NAG Library Function Document

nag_matop_complex_gen_matrix_cond_exp (f01kgc)

+− 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_complex_gen_matrix_cond_exp (f01kgc) computes an estimate of the relative condition number

κ_{\exp} (A)

of the exponential of a complex

n

n

matrix

A

, in the

1

-norm. The matrix exponential

e^{A}

is also returned.

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_matop_complex_gen_matrix_cond_exp (Integer n, Complex a[], Integer pda, double condea, 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}

The relative condition number of the matrix exponential can be defined by

κ_{\exp} (A) = \frac{‖L (A)‖ ‖A‖}{‖\exp (A)‖},

where

‖L (A)‖

is the norm of the Fréchet derivative of the matrix exponential at

A

To obtain the estimate of

κ_{\exp} (A)

, nag_matop_complex_gen_matrix_cond_exp (f01kgc) first estimates

‖L (A)‖

by computing an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

The algorithms used to compute

κ_{\exp} (A)

are detailed in the Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).

The matrix exponential

e^{A}

is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives

L (A, E)

which are used to estimate the condition number.

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[ $\dim$ ] – ComplexInput/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: condea – double *Output: On exit: an estimate of the relative condition number of the matrix exponential $κ_{\exp} (A)$ .
5: 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: $pda \geq n$ .
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

nag_matop_complex_gen_matrix_cond_exp (f01kgc) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zdc) to produce an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zdc).

For a normal matrix

A

(for which

A^{H} A = A A^{H}

) 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) for details and further discussion.

For further discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008).

8 Parallelism and Performance

nag_matop_complex_gen_matrix_cond_exp (f01kgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_complex_gen_matrix_cond_exp (f01kgc) 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

nag_matop_complex_gen_matrix_cond_std (f01kac) uses a similar algorithm to nag_matop_complex_gen_matrix_cond_exp (f01kgc) to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of

‖A‖ / ‖\exp (A)‖

). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_complex_gen_matrix_cond_exp (f01kgc) and so its use is recommended over nag_matop_complex_gen_matrix_cond_std (f01kac).

The cost of the algorithm is

O (n^{3})

and the complex allocatable memory required is approximately

15 n^{2}

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

If the matrix exponential alone is required, without an estimate of the condition number, then nag_matop_complex_gen_matrix_exp (f01fcc) should be used. If the Fréchet derivative of the matrix exponential is required then nag_matop_complex_gen_matrix_frcht_exp (f01khc) 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 estimates the relative condition number of the matrix exponential

e^{A}

, where

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

NAG Library Function Documentnag_matop_complex_gen_matrix_cond_exp (f01kgc)