NAG Library Function Document
nag_matop_complex_gen_matrix_log (f01fjc) computes the principal matrix logarithm, , of a complex by matrix , with no eigenvalues on the closed negative real line.
||nag_matop_complex_gen_matrix_log (Nag_OrderType order,
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 nag_matop_complex_gen_matrix_log (f01fjc) will return a non-principal logarithm.
is computed using the inverse scaling and squaring algorithm for the matrix logarithm described in Al–Mohy and Higham (2011)
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
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
order – Nag_OrderTypeInput
: the order
argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See Section 188.8.131.52
in the Essential Introduction for a more detailed explanation of the use of this argument.
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
principal matrix logarithm,
, unless NE_EIGENVALUES
, in which case a non-principal logarithm is returned.
pda – IntegerInput
: the stride separating row or column elements (depending on the value of order
) in the array a
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Allocation of memory failed.
On entry, argument had an illegal value.
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.
On entry, .
On entry, and .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
is singular so the logarithm cannot be computed.
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
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 Al–Mohy and Higham (2011)
and Section 9.4 of Higham (2008)
for details and further discussion.
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.
If estimates of the condition number of the matrix logarithm are required then nag_matop_complex_gen_matrix_cond_log (f01kjc)
should be used.
8 Parallelism and Performance
nag_matop_complex_gen_matrix_log (f01fjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_gen_matrix_log (f01fjc) 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 Users' Note
for your implementation for any additional implementation-specific information.
The cost of the algorithm is
floating-point operations (see Al–Mohy and Higham (2011)
). The Complex allocatable memory required is approximately
If the Fréchet derivative of the matrix logarithm is required then nag_matop_complex_gen_matrix_frcht_log (f01kkc)
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
10.1 Program Text
Program Text (f01fjce.c)
10.2 Program Data
Program Data (f01fjce.d)
10.3 Program Results
Program Results (f01fjce.r)