The function may be called by the names: s14ccc, nag_specfun_beta_incomplete or nag_incomplete_beta.
3Description
s14ccc evaluates the regularized incomplete beta function and its complement in the normalized form
with
,
and ,
and the beta function is defined as where is the gamma function.
Several methods are used to evaluate the functions depending on the arguments , and . The methods include Wise's asymptotic expansion (see Wise (1950)) when , continued fraction derived by DiDonato and Morris (1992) when , , and power series when or . When both and are large, specifically , , the DiDonato and Morris (1992) asymptotic expansion is employed for greater efficiency.
Once either or is computed, the other is obtained by subtraction from . In order to avoid loss of relative precision in this subtraction, the smaller of and is computed first.
DiDonato A R and Morris A H (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios ACM Trans. Math. Software18 360–373
Wise M E (1950) The incomplete beta function as a contour integral and a quickly converging series for its inverse Biometrika37 208–218
5Arguments
1: – doubleInput
On entry: the argument of the function.
Constraint:
.
2: – doubleInput
On entry: the argument of the function.
Constraints:
;
either or .
3: – doubleInput
On entry: , upper limit of integration.
Constraints:
;
either or ;
either or .
4: – double *Output
On exit: the value of the incomplete beta function evaluated from zero to .
5: – double *Output
On exit: the value of the complement of the incomplete beta function , i.e., the incomplete beta function evaluated from to one.
6: – 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.
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_REAL
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
NE_REAL_2
On entry, and b were zero.
Constraint: or b must be nonzero.
On entry, a and b were zero.
Constraint: a or b must be nonzero.
On entry, x and a were zero.
Constraint: x or a must be nonzero.
7Accuracy
s14ccc 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
Background information to multithreading can be found in the Multithreading documentation.
s14ccc is not threaded in any implementation.
9Further Comments
None.
10Example
This example reads values of the arguments and from a file, evaluates the function and its complement for different values of and prints the results.