NAG CL Interfacef01jkc (real_​gen_​matrix_​frcht_​log)

1Purpose

f01jkc computes the Fréchet derivative $L\left(A,E\right)$ of the matrix logarithm of the real $n$ by $n$ matrix $A$ applied to the real $n$ by $n$ matrix $E$. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is also returned.

2Specification

 #include
 void f01jkc (Integer n, double a[], Integer pda, double e[], Integer pde, NagError *fail)
The function may be called by the names: f01jkc or nag_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 $E⟼L\left(A,E\right)$ such that for any matrix $E$
 $logA+E - logA - LA,E = oE .$
The derivative describes the first order effect of perturbations in $A$ on the logarithm $\mathrm{log}\left(A\right)$.
f01jkc uses the algorithm of Al–Mohy et al. (2012) to compute $\mathrm{log}\left(A\right)$ and $L\left(A,E\right)$. 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 $L\left(A,E\right)$.

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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{a}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$.
3: $\mathbf{pda}$Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
4: $\mathbf{e}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array e must be at least ${\mathbf{pde}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $E$ is stored in ${\mathbf{e}}\left[\left(j-1\right)×{\mathbf{pde}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $E$
On exit: the Fréchet derivative $L\left(A,E\right)$
5: $\mathbf{pde}$Integer Input
On entry: the stride separating matrix row elements in the array e.
Constraint: ${\mathbf{pde}}\ge {\mathbf{n}}$.
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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pde}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pde}}\ge {\mathbf{n}}$.
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; f01kkc can be used to return a complex, non-principal log.
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 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 $L\left(A,E\right)$ 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), Al–Mohy et al. (2012) and Section 11.2 of Higham (2008) for details and further discussion.

8Parallelism and Performance

f01jkc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jkc 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.

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 f01ejc should be used. If the condition number of the matrix logarithm is required then f01jjc should be used. If $A$ has negative real eigenvalues then f01kkc can be used to return a complex, non-principal matrix logarithm and its Fréchet derivative $L\left(A,E\right)$.

10Example

This example finds the principal matrix logarithm $\mathrm{log}\left(A\right)$ and the Fréchet derivative $L\left(A,E\right)$, where
 $A = 4 2 0 2 3 3 1 1 3 2 1 0 3 3 1 2 and ​ E = 1 2 2 2 0 0 3 1 1 2 1 2 1 3 1 1 .$

10.1Program Text

Program Text (f01jkce.c)

10.2Program Data

Program Data (f01jkce.d)

10.3Program Results

Program Results (f01jkce.r)