f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_real_gen_matrix_exp (f01ecc)

## 1  Purpose

nag_real_gen_matrix_exp (f01ecc) computes the matrix exponential, ${e}^{A}$, of a real $n$ by $n$ matrix $A$.

## 2  Specification

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

## 3  Description

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

## 4  References

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

## 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 $\mathrm{Nag_ColMajor}$.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     a[$\mathit{dim}$]doubleInput/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.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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: ${\mathbf{n}}\ge 0$.
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
The linear equations to be solved are nearly singular and the Padé approximant probably has no correct figures; it is likely that this function has been called incorrectly.
The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.
NW_SOME_PRECISION_LOSS
${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

## 7  Accuracy

For a normal matrix $A$ (for which ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$) 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 nag_matop_real_gen_matrix_cond_exp (f01jgc) should be used.

## 8  Parallelism and Performance

nag_real_gen_matrix_exp (f01ecc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_gen_matrix_exp (f01ecc) 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.

The cost of the algorithm is $O\left({n}^{3}\right)$; see Section 5 of of Al–Mohy and Higham (2009). The real allocatable memory required is approximately $6×{n}^{2}$.
If the Fréchet derivative of the matrix exponential is required then nag_matop_real_gen_matrix_frcht_exp (f01jhc) 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).

## 10  Example

This example finds the matrix exponential of the matrix
 $A = 1 2 2 2 3 1 1 2 3 2 1 2 3 3 3 1 .$

### 10.1  Program Text

Program Text (f01ecce.c)

### 10.2  Program Data

Program Data (f01ecce.d)

### 10.3  Program Results

Program Results (f01ecce.r)