f01kjc computes an estimate of the relative condition number ${\kappa}_{\mathrm{log}}\left(A\right)$ of the logarithm of a complex $n\times n$ matrix $A$, in the $1$-norm. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is also returned.
The function may be called by the names: f01kjc or nag_matop_complex_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 ${\kappa}_{\mathrm{log}}\left(A\right)$, f01kjc 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).
If $A$ is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but f01kjc will return a non-principal logarithm and its condition number.
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
Note: the dimension, dim, of the array a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $(i,j)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[(j-1)\times {\mathbf{pda}}+i-1\right]$.
On entry: the $n\times n$ matrix $A$.
On exit: the $n\times n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$. Alternatively, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_NEGATIVE_EIGVAL, a non-principal logarithm is returned.
3: $\mathbf{pda}$ – IntegerInput
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, NE_NEGATIVE_EIGVAL 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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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, so a non-principal logarithm was returned.
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.
7Accuracy
f01kjc uses the norm estimation function f04zdc 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 f04zdc.
For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$), 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.
f01kjc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01kjc 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 implementation-specific information.
9Further Comments
f01kac uses a similar algorithm to f01kjc 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 f01kjc and so its use is recommended over f01kac.
The amount of complex 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 f01fjc should be used. If the Fréchet derivative of the matrix logarithm is required then f01kkc should be used. The real analogue of this function is f01jjc.
10Example
This example estimates the relative condition number of the matrix logarithm $\mathrm{log}\left(A\right)$, where