NAG Library Routine Document
F01EJF
1 Purpose
F01EJF 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
INTEGER 
N, LDA, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,*), IMNORM 

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 MIMS EPrint 2012.72
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
A
must be at least
${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$.
 3: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01EJF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{IMNORM}$ – REAL (KIND=nag_wp)Output

On exit: if the routine has given a reliable answer then
${\mathbf{IMNORM}}=0.0$. If
IMNORM differs from
$0.0$ by more than unit roundoff (as returned by
X02AJF) then the computed matrix logarithm is unreliable.
 5: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

$A$ is singular so the logarithm cannot be computed.
 ${\mathbf{IFAIL}}=2$

$A$ was found to have eigenvalues on the negative real line. The principal logarithm is not defined in this case.
F01FJF can be used to find a complex nonprincipal logarithm.
 ${\mathbf{IFAIL}}=3$

$\mathrm{log}\left(A\right)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
 ${\mathbf{IFAIL}}=4$

An unexpected internal error occurred. Please contact
NAG.
 ${\mathbf{IFAIL}}=1$

On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}\ge 0$.
 ${\mathbf{IFAIL}}=3$

On entry, ${\mathbf{LDA}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
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
F01JJF should be used.
8 Parallelism and Performance
F01EJF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01EJF 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 routine. 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 real allocatable memory required is approximately
$3\times {n}^{2}$.
If the Fréchet derivative of the matrix logarithm is required then
F01JKF should be used.
F01FJF 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
Program Text (f01ejfe.f90)
10.2 Program Data
Program Data (f01ejfe.d)
10.3 Program Results
Program Results (f01ejfe.r)