NAG Library Function Document

nag_matop_real_gen_matrix_frcht_exp (f01jhc)

+− 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

1 Purpose

nag_matop_real_gen_matrix_frcht_exp (f01jhc) computes the Fréchet derivative

L (A, E)

of the matrix exponential of a real

n

n

matrix

A

applied to the real

n

n

matrix

E

. The matrix exponential

e^{A}

is also returned.

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_matop_real_gen_matrix_frcht_exp (Integer n, double a[], Integer pda, double e[], Integer pde, NagError *fail)

3 Description

The Fréchet derivative of the matrix exponential of

A

is the unique linear mapping

E ⟼ L (A, E)

such that for any matrix

E

e^{A + E} - e^{A} - L (A, E) = o (‖E‖) .

The derivative describes the first-order effect of perturbations in

A

on the exponential

e^{A}

nag_matop_real_gen_matrix_frcht_exp (f01jhc) uses the algorithms of Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) to compute

e^{A}

and

L (A, E)

. The matrix exponential

e^{A}

is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative

L (A, E)

4 References

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

Al–Mohy A H and Higham N J (2009b) Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation SIAM J. Matrix Anal. Appl. 30(4) 1639–1657

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: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 0$ .
2: a[ $\dim$ ] – doubleInput/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 $a [(j - 1) \times pda + i - 1]$ .
On entry: the $n$ by $n$ matrix $A$ .

On exit: the $n$ by $n$ matrix exponential $e^{A}$ .
3: pda – IntegerInput: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: e[ $\dim$ ] – doubleInput/Output: Note: the dimension, dim, of the array e must be at least $pde \times n$ .

The $(i, j)$ th element of the matrix $E$ is stored in $e [(j - 1) \times pde + i - 1]$ .
On entry: the $n$ by $n$ matrix $E$

On exit: the Fréchet derivative $L (A, E)$
5: pde – IntegerInput: On entry: the stride separating matrix row elements in the array e.

Constraint: $pde \geq n$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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: $pda \geq n$ .
On entry, $pde = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pde \geq n$ .
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.
NE_SINGULAR: 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 Section 10.3 of Higham (2008), Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) for details and further discussion.

8 Parallelism and Performance

nag_matop_real_gen_matrix_frcht_exp (f01jhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_real_gen_matrix_frcht_exp (f01jhc) 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

O (n^{3})

and the real allocatable memory required is approximately

9 n^{2}

; see Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).

If the matrix exponential alone is required, without the Fréchet derivative, then nag_real_gen_matrix_exp (f01ecc) should be used.

If the condition number of the matrix exponential is required then nag_matop_real_gen_matrix_cond_exp (f01jgc) should be used.

As well as the excellent book Higham (2008), the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).

10 Example

This example finds the matrix exponential

e^{A}

and the Fréchet derivative

L (A, E)

, where

A = (\begin{array}{r} 1 & 2 & 2 & 2 \\ 3 & 1 & 1 & 2 \\ 3 & 2 & 1 & 2 \\ 3 & 3 & 3 & 1 \end{array}) and E = (\begin{array}{r} 1 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \\ 4 & 2 & 1 & 2 \\ 0 & 3 & 2 & 1 \end{array}) .

NAG Library Function Documentnag_matop_real_gen_matrix_frcht_exp (f01jhc)