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NAG Toolbox: nag_lapack_dsyev (f08fa)

Purpose

nag_lapack_dsyev (f08fa) computes all the eigenvalues and, optionally, all the eigenvectors of a real n$n$ by n$n$ symmetric matrix A$A$.

Syntax

[a, w, info] = f08fa(jobz, uplo, a, 'n', n)
[a, w, info] = nag_lapack_dsyev(jobz, uplo, a, 'n', n)

Description

The symmetric matrix A$A$ is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the QR$QR$ algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

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

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then a contains the orthonormal eigenvectors of the matrix A$A$.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, then on exit the lower triangle (if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$) or the upper triangle (if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$) of a, including the diagonal, is overwritten.
2:     w(n) – double array
The eigenvalues in ascending order.
3:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: jobz, 2: uplo, 3: n, 4: a, 5: lda, 6: w, 7: work, 8: lwork, 9: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the algorithm failed to converge; i$i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (A + E)$\left(A+E\right)$, where
 ‖E‖2 = O(ε) ‖A‖2 , $‖E‖2 = O(ε) ‖A‖2 ,$
and ε$\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

Further Comments

The total number of floating point operations is proportional to n3${n}^{3}$.
The complex analogue of this function is nag_lapack_zheev (f08fn).

Example

```function nag_lapack_dsyev_example
jobz = 'Vectors';
uplo = 'Upper';
a = [1, 2, 3, 4;
0, 2, 3, 4;
0, 0, 3, 4;
0, 0, 0, 4];
[aOut, w, info] = nag_lapack_dsyev(jobz, uplo, a)
```
```

aOut =

0.7003   -0.5144    0.2767    0.4103
0.3592    0.4851   -0.6634    0.4422
-0.1569    0.5420    0.6504    0.5085
-0.5965   -0.4543   -0.2457    0.6144

w =

-2.0531
-0.5146
-0.2943
12.8621

info =

0

```
```function f08fa_example
jobz = 'Vectors';
uplo = 'Upper';
a = [1, 2, 3, 4;
0, 2, 3, 4;
0, 0, 3, 4;
0, 0, 0, 4];
[aOut, w, info] = f08fa(jobz, uplo, a)
```
```

aOut =

0.7003   -0.5144    0.2767    0.4103
0.3592    0.4851   -0.6634    0.4422
-0.1569    0.5420    0.6504    0.5085
-0.5965   -0.4543   -0.2457    0.6144

w =

-2.0531
-0.5146
-0.2943
12.8621

info =

0

```

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