The function may be called by the names: f01ecc, nag_matop_real_gen_matrix_exp or nag_real_gen_matrix_exp.
3Description
${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method described in Al–Mohy and Higham (2009).
4References
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, twenty-five years later SIAM Rev.45 3–49
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 $n\times n$ 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 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.
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 8 in the Introduction to the NAG Library CL Interface 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.
7Accuracy
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 non-normal 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 f01jgc should be used.
8Parallelism and Performance
f01ecc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 implementation-specific information.
9Further Comments
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 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).
10Example
This example finds the matrix exponential of the matrix