NAG Library Routine Document

1Purpose

s22bbf returns a value for the confluent hypergeometric function ${}_{1}F_{1}\left(a;b;x\right)$, with real parameters $a$ and $b$ and real argument $x$. The solution is returned in the scaled form ${}_{1}F_{1}\left(a;b;x\right)={m}_{f}×{2}^{{m}_{s}}$. This function is sometimes also known as Kummer's function $M\left(a,b,x\right)$.

2Specification

Fortran Interface
 Subroutine s22bbf ( ani, adr, bni, bdr, x, frm, scm,
 Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: scm Real (Kind=nag_wp), Intent (In) :: ani, adr, bni, bdr, x Real (Kind=nag_wp), Intent (Out) :: frm
#include nagmk26.h
 void s22bbf_ (const double *ani, const double *adr, const double *bni, const double *bdr, const double *x, double *frm, Integer *scm, Integer *ifail)

3Description

s22bbf returns a value for the confluent hypergeometric function ${}_{1}F_{1}\left(a;b;x\right)$, with real parameters $a$ and $b$ and real argument $x$, in the scaled form ${}_{1}F_{1}\left(a;b;x\right)={m}_{f}×{2}^{{m}_{s}}$, where ${m}_{f}$ is the real scaled component and ${m}_{s}$ is the integer power of two scaling. This function is unbounded or not uniquely defined for $b$ equal to zero or a negative integer.
The confluent hypergeometric function is defined by the confluent series,
 $F1 1 a;b;x = Ma,b,x = ∑ s=0 ∞ as xs bs s! = 1 + a b x + aa+1 bb+1 2! x2 + ⋯$
where ${\left(a\right)}_{s}=1\left(a\right)\left(a+1\right)\left(a+2\right)\dots \left(a+s-1\right)$ is the rising factorial of $a$. $M\left(a,b,x\right)$ is a solution to the second order ODE (Kummer's Equation):
 $x d2M dx2 + b-x dM dx - a M = 0 .$ (1)
Given the parameters and argument $\left(a,b,x\right)$, this routine determines a set of safe values $\left\{\left({\alpha }_{i},{\beta }_{i},{\zeta }_{i}\right)\mid i\le 2\right\}$ and selects an appropriate algorithm to accurately evaluate the functions ${M}_{i}\left({\alpha }_{i},{\beta }_{i},{\zeta }_{i}\right)$. The result is then used to construct the solution to the original problem $M\left(a,b,x\right)$ using, where necessary, recurrence relations and/or continuation.
For improved precision in the final result, this routine accepts $a$ and $b$ split into an integral and a decimal fractional component. Specifically $a={a}_{i}+{a}_{r}$, where $\left|{a}_{r}\right|\le 0.5$ and ${a}_{i}=a-{a}_{r}$ is integral. $b$ is similarly deconstructed.
Additionally, an artificial bound, $\mathit{arbnd}$ is placed on the magnitudes of ${a}_{i}$, ${b}_{i}$ and $x$ to minimize the occurrence of overflow in internal calculations. $\mathit{arbnd}=0.0001×{I}_{\mathrm{max}}$, where ${I}_{\mathrm{max}}={\mathbf{x02bbf}}$. It should, however, not be assumed that this routine will produce an accurate result for all values of ${a}_{i}$, ${b}_{i}$ and $x$ satisfying this criterion.
Please consult the NIST Digital Library of Mathematical Functions for a detailed discussion of the confluent hypergeometric function including special cases, transformations, relations and asymptotic approximations.

4References

NIST Digital Library of Mathematical Functions
Pearson J (2009) Computation of hypergeometric functions MSc Dissertation, Mathematical Institute, University of Oxford

