NAG Library Function Document
nag_matop_real_gen_matrix_frcht_log (f01jkc)
1
Purpose
nag_matop_real_gen_matrix_frcht_log (f01jkc) computes the Fréchet derivative of the matrix logarithm of the real by matrix applied to the real by matrix . The principal matrix logarithm is also returned.
2
Specification
#include <nag.h> |
#include <nagf01.h> |
void |
nag_matop_real_gen_matrix_frcht_log (Integer n,
double a[],
Integer pda,
double e[],
Integer pde,
NagError *fail) |
|
3
Description
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 .
nag_matop_real_gen_matrix_frcht_log (f01jkc) uses the algorithm of
Al–Mohy et al. (2012) to compute
and
. The principal matrix logarithm
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
.
4
References
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
5
Arguments
- 1:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 2:
– doubleInput/Output
-
Note: 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 .
On exit: the by principal matrix logarithm, .
- 3:
– IntegerInput
-
On entry: the stride separating matrix row elements in the array
a.
Constraint:
.
- 4:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
e
must be at least
.
The th element of the matrix is stored in .
On entry: the by matrix
On exit: the Fréchet derivative
- 5:
– IntegerInput
-
On entry: the stride separating matrix row elements in the array
e.
Constraint:
.
- 6:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
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 for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NEGATIVE_EIGVAL
-
has eigenvalues on the negative real line. The principal logarithm is not defined in this case;
nag_matop_complex_gen_matrix_frcht_log (f01kkc) can be used to return a complex, non-principal log.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_SINGULAR
-
is singular so the logarithm cannot be computed.
- NW_SOME_PRECISION_LOSS
-
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
7
Accuracy
For a normal matrix
(for which
), 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
and
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),
Al–Mohy et al. (2012) and Section 11.2 of
Higham (2008) for details and further discussion.
8
Parallelism and Performance
nag_matop_real_gen_matrix_frcht_log (f01jkc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_frcht_log (f01jkc) 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The cost of the algorithm is
floating-point operations. The real allocatable memory required is approximately
; see
Al–Mohy et al. (2012) for further details.
If the matrix logarithm alone is required, without the Fréchet derivative, then
nag_matop_real_gen_matrix_log (f01ejc) should be used. If the condition number of the matrix logarithm is required then
nag_matop_real_gen_matrix_cond_log (f01jjc) should be used. If
has negative real eigenvalues then
nag_matop_complex_gen_matrix_frcht_log (f01kkc) can be used to return a complex, non-principal matrix logarithm and its Fréchet derivative
.
10
Example
This example finds the principal matrix logarithm
and the Fréchet derivative
, where
10.1
Program Text
Program Text (f01jkce.c)
10.2
Program Data
Program Data (f01jkce.d)
10.3
Program Results
Program Results (f01jkce.r)