f01jjf computes an estimate of the relative condition number ${\kappa}_{\mathrm{log}}\left(A\right)$ of the logarithm of a real $n\times n$ matrix $A$, in the $1$-norm. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is also returned.
The routine may be called by the names f01jjf or nagf_matop_real_gen_matrix_cond_log.
3Description
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 $E\u27fcL(A,E)$ such that for any matrix $E$
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 ${\kappa}_{\mathrm{log}}\left(A\right)$, f01jjf first estimates $\Vert L\left(A\right)\Vert $ by computing an estimate $\gamma $ of a quantity $K\in [{n}^{\mathrm{-1}}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}]$, 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).
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: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
2: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
${\mathbf{n}}$.
On entry: the $n\times n$ matrix $A$.
On exit: the $n\times n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$.
3: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01jjf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
4: $\mathbf{condla}$ – Real (Kind=nag_wp)Output
On exit: with ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{3}}$, an estimate of the relative condition number of the matrix logarithm, ${\kappa}_{\mathrm{log}}\left(A\right)$. Alternatively, if ${\mathbf{ifail}}={\mathbf{4}}$, contains the absolute condition number of the matrix logarithm.
5: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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.
${\mathbf{ifail}}=2$
$A$ has eigenvalues on the negative real line. The principal logarithm is not defined in this case; f01kjf can be used to return a complex, non-principal log.
${\mathbf{ifail}}=3$
$\mathrm{log}\left(A\right)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
${\mathbf{ifail}}=4$
The relative condition number is infinite. The absolute condition number was returned instead.
${\mathbf{ifail}}=5$
An unexpected internal error occurred. This failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
f01jjf uses the norm estimation routine f04ydf to produce an estimate $\gamma $ of a quantity $K\in [{n}^{\mathrm{-1}}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}]$, such that $\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for f04ydf.
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01jjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jjf 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 implementation-specific information.
9Further Comments
f01jaf uses a similar algorithm to f01jjf 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 f01jjf and so its use is recommended over f01jaf.
The amount of real allocatable memory required by the algorithm is typically of the order $10{n}^{2}$.
The cost of the algorithm is $O\left({n}^{3}\right)$ floating-point operations; see Al–Mohy et al. (2012).
If the matrix logarithm alone is required, without an estimate of the condition number, then f01ejf should be used. If the Fréchet derivative of the matrix logarithm is required then f01jkf should be used. If $A$ has negative real eigenvalues then f01kjf can be used to return a complex, non-principal matrix logarithm and its condition number.
10Example
This example estimates the relative condition number of the matrix logarithm $\mathrm{log}\left(A\right)$, where