NAG Library Function Document

1Purpose

nag_matop_complex_herm_matrix_exp (f01fdc) computes the matrix exponential, ${e}^{A}$, of a complex Hermitian $n$ by $n$ matrix $A$.

2Specification

 #include #include
 void nag_matop_complex_herm_matrix_exp (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, NagError *fail)

3Description

${e}^{A}$ is computed using a spectral factorization of $A$
 $A = Q D QH ,$
where $D$ is the diagonal matrix whose diagonal elements, ${d}_{i}$, are the eigenvalues of $A$, and $Q$ is a unitary matrix whose columns are the eigenvectors of $A$. ${e}^{A}$ is then given by
 $eA = Q eD QH ,$
where ${e}^{D}$ is the diagonal matrix whose $i$th diagonal element is ${e}^{{d}_{i}}$. See for example Section 4.5 of Higham (2008).

4References

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

5Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{a}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the upper or lower triangular part of the $n$ by $n$ matrix exponential, ${e}^{A}$.
5:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
The computation of the spectral factorization failed to converge.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

For an Hermitian matrix $A$, the matrix ${e}^{A}$, has the relative condition number
 $κA = A2 ,$
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

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 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 $\left({\mathbf{n}}+\mathit{nb}+1\right)×{\mathbf{n}}$, where nb is the block size required by nag_zheev (f08fnc).
The cost of the algorithm is $O\left({n}^{3}\right)$.
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).

10Example

This example finds the matrix exponential of the Hermitian matrix
 $A = 1 2+2i 3+2i 4+3i 2-2i 1 2+2i 3+2i 3-2i 2-2i 1 2+2i 4-3i 3-2i 2-2i 1 .$

10.1Program Text

Program Text (f01fdce.c)

10.2Program Data

Program Data (f01fdce.d)

10.3Program Results

Program Results (f01fdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017