Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	scf
Real (Kind=nag_wp), Intent (In)	::	ani, adr, bni, bdr, cni, cdr, x
Real (Kind=nag_wp), Intent (Out)	::	frf

C Header Interface

#include <nag.h>

void	s22bff_ (const double ani, const double adr, const double bni, const double bdr, const double cni, const double cdr, const double x, double frf, Integer scf, Integer ifail)

The routine may be called by the names s22bff or nagf_specfun_hyperg_gauss_real_scaled.

3 Description

s22bff returns a value for the Gauss hypergeometric function

{}_{2}F_{1} (a, b; c; x)

for real parameters

a

b

and

c

, and for real argument

x

The Gauss hypergeometric function is a solution to the hypergeometric differential equation,

x (1 - x) \frac{d^{2} f}{d x^{2}} + (c - (a + b + 1) x) \frac{d f}{d x} - a b f = 0 .

(1)

For

| x | < 1

, it may be defined by the Gauss series,

{}_{2}F_{1} (a, b; c; x) = \sum_{s = 0}^{\infty} \frac{{(a)}_{s} {(b)}_{s}}{{(c)}_{s} s!} x^{s} = 1 + \frac{a b}{c} x + \frac{a (a + 1) b (b + 1)}{c (c + 1) 2!} x^{2} + \dots,

(2)

where

{(a)}_{s} = 1 (a) (a + 1) (a + 2) \dots (a + s - 1)

is the rising factorial of

a

{}_{2}F_{1} (a, b; c; x)

is undefined for

c = 0

c

a negative integer.

For

| x | < 1

, the series is absolutely convergent and

{}_{2}F_{1} (a, b; c; x)

is finite.

For

x < 1

, linear transformations of the form,

{}_{2}F_{1} (a, b; c; x) = C_{1} (a_{1}, b_{1}, c_{1}, x_{1}) {}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1}) + C_{2} (a_{2}, b_{2}, c_{2}, x_{2}) {}_{2}F_{1} (a_{2}, b_{2}; c_{2}; x_{2})

(3)

exist, where

x_{1}

x_{2} \in (0, 1]

C_{1}

and

C_{2}

are real valued functions of the parameters and argument, typically involving products of gamma functions. When these are degenerate, finite limiting cases exist. Hence for

x < 0

{}_{2}F_{1} (a, b; c; x)

is defined by analytic continuation, and for

x < 1

{}_{2}F_{1} (a, b; c; x)

is real and finite.

For

x = 1

, the following apply:

If $c > a + b$ , ${}_{2}F_{1} (a, b; c; 1) = \frac{Γ (c) Γ (c - a - b)}{Γ (c - a) Γ (c - b)}$ , and hence is finite. Solutions also exist for the degenerate cases where $c - a$ or $c - b$ are negative integers or zero.
If $c \leq a + b$ , ${}_{2}F_{1} (a, b; c; 1)$ is infinite, and the sign of ${}_{2}F_{1} (a, b; c; 1)$ is determinable as $x$ approaches $1$ from below.

In the complex plane, the principal branch of

{}_{2}F_{1} (a, b; c; z)

is taken along the real axis from

x = 1.0

increasing.

{}_{2}F_{1} (a, b; c; z)

is multivalued along this branch, and for real parameters

a, b

and

c

is typically not real valued. As such, this routine will not compute a solution when

x > 1

The solution strategy used by this routine is primarily dependent upon the value of the argument

x

. Once trivial cases and the case

x = 1.0

are eliminated, this proceeds as follows.

For

0 < x \leq 0.5

, sets of safe parameters

{α_{i, j}, β_{i, j}, ζ_{i, j}, χ_{j} | 1 \leq j \leq 2 |, 1 \leq i \leq 4}

are determined, such that the values of

{}_{2}F_{1} (a_{j}, b_{j}; c_{j}; x_{j})

required for an appropriate transformation of the type (3) may be calculated either directly or using recurrence relations from the solutions of

{}_{2}F_{1} (α_{i, j}, β_{i, j}; ζ_{i, j}; χ_{j})

. If

c

is positive, then only transformations with

C_{2} = 0.0

will be used, implying only

{}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1})

will be required, with the transformed argument

x_{1} = x

. If

c

is negative, in some cases a transformation with

C_{2} \neq 0.0

will be used, with the argument

x_{2} = 1.0 - x

. The routine then cycles through these sets until acceptable solutions are generated. If no computation produces an accurate answer, the least inaccurate answer is selected to complete the computation. See Section 7.