5Arguments

1:     $\mathbf{ani}$ – Real (Kind=nag_wp)Input
On entry: ${a}_{i}$, the nearest integer to $a$, satisfying ${a}_{i}=a-{a}_{r}$.
Constraints:
• ${\mathbf{ani}}=⌊{\mathbf{ani}}⌋$;
• $\left|{\mathbf{ani}}\right|\le \mathit{arbnd}$.
2:     $\mathbf{adr}$ – Real (Kind=nag_wp)Input
On entry: ${a}_{r}$, the signed decimal remainder satisfying ${a}_{r}=a-{a}_{i}$ and $\left|{a}_{r}\right|\le 0.5$.
Constraint: $\left|{\mathbf{adr}}\right|\le 0.5$.
Note: if $\left|{\mathbf{adr}}\right|<100.0\epsilon$, ${a}_{r}=0.0$ will be used, where $\epsilon$ is the machine precision as returned by x02ajf.
3:     $\mathbf{bni}$ – Real (Kind=nag_wp)Input
On entry: ${b}_{i}$, the nearest integer to $b$, satisfying ${b}_{i}=b-{b}_{r}$.
Constraints:
• ${\mathbf{bni}}=⌊{\mathbf{bni}}⌋$;
• $\left|{\mathbf{bni}}\right|\le \mathit{arbnd}$;
• if ${\mathbf{bdr}}=0.0$, ${\mathbf{bni}}>0$.
4:     $\mathbf{bdr}$ – Real (Kind=nag_wp)Input
On entry: ${b}_{r}$, the signed decimal remainder satisfying ${b}_{r}=b-{b}_{i}$ and $\left|{b}_{r}\right|\le 0.5$.
Constraint: $\left|{\mathbf{bdr}}\right|\le 0.5$.
Note: if $\left|{\mathbf{bdr}}-{\mathbf{adr}}\right|<100.0\epsilon$, ${a}_{r}={b}_{r}$ will be used, where $\epsilon$ is the machine precision as returned by x02ajf.
5:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: $\left|{\mathbf{x}}\right|\le \mathit{arbnd}$.
6:     $\mathbf{frm}$ – Real (Kind=nag_wp)Output
On exit: ${m}_{f}$, the scaled real component of the solution satisfying ${m}_{f}=M\left(a,b,x\right)×{2}^{-{m}_{s}}$.
Note: if overflow occurs upon completion, as indicated by ${\mathbf{ifail}}={\mathbf{2}}$, the value of ${m}_{f}$ returned may still be correct. If overflow occurs in a subcalculation, as indicated by ${\mathbf{ifail}}={\mathbf{5}}$, this should not be assumed.
7:     $\mathbf{scm}$ – IntegerOutput
On exit: ${m}_{s}$, the scaling power of two, satisfying ${m}_{s}={\mathrm{log}}_{2}\left(\frac{M\left(a,b,x\right)}{{m}_{f}}\right)$.
Note: if overflow occurs upon completion, as indicated by ${\mathbf{ifail}}={\mathbf{2}}$, then ${m}_{s}\ge {I}_{\mathrm{max}}$, where ${I}_{\mathrm{max}}$ is the largest representable integer (see x02bbf). If overflow occurs during a subcalculation, as indicated by ${\mathbf{ifail}}={\mathbf{5}}$, ${m}_{s}$ may or may not be greater than ${I}_{\mathrm{max}}$. In either case, ${\mathbf{scm}}={\mathbf{x02bbf}}$ will have been returned.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
Underflow occurred during the evaluation of $M\left(a,b,x\right)$.
The returned value may be inaccurate.
${\mathbf{ifail}}=2$
On completion, overflow occurred in the evaluation of $M\left(a,b,x\right)$.
${\mathbf{ifail}}=3$
All approximations have completed, and the final residual estimate indicates some precision may have been lost.
Relative residual $\text{}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=4$
All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.
Relative residual $\text{}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=5$
Overflow occurred in a subcalculation of $M\left(a,b,x\right)$.
The answer may be completely incorrect.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{ani}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{ani}}\right|\le \mathit{arbnd}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=13$
ani is non-integral.
On entry, ${\mathbf{ani}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ani}}=⌊{\mathbf{ani}}⌋$.
${\mathbf{ifail}}=21$
On entry, ${\mathbf{adr}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{adr}}\right|\le 0.5$.
${\mathbf{ifail}}=31$
On entry, ${\mathbf{bni}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{bni}}\right|\le \mathit{arbnd}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=32$
On entry, $b={\mathbf{bni}}+{\mathbf{bdr}}=〈\mathit{\text{value}}〉$.
$M\left(a,b,x\right)$ is undefined when $b$ is zero or a negative integer.
${\mathbf{ifail}}=33$
bni is non-integral.
On entry, ${\mathbf{bni}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{bni}}=⌊{\mathbf{bni}}⌋$.
${\mathbf{ifail}}=41$
On entry, ${\mathbf{bdr}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{bdr}}\right|\le 0.5$.
${\mathbf{ifail}}=51$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\le \mathit{arbnd}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

In general, if ${\mathbf{ifail}}={\mathbf{0}}$, the value of $M$ may be assumed accurate, with the possible loss of one or two decimal places. Assuming the result does not under or overflow, an error estimate $\mathit{res}$ is made internally using equation (1). If the magnitude of $\mathit{res}$ is sufficiently large a nonzero ifail will be returned. Specifically,
 ${\mathbf{ifail}}={\mathbf{0}}$ $\mathit{res}\le 1000\epsilon$ ${\mathbf{ifail}}={\mathbf{3}}$ $1000\epsilon <\mathit{res}\le 0.1$ ${\mathbf{ifail}}={\mathbf{4}}$ $\mathit{res}>0.1$
A further estimate of the residual can be constructed using equation (1), and the differential identity,
 $d Ma,b,x dx = ab M a+1,b+1,x , d2 Ma,b,x dx2 = aa+1 bb+1 M a+2,b+2,x .$
This estimate is however dependent upon the error involved in approximating $M\left(a+1,b+1,x\right)$ and $M\left(a+2,b+2,x\right)$.

8Parallelism and Performance

s22bbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
s22bbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The values of ${m}_{f}$ and ${m}_{s}$ are implementation dependent. In most cases, if ${}_{1}F_{1}\left(a;b;x\right)=0$, ${m}_{f}=0$ and ${m}_{s}=0$ will be returned, and if ${}_{1}F_{1}\left(a;b;x\right)=0$ is finite, the fractional component will be bound by $0.5\le \left|{m}_{f}\right|<1$, with ${m}_{s}$ chosen accordingly.
The values returned in frm (${m}_{f}$) and scm (${m}_{s}$) may be used to explicitly evaluate $M\left(a,b,x\right)$, and may also be used to evaluate products and ratios of multiple values of $M$ as follows,
 $Ma,b,x = mf × 2ms M a1,b1,x1 × M a2,b2,x2 = mf1 × mf2 × 2 ms1 + ms2 M a1,b1,x1 M a2,b2,x2 = mf1 mf2 × 2 ms1 - ms2 ln M a,b,x = lnmf + ms × ln2 .$

10Example

This example evaluates the confluent hypergeometric function at two points in scaled form using s22bbf, and subsequently calculates their product and ratio without having to explicitly construct $M$.

10.1Program Text

Program Text (s22bbfe.f90)

None.

10.3Program Results

Program Results (s22bbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017