NAG CL Interface
f01ejc (real_gen_matrix_log)
1
Purpose
f01ejc computes the principal matrix logarithm, $\mathrm{log}\left(A\right)$, of a real $n$ by $n$ matrix $A$, with no eigenvalues on the closed negative real line.
2
Specification
void 
f01ejc (Nag_OrderType order,
Integer n,
double a[],
Integer pda,
double *imnorm,
NagError *fail) 

The function may be called by the names: f01ejc or nag_matop_real_gen_matrix_log.
3
Description
Any nonsingular matrix $A$ has infinitely many logarithms. For a matrix with no eigenvalues on the closed negative real line, the principal logarithm is the unique logarithm whose spectrum lies in the strip $\left\{z:\pi <\mathrm{Im}\left(z\right)<\pi \right\}$.
$\mathrm{log}\left(A\right)$ is computed using the inverse scaling and squaring algorithm for the matrix logarithm described in
Al–Mohy and Higham (2011), adapted to real matrices by
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:
$\mathbf{order}$ – Nag_OrderType
Input

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:
$\mathbf{n}$ – Integer
Input

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

3:
$\mathbf{a}\left[\mathit{dim}\right]$ – 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
 ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(i1\right)\times {\mathbf{pda}}+j1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$.

4:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.

5:
$\mathbf{imnorm}$ – double *
Output

On exit: if the function has given a reliable answer then
${\mathbf{imnorm}}=0.0$. If
imnorm differs from
$0.0$ by more than unit roundoff (as returned by
X02AJC) then the computed matrix logarithm is unreliable.

6:
$\mathbf{fail}$ – 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_EIGENVALUES

$A$ was found to have eigenvalues on the negative real line. The principal logarithm is not defined in this case.
f01fjc can be used to find a complex nonprincipal logarithm.
 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_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

$A$ 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
For a normal matrix
$A$ (for which
${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), the Schur decomposition is diagonal and the algorithm 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. See
Al–Mohy and Higham (2011) and Section 9.4 of
Higham (2008) for details and further discussion.
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.
If estimates of the condition number of the matrix logarithm are required then
f01jjc should be used.
8
Parallelism and Performance
f01ejc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01ejc 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.
The cost of the algorithm is
$O\left({n}^{3}\right)$ floatingpoint operations (see
Al–Mohy and Higham (2011)). The double allocatable memory required is approximately
$3\times {n}^{2}$.
If the Fréchet derivative of the matrix logarithm is required then
f01jkc should be used.
f01fjc can be used to find the principal logarithm of a complex matrix. It can also be used to return a complex, nonprincipal logarithm if a real matrix has no principal logarithm due to the presence of negative eigenvalues.
10
Example
This example finds the principal matrix logarithm of the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results