For

0.5 < x < 1.0

, an identical approach is first used with the argument

x

. Should this fail, a linear transformation resulting in both transformed arguments satisfying

x_{j} = 1.0 - x

is employed, and the above strategy for

0 < x \leq 0.5

is utilized on both components. Further transformations in these sub-computations are however, limited to single terms with no argument transformation.

For

x < 0

, a linear transformation mapping the argument

x

to the interval

(0, 0.5]

is first employed. The strategy for

0 < x \leq 0.5

is then used on each component, including possible further two term transforms. To avoid some degenerate cases, a transform mapping the argument

x

[0.5, 1)

may also be used.

For improved precision in the final result, this routine accepts

a, b

and

c

split into an integral and a decimal fractional component. Specifically,

a = a_{i} + a_{r}

, where

| a_{r} | \leq 0.5

and

a_{i} = a - a_{r}

is integral. The other parameters

b

and

c

are similarly deconstructed.

In addition to the above restrictions on

c

and

x

, an artificial bound, arbnd, is placed on the magnitudes of

a, b, c

and

x

to minimize the occurrence of overflow in internal calculations, particularly those involving real to integer conversions.

arbnd = 0.0001 \times I_{\max}

, where

I_{\max}

is the largest machine integer (see x02bbf). It should however, not be assumed that this routine will produce accurate answers for all values of

a, b, c

and

x

satisfying this criterion.

This routine also tests for non-finite values of the parameters and argument on entry, and assigns non-finite values upon completion if appropriate. See Section 9 and Chapter X07.

Please consult the NIST Digital Library of Mathematical Functions for a detailed discussion of the Gauss hypergeometric function including special cases, transformations, relations and asymptotic approximations.

4 References

NIST Digital Library of Mathematical Functions

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

5 Arguments

1: $ani$ – Real (Kind=nag_wp) Input

On entry:

a_{i}

, the nearest integer to

a

, satisfying

a_{i} = a - a_{r}

Constraints:

$ani = ⌊ ani ⌋$ ;
$| ani | \leq arbnd$ .

2: $adr$ – Real (Kind=nag_wp) Input

On entry:

a_{r}

, the signed decimal remainder satisfying

a_{r} = a - a_{i}

and

| a_{r} | \leq 0.5

Constraint:

| adr | \leq 0.5

3: $bni$ – Real (Kind=nag_wp) Input

On entry:

b_{i}

, the nearest integer to

b

, satisfying

b_{i} = b - b_{r}

Constraints:

$bni = ⌊ bni ⌋$ ;
$| bni | \leq arbnd$ .

4: $bdr$ – Real (Kind=nag_wp) Input

On entry:

b_{r}

, the signed decimal remainder satisfying

b_{r} = b - b_{i}

and

| b_{r} | \leq 0.5

Constraint:

| bdr | \leq 0.5

5: $cni$ – Real (Kind=nag_wp) Input

On entry:

c_{i}

, the nearest integer to

c

, satisfying

c_{i} = c - c_{r}

Constraints:

$cni = ⌊ cni ⌋$ ;
$| cni | \leq arbnd$ ;
if $| cdr | < 16.0 ε$ , $cni \geq 1.0$ .

6: $cdr$ – Real (Kind=nag_wp) Input

On entry:

c_{r}

, the signed decimal remainder satisfying

c_{r} = c - c_{i}

and

| c_{r} | \leq 0.5

Constraint:

| cdr | \leq 0.5

7: $x$ – Real (Kind=nag_wp) Input

On entry: the argument

x

Constraint:

- arbnd < x \leq 1

8: $frf$ – Real (Kind=nag_wp) Output

On exit:

f_{fr}

, the scaled real component of the solution satisfying

f_{fr} = {}_{2}F_{1} (a, b; c; x) \times 2^{- f_{sc}}

, i.e.,

{}_{2}F_{1} (a, b; c; x) = f_{fr} \times 2^{f_{sc}}

. See Section 9 for the behaviour of

f_{fr}

when a finite or non-finite answer is returned.

