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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dspgv (f08ta)

## Purpose

nag_lapack_dspgv (f08ta) computes all the eigenvalues and, optionally, all the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
 Az = λBz ,   ABz = λz   or   BAz = λz , $Az=λBz , ABz=λz or BAz=λz ,$
where A$A$ and B$B$ are symmetric, stored in packed format, and B$B$ is also positive definite.

## Syntax

[ap, bp, w, z, info] = f08ta(itype, jobz, uplo, n, ap, bp)
[ap, bp, w, z, info] = nag_lapack_dspgv(itype, jobz, uplo, n, ap, bp)

## Description

nag_lapack_dspgv (f08ta) first performs a Cholesky factorization of the matrix B$B$ as B = UTU $B={U}^{\mathrm{T}}U$, when uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or B = LLT $B=L{L}^{\mathrm{T}}$, when uplo = 'L'${\mathbf{uplo}}=\text{'L'}$. The generalized problem is then reduced to a standard symmetric eigenvalue problem
 Cx = λx , $Cx=λx ,$
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az = λBz $Az=\lambda Bz$, the eigenvectors are normalized so that the matrix of eigenvectors, Z$Z$, satisfies
 ZT A Z = Λ   and   ZT B Z = I , $ZT A Z = Λ and ZT B Z = I ,$
where Λ $\Lambda$ is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z $ABz=\lambda z$ we correspondingly have
 Z − 1 A Z − T = Λ   and   ZT B Z = I , $Z-1 A Z-T = Λ and ZT B Z = I ,$
and for B A z = λ z $BAz=\lambda z$ we have
 ZT A Z = Λ   and   ZT B − 1 Z = I . $ZT A Z = Λ and ZT B-1 Z = I .$

## 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:     itype – int64int32nag_int scalar
Specifies the problem type to be solved.
itype = 1${\mathbf{itype}}=1$
Az = λBz$Az=\lambda Bz$.
itype = 2${\mathbf{itype}}=2$
ABz = λz$ABz=\lambda z$.
itype = 3${\mathbf{itype}}=3$
BAz = λz$BAz=\lambda z$.
Constraint: itype = 1${\mathbf{itype}}=1$, 2$2$ or 3$3$.
2:     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'$\text{'V'}$.
3:     uplo – string (length ≥ 1)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangles of A$A$ and B$B$ are stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangles of A$A$ and B$B$ are stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
4:     n – int64int32nag_int scalar
n$n$, the order of the matrices A$A$ and B$B$.
Constraint: n0${\mathbf{n}}\ge 0$.
5:     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$i\ge j$.
6:     bp( : $:$) – double array
Note: the dimension of the array bp 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 B$B$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of B$B$ must be stored with element Bij${B}_{ij}$ in bp(i + j(j1) / 2)${\mathbf{bp}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of B$B$ must be stored with element Bij${B}_{ij}$ in bp(i + (2nj)(j1) / 2)${\mathbf{bp}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.

None.

ldz work

### Output Parameters

1:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The contents of ap are destroyed.
2:     bp( : $:$) – double array
Note: the dimension of the array bp 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 triangular factor U$U$ or L$L$ from the Cholesky factorization B = UTU$B={U}^{\mathrm{T}}U$ or B = LLT$B=L{L}^{\mathrm{T}}$, in the same storage format as B$B$.
3:     w(n) – double array
The eigenvalues in ascending order.
4:     z(ldz, : $:$) – double array
The first dimension, ldz, of the array z will be
• if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, ldz max (1,n) $\mathit{ldz}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise 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 jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, z contains the matrix Z$Z$ of eigenvectors. The eigenvectors are normalized as follows:
• if itype = 1${\mathbf{itype}}=1$ or 2$2$, ZTBZ = I${Z}^{\mathrm{T}}BZ=I$;
• if itype = 3${\mathbf{itype}}=3$, ZTB1Z = I${Z}^{\mathrm{T}}{B}^{-1}Z=I$.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
5:     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: itype, 2: jobz, 3: uplo, 4: n, 5: ap, 6: bp, 7: w, 8: z, 9: ldz, 10: work, 11: 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${\mathbf{INFO}}>0$
nag_lapack_dpptrf (f07gd) or nag_lapack_dspev (f08ga) returned an error code:
 ≤ n$\le {\mathbf{n}}$ if info = i${\mathbf{info}}=i$, nag_lapack_dspev (f08ga) failed to converge; i$i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > n$>{\mathbf{n}}$ if info = n + i${\mathbf{info}}={\mathbf{n}}+i$, for 1 ≤ i ≤ n$1\le i\le {\mathbf{n}}$, then the leading minor of order i$i$ of B$B$ is not positive definite. The factorization of B$B$ could not be completed and no eigenvalues or eigenvectors were computed.

## Accuracy

If B$B$ is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of B$B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B$B$ would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
The example program below illustrates the computation of approximate error bounds.

The total number of floating point operations is proportional to n3 ${n}^{3}$.
The complex analogue of this function is nag_lapack_zhpgv (f08tn).

## Example

```function nag_lapack_dspgv_example
itype = int64(1);
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [0.24;
0.39;
-0.11;
0.42;
0.79;
-0.25;
-0.16;
0.63;
0.48;
-0.03];
bp = [4.16;
-3.12;
5.03;
0.56;
-0.83;
0.76;
-0.1;
1.09;
0.34;
1.18];
[apOut, bpOut, w, z, info] = nag_lapack_dspgv(itype, jobz, uplo, n, ap, bp)
```
```

apOut =

0.0875
0.4683
0.5244
0.4892
-0.6812
-0.3775
-0.4476
-0.4576
-0.9487
-1.6875

bpOut =

2.0396
-1.5297
1.6401
0.2746
-0.2500
0.7887
-0.0490
0.6189
0.6443
0.6161

w =

-2.2254
-0.4548
0.1001
1.1270

z =

0

info =

0

```
```function f08ta_example
itype = int64(1);
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [0.24;
0.39;
-0.11;
0.42;
0.79;
-0.25;
-0.16;
0.63;
0.48;
-0.03];
bp = [4.16;
-3.12;
5.03;
0.56;
-0.83;
0.76;
-0.1;
1.09;
0.34;
1.18];
[apOut, bpOut, w, z, info] = f08ta(itype, jobz, uplo, n, ap, bp)
```
```

apOut =

0.0875
0.4683
0.5244
0.4892
-0.6812
-0.3775
-0.4476
-0.4576
-0.9487
-1.6875

bpOut =

2.0396
-1.5297
1.6401
0.2746
-0.2500
0.7887
-0.0490
0.6189
0.6443
0.6161

w =

-2.2254
-0.4548
0.1001
1.1270

z =

0

info =

0

```