The routine may be called by the names f01jhf or nagf_matop_real_gen_matrix_frcht_exp.
The Fréchet derivative of the matrix exponential of is the unique linear mapping such that for any matrix
The derivative describes the first-order effect of perturbations in on the exponential .
f01jhf uses the algorithms of Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) to compute and . The matrix exponential is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative .
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: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the matrix .
On exit: the matrix exponential .
3: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01jhf is called.
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array e
must be at least
On entry: the matrix
On exit: the Fréchet derivative
5: – IntegerInput
On entry: the first dimension of the array e as declared in the (sub)program from which f01jhf is called.
6: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
The linear equations to be solved for the Padé approximant are singular; it is likely that this routine has been called incorrectly.
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
An unexpected internal error has occurred. Please contact NAG.
On entry, .
On entry, and .
On entry, and . Constraint: .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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), Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) for details and further discussion.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01jhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jhf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.