NAG Library Routine Document
F01KGF
1 Purpose
F01KGF computes an estimate of the relative condition number ${\kappa}_{\mathrm{exp}}\left(A\right)$ of the exponential of a complex $n$ by $n$ matrix $A$, in the $1$norm. The matrix exponential ${e}^{A}$ is also returned.
2 Specification
INTEGER 
N, LDA, IFAIL 
REAL (KIND=nag_wp) 
CONDEA 
COMPLEX (KIND=nag_wp) 
A(LDA,*) 

3 Description
The Fréchet derivative of the matrix exponential of
$A$ is the unique linear mapping
$E\u27fcL\left(A,E\right)$ such that for any matrix
$E$
The derivative describes the firstorder effect of perturbations in $A$ on the exponential ${e}^{A}$.
The relative condition number of the matrix exponential can be defined by
where
$\Vert L\left(A\right)\Vert $ is the norm of the Fréchet derivative of the matrix exponential at
$A$.
To obtain the estimate of ${\kappa}_{\mathrm{exp}}\left(A\right)$, F01KGF first estimates $\Vert L\left(A\right)\Vert $ by computing an estimate $\gamma $ of a quantity $K\in \left[{n}^{1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}\right]$, such that $\gamma \le K$.
The algorithms used to compute
${\kappa}_{\mathrm{exp}}\left(A\right)$ are detailed in the
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b).
The matrix exponential ${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives $L\left(A,E\right)$ which are used to estimate the condition number.
4 References
Al–Mohy A H and Higham N J (2009a) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Al–Mohy A H and Higham N J (2009b) Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation SIAM J. Matrix Anal. Appl. 30(4) 1639–1657
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, twentyfive years later SIAM Rev. 45 3–49
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
A
must be at least
${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix exponential ${e}^{A}$.
 3: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01KGF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{CONDEA}$ – REAL (KIND=nag_wp)Output

On exit: an estimate of the relative condition number of the matrix exponential ${\kappa}_{\mathrm{exp}}\left(A\right)$.
 5: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

The linear equations to be solved for the Padé approximant are singular; it is likely that this routine has been called incorrectly.
 ${\mathbf{IFAIL}}=2$

${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
 ${\mathbf{IFAIL}}=3$

An unexpected internal error has occurred. Please contact
NAG.
 ${\mathbf{IFAIL}}=1$

On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}\ge 0$.
 ${\mathbf{IFAIL}}=3$

On entry, ${\mathbf{LDA}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
F01KGF uses the norm estimation routine
F04ZDF to produce an estimate
$\gamma $ of a quantity
$K\in \left[{n}^{1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}\right]$, such that
$\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for
F04ZDF.
For a normal matrix
$A$ (for which
${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$) the computed matrix,
${e}^{A}$, is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for nonnormal matrices. See Section 10.3 of
Higham (2008) for details and further discussion.
For further discussion of the condition of the matrix exponential see Section 10.2 of
Higham (2008).
8 Parallelism and Performance
F01KGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01KGF 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 implementationspecific information.
F01KAF uses a similar algorithm to F01KGF to compute an estimate of the
absolute condition number (which is related to the relative condition number by a factor of
$\Vert A\Vert /\Vert \mathrm{exp}\left(A\right)\Vert $). However, the required Fréchet derivatives are computed in a more efficient and stable manner by F01KGF and so its use is recommended over
F01KAF.
The cost of the algorithm is
$O\left({n}^{3}\right)$ and the complex allocatable memory required is approximately
$15{n}^{2}$; see
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b) for further details.
If the matrix exponential alone is required, without an estimate of the condition number, then
F01FCF should be used. If the Fréchet derivative of the matrix exponential is required then
F01KHF should be used.
As well as the excellent book
Higham (2008), the classic reference for the computation of the matrix exponential is
Moler and Van Loan (2003).
10 Example
This example estimates the relative condition number of the matrix exponential
${e}^{A}$, where
10.1 Program Text
Program Text (f01kgfe.f90)
10.2 Program Data
Program Data (f01kgfe.d)
10.3 Program Results
Program Results (f01kgfe.r)