NAG Library Function Document

nag_matop_real_gen_matrix_pow (f01eqc)

+− 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_pow (f01eqc) computes the principal real power

A^{p}

, for arbitrary

p

, of a real

n

n

matrix

A

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_matop_real_gen_matrix_pow (Integer n, double a[], Integer pda, double p, NagError *fail)

3 Description

For a matrix

A

with no eigenvalues on the closed negative real line,

A^{p}

(

p \in ℝ

) can be defined as

A^{p} = \exp (p \log (A))

where

\log (A)

is the principal logarithm of

A

(the unique logarithm whose spectrum lies in the strip

\{z : - π < Im (z) < π\}

A^{p}

is computed using the real version of the Schur–Padé algorithm described in Higham and Lin (2011) and Higham and Lin (2013).

The real number

p

is expressed as

p = q + r

where

q \in (- 1, 1)

and

r \in ℤ

. Then

A^{p} = A^{q} A^{r}

. The integer power

A^{r}

is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power

A^{q}

is computed, entirely in real arithmetic, using a real Schur decomposition and a Padé approximant.

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 MIMS Eprint 2013.1 Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester http://eprints.ma.man.ac.uk/

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 $p$ th power, $A^{p}$ .
3: pda – IntegerInput: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: p – doubleInput: On entry: the required power of $A$ .
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_NEGATIVE_EIGVAL: $A$ has eigenvalues on the negative real line. The principal $p$ th power is not defined. nag_matop_complex_gen_matrix_pow (f01fqc) can be used to find a complex, non-principal $p$ th power.
NE_SINGULAR: $A$ is singular so the $p$ th power cannot be computed.
NW_SOME_PRECISION_LOSS: $A^{p}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

For positive integer

p

, the algorithm reduces to a sequence of matrix multiplications. For negative integer

p

, the algorithm consists of a combination of matrix inversion and matrix multiplications.

For a normal matrix

A

(for which

A^{T} A = A A^{T}

) and non-integer

p

, the Schur decomposition is diagonal and the algorithm reduces to evaluating powers of the eigenvalues of

A

and then constructing

A^{p}

using the Schur vectors. This should give a very accurate result. In general however, no error bounds are available for the algorithm.

8 Parallelism and Performance

nag_matop_real_gen_matrix_pow (f01eqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_real_gen_matrix_pow (f01eqc) 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})

. The exact cost depends on the matrix

A

but if

p \in (- 1, 1)

then the cost is independent of

p

O (4 \times n^{2})

of real allocatable memory is required by the function.

If estimates of the condition number of

A^{p}

are required then nag_matop_real_gen_matrix_cond_pow (f01jec) should be used.

10 Example

This example finds

A^{p}

where

p = 0.2

and

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

NAG Library Function Documentnag_matop_real_gen_matrix_pow (f01eqc)

+− 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

NAG Library Function Document

nag_matop_real_gen_matrix_pow (f01eqc)