NAG CL Interface
f01jhc (real_gen_matrix_frcht_exp)
1
Purpose
f01jhc computes the Fréchet derivative $L\left(A,E\right)$ of the matrix exponential of a real $n$ by $n$ matrix $A$ applied to the real $n$ by $n$ matrix $E$. The matrix exponential ${e}^{A}$ is also returned.
2
Specification
void 
f01jhc (Integer n,
double a[],
Integer pda,
double e[],
Integer pde,
NagError *fail) 

The function may be called by the names: f01jhc or nag_matop_real_gen_matrix_frcht_exp.
3
Description
The Fréchet derivative of the matrix exponential of
$A$ is the unique linear mapping
$E\u27fcL\left(A,E\right)$ such that for any matrix
$E$
The derivative describes the firstorder effect of perturbations in $A$ on the exponential ${e}^{A}$.
f01jhc uses the algorithms of
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b) to compute
${e}^{A}$ and
$L\left(A,E\right)$. The matrix exponential
${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative
$L\left(A,E\right)$.
4
References
Al–Mohy A H and Higham N J (2009a) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Al–Mohy A H and Higham N J (2009b) Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation SIAM J. Matrix Anal. Appl. 30(4) 1639–1657
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twentyfive years later SIAM Rev. 45 3–49
5
Arguments

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
${\mathbf{pda}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix exponential ${e}^{A}$.

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
${\mathbf{pde}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $E$ is stored in ${\mathbf{e}}\left[\left(j1\right)\times {\mathbf{pde}}+i1\right]$.
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).
6
Error 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pde}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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_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

The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.
 NW_SOME_PRECISION_LOSS

${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
7
Accuracy
For a normal matrix
$A$ (for which
${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$) the computed matrix,
${e}^{A}$, is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for nonnormal matrices. See Section 10.3 of
Higham (2008),
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b) for details and further discussion.
8
Parallelism and Performance
f01jhc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jhc 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 implementationspecific information.
The cost of the algorithm is
$O\left({n}^{3}\right)$ and the real allocatable memory required is approximately
$9{n}^{2}$; see
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b).
If the matrix exponential alone is required, without the Fréchet derivative, then
f01ecc should be used.
If the condition number of the matrix exponential is required then
f01jgc should be used.
As well as the excellent book
Higham (2008), the classic reference for the computation of the matrix exponential is
Moler and Van Loan (2003).
10
Example
This example finds the matrix exponential
${e}^{A}$ and the Fréchet derivative
$L\left(A,E\right)$, where
10.1
Program Text
10.2
Program Data
10.3
Program Results