NAG CL Interface
f01khc (complex_​gen_​matrix_​frcht_​exp)

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1 Purpose

f01khc computes the Fréchet derivative L(A,E) of the matrix exponential of a complex n×n matrix A applied to the complex n×n matrix E. The matrix exponential eA is also returned.

2 Specification

#include <nag.h>
void  f01khc (Integer n, Complex a[], Integer pda, Complex e[], Integer pde, NagError *fail)
The function may be called by the names: f01khc or nag_matop_complex_gen_matrix_frcht_exp.

3 Description

The Fréchet derivative of the matrix exponential of A is the unique linear mapping EL(A,E) such that for any matrix E
eA+E - e A - L(A,E) = o(E) .  
The derivative describes the first-order effect of perturbations in A on the exponential eA.
f01khc uses the algorithms of Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) to compute eA and L(A,E). The matrix exponential eA 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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The (i,j)th element of the matrix A is stored in a[(j-1)×pda+i-1].
On entry: the n×n matrix A.
On exit: the n×n matrix exponential eA.
3: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4: e[dim] Complex Input/Output
Note: the dimension, dim, of the array e must be at least pde×n.
The (i,j)th element of the matrix E is stored in e[(j-1)×pde+i-1].
On entry: the n×n matrix E
On exit: the Fréchet derivative L(A,E)
5: pde Integer Input
On entry: the stride separating matrix row elements in the array e.
Constraint: pden.
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

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: n0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdan.
On entry, pde=value and n=value.
Constraint: pden.
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.
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 for the Padé approximant are singular; it is likely that this function has been called incorrectly.
NW_SOME_PRECISION_LOSS
eA 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 AHA=AAH) 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), Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) for details and further discussion.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01khc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01khc 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 cost of the algorithm is O(n3) and the complex allocatable memory required is approximately 9n2; see Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).
If the matrix exponential alone is required, without the Fréchet derivative, then f01fcc should be used.
If the condition number of the matrix exponential is required then f01kgc 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 eA and the Fréchet derivative L(A,E), where
A = ( 1+0i 2+0i 2+0i 2+i 3+2i 1i+0 1i+0 2+i 3+2i 2+0i 1i+0 2+i 3+2i 3+2i 3+2i 1+i )   and   E = ( 1i+0 2+0i 2i+0 4+i 3+2i 0i+0 1i+0 0+i 0+2i 0+0i 1i+0 0i+ 1+0i 2+2i 0+3i 1i+ ) .  

10.1 Program Text

Program Text (f01khce.c)

10.2 Program Data

Program Data (f01khce.d)

10.3 Program Results

Program Results (f01khce.r)