9: $scf$ – Integer Output

On exit:

f_{sc}

, the scaling power of two, satisfying

f_{sc} = \log_{2} (\frac{{}_{2}F_{1} (a, b; c; x)}{f_{fr}})

, i.e.,

{}_{2}F_{1} (a, b; c; x) = f_{fr} \times 2^{f_{sc}}

. See Section 9 for the behaviour of

f_{sc}

when a non-finite answer is returned.

10: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: Underflow occurred during the evaluation of ${}_{2}F_{1} (a, b; c; x)$ . The returned value may be inaccurate.

$ifail = 2$: All approximations have completed, and the final residual estimate indicates some precision may have been lost.
$Relative residual = ⟨ value ⟩$ .

$ifail = 3$: All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.
$Relative residual = ⟨ value ⟩$ .

$ifail = 4$: On entry, $x = ⟨ value ⟩$ , $c = ⟨ value ⟩$ , $a + b = ⟨ value ⟩$ .
${}_{2}F_{1} (a, b; c; 1)$ is infinite in the case $c \leq a + b$ .

$ifail = 5$: On completion, overflow occurred in the evaluation of ${}_{2}F_{1} (a, b; c; x)$ .

$ifail = 6$: Overflow occurred in a subcalculation of ${}_{2}F_{1} (a, b; c; x)$ . The answer may be completely incorrect.

$ifail = 9$: An internal calculation has resulted in an undefined result.

$ifail = 11$: On entry, ani does not satisfy $| ani | \leq arbnd = ⟨ value ⟩$ .

$ifail = 13$: ani is non-integral.
On entry, $ani = ⟨ value ⟩$ .
Constraint: $ani = ⌊ ani ⌋$ .

$ifail = 21$: On entry, adr does not satisfy $| adr | \leq 0.5$ .

$ifail = 31$: On entry, bni does not satisfy $| bni | \leq arbnd = ⟨ value ⟩$ .

$ifail = 33$: bni is non-integral.
On entry, $bni = ⟨ value ⟩$ .
Constraint: $bni = ⌊ bni ⌋$ .

$ifail = 41$: On entry, bdr does not satisfy $| bdr | \leq 0.5$ .

$ifail = 51$: On entry, cni does not satisfy $| cni | \leq arbnd = ⟨ value ⟩$ .

$ifail = 52$: On entry, $c = cni + cdr = ⟨ value ⟩$ .
${}_{2}F_{1} (a, b; c; x)$ is undefined when $c$ is zero or a negative integer.

$ifail = 53$: cni is non-integral.
On entry, $cni = ⟨ value ⟩$ .
Constraint: $cni = ⌊ cni ⌋$ .

$ifail = 61$: On entry, cdr does not satisfy $| cdr | \leq 0.5$ .

$ifail = 71$: On entry, x does not satisfy $| x | \leq arbnd = ⟨ value ⟩$ .

$ifail = 72$: On entry, $x = ⟨ value ⟩$ .
In general, ${}_{2}F_{1} (a, b; c; x)$ is not real valued when $x > 1$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

In general, if

ifail = 0

, the value of

{}_{2}F_{1} (a, b; c; x)

may be assumed accurate, with the possible loss of one or two decimal places. Assuming the result does not overflow, an error estimate

res

is made internally using equation (1). If the magnitude of this residual

res

is sufficiently large, a nonzero ifail will be returned. Specifically,

$ifail = 0$ or $1$	$res \leq 1000 ε$
$ifail = 2$	$1000 ε < res \leq 0.1$
$ifail = 3$	$res > 0.1$

where

ε

is the machine precision as returned by x02ajf. Note that underflow may also have occurred if

ifail = 2

3

A further estimate of the residual can be constructed using equation (1), and the differential identity,

\begin{matrix} \frac{d ({}_{2}F_{1} (a, b; c; x))}{d x} = \frac{a b}{c} {}_{2}F_{1} (a + 1, b + 1; c + 1; x) \\ \frac{d^{2} ({}_{2}F_{1} (a, b; c; x))}{d x^{2}} = \frac{a (a + 1) b (b + 1)}{c (c + 1)} {}_{2}F_{1} (a + 2, b + 2; c + 2; x) \end{matrix}

(4)

This estimate is however, dependent upon the error involved in approximating

{}_{2}F_{1} (a + 1, b + 1; c + 1; x)

and

{}_{2}F_{1} (a + 2, b + 2; c + 2; x)

8 Parallelism and Performance

s22bff is not threaded in any implementation.

