NAG Library Function Document
nag_real_gen_matrix_exp (f01ecc)
1
Purpose
nag_real_gen_matrix_exp (f01ecc) computes the matrix exponential, ${e}^{A}$, of a real $n$ by $n$ matrix $A$.
2
Specification
#include <nag.h> 
#include <nagf01.h> 
void 
nag_real_gen_matrix_exp (Nag_OrderType order,
Integer n,
double a[],
Integer pda,
NagError *fail) 

3
Description
${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method described in
Al–Mohy and Higham (2009).
4
References
Al–Mohy A H and Higham N J (2009) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl. 26(4) 1179–1193
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:
$\mathbf{order}$ – Nag_OrderTypeInput

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.
 2:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
 3:
$\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/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
 ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(i1\right)\times {\mathbf{pda}}+j1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix exponential ${e}^{A}$.
 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:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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: ${\mathbf{pda}}\ge {\mathbf{n}}$.
 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
An unexpected internal error has occurred. Please contact
NAG.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_SINGULAR

The linear equations to be solved are nearly singular and the Padé approximant probably has no correct figures; it is likely that this function has been called incorrectly.
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
Al–Mohy and Higham (2009) and Section 10.3 of
Higham (2008) for details and further discussion.
If estimates of the condition number of the matrix exponential are required then
nag_matop_real_gen_matrix_cond_exp (f01jgc) should be used.
8
Parallelism and Performance
nag_real_gen_matrix_exp (f01ecc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_gen_matrix_exp (f01ecc) 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 Integer allocatable memory required is
n, and the double allocatable memory required is approximately
$6\times {{\mathbf{n}}}^{2}$.
The cost of the algorithm is
$O\left({n}^{3}\right)$; see Section 5 of of
Al–Mohy and Higham (2009). The real allocatable memory required is approximately
$6\times {n}^{2}$.
If the Fréchet derivative of the matrix exponential is required then
nag_matop_real_gen_matrix_frcht_exp (f01jhc) should be used.
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is
Moler and Van Loan (2003).
10
Example
This example finds the matrix exponential of the matrix
10.1
Program Text
Program Text (f01ecce.c)
10.2
Program Data
Program Data (f01ecce.d)
10.3
Program Results
Program Results (f01ecce.r)