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: $n \geq 0$ .
2: $a [\dim]$ – Complex 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 exponential $e^{A}$ .
3: $pda$ – Integer Input: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: $condea$ – double * Output: On exit: an estimate of the relative condition number of the matrix exponential $κ_{\exp} (A)$ .
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_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: The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.
NW_SOME_PRECISION_LOSS: $e^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

f01kgc uses the norm estimation function f04zdc 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 f04zdc.

For a normal matrix

A

(for which

A^{H} A = A A^{H}

) the computed matrix,

e^{A}

, 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) for details and further discussion.

For further discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008).

8 Parallelism and Performance

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

f01kgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f01kgc 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

f01kac uses a similar algorithm to f01kgc to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of

‖ A ‖ / ‖ \exp (A) ‖

). However, the required Fréchet derivatives are computed in a more efficient and stable manner by f01kgc and so its use is recommended over f01kac.

The cost of the algorithm is

O (n^{3})

and the complex allocatable memory required is approximately

15 n^{2}

; see Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) for further details.

If the matrix exponential alone is required, without an estimate of the condition number, then f01fcc should be used. If the Fréchet derivative of the matrix exponential is required then f01khc 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 estimates the relative condition number of the matrix exponential

e^{A}

, where

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

f01kg: FL CL CPP AD PY MB

NAG CL Interfacef01kgc (complex_​gen_​matrix_​cond_​exp)

▸▿ 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
f01kgc (complex_gen_matrix_cond_exp)