f01jkf computes the Fréchet derivative $L(A,E)$ of the matrix logarithm of the real $n\times n$ matrix $A$ applied to the real $n\times n$ matrix $E$. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is also returned.
The routine may be called by the names f01jkf or nagf_matop_real_gen_matrix_frcht_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 $E$
The derivative describes the first order effect of perturbations in $A$ on the logarithm $\mathrm{log}\left(A\right)$.
f01jkf uses the algorithm of Al–Mohy et al. (2012) to compute $\mathrm{log}\left(A\right)$ and $L(A,E)$. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is computed using a Schur decomposition, a Padé approximant and the inverse scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative $L(A,E)$.
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 f01jkf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
4: $\mathbf{e}({\mathbf{lde}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array e
must be at least
${\mathbf{n}}$.
On entry: the $n\times n$ matrix $E$
On exit: the Fréchet derivative $L(A,E)$
5: $\mathbf{lde}$ – IntegerInput
On entry: the first dimension of the array e as declared in the (sub)program from which f01jkf is called.
Constraint:
${\mathbf{lde}}\ge {\mathbf{n}}$.
6: $\mathbf{ifail}$ – 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; f01kkf 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.
${\mathbf{ifail}}=4$
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: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-5$
On entry, ${\mathbf{lde}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{lde}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
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
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 $\mathrm{log}\left(A\right)$ and $L(A,E)$ 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), Al–Mohy et al. (2012) 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.
f01jkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jkf 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
The cost of the algorithm is $O\left({n}^{3}\right)$ floating-point operations. The real allocatable memory required is approximately $5{n}^{2}$; see Al–Mohy et al. (2012) for further details.
If the matrix logarithm alone is required, without the Fréchet derivative, then f01ejf should be used. If the condition number of the matrix logarithm is required then f01jjf should be used. If $A$ has negative real eigenvalues then f01kkf can be used to return a complex, non-principal matrix logarithm and its Fréchet derivative $L(A,E)$.
10Example
This example finds the principal matrix logarithm $\mathrm{log}\left(A\right)$ and the Fréchet derivative $L(A,E)$, where