The routine may be called by the names f01fdf or nagf_matop_complex_herm_matrix_exp.
3Description
${e}^{A}$ is computed using a spectral factorization of $A$
$$A=QD{Q}^{\mathrm{H}}\text{,}$$
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
$${e}^{A}=Q{e}^{D}{Q}^{\mathrm{H}}\text{,}$$
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: $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
Note: the second dimension of the array a
must be at least
${\mathbf{n}}$.
On entry: the $n\times n$ Hermitian matrix $A$.
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, the upper or lower triangular part of the $n\times n$ matrix exponential, ${e}^{A}$.
4: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01fdf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
5: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}>0$
The computation of the spectral factorization failed to converge.
The value of ifail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08fnf).
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{uplo}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{uplo}}=\text{'L'}$ or $\text{'U'}$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
An internal error occurred when computing the spectral factorization. Please contact NAG.
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
For an Hermitian matrix $A$, the matrix ${e}^{A}$, 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
f01fdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fdf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The integer allocatable memory required is n, the real allocatable memory required is n and the complex allocatable memory required is approximately $({\mathbf{n}}+\mathit{nb}+1)\times {\mathbf{n}}$, where nb is the block size required by f08fnf.
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