NAG Library Routine Document
F01FDF computes the matrix exponential, , of a complex Hermitian by matrix .
||N, LDA, IFAIL
is computed using a spectral factorization of
is the diagonal matrix whose diagonal elements,
, are the eigenvalues of
is a unitary matrix whose columns are the eigenvectors of
is then given by
is the diagonal matrix whose
th diagonal element is
. See for example Section 4.5 of Higham (2008)
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: UPLO – CHARACTER(1)Input
, the upper triangle of the matrix
If , the lower triangle of the matrix is stored.
- 2: N – INTEGERInput
On entry: , the order of the matrix .
- 3: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
the second dimension of the array A
must be at least
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if , the upper or lower triangular part of the by matrix exponential, .
- 4: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F01FDF is called.
- 5: 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.
Allocation of memory failed. The INTEGER allocatable memory required is N
, the real allocatable memory required is N
allocatable memory required is approximately
, where nb
is the block size required by F08FNF (ZHEEV)
The algorithm to compute the spectral factorization failed to converge;
off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see F08FNF (ZHEEV)
Note: this failure is unlikely to occur.
For an Hermitian matrix
, the matrix
, has the relative condition number
which is the minimal possible for the matrix exponential and so the computed matrix exponential is guaranteed to be close to the exact matrix. See Section 10.2 of Higham (2008)
for details and further discussion.
The cost of the algorithm is .
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 finds the matrix exponential of the Hermitian matrix
9.1 Program Text
Program Text (f01fdfe.f90)
9.2 Program Data
Program Data (f01fdfe.d)
9.3 Program Results
Program Results (f01fdfe.r)