NAG Library Function Document

nag_matop_real_gen_matrix_cond_pow (f01jec)

+− 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_cond_pow (f01jec) computes an estimate of the relative condition number

κ_{A^{p}}

of the

p

th power (where

p

is real) of a real

n

n

matrix

A

, in the

1

-norm. The principal matrix power

A^{p}

is also returned.

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_matop_real_gen_matrix_cond_pow (Integer n, double a[], Integer pda, double p, double condpa, 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\}

The Fréchet derivative of the matrix

p

th power of

A

is the unique linear mapping

E ⟼ L (A, E)

such that for any matrix

E

{(A + E)}^{p} - A^{p} - L (A, E) = o (‖E‖) .

The derivative describes the first-order effect of perturbations in

A

on the matrix power

A^{p}

The relative condition number of the matrix

p

th power can be defined by

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

where

‖L (A)‖

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

A

nag_matop_real_gen_matrix_cond_pow (f01jec) uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute

κ_{A^{p}}

and

A^{p}

. 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 using a Schur decomposition, a Padé approximant and the scaling and squaring method.

To obtain an estimate of

κ_{A^{p}}

, nag_matop_real_gen_matrix_cond_pow (f01jec) 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

. This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of

A^{q}

are obtained by differentiating the Padé approximant. Fréchet derivatives of

A^{p}

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 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 $n$ principal 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: condpa – double *Output: On exit: if $fail . code =$ NE_NOERROR or NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix $p$ th power, $κ_{A^{p}}$ . Alternatively, if $fail . code =$ NE_RCOND, the absolute condition number of the matrix $p$ th power.
6: 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 in this case; nag_matop_complex_gen_matrix_cond_pow (f01kec) can be used to find a complex, non-principal $p$ th power.
NE_RCOND: The relative condition number is infinite. The absolute condition number was returned instead.
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

nag_matop_real_gen_matrix_cond_pow (f01jec) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04ydc) 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_real_gen_norm_rcomm (f04ydc).

For a normal matrix

A

(for which

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

), 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

A^{p}

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

nag_matop_real_gen_matrix_cond_pow (f01jec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_real_gen_matrix_cond_pow (f01jec) 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 amount of real allocatable memory required by the algorithm is typically of the order

10 \times n^{2}

The cost of the algorithm is

O (n^{3})

floating-point operations; see Higham and Lin (2013).

If the matrix

p

th power alone is required, without an estimate of the condition number, then nag_matop_real_gen_matrix_pow (f01eqc) should be used. If the Fréchet derivative of the matrix power is required then nag_matop_real_gen_matrix_frcht_pow (f01jfc) should be used. If

A

has negative real eigenvalues then nag_matop_complex_gen_matrix_cond_pow (f01kec) can be used to return a complex, non-principal

p

th power and its condition number.

10 Example

This example estimates the relative condition number of the matrix power

A^{p}

, where

p = 0.2

and

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

NAG Library Function Documentnag_matop_real_gen_matrix_cond_pow (f01jec)

+− 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_cond_pow (f01jec)