\mathbf{b}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$;
• ${\mathbf{b}}\left[\mathit{i}-1\right]\ne 0.0$ or ${\mathbf{a}}\left[\mathit{i}-1\right]\ne 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
4: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: ${x}_{\mathit{i}}$, the upper limit of integration, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraints:
• ${\mathbf{x}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$;
• ${\mathbf{x}}\left[\mathit{i}-1\right]\ne 0.0$ or ${\mathbf{a}}\left[\mathit{i}-1\right]\ne 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$;
• $1-{\mathbf{x}}\left[\mathit{i}-1\right]\ne 0.0$ or ${\mathbf{b}}\left[\mathit{i}-1\right]\ne 0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
5: $\mathbf{w}\left[{\mathbf{n}}\right]$double Output
On exit: the values of the incomplete beta function ${I}_{{x}_{i}}\left({a}_{i},{b}_{i}\right)$ evaluated from zero to ${x}_{i}$.
6: $\mathbf{w1}\left[{\mathbf{n}}\right]$double Output
On exit: the values of the complement of the incomplete beta function $1-{I}_{{x}_{i}}\left({a}_{i},{b}_{i}\right)$, i.e., the incomplete beta function evaluated from ${x}_{i}$ to one.
7: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for the $\mathit{i}$th evaluation, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${a}_{i}\text{​ or ​}{b}_{i}<0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
Both ${a}_{i}\text{​ and ​}{b}_{i}=0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
${x}_{i}\notin \left[0,1\right]$.
${\mathbf{ivalid}}\left[i-1\right]=4$
Both ${x}_{i}\text{​ and ​}{a}_{i}=0$.
${\mathbf{ivalid}}\left[i-1\right]=5$
Both $1-{x}_{i}\text{​ and ​}{b}_{i}=0$.
8: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one argument had an invalid value.

7Accuracy

s14cqc is designed to maintain relative accuracy for all arguments. For very tiny results (of the order of machine precision or less) some relative accuracy may be lost – loss of three or four decimal places has been observed in experiments. For other arguments full relative accuracy may be expected.

8Parallelism and Performance

s14cqc is not threaded in any implementation.

None.

10Example

This example reads $10$ values for each vector argument $a$, $b$ and $x$ from a file. It then evaluates the function and its complement for each set of values.

10.1Program Text

Program Text (s14cqce.c)

10.2Program Data

Program Data (s14cqce.d)

10.3Program Results

Program Results (s14cqce.r)