The routine may be called by the names s14aef or nagf_specfun_psi_deriv_real.
s14aef evaluates an approximation to the th derivative of the psi function given by
where is real with and . For negative noninteger values of , the recurrence relationship
is used. The value of is obtained by a call to s14adf, which is based on the routine PSIFN in Amos (1983).
Note that is also known as the polygamma function. Specifically, is often referred to as the digamma function and as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).
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. Software9 494–502
1: – Real (Kind=nag_wp)Input
On entry: the argument of the function.
must not be ‘too close’ (see Section 6) to a non-positive integer.
2: – IntegerInput
On entry: the function to be evaluated.
3: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or 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).
6Error 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:
On entry, .
On entry, .
On entry, x is ‘too close’ to a non-positive integer: and .
Evaluation abandoned due to likelihood of underflow.
Evaluation abandoned due to likelihood of overflow.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
All constants in s14adf are given to approximately digits of precision. If denotes the number of digits of precision in the floating-point arithmetic being used, then clearly the maximum number in the results obtained is limited by . Empirical tests by Amos (1983) have shown that the maximum relative error is a loss of approximately two decimal places of precision. Further tests with the function have shown somewhat improved accuracy, except at points near the positive zero of at , where only absolute accuracy can be obtained.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s14aef is not threaded in any implementation.
This example evaluates at , and prints the results.