f01ejc computes the principal matrix logarithm, $\mathrm{log}\left(A\right)$, of a real $n\times n$ matrix $A$, with no eigenvalues on the closed negative real line.
The function may be called by the names: f01ejc or nag_matop_real_gen_matrix_log.
3Description
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).
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: $\mathbf{order}$ – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major 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 $\mathrm{Nag\_ColMajor}$.
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
${\mathbf{a}}\left[(j-1)\times {\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
${\mathbf{a}}\left[(i-1)\times {\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: the $n\times n$ matrix $A$.
On exit: the $n\times n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$.
4: $\mathbf{pda}$ – IntegerInput
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).
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_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 non-principal logarithm.
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_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.
7Accuracy
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.
8Parallelism 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 implementation-specific information.
9Further Comments
The cost of the algorithm is $O\left({n}^{3}\right)$ floating-point 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, non-principal logarithm if a real matrix has no principal logarithm due to the presence of negative eigenvalues.
10Example
This example finds the principal matrix logarithm of the matrix