NAG Library Routine Document

f01kkf (complex_gen_matrix_frcht_log)


f01kkf computes the Fréchet derivative LA,E of the matrix logarithm of the complex n by n matrix A applied to the complex n by n matrix E. The principal matrix logarithm logA is also returned.


Fortran Interface
Subroutine f01kkf ( n, a, lda, e, lde, ifail)
Integer, Intent (In):: n, lda, lde
Integer, Intent (Inout):: ifail
Complex (Kind=nag_wp), Intent (Inout):: a(lda,*), e(lde,*)
C Header Interface
#include <nagmk26.h>
void  f01kkf_ (const Integer *n, Complex a[], const Integer *lda, Complex e[], const Integer *lde, Integer *ifail)


For a matrix with no eigenvalues on the closed negative real line, the principal matrix logarithm logA is the unique logarithm whose spectrum lies in the strip z:-π<Imz<π.
The Fréchet derivative of the matrix logarithm of A is the unique linear mapping ELA,E such that for any matrix E 
logA+E - logA - LA,E = oE .  
The derivative describes the first order effect of perturbations in A on the logarithm logA.
f01kkf uses the algorithm of Al–Mohy et al. (2012) to compute logA and LA,E. The principal matrix logarithm logA is computed using a Schur decomposition, a Padé approximant and the inverse scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative LA,E. If A is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but f01kkf will return a non-principal logarithm and Fréchet derivative.


Al–Mohy A H and Higham N J (2011) Improved inverse scaling and squaring algorithms for the matrix logarithm SIAM J. Sci. Comput. 34(4) C152–C169
Al–Mohy A H, Higham N J and Relton S D (2012) Computing the Fréchet derivative of the matrix logarithm and estimating the condition number SIAM J. Sci. Comput. 35(4) C394–C410
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA


1:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     alda* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least n.
On entry: the n by n matrix A.
On exit: the n by n principal matrix logarithm, logA. Alterntively, if ifail=2, a non-principal logarithm is returned.
3:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01kkf is called.
Constraint: ldan.
4:     elde* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array e must be at least n.
On entry: the n by n matrix E
On exit: with ifail=0, 2 or 3, the Fréchet derivative LA,E
5:     lde – IntegerInput
On entry: the first dimension of the array e as declared in the (sub)program from which f01kkf is called.
Constraint: lden.
6:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
A is singular so the logarithm cannot be computed.
A has eigenvalues on the negative real line. The principal logarithm is not defined in this case, so a non-principal logarithm was returned.
logA has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
An unexpected internal error occurred. This failure should not occur and suggests that the routine has been called incorrectly.
On entry, n=value.
Constraint: n0.
On entry, lda=value and n=value.
Constraint: ldan.
On entry, lde=value and n=value.
Constraint: lden.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


For a normal matrix A (for which AHA=AAH), the Schur decomposition is diagonal and the computation of the matrix logarithm reduces to evaluating the logarithm of the eigenvalues of A and then constructing logA using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. The sensitivity of the computation of logA and LA,E is worst when A has an eigenvalue of very small modulus or has a complex conjugate pair of eigenvalues lying close to the negative real axis. See Al–Mohy and Higham (2011), Al–Mohy et al. (2012) and Section 11.2 of Higham (2008) for details and further discussion.

Parallelism and Performance

f01kkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kkf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

The cost of the algorithm is On3 floating-point operations. The complex allocatable memory required is approximately 5n2; see Al–Mohy et al. (2012) for further details.
If the matrix logarithm alone is required, without the Fréchet derivative, then f01fjf should be used. If the condition number of the matrix logarithm is required then f01kjf should be used. The real analogue of this routine is f01jkf.


This example finds the principal matrix logarithm logA and the Fréchet derivative LA,E, where
A = 1+4i 3i 0i 2i+ 2i 3i+0 1i+0 1+i 0i 2+0i 2i+0 i 1+2i 3+2i 1+2i 3+i   and   E = 1i+0 1+2i 2i+0 2+i 1+3i 0i 1i+0 0i+ 2i 4+0i 1i+0 1i+ 1i+0 2+2i 3i 1i+ .  

Program Text

Program Text (f01kkfe.f90)

Program Data

Program Data (f01kkfe.d)

Program Results

Program Results (f01kkfe.r)