Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dspevd (f08gc)

## Purpose

nag_lapack_dspevd (f08gc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix held in packed storage. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL$QL$ or QR$QR$ algorithm.

## Syntax

[ap, w, z, info] = f08gc(job, uplo, n, ap)
[ap, w, z, info] = nag_lapack_dspevd(job, uplo, n, ap)

## Description

nag_lapack_dspevd (f08gc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix A$A$ (held in packed storage). In other words, it can compute the spectral factorization of A$A$ as
 A = ZΛZT, $A=ZΛZT,$
where Λ$\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues λi${\lambda }_{i}$, and Z$Z$ is the orthogonal matrix whose columns are the eigenvectors zi${z}_{i}$. Thus
 Azi = λizi,  i = 1,2, … ,n. $Azi=λizi, i=1,2,…,n.$

## 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:     job – string (length ≥ 1)
Indicates whether eigenvectors are computed.
job = 'N'${\mathbf{job}}=\text{'N'}$
Only eigenvalues are computed.
job = 'V'${\mathbf{job}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: job = 'N'${\mathbf{job}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A$A$ is stored.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored.
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:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
4:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The upper or lower triangle of the n$n$ by n$n$ symmetric matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.

None.

### Input Parameters Omitted from the MATLAB Interface

ldz work lwork iwork liwork

### Output Parameters

1:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
ap stores the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A$A$.
2:     w( : $:$) – double array
Note: the dimension of the array w must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The eigenvalues of the matrix A$A$ in ascending order.
3:     z(ldz, : $:$) – double array
The first dimension, ldz, of the array z will be
• if job = 'V'${\mathbf{job}}=\text{'V'}$, ldz max (1,n) $\mathit{ldz}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if job = 'N'${\mathbf{job}}=\text{'N'}$, ldz1$\mathit{ldz}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if job = 'V'${\mathbf{job}}=\text{'V'}$ and at least 1$1$ if job = 'N'${\mathbf{job}}=\text{'N'}$
If job = 'V'${\mathbf{job}}=\text{'V'}$, z stores the orthogonal matrix Z$Z$ which contains the eigenvectors of A$A$.
If job = 'N'${\mathbf{job}}=\text{'N'}$, z is not referenced.
4:     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: job, 2: uplo, 3: n, 4: ap, 5: w, 6: z, 7: ldz, 8: work, 9: lwork, 10: iwork, 11: liwork, 12: 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${\mathbf{info}}=i$ and job = 'N'${\mathbf{job}}=\text{'N'}$, the algorithm failed to converge; i$i$ elements of an intermediate tridiagonal form did not converge to zero; if info = i${\mathbf{info}}=i$ and job = 'V'${\mathbf{job}}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column i / (n + 1)$i/\left({\mathbf{n}}+1\right)$ through i  mod  (n + 1).

## 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.

The complex analogue of this function is nag_lapack_zhpevd (f08gq).

## Example

```function nag_lapack_dspevd_example
job = 'V';
uplo = 'L';
n = int64(4);
ap = [1;
2;
3;
4;
2;
3;
4;
3;
4;
4];
[apOut, w, z, info] = nag_lapack_dspevd(job, uplo, n, ap)
```
```

apOut =

1.0000
-5.3852
0.4062
0.5416
10.1724
-1.8302
0.7276
-0.8160
0.1223
-0.3564

w =

-2.0531
-0.5146
-0.2943
12.8621

z =

-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

info =

0

```
```function f08gc_example
job = 'V';
uplo = 'L';
n = int64(4);
ap = [1;
2;
3;
4;
2;
3;
4;
3;
4;
4];
[apOut, w, z, info] = f08gc(job, uplo, n, ap)
```
```

apOut =

1.0000
-5.3852
0.4062
0.5416
10.1724
-1.8302
0.7276
-0.8160
0.1223
-0.3564

w =

-2.0531
-0.5146
-0.2943
12.8621

z =

-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

info =

0

```