NAG CL Interface
s22bac (hyperg_confl_real)
1
Purpose
s22bac 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 $x$. This function is sometimes also known as Kummer's function $M\left(a,b,x\right)$.
2
Specification
void 
s22bac (double a,
double b,
double x,
double *m,
NagError *fail) 

The function may be called by the names: s22bac, nag_specfun_hyperg_confl_real or nag_specfun_1f1_real.
3
Description
s22bac 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 $x$. This function is unbounded or not uniquely defined for $b$ equal to zero or a negative integer.
The associated function
s22bbc performs the same operations, but returns
$M$ in the scaled form
$M={m}_{f}\times {2}^{{m}_{s}}$ to allow calculations to be performed when
$M$ is not representable as a single working precision number. It also accepts the parameters
$a$ and
$b$ as summations of an integer and a decimal fraction, giving higher accuracy when
$a$ or
$b$ are close to an integer. In such cases,
s22bbc should be used when high accuracy is required.
The confluent hypergeometric function is defined by the confluent series
where
${\left(a\right)}_{s}=1\left(a\right)\left(a+1\right)\left(a+2\right)\dots \left(a+s1\right)$ is the rising factorial of
$a$.
$M\left(a,b,x\right)$ is a solution to the second order ODE (Kummer's Equation):
Given the parameters and argument $\left(a,b,x\right)$, this function 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.
Additionally, an artificial bound, $\mathit{arbnd}$ is placed on the magnitudes of $a$, $b$ and $x$ to minimize the occurrence of overflow in internal calculations. $\mathit{arbnd}=0.0001\times {I}_{\mathrm{max}}$, where ${I}_{\mathrm{max}}={\mathbf{X02BBC}}$. It should, however, not be assumed that this function will produce an accurate result for all values of $a$, $b$ 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.
4
References
Pearson J (2009) Computation of hypergeometric functions MSc Dissertation, Mathematical Institute, University of Oxford
5
Arguments

1:
$\mathbf{a}$ – double
Input

On entry: the parameter $a$ of the function.
Constraint:
$\left{\mathbf{a}}\right\le \mathit{arbnd}$.

2:
$\mathbf{b}$ – double
Input

On entry: the parameter $b$ of the function.
Constraint:
$\left{\mathbf{b}}\right\le \mathit{arbnd}$.

3:
$\mathbf{x}$ – double
Input

On entry: the argument $x$ of the function.
Constraint:
$\left{\mathbf{x}}\right\le \mathit{arbnd}$.

4:
$\mathbf{m}$ – double *
Output

On exit: the solution
$M\left(a,b,x\right)$.
Note: if overflow occurs upon completion, as indicated by
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_OVERFLOW_WARN,
$\leftM\left(a,b,x\right)\right$ may be assumed to be too large to be representable.
m will be returned as
$\pm {R}_{\mathrm{max}}$, where
${R}_{\mathrm{max}}$ is the largest representable real number (see
X02ALC). The sign of
m should match the sign of
$M\left(a,b,x\right)$. If overflow occurs during a subcalculation, as indicated by
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_OVERFLOW, the sign may be incorrect, and the true value of
$M\left(a,b,x\right)$ may or may not be greater than
${R}_{\mathrm{max}}$. In either case it is advisable to subsequently use
s22bbc.

5:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error 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.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 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.
 NE_OVERFLOW

Overflow occurred in a subcalculation of $M\left(a,b,x\right)$.
The answer may be completely incorrect.
 NE_REAL

On entry, ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
$M\left(a,b,x\right)$ is undefined when $b$ is zero or a negative integer.
 NE_REAL_RANGE_CONS

On entry, ${\mathbf{a}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $\left{\mathbf{a}}\right\le \mathit{arbnd}=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $\left{\mathbf{b}}\right\le \mathit{arbnd}=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{x}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $\left{\mathbf{x}}\right\le \mathit{arbnd}=\u2329\mathit{\text{value}}\u232a$.
 NE_TOTAL_PRECISION_LOSS

All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.
Relative residual $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NW_OVERFLOW_WARN

On completion, overflow occurred in the evaluation of $M\left(a,b,x\right)$.
 NW_SOME_PRECISION_LOSS

All approximations have completed, and the final residual estimate indicates some precision may have been lost.
Relative residual $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NW_UNDERFLOW_WARN

Underflow occurred during the evaluation of $M\left(a,b,x\right)$.
The returned value may be inaccurate.
7
Accuracy
In general, if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, 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
different
fail.code
will be returned. Specifically,
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR 
$\mathit{res}\le 1000\epsilon $ 
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NW_SOME_PRECISION_LOSS 
$1000\epsilon <\mathit{res}\le 0.1$ 
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOTAL_PRECISION_LOSS 
$\mathit{res}>0.1$ 
where
$\epsilon $ is the
machine precision as returned by
X02AJC.
A further estimate of the residual can be constructed using equation
(1), and the differential identity,
This estimate is however, dependent upon the error involved in approximating $M\left(a+1,b+1,x\right)$ and $M\left(a+2,b+2,x\right)$.
Furthermore, the accuracy of the solution, and the error estimate, can be dependent upon the accuracy of the decimal fraction of the input parameters
$a$ and
$b$. For example, if
$b={b}_{i}+{b}_{r}=100+\text{1.0e\u22126}$, then on a machine with
$16$ decimal digits of precision, the internal calculation of
${b}_{r}$ will only be accurate to
$8$ decimal places. This can subsequently pollute the final solution by several decimal places without affecting the residual estimate as greatly. Should you require higher accuracy in such regions, then you should use
s22bbc, which requires you to supply the correct decimal fraction.
8
Parallelism and Performance
s22bac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
s22bac 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 function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
None.
10
Example
This example prints the results returned by s22bac called using parameters $a=13.6$ and $b=14.2$ with $11$ differing values of argument $x$.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results