f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_matop_real_gen_matrix_fun_std (f01ekc)

## 1  Purpose

nag_matop_real_gen_matrix_fun_std (f01ekc) computes the matrix exponential, sine, cosine, sinh or cosh, of a real $n$ by $n$ matrix $A$ using the Schur–Parlett algorithm.

## 2  Specification

 #include #include
 void nag_matop_real_gen_matrix_fun_std (Nag_OrderType order, Nag_MatFunType fun, Integer n, double a[], Integer pda, double *imnorm, NagError *fail)

## 3  Description

$f\left(A\right)$, 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).

## 4  References

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

## 5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     funNag_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 $\mathrm{Nag_Cosh}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]doubleInput/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
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
5:     pdaIntegerInput
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:     imnormdouble *Output
On exit: if $A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_std (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, nag_matop_real_gen_matrix_fun_std (f01ekc) uses real arithmetic and ${\mathbf{imnorm}}=0$.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Allocation of memory failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
A Taylor series failed to converge.
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: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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.
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_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.

## 7  Accuracy

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).

## 8  Parallelism and Performance

nag_matop_real_gen_matrix_fun_std (f01ekc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_fun_std (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.
In these implementations, this may make calls to the user supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions should be avoided, unless you are using the same OpenMP runtime library (which normally means using the same compiler) as that used to build your NAG Library implementation, as listed in the Installers' Note.

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 nag_real_gen_matrix_exp (f01ecc) be used. nag_real_gen_matrix_exp (f01ecc) uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by nag_matop_real_gen_matrix_fun_std (f01ekc).
If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_std (f01jac) should be used.
nag_matop_complex_gen_matrix_fun_std (f01fkc) can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.

## 10  Example

This example finds the matrix cosine of the matrix
 $A = 2 0 1 0 0 2 -2 1 0 2 3 1 1 4 0 0 .$

### 10.1  Program Text

Program Text (f01ekce.c)

### 10.2  Program Data

Program Data (f01ekce.d)

### 10.3  Program Results

Program Results (f01ekce.r)