NAG CL Interface
f01jjc (real_​gen_​matrix_​cond_​log)

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1 Purpose

f01jjc computes an estimate of the relative condition number κlog(A) of the logarithm of a real n×n matrix A, in the 1-norm. The principal matrix logarithm log(A) is also returned.

2 Specification

#include <nag.h>
void  f01jjc (Integer n, double a[], Integer pda, double *condla, NagError *fail)
The function may be called by the names: f01jjc or nag_matop_real_gen_matrix_cond_log.

3 Description

For a matrix with no eigenvalues on the closed negative real line, the principal matrix logarithm log(A) is the unique logarithm whose spectrum lies in the strip {z:-π<Im(z)<π}.
The Fréchet derivative of the matrix logarithm of A is the unique linear mapping EL(A,E) such that for any matrix E
log(A+E) - log(A) - L(A,E) = o(E) .  
The derivative describes the first order effect of perturbations in A on the logarithm log(A).
The relative condition number of the matrix logarithm can be defined by
κlog(A) = L(A) A log(A) ,  
where L(A) is the norm of the Fréchet derivative of the matrix logarithm at A.
To obtain the estimate of κlog(A), f01jjc first estimates L(A) by computing an estimate γ of a quantity K[n−1L(A)1,nL(A)1], such that γK.
The algorithms used to compute κlog(A) and log(A) 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).

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: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The (i,j)th element of the matrix A is stored in a[(j-1)×pda+i-1].
On entry: the n×n matrix A.
On exit: the n×n principal matrix logarithm, log(A).
3: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4: condla double * Output
On exit: with fail.code= NE_NOERROR or NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix logarithm, κlog(A). Alternatively, if fail.code= NE_RCOND, contains the absolute condition number of the matrix logarithm.
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdan.
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
A has eigenvalues on the negative real line. The principal logarithm is not defined in this case; f01kjc 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_RCOND
The relative condition number is infinite. The absolute condition number was returned instead.
NE_SINGULAR
A is singular so the logarithm cannot be computed.
NW_SOME_PRECISION_LOSS
log(A) has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

f01jjc uses the norm estimation function f04ydc to produce an estimate γ of a quantity K[n−1L(A)1,nL(A)1], such that γK. For further details on the accuracy of norm estimation, see the documentation for f04ydc.
For a normal matrix A (for which ATA=AAT), the Schur decomposition is diagonal and the computation of the matrix logarithm reduces to evaluating the logarithm of the eigenvalues of A and then constructing log(A) 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 log(A) is worst when A 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

f01jjc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jjc 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.

9 Further Comments

f01jac uses a similar algorithm to f01jjc to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of A/log(A)). However, the required Fréchet derivatives are computed in a more efficient and stable manner by f01jjc and so its use is recommended over f01jac.
The amount of real allocatable memory required by the algorithm is typically of the order 10n2.
The cost of the algorithm is O(n3) floating-point operations; see Al–Mohy et al. (2012).
If the matrix logarithm alone is required, without an estimate of the condition number, then f01ejc should be used. If the Fréchet derivative of the matrix logarithm is required then f01jkc should be used. If A has negative real eigenvalues then f01kjc can be used to return a complex, non-principal matrix logarithm and its condition number.

10 Example

This example estimates the relative condition number of the matrix logarithm log(A), where
A = ( 4 −1 0 1 2 5 −2 2 1 1 3 −1 2 0 2 8 ) .  

10.1 Program Text

Program Text (f01jjce.c)

10.2 Program Data

Program Data (f01jjce.d)

10.3 Program Results

Program Results (f01jjce.r)