# NAG CL Interfacef01edc (real_​symm_​matrix_​exp)

Settings help

CL Name Style:

## 1Purpose

f01edc computes the matrix exponential, ${e}^{A}$, of a real symmetric $n×n$ matrix $A$.

## 2Specification

 #include
 void f01edc (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, NagError *fail)
The function may be called by the names: f01edc, nag_matop_real_symm_matrix_exp or nag_real_symm_matrix_exp.

## 3Description

${e}^{A}$ is computed using a spectral factorization of $A$
 $A = Q D QT ,$
where $D$ is the diagonal matrix whose diagonal elements, ${d}_{i}$, are the eigenvalues of $A$, and $Q$ is an orthogonal matrix whose columns are the eigenvectors of $A$. ${e}^{A}$ is then given by
 $eA = Q eD QT ,$
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 (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: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{uplo}$Nag_UploType Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{a}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
On entry: the $n×n$ symmetric matrix $A$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the upper or lower triangular part of the $n×n$ matrix exponential, ${e}^{A}$.
5: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

The value of fail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08fac).
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
The computation of the spectral factorization failed to converge.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

For a symmetric matrix $A$, the matrix ${e}^{A}$, has the relative condition number
 $κ(A) = ‖A‖2 ,$
which is the minimum 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

f01edc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01edc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The Integer allocatable memory required is n, and the double allocatable memory required is approximately $\left({\mathbf{n}}+\mathit{nb}+4\right)×{\mathbf{n}}$, where nb is the block size required by f08fac.
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 symmetric matrix
 $A = ( 1 2 3 4 2 1 2 3 3 2 1 2 4 3 2 1 )$

### 10.1Program Text

Program Text (f01edce.c)

### 10.2Program Data

Program Data (f01edce.d)

### 10.3Program Results

Program Results (f01edce.r)