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: $n \geq 0$ .
2: $a [\dim]$ – double Input/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 $n \times n$ matrix $A$ .

On exit: the $n \times n$ matrix $p$ th power, $A^{p}$ .
3: $pda$ – Integer Input: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: $p$ – double Input: On entry: the required power of $A$ .
5: $fail$ – 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: $n \geq 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NEGATIVE_EIGVAL: $A$ has eigenvalues on the negative real line. The principal $p$ th power is not defined. f01fqc can be used to find a complex, non-principal $p$ th power.
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: $A$ 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

Background information to multithreading can be found in the Multithreading documentation.

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

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 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 (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 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}) .

f01eq: FL CL CPP AD PY MB

NAG CL Interfacef01eqc (real_​gen_​matrix_​pow)

▸▿ 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 CL Interface
f01eqc (real_gen_matrix_pow)