# NAG Library Routine Document

## 1Purpose

f01fcf computes the matrix exponential, ${e}^{A}$, of a complex $n$ by $n$ matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f01fcf ( n, a, lda,
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*)
#include nagmk26.h
 void f01fcf_ (const Integer *n, Complex a[], const Integer *lda, Integer *ifail)

## 3Description

${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method described in Al–Mohy and Higham (2009).

## 4References

Al–Mohy A H and Higham N J (2009) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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: with ${\mathbf{ifail}}={\mathbf{0}}$, the $n$ by $n$ matrix exponential ${e}^{A}$.
3:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01fcf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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.
${\mathbf{ifail}}=2$
The linear equations to be solved are nearly singular and the Padé approximant probably has no correct figures; it is likely that this routine has been called incorrectly.
${\mathbf{ifail}}=3$
${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
${\mathbf{ifail}}=4$
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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 non-normal matrices. See Al–Mohy and Higham (2009) and Section 10.3 of Higham (2008) for details and further discussion.
If estimates of the condition number of the matrix exponential are required then f01kgf should be used.

## 8Parallelism and Performance

f01fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fcf 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.

The integer allocatable memory required is n, and the complex allocatable memory required is approximately $6×{{\mathbf{n}}}^{2}$.
The cost of the algorithm is $O\left({n}^{3}\right)$; see Section 5 of Al–Mohy and Higham (2009). The complex allocatable memory required is approximately $6×{n}^{2}$.
If the Fréchet derivative of the matrix exponential is required then f01khf should be used.
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 matrix
 $A = 1+2i 2+2i 2+2i 2+i 3+2i 1 1 2+i 3+2i 2+2i 1 2+i 3+2i 3+2i 3+2i 1+i .$

### 10.1Program Text

Program Text (f01fcfe.f90)

### 10.2Program Data

Program Data (f01fcfe.d)

### 10.3Program Results

Program Results (f01fcfe.r)

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