9 Further Comments

s22bff returns non-finite values when appropriate. See Chapter X07 for more information on the definitions of non-finite values.

Should a non-finite value be returned, this will be indicated in the value of ifail, as detailed in the following cases.

ifail = 0

ifail = 1

2

3

, a finite value will have been returned with approximate accuracy as detailed in Section 7.

The values of

f_{f r}

and

f_{s c}

are implementation dependent. In most cases, if

{}_{2}F_{1} (a, b; c; x) = 0

f_{f r} = 0

and

f_{s c} = 0

will be returned, and if

{}_{2}F_{1} (a, b; c; x)

is finite, the fractional component will be bound by

0.5 \leq | f_{f r} | < 1

, with

f_{s c}

chosen accordingly.

The values returned in frf (

f_{fr}

) and scf (

f_{sc}

) may be used to explicitly evaluate

{}_{2}F_{1} (a, b; c; x)

, and may also be used to evaluate products and ratios of multiple values of

{}_{2}F_{1}

as follows,

\begin{matrix} {}_{2}F_{1} (a, b; c; x) = f_{fr} \times 2^{f_{sc}} \\ {}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1}) \times {}_{2}F_{1} (a_{2}, b_{2}; c_{2}; x_{2}) = (f_{fr 1} \times f_{fr 2}) \times 2^{(f_{sc 1} + f_{sc 2})} \\ \frac{{}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1})}{{}_{2}F_{1} (a_{2}, b_{2}; c_{2}; x_{2})} = \frac{f_{fr 1}}{f_{fr 2}} \times 2^{(f_{sc 1} - f_{sc 2})} \\ \ln | {}_{2}F_{1} (a, b; c; x) | = \ln | f_{fr} | + f_{sc} \times \ln (2) . \end{matrix}

ifail = 4

then

{}_{2}F_{1} (a, b; c; x)

is infinite. A signed infinity will have been returned for frf, and

scf = 0

. The sign of frf should be correct when taking the limit as

x

approaches

1

from below.

ifail = 5

then upon completion,

| {}_{2}F_{1} (a, b; c; x) | > 2^{I_{\max}}

, where

I_{\max}

is given by x02bbf, and hence is too large to be representable even in the scaled form. The scaled real component returned in frf may still be correct, whilst

scf = I_{\max}

will have been returned.

ifail = 6

then overflow occurred during a subcalculation of

{}_{2}F_{1} (a, b; c; x)

. The same result as for

ifail = 5

will have been returned, however, there is no guarantee that this is representative of either the magnitude of the scaling power

f_{sc}

, or the scaled component

f_{fr}

{}_{2}F_{1} (a, b; c; x)

For all other error exits,

scf = 0

will be returned and frf will be returned as a signalling NaN (see x07bbf).

ifail = 9

an internal computation produced an undefined result. This may occur when two terms overflow with opposite signs, and the result is dependent upon their summation for example.

ifail = 52

then

c

is too close to a negative integer or zero on entry, and

{}_{2}F_{1} (a, b; c; x)

is undefined. Note, this will also be the case when

c

is a negative integer, and a (possibly trivial) linear transformation of the form (3) would result in either:

(i)all $c_{j}$ not being negative integers,
(ii)for any $c_{j}$ which remain as negative integers, one of the corresponding parameters $a_{j}$ or $b_{j}$ is a negative integer of magnitude less than $c_{j}$ .

In the first case, the transformation coefficients

C_{j} (a_{j}, b_{j}, c_{j}, x_{j})

are typically either infinite or undefined, preventing a solution being constructed. In the second case, the series (2) will terminate before the degenerate term, resulting in a polynomial of fixed degree, and hence potentially a finite solution.

ifail = 11

31

51

71

then no computation will have been performed due to the risk of integer overflow. The actual solution may however, be finite.

ifail = 72

indicates

x > 1

, and hence the requested solution is on the boundary of the principal branch of

{}_{2}F_{1} (a, b; c; x)

. Hence it is multivalued, typically with a nonzero imaginary component. It is however, strictly finite.

10 Example

This example evaluates the Gauss hypergeometric function at two points in scaled form using s22bff, and subsequently calculates their product and ratio implicitly.