nag_real_gen_matrix_exp (f01ecc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_real_gen_matrix_exp (f01ecc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_gen_matrix_exp (f01ecc) computes the matrix exponential, eA, 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

eA is computed using a Padé approximant and the scaling and squaring method described in Higham (2005) and Higham (2008).
If A has a full set of eigenvectors V then A can be factorized as
A = V D V-1 ,
where D is the diagonal matrix whose diagonal elements, di, are the eigenvalues of A. eA is then given by
eA = V eD V-1 ,
where eD is the diagonal matrix whose ith diagonal element is edi.
Note that eA is not computed this way as to do so would, in general, be unstable.

4  References

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

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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     a[pda×n]doubleInput/Output
Note: the i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the n by n matrix A.
On exit: the n by n matrix exponential eA.
4:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
  • if order=Nag_ColMajor, pdamax1,n;
  • if order=Nag_RowMajor, pdan.
5:     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.
On entry, n=value.
Constraint: n0.
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 processing the Padé approximant. Please contact NAG.
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.
The arithmetic precision is higher than that used for the Padé approximant computed matrix exponential.

7  Accuracy

For a normal matrix A (for which ATA=AAT) the computed matrix, eA, 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 Section 10.3 of Higham (2008) for details and further discussion.
For discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008).

8  Further Comments

The cost of the algorithm is On3; see Algorithm 10.20 of Higham (2008).
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).

9  Example

This example finds the matrix exponential of the matrix
A = 1 2 2 2 3 1 1 2 3 2 1 2 3 3 3 1 .

9.1  Program Text

Program Text (f01ecce.c)

9.2  Program Data

Program Data (f01ecce.d)

9.3  Program Results

Program Results (f01ecce.r)

nag_real_gen_matrix_exp (f01ecc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012