NAG CL Interface
f01edc (real_​symm_​matrix_​exp)

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1 Purpose

f01edc computes the matrix exponential, eA, of a real symmetric n×n matrix A.

2 Specification

#include <nag.h>
void  f01edc (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, NagError *fail)
The function may be called by the names: f01edc, nag_matop_real_symm_matrix_exp or nag_real_symm_matrix_exp.

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: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
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: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least pda×n.
On entry: the n×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×n matrix exponential, eA.
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax(1,n).
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

The value of fail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08fac).
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
An internal error occurred when computing the spectral factorization. Please contact NAG.
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.

7 Accuracy

For a symmetric matrix A, the matrix eA, has the relative condition number
κ(A) = 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

f01edc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 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.

9 Further Comments

The Integer allocatable memory required is n, and the double allocatable memory required is approximately (n+nb+4)×n, where nb is the block size required by f08fac.
The cost of the algorithm is O(n3).
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)