NAG FL Interface
f01jef (real_​gen_​matrix_​cond_​pow)

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1 Purpose

f01jef computes an estimate of the relative condition number κAp of the pth power (where p is real) of a real n×n matrix A, in the 1-norm. The principal matrix power Ap is also returned.

2 Specification

Fortran Interface
Subroutine f01jef ( n, a, lda, p, condpa, ifail)
Integer, Intent (In) :: n, lda
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: p
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Real (Kind=nag_wp), Intent (Out) :: condpa
C Header Interface
#include <nag.h>
void  f01jef_ (const Integer *n, double a[], const Integer *lda, const double *p, double *condpa, Integer *ifail)
The routine may be called by the names f01jef or nagf_matop_real_gen_matrix_cond_pow.

3 Description

For a matrix A with no eigenvalues on the closed negative real line, Ap (p) can be defined as
Ap= exp(plog(A))  
where log(A) is the principal logarithm of A (the unique logarithm whose spectrum lies in the strip {z:-π<Im(z)<π}).
The Fréchet derivative of the matrix pth power of A is the unique linear mapping EL(A,E) such that for any matrix E
(A+E)p - Ap - L(A,E) = o(E) .  
The derivative describes the first-order effect of perturbations in A on the matrix power Ap.
The relative condition number of the matrix pth power can be defined by
κAp = L(A) A Ap ,  
where L(A) is the norm of the Fréchet derivative of the matrix power at A.
f01jef uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute κAp and Ap. The real number p is expressed as p=q+r where q(−1,1) and r. Then Ap=AqAr. The integer power Ar is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power Aq is computed using a Schur decomposition, a Padé approximant and the scaling and squaring method.
To obtain an estimate of κAp, f01jef first estimates L(A) by computing an estimate γ of a quantity K[n-1L(A)1,nL(A)1], such that γK. This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of Aq are obtained by differentiating the Padé approximant. Fréchet derivatives of Ap are then computed using a combination of the chain rule and the product rule for Fréchet derivatives.

4 References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Higham N J and Lin L (2011) A Schur–Padé algorithm for fractional powers of a matrix SIAM J. Matrix Anal. Appl. 32(3) 1056–1078
Higham N J and Lin L (2013) An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives SIAM J. Matrix Anal. Appl. 34(3) 1341–1360

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a(lda,*) Real (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 principal matrix pth power, Ap.
3: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01jef is called.
Constraint: ldan.
4: p Real (Kind=nag_wp) Input
On entry: the required power of A.
5: condpa Real (Kind=nag_wp) Output
On exit: if ifail=0 or 3, an estimate of the relative condition number of the matrix pth power, κAp. Alternatively, if ifail=4, the absolute condition number of the matrix pth power.
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:
A has eigenvalues on the negative real line. The principal pth power is not defined in this case; f01kef can be used to find a complex, non-principal pth power.
A is singular so the pth power cannot be computed.
Ap has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
The relative condition number is infinite. The absolute condition number was returned instead.
An unexpected internal error occurred. This failure should not occur and suggests that the routine has been called incorrectly.
On entry, n=value.
Constraint: n0.
On entry, lda=value and n=value.
Constraint: ldan.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

f01jef uses the norm estimation routine f04ydf to produce an estimate γ of a quantity K[n−1L(A)1,nL(A)1], such that γK. For further details on the accuracy of norm estimation, see the documentation for f04ydf.
For a normal matrix A (for which ATA=AAT), the Schur decomposition is diagonal and the computation of the fractional part of the matrix power reduces to evaluating powers of the eigenvalues of A and then constructing Ap using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Higham and Lin (2011) and Higham and Lin (2013) for details and further discussion.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01jef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jef 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 amount of real allocatable memory required by the algorithm is typically of the order 10×n2.
The cost of the algorithm is O(n3) floating-point operations; see Higham and Lin (2013).
If the matrix pth power alone is required, without an estimate of the condition number, then f01eqf should be used. If the Fréchet derivative of the matrix power is required then f01jff should be used. If A has negative real eigenvalues then f01kef can be used to return a complex, non-principal pth power and its condition number.

10 Example

This example estimates the relative condition number of the matrix power Ap, where p=0.2 and
A = ( 3 3 2 1 1 1 0 2 1 4 4 2 3 1 3 1 ) .  

10.1 Program Text

Program Text (f01jefe.f90)

10.2 Program Data

Program Data (f01jefe.d)

10.3 Program Results

Program Results (f01jefe.r)