NAG FL Interface
f01khf (complex_​gen_​matrix_​frcht_​exp)

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1 Purpose

f01khf 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

Fortran Interface
Subroutine f01khf ( n, a, lda, e, lde, ifail)
Integer, Intent (In) :: n, lda, lde
Integer, Intent (Inout) :: ifail
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), e(lde,*)
C Header Interface
#include <nag.h>
void  f01khf_ (const Integer *n, Complex a[], const Integer *lda, Complex e[], const Integer *lde, Integer *ifail)
The routine may be called by the names f01khf or nagf_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.
f01khf 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(lda,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least n.
On entry: the n×n matrix A.
On exit: the n×n matrix exponential eA.
3: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01khf is called.
Constraint: ldan.
4: e(lde,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array e must be at least n.
On entry: the n×n matrix E
On exit: the Fréchet derivative L(A,E)
5: lde Integer Input
On entry: the first dimension of the array e as declared in the (sub)program from which f01khf is called.
Constraint: lden.
6: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
The linear equations to be solved for the Padé approximant are singular; it is likely that this routine has been called incorrectly.
ifail=2
eA has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
ifail=3
An unexpected internal error has occurred. Please contact NAG.
ifail=-1
On entry, n=value.
Constraint: n0.
ifail=-3
On entry, lda=value and n=value.
Constraint: ldan.
ifail=-5
On entry, lde=value and n=value.
Constraint: lden.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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.
f01khf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01khf 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 routine. 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 f01fcf should be used.
If the condition number of the matrix exponential is required then f01kgf 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 (f01khfe.f90)

10.2 Program Data

Program Data (f01khfe.d)

10.3 Program Results

Program Results (f01khfe.r)