NAG Library Function Document
nag_matop_complex_herm_matrix_exp (f01fdc) computes the matrix exponential, , of a complex Hermitian by matrix .
||nag_matop_complex_herm_matrix_exp (Nag_OrderType order,
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
order – Nag_OrderTypeInput
: the order
argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See Section 220.127.116.11
in the Essential Introduction for a more detailed explanation of the use of this argument.
uplo – Nag_UploTypeInput
, the upper triangle of the matrix
If , the lower triangle of the matrix is stored.
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
If , is stored in .
If , is stored in .
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: the upper or lower triangular part of the by matrix exponential, .
pda – IntegerInput
: the stride separating row or column elements (depending on the value of order
) of the matrix
in the array
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
The computation of the spectral factorization failed to converge.
On entry, .
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 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.
8 Parallelism and Performance
nag_matop_complex_herm_matrix_exp (f01fdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_herm_matrix_exp (f01fdc) 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 Users' Note
for your implementation for any additional implementation-specific information.
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
10.1 Program Text
Program Text (f01fdce.c)
10.2 Program Data
Program Data (f01fdce.d)
10.3 Program Results
Program Results (f01fdce.r)