NAG CL Interface
f01jjc (real_gen_matrix_cond_log)
1
Purpose
f01jjc computes an estimate of the relative condition number ${\kappa}_{\mathrm{log}}\left(A\right)$ of the logarithm of a real $n$ by $n$ matrix $A$, in the $1$norm. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is also returned.
2
Specification
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 $\mathrm{log}\left(A\right)$ is the unique logarithm whose spectrum lies in the strip $\left\{z:\pi <\mathrm{Im}\left(z\right)<\pi \right\}$.
The Fréchet derivative of the matrix logarithm of
$A$ is the unique linear mapping
$E\u27fcL\left(A,E\right)$ such that for any matrix
$E$
The derivative describes the first order effect of perturbations in $A$ on the logarithm $\mathrm{log}\left(A\right)$.
The relative condition number of the matrix logarithm can be defined by
where
$\Vert L\left(A\right)\Vert $ is the norm of the Fréchet derivative of the matrix logarithm at
$A$.
To obtain the estimate of ${\kappa}_{\mathrm{log}}\left(A\right)$, f01jjc first estimates $\Vert L\left(A\right)\Vert $ by computing an estimate $\gamma $ of a quantity $K\in \left[{n}^{1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}\right]$, such that $\gamma \le K$.
The algorithms used to compute
${\kappa}_{\mathrm{log}}\left(A\right)$ and
$\mathrm{log}\left(A\right)$ 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:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

2:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$.

3:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.

4:
$\mathbf{condla}$ – double *
Output

On exit: with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or
NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix logarithm,
${\kappa}_{\mathrm{log}}\left(A\right)$. Alternatively, if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_RCOND, contains the absolute condition number of the matrix logarithm.

5:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
 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, nonprincipal 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

$\mathrm{log}\left(A\right)$ 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
$\gamma $ of a quantity
$K\in \left[{n}^{1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}\right]$, such that
$\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for
f04ydc.
For a normal matrix
$A$ (for which
${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), 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
$\mathrm{log}\left(A\right)$ 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
$\mathrm{log}\left(A\right)$ 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 implementationspecific information.
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
$\Vert A\Vert /\Vert \mathrm{log}\left(A\right)\Vert $). 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 $10{n}^{2}$.
The cost of the algorithm is
$O\left({n}^{3}\right)$ floatingpoint 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, nonprincipal matrix logarithm and its condition number.
10
Example
This example estimates the relative condition number of the matrix logarithm
$\mathrm{log}\left(A\right)$, where
10.1
Program Text
10.2
Program Data
10.3
Program Results