f01jkc computes the Fréchet derivative of the matrix logarithm of the real matrix applied to the real matrix . The principal matrix logarithm is also returned.
The function may be called by the names: f01jkc or nag_matop_real_gen_matrix_frcht_log.
3Description
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 .
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 .
4References
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
5Arguments
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 matrix .
On exit: the 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 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 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NEGATIVE_EIGVAL
has eigenvalues on the negative real line. The principal logarithm is not defined in this case; 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 8 in the Introduction to the NAG Library CL Interface 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.
7Accuracy
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01jkc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.
9Further Comments
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 f01ejc should be used. If the condition number of the matrix logarithm is required then f01jjc should be used. If has negative real eigenvalues then f01kkc can be used to return a complex, non-principal matrix logarithm and its Fréchet derivative .
10Example
This example finds the principal matrix logarithm and the Fréchet derivative , where