The function may be called by the names: f01fdc or nag_matop_complex_herm_matrix_exp.
is computed using a spectral factorization of
where is the diagonal matrix whose diagonal elements, , are the eigenvalues of , and is a unitary matrix whose columns are the eigenvectors of . is then given by
where 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: – Nag_OrderTypeInput
On entry: 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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_UploTypeInput
On entry: if , the upper triangle of the matrix is stored.
If , the lower triangle of the matrix is stored.
3: – IntegerInput
On entry: , the order of the matrix .
4: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
On entry: the Hermitian matrix .
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: if NE_NOERROR, the upper or lower triangular part of the matrix exponential, .
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
6: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
The value of fail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08fnc).
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.
The computation of the spectral factorization failed to converge.
On entry, .
On entry, . Constraint: .
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.
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01fdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 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.
The Integer allocatable memory required is n, the double allocatable memory required is n and the Complex allocatable memory required is approximately , where nb is the block size required by f08fnc.
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