NAG Library Routine Document
F01FCF computes the matrix exponential, , of a complex by matrix .
||N, LDA, IFAIL
is computed using a Padé approximant and the scaling and squaring method described in Higham (2005)
and Higham (2008)
has a full set of eigenvectors
can be factorized as
is the diagonal matrix whose diagonal elements,
, are the eigenvalues of
is then given by
is the diagonal matrix whose
th diagonal element is
Note that is not computed this way as to do so would, in general, be unstable.
Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl. 26(4) 1179–1193
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: N – INTEGERInput
On entry: , the order of the matrix .
- 2: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
the second dimension of the array A
must be at least
On entry: the by matrix .
On exit: with , the by matrix exponential, .
- 3: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F01FCF is called.
- 4: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
If , the th argument had an illegal value.
Internal memory allocation failed.
The INTEGER allocatable memory required is N
, and the
allocatable memory required is approximately
Note: these failures should not occur, and suggest that the routine has been called incorrectly.
If , the linear equations to be solved for the Padé approximant are singular.
If , the linear equations to be solved are nearly singular and the Padé approximant probably has no correct figures.
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
For a normal matrix
) 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 discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008)
The cost of the algorithm is
; see Algorithm 10.20 in Higham (2008)
If estimates of the condition number of the matrix exponential are required then F01KAF
should be used.
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003)
This example find the matrix exponential of the matrix
9.1 Program Text
Program Text (f01fcfe.f90)
9.2 Program Data
Program Data (f01fcfe.d)
9.3 Program Results
Program Results (f01fcfe.r)