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# NAG Toolbox: nag_matop_real_symm_matrix_exp (f01ed)

## Purpose

nag_matop_real_symm_matrix_exp (f01ed) computes the matrix exponential, ${e}^{A}$, of a real symmetric $n$ by $n$ matrix $A$.

## Syntax

[a, ifail] = f01ed(uplo, a, 'n', n)
[a, ifail] = nag_matop_real_symm_matrix_exp(uplo, a, 'n', n)

## Description

${e}^{A}$ 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).

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least ${\mathbf{n}}$.
The $n$ by $n$ symmetric matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be ${\mathbf{n}}$.
If ${\mathbf{ifail}}={\mathbf{0}}$, the upper or lower triangular part of the $n$ by $n$ matrix exponential, ${e}^{A}$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}>0$
The computation of the spectral factorization failed to converge.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{uplo}}=\text{'L'}$ or $\text{'U'}$.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
An internal error occurred when computing the spectral factorization. Please contact NAG.
${\mathbf{ifail}}=-4$
Constraint: $\mathit{lda}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

For a symmetric matrix $A$, the matrix ${e}^{A}$, has the relative condition number
 $κA = A2 ,$
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.

## Further Comments

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 nag_lapack_dsyev (f08fa).
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).

## Example

This example finds the matrix exponential of the symmetric matrix
 $A = 1 2 3 4 2 1 2 3 3 2 1 2 4 3 2 1$
```function f01ed_example

fprintf('f01ed example results\n\n');

uplo = 'u';
a =  [1, 2, 3, 4;
0, 1, 2, 3;
0, 0, 1, 2;
0, 0, 0, 1];

% Compute exp(a)
[expa, ifail] = f01ed(uplo, a);

% Display results
[ifail] = x04ca(uplo, 'n', expa, 'Symmetric Exp(a)');

```
```f01ed example results

Symmetric Exp(a)
1          2          3          4
1   2675.3899  2193.0210  2193.2062  2675.2803
2              1798.3297  1797.8497  2193.2062
3                         1798.3297  2193.0210
4                                    2675.3899
```

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