NAG CL Interface
s22bbc (hyperg_confl_real_scaled)
1
Purpose
s22bbc returns a value for the confluent hypergeometric function , with real parameters and and real argument . The solution is returned in the scaled form . This function is sometimes also known as Kummer's function .
2
Specification
The function may be called by the names: s22bbc, nag_specfun_hyperg_confl_real_scaled or nag_specfun_1f1_real_scaled.
3
Description
s22bbc returns a value for the confluent hypergeometric function , with real parameters and and real argument , in the scaled form , where is the real scaled component and is the integer power of two scaling. This function is unbounded or not uniquely defined for equal to zero or a negative integer.
The confluent hypergeometric function is defined by the confluent series,
where
is the rising factorial of
.
is a solution to the second order ODE (Kummer's Equation):
Given the parameters and argument , this function determines a set of safe values and selects an appropriate algorithm to accurately evaluate the functions . The result is then used to construct the solution to the original problem using, where necessary, recurrence relations and/or continuation.
For improved precision in the final result, this function accepts and split into an integral and a decimal fractional component. Specifically , where and is integral. is similarly deconstructed.
Additionally, an artificial bound, is placed on the magnitudes of , and to minimize the occurrence of overflow in internal calculations. , where . It should, however, not be assumed that this function will produce an accurate result for all values of , and 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:
– double
Input
-
On entry: , the nearest integer to , satisfying .
-
2:
– double
Input
-
On entry: , the signed decimal remainder satisfying and .
Constraint:
.
Note: if
,
will be used, where
is the
machine precision as returned by
X02AJC.
-
3:
– double
Input
-
On entry: , the nearest integer to , satisfying .
Constraints:
- ;
- ;
- if , .
-
4:
– double
Input
-
On entry: , the signed decimal remainder satisfying and .
Constraint:
.
Note: if
,
will be used, where
is the
machine precision as returned by
X02AJC.
-
5:
– double
Input
-
On entry: the argument of the function.
Constraint:
.
-
6:
– double *
Output
-
On exit:
, the scaled real component of the solution satisfying
.
Note: if overflow occurs upon completion, as indicated by
NW_OVERFLOW_WARN, the value of
returned may still be correct. If overflow occurs in a subcalculation, as indicated by
NE_OVERFLOW, this should not be assumed.
-
7:
– Integer *
Output
-
On exit:
, the scaling power of two, satisfying
.
Note: if overflow occurs upon completion, as indicated by
NW_OVERFLOW_WARN, then
, where
is the largest representable integer (see
X02BBC). If overflow occurs during a subcalculation, as indicated by
NE_OVERFLOW,
may or may not be greater than
. In either case,
will have been returned.
-
8:
– 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 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 .
The answer may be completely incorrect.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_REAL_2
-
On entry, .
is undefined when is zero or a negative integer.
- NE_REAL_ARG_NON_INTEGRAL
-
ani is non-integral.
On entry,
.
Constraint:
.
bni is non-integral.
On entry,
.
Constraint:
.
- NE_REAL_RANGE_CONS
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_TOTAL_PRECISION_LOSS
-
All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.
Relative residual .
- NW_OVERFLOW_WARN
-
On completion, overflow occurred in the evaluation of .
- NW_SOME_PRECISION_LOSS
-
All approximations have completed, and the final residual estimate indicates some precision may have been lost.
Relative residual .
- NW_UNDERFLOW_WARN
-
Underflow occurred during the evaluation of .
The returned value may be inaccurate.
7
Accuracy
In general, if
NE_NOERROR, the value of
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
is made internally using equation
(1). If the magnitude of
is sufficiently large a
different
fail.code
will be returned. Specifically,
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 and .
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
s22bbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
s22bbc 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 implementation-specific information.
The values of and are implementation dependent. In most cases, if , and will be returned, and if is finite, the fractional component will be bound by , with chosen accordingly.
The values returned in
frm (
) and
scm (
) may be used to explicitly evaluate
, and may also be used to evaluate products and ratios of multiple values of
as follows,
10
Example
This example evaluates the confluent hypergeometric function at two points in scaled form using s22bbc, and subsequently calculates their product and ratio without having to explicitly construct .
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results