The matrix exponential is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives which are used to estimate the condition number.
Al–Mohy A H and Higham N J (2009a) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal.31(3) 970–989
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
1: – IntegerInput
On entry: , the order of the matrix .
2: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
The th element of the matrix is stored in .
On entry: the matrix .
On exit: the matrix exponential .
3: – IntegerInput
On entry: the stride separating matrix row elements in the array a.
4: – double *Output
On exit: an estimate of the relative condition number of the matrix exponential .
5: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, and .
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.
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.
The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
f01jgc uses the norm estimation function f04ydc to produce an estimate of a quantity , such that . For further details on the accuracy of norm estimation, see the documentation for f04ydc.
For a normal matrix (for which ) the computed matrix, , 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).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01jgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jgc 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.
f01jac uses a similar algorithm to f01jgc to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of ). However, the required Fréchet derivatives are computed in a more efficient and stable manner by f01jgc and so its use is recommended over f01jac.
If the matrix exponential alone is required, without an estimate of the condition number, then f01ecc should be used. If the Fréchet derivative of the matrix exponential is required then f01jhc should be used.