f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_matop_complex_gen_matrix_exp (f01fcc)

## 1  Purpose

nag_matop_complex_gen_matrix_exp (f01fcc) computes the matrix exponential, ${e}^{A}$, of a complex $n$ by $n$ matrix $A$.

## 2  Specification

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

## 3  Description

${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method described in Higham (2005) and Higham (2008).
If $A$ has a full set of eigenvectors $V$ then $A$ can be factorized as
 $A = V D V-1 ,$
where $D$ is the diagonal matrix whose diagonal elements, ${d}_{i}$, are the eigenvalues of $A$. ${e}^{A}$ is then given by
 $eA = V eD V-1 ,$
where ${e}^{D}$ is the diagonal matrix whose $i$th diagonal element is ${e}^{{d}_{i}}$.
Note that ${e}^{A}$ is not computed this way as to do so would, in general, be unstable.

## 4  References

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

## 5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix exponential, ${e}^{A}$.
4:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge {\mathbf{n}}$.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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.
NE_SINGULAR
Linear equations are nearly singular, Padé approximant has no accuracy. This error should not occur, check input arguments.
Linear equations are singular, Padé approximant cannot be computed. This error should not occur, check input arguments.
NW_SOME_PRECISION_LOSS
Warning: matrix exponential not computed to full working precision.

## 7  Accuracy

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 Section 10.3 of Higham (2008) for details and further discussion.
For discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008).

The cost of the algorithm is $O\left({n}^{3}\right)$; see Algorithm 10.20 in Higham (2008).
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).

## 9  Example

This example find the matrix exponential of the matrix
 $A = 1+2i 2+2i 2+2i 2+i 3+2i 1 1 2+i 3+2i 2+2i 1 2+i 3+2i 3+2i 3+2i 1+i .$

### 9.1  Program Text

Program Text (f01fcce.c)

### 9.2  Program Data

Program Data (f01fcce.d)

### 9.3  Program Results

Program Results (f01fcce.r)