NAG Library Function Document
nag_matop_complex_gen_matrix_cond_log (f01kjc) computes an estimate of the relative condition number of the logarithm of a complex by matrix , in the -norm. The principal matrix logarithm is also returned.
||nag_matop_complex_gen_matrix_cond_log (Integer n,
For a matrix with no eigenvalues on the closed negative real line, the principal matrix logarithm is the unique logarithm whose spectrum lies in the strip .
The Fréchet derivative of the matrix logarithm of
is the unique linear mapping
such that for any matrix
The derivative describes the first order effect of perturbations in on the logarithm .
The relative condition number of the matrix logarithm can be defined by
is the norm of the Fréchet derivative of the matrix logarithm at
To obtain the estimate of , nag_matop_complex_gen_matrix_cond_log (f01kjc) first estimates by computing an estimate of a quantity , such that .
The algorithms used to compute
are based on a Schur decomposition, the inverse scaling and squaring method and Padé approximants. Further details can be found in Al–Mohy and Higham (2011)
and Al–Mohy et al. (2012)
If is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but nag_matop_complex_gen_matrix_cond_log (f01kjc) will return a non-principal logarithm and its condition number.
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 MIMS EPrint 2012.72
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
The th element of the matrix is stored in .
On entry: the by matrix .
principal matrix logarithm,
. Alternatively, if NE_NEGATIVE_EIGVAL
, a non-principal logarithm is returned.
pda – IntegerInput
: the stride separating matrix row elements in the array a
condla – double *Output
, an estimate of the relative condition number of the matrix logarithm,
. Alternatively, if NE_RCOND
, contains the absolute condition number of the matrix logarithm.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
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
has eigenvalues on the negative real line. The principal logarithm is not defined in this case, so a non-principal logarithm was returned.
The relative condition number is infinite. The absolute condition number was returned instead.
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.
nag_matop_complex_gen_matrix_cond_log (f01kjc) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zdc)
to produce an estimate
of a quantity
, such that
. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zdc)
For a normal matrix
), the Schur decomposition is diagonal and the computation of the matrix logarithm 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. 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. See Al–Mohy and Higham (2011)
and Section 11.2 of Higham (2008)
for details and further discussion.
8 Parallelism and Performance
nag_matop_complex_gen_matrix_cond_log (f01kjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_gen_matrix_cond_log (f01kjc) 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.
uses a similar algorithm to nag_matop_complex_gen_matrix_cond_log (f01kjc) to compute an estimate of the absolute
condition number (which is related to the relative condition number by a factor of
). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_complex_gen_matrix_cond_log (f01kjc) and so its use is recommended over nag_matop_complex_gen_matrix_cond_std (f01kac)
The amount of complex allocatable memory required by the algorithm is typically of the order .
The cost of the algorithm is
floating-point operations; see Al–Mohy et al. (2012)
If the matrix logarithm alone is required, without an estimate of the condition number, then nag_matop_complex_gen_matrix_log (f01fjc)
should be used. If the Fréchet derivative of the matrix logarithm is required then nag_matop_complex_gen_matrix_frcht_log (f01kkc)
should be used. The real analogue of this function is nag_matop_real_gen_matrix_cond_log (f01jjc)
This example estimates the relative condition number of the matrix logarithm
10.1 Program Text
Program Text (f01kjce.c)
10.2 Program Data
Program Data (f01kjce.d)
10.3 Program Results
Program Results (f01kjce.r)