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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_psi_deriv_real (s14ae)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_psi_deriv_real (s14ae) returns the value of the kth derivative of the psi function ψx for real x and k=0,1,,6, via the function name.


[result, ifail] = s14ae(x, k)
[result, ifail] = nag_specfun_psi_deriv_real(x, k)


nag_specfun_psi_deriv_real (s14ae) evaluates an approximation to the kth derivative of the psi function ψx given by
ψ k x=dkdxk ψx=dkdxk ddx logeΓx ,  
where x is real with x0,-1,-2, and k=0,1,,6. For negative noninteger values of x, the recurrence relationship
ψ k x+1=ψ k x+dkdxk 1x  
is used. The value of -1k+1ψ k x k!  is obtained by a call to nag_specfun_polygamma_deriv (s14ad), which is based on the function PSIFN in Amos (1983).
Note that ψ k x is also known as the polygamma function. Specifically, ψ 0 x is often referred to as the digamma function and ψ 1 x 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. Software 9 494–502


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.
Constraint: x must not be ‘too close’ (see Error Indicators and Warnings) to a non-positive integer.
2:     k int64int32nag_int scalar
The function ψkx to be evaluated.
Constraint: 0k6.

Optional Input Parameters


Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,k<0,
orx is ‘too close’ to a non-positive integer. That is, abs x - nintx < machine precision × nintabsx .
The evaluation has been abandoned due to the likelihood of underflow. The result is returned as zero.
The evaluation has been abandoned due to the likelihood of overflow. The result is returned as zero.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


All constants in nag_specfun_polygamma_deriv (s14ad) are given to approximately 18 digits of precision. If t 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 p=mint,18. 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 -ψ 0 x have shown somewhat improved accuracy, except at points near the positive zero of ψ 0 x at x=1.46, where only absolute accuracy can be obtained.

Further Comments



This example evaluates ψ 2 x at x=2.5, and prints the results.
function s14ae_example

fprintf('s14ae example results\n\n');

x = 2.5;
k = int64(2);
[result, ifail] = s14ae(x, k);

disp('    x     k   (d^K/dx^K)psi(x)');
fprintf('%6.1f%5d    %12.4e\n',x,k,result);

s14ae example results

    x     k   (d^K/dx^K)psi(x)
   2.5    2     -2.3620e-01

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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