NAG Library Routine Document
F01FJF computes the principal matrix logarithm, , of a complex by matrix , with no eigenvalues on the closed negative real line.
||N, LDA, IFAIL
Any nonsingular matrix has infinitely many logarithms. For a matrix with no eigenvalues on the closed negative real line, the principal logarithm is the unique logarithm whose spectrum lies in the strip . If is nonsingular but has eigenvalues on the negative real line, the principal logarithm is not defined, but F01FJF will return a non-principal logarithm.
is computed using the Schur–Parlett algorithm for the matrix logarithm described in Higham (2008)
and Davies and Higham (2003)
Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
- 1: N – INTEGERInput
On entry: , the order of the matrix .
- 2: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
the second dimension of the array A
must be at least
On entry: the by matrix .
On exit: the by principal matrix logarithm, , unless , in which case a non-principal logarithm is returned.
- 3: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F01FJF is called.
- 4: IFAIL – INTEGERInput/Output
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.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
is singular so the logarithm cannot be computed.
was found to have eigenvalues on the negative real line.
The principal logarithm is not defined in this case,
so a non-principal logarithm was returned.
The arithmetic precision is higher than that used for the Padé approximant computed matrix logarithm.
An unexpected internal error has occurred. Please contact NAG
On entry, .
On entry, and .
Allocation of memory failed. The
allocatable memory required is approximately .
For a normal matrix
), the Schur decomposition is diagonal and the algorithm reduces to evaluating the logarithm of the eigenvalues of
and then constructing
using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Section 9.4 of Higham (2008)
for details and further discussion.
For discussion of the condition of the matrix logarithm see Section 11.2 of Higham (2008)
In particular, the condition number of the matrix logarithm at
, which is a measure of the sensitivity of the computed logarithm to perturbations in the matrix
is the condition number of
. Further, the sensitivity of the computation of
is worst when
has an eigenvalue of very small modulus, or has a complex conjugate pair of eigenvalues lying close to the negative real axis.
Up to of
allocatable memory may be required.
The cost of the algorithm is
floating point operations. The exact cost depends on the eigenvalue distribution of
; see Algorithm 11.11 of Higham (2008)
If estimates of the condition number of the matrix logarithm are required then F01KAF
should be used.
can be used to find the principal logarithm of a real matrix.
This example finds the principal matrix logarithm of the matrix
9.1 Program Text
Program Text (f01fjfe.f90)
9.2 Program Data
Program Data (f01fjfe.d)
9.3 Program Results
Program Results (f01fjfe.r)