f01ec: FL CL CPP AD

NAG CL Interface
f01ecc (real_gen_matrix_exp)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

F01 (Matop) Chapter Contents

F01 (Matop) Chapter Introduction

f01ec: FL CL CPP AD

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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1 Purpose

f01ecc computes the matrix exponential,

e^{A}

, of a real

n \times n

matrix

A

2 Specification

#include <nag.h>

void	f01ecc (Nag_OrderType order, Integer n, double a[], Integer pda, NagError *fail)

The function may be called by the names: f01ecc, nag_matop_real_gen_matrix_exp or nag_real_gen_matrix_exp.

3 Description

e^{A}

is computed using a Padé approximant and the scaling and squaring method described in Al–Mohy and Higham (2009).

4 References

Al–Mohy A H and Higham N J (2009) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989

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: $order$ – 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

Nag_ColMajor

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

pda \times n

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n \times n

matrix

A

On exit: the

n \times n

matrix exponential

e^{A}

4: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq n

5: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

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 $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq n$ .
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 has occurred. Please contact NAG.
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 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.
NW_SOME_PRECISION_LOSS: $e^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

For a normal matrix

A

(for which

A^{T} A = A A^{T}

) the computed matrix,

e^{A}

, 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 Al–Mohy and Higham (2009) and Section 10.3 of Higham (2008) for details and further discussion.

If estimates of the condition number of the matrix exponential are required then f01jgc should be used.

8 Parallelism and Performance

f01ecc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f01ecc 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

6 \times n^{2}

The cost of the algorithm is

O (n^{3})

; see Section 5 of of Al–Mohy and Higham (2009). The real allocatable memory required is approximately

6 \times n^{2}

If the Fréchet derivative of the matrix exponential is required then f01jhc should be used.

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 matrix

A = (\begin{array}{r} 1 & 2 & 2 & 2 \\ 3 & 1 & 1 & 2 \\ 3 & 2 & 1 & 2 \\ 3 & 3 & 3 & 1 \end{array}) .

f01ec: FL CL CPP AD

NAG CL Interfacef01ecc (real_​gen_​matrix_​exp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f01ecc (real_gen_matrix_exp)