nag_real_symm_matrix_exp (f01edc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_real_symm_matrix_exp (f01edc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_symm_matrix_exp (f01edc) computes the matrix exponential, eA, of a real symmetric n by n matrix A.

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_real_symm_matrix_exp (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, NagError *fail)

3  Description

eA is computed using a spectral factorization of A 
A = Q D QT ,
where D is the diagonal matrix whose diagonal elements, di, are the eigenvalues of A, and Q is an orthogonal matrix whose columns are the eigenvectors of A. eA is then given by
eA = Q eD QT ,
where eD is the diagonal matrix whose ith diagonal element is edi. See for example Section 4.5 of Higham (2008).

4  References

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

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 order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored.
The upper triangular part of A is stored.
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least pda×n.
On entry: the n by n symmetric matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if fail.code= NE_NOERROR, the upper or lower triangular part of the n by n matrix exponential, eA.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
The computation of the spectral factorization failed to converge.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and n=value.
Constraint: pdan.
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 internal error occurred when computing the spectral factorization. Please contact NAG.

7  Accuracy

For a symmetric matrix A, the matrix eA, has the relative condition number
κA = A2 ,
which is the minimum possible for the matrix exponential and so the computed matrix exponential is guaranteed to be close to the exact matrix. See Section 10.2 of Higham (2008) for details and further discussion.

8  Parallelism and Performance

nag_real_symm_matrix_exp (f01edc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_symm_matrix_exp (f01edc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The cost of the algorithm is On3.
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 symmetric matrix
A = 1 2 3 4 2 1 2 3 3 2 1 2 4 3 2 1

10.1  Program Text

Program Text (f01edce.c)

10.2  Program Data

Program Data (f01edce.d)

10.3  Program Results

Program Results (f01edce.r)

nag_real_symm_matrix_exp (f01edc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014