NAG Library Routine Document
S14ADF
1 Purpose
S14ADF returns a sequence of values of scaled derivatives of the psi function (also known as the digamma function).
2 Specification
INTEGER |
N, M, IFAIL |
REAL (KIND=nag_wp) |
X, ANS(M) |
|
3 Description
S14ADF computes
values of the function
for
,
,
, where
is the psi function
and
denotes the
th derivative of
.
The routine is derived from the routine PSIFN in
Amos (1983). The basic method of evaluation of
is the asymptotic series
for large
greater than a machine-dependent value
, followed by backward recurrence using
for smaller values of
, where
when
,
when
, and
,
, are the Bernoulli numbers.
When
is large, the above procedure may be inefficient, and the expansion
which converges rapidly for large
, is used instead.
4 References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502
5 Parameters
- 1: X – REAL (KIND=nag_wp)Input
On entry: the argument of the function.
Constraint:
.
- 2: N – INTEGERInput
On entry: the index of the first member of the sequence of functions.
Constraint:
.
- 3: M – INTEGERInput
On entry: the number of members required in the sequence
, for .
Constraint:
.
- 4: ANS(M) – REAL (KIND=nag_wp) arrayOutput
On exit: the first
elements of
ANS contain the required values
, for
.
- 5: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
,
. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
.
When the value is used it is essential to test the value of IFAIL on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
No results are returned because underflow is likely. Either
X or
is too large. If possible, reduce the value of
M and call S14ADF again.
No results are returned because overflow is likely. Either
X is too small, or
is too large. If possible, reduce the value of
M and call S14ADF again.
No results are returned because there is not enough internal workspace to continue computation.
may be too large. If possible, reduce the value of
M and call S14ADF again.
7 Accuracy
All constants in S14ADF are given to approximately digits of precision. Calling the number of digits of precision in the floating point arithmetic being used , then clearly the maximum number of correct digits in the results obtained is limited by . Empirical tests of S14ADF, taking values of in the range , and in the range , have shown that the maximum relative error is a loss of approximately two decimal places of precision. Tests with , i.e., testing the function , have shown somewhat better accuracy, except at points close to the zero of , , where only absolute accuracy can be obtained.
The time taken for a call of S14ADF is approximately proportional to , plus a constant. In general, it is much cheaper to call S14ADF with greater than to evaluate the function , for , rather than to make separate calls of S14ADF.
9 Example
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s14adfe.f90)
9.2 Program Data
Program Data (s14adfe.d)
9.3 Program Results
Program Results (s14adfe.r)