The function may be called by the names: f01ekc or nag_matop_real_gen_matrix_fun_std.
3Description
$f\left(A\right)$, where $f$ is either the exponential, sine, cosine, sinh or cosh, is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).
4References
Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions SIAM J. Matrix Anal. Appl.25(2) 464–485
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 $\mathrm{Nag\_ColMajor}$.
2: $\mathbf{fun}$ – Nag_MatFunTypeInput
On entry: indicates which matrix function will be computed.
${\mathbf{fun}}=\mathrm{Nag\_Exp}$
The matrix exponential, ${e}^{A}$, will be computed.
${\mathbf{fun}}=\mathrm{Nag\_Sin}$
The matrix sine, $\mathrm{sin}\left(A\right)$, will be computed.
${\mathbf{fun}}=\mathrm{Nag\_Cos}$
The matrix cosine, $\mathrm{cos}\left(A\right)$, will be computed.
${\mathbf{fun}}=\mathrm{Nag\_Sinh}$
The hyperbolic matrix sine, $\mathrm{sinh}\left(A\right)$, will be computed.
${\mathbf{fun}}=\mathrm{Nag\_Cosh}$
The hyperbolic matrix cosine, $\mathrm{cosh}\left(A\right)$, will be computed.
Constraint:
${\mathbf{fun}}=\mathrm{Nag\_Exp}$, $\mathrm{Nag\_Sin}$, $\mathrm{Nag\_Cos}$, $\mathrm{Nag\_Sinh}$ or $\mathrm{Nag\_Cosh}$.
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, $f\left(A\right)$.
5: $\mathbf{pda}$ – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.
6: $\mathbf{imnorm}$ – double *Output
On exit: if $A$ has complex eigenvalues, f01ekc will use complex arithmetic to compute the matrix function. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the $1$-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.
If $A$ has real eigenvalues, f01ekc uses real arithmetic and ${\mathbf{imnorm}}=0$.
7: $\mathbf{fail}$ – 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_CONVERGENCE
A Taylor series failed to converge.
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 occurred when evaluating the function at a point. Please contact NAG.
An unexpected internal error occurred when ordering the eigenvalues of $A$. Please contact NAG.
The function was unable to compute the Schur decomposition of $A$. Note: this failure should not occur and suggests that the function has been called incorrectly.
There was an error whilst reordering the Schur form of $A$. Note: this failure should not occur and suggests that the function has been called incorrectly.
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 used to compute the exponential may have no correct figures. Note: this failure should not occur and suggests that the function has been called incorrectly.
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 $f$ at the eigenvalues of $A$ and then constructing $f\left(A\right)$ using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm.
For further discussion of the Schur–Parlett algorithm see Section 9.4 of Higham (2008).
8Parallelism and Performance
f01ekc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01ekc 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 $n$. If $A$ has real eigenvalues then up to $9{n}^{2}$ of double allocatable memory may be required. If $A$ has complex eigenvalues then up to $9{n}^{2}$ of Complex allocatable memory may be required.
The cost of the Schur–Parlett algorithm depends on the spectrum of $A$, but is roughly between $28{n}^{3}$ and ${n}^{4}/3$ floating-point operations; see Algorithm 9.6 of Higham (2008).
If the matrix exponential is required then it is recommended that f01ecc be used. f01ecc uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by f01ekc.
If estimates of the condition number of the matrix function are required then f01jac should be used.
f01fkc can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.
10Example
This example finds the matrix cosine of the matrix