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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsygvx (f08sb)

## Purpose

nag_lapack_dsygvx (f08sb) computes selected eigenvalues and, optionally, 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 and B$B$ is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## Syntax

[a, b, m, w, z, jfail, info] = f08sb(itype, jobz, range, uplo, a, b, vl, vu, il, iu, abstol, 'n', n)
[a, b, m, w, z, jfail, info] = nag_lapack_dsygvx(itype, jobz, range, uplo, a, b, vl, vu, il, iu, abstol, 'n', n)

## Description

nag_lapack_dsygvx (f08sb) 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 desired eigenvalues and 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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
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:     range – string (length ≥ 1)
If range = 'A'${\mathbf{range}}=\text{'A'}$, all eigenvalues will be found.
If range = 'V'${\mathbf{range}}=\text{'V'}$, all eigenvalues in the half-open interval (vl,vu]$\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If range = 'I'${\mathbf{range}}=\text{'I'}$, the ilth to iuth eigenvalues will be found.
Constraint: range = 'A'${\mathbf{range}}=\text{'A'}$, 'V'$\text{'V'}$ or 'I'$\text{'I'}$.
4:     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'$\text{'L'}$.
5:     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.
6:     b(ldb, : $:$) – double array
The first dimension of the array b 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 B$B$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of b$b$ 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 b$b$ must be stored and the elements of the array above the diagonal are not referenced.
7:     vl – double scalar
8:     vu – double scalar
If range = 'V'${\mathbf{range}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'I'$\text{'I'}$, vl and vu are not referenced.
Constraint: if range = 'V'${\mathbf{range}}=\text{'V'}$, vl < vu${\mathbf{vl}}<{\mathbf{vu}}$.
9:     il – int64int32nag_int scalar
10:   iu – int64int32nag_int scalar
If range = 'I'${\mathbf{range}}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'V'$\text{'V'}$, il and iu are not referenced.
Constraints:
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n = 0${\mathbf{n}}=0$, il = 1${\mathbf{il}}=1$ and iu = 0${\mathbf{iu}}=0$;
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n > 0${\mathbf{n}}>0$, 1 il iu n $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
11:   abstol – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] $\left[a,b\right]$ of width less than or equal to
 abstol + ε max (|a|,|b|) , $abstol+ε max(|a|,|b|) ,$
where ε $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then ε T1 $\epsilon {‖T‖}_{1}$ will be used in its place, where T$T$ is the tridiagonal matrix obtained by reducing C$C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am (   ) , not zero. If this function returns with INFO = 1ton${\mathbf{INFO}}={\mathbf{1}} \text{to} {\mathbf{n}}$, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am (   ) . See Demmel and Kahan (1990).

### Optional Input Parameters

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

### Input Parameters Omitted from the MATLAB Interface

lda ldb ldz work lwork iwork

### 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)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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:     b(ldb, : $:$) – double array
The first dimension of the array b 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The triangular factor U$U$ or L$L$ from the Cholesky factorization B = UTU$B={U}^{\mathrm{T}}U$ or B = LLT$B=L{L}^{\mathrm{T}}$.
3:     m – int64int32nag_int scalar
The total number of eigenvalues found. 0mn$0\le {\mathbf{m}}\le {\mathbf{n}}$.
If range = 'A'${\mathbf{range}}=\text{'A'}$, m = n${\mathbf{m}}={\mathbf{n}}$.
If range = 'I'${\mathbf{range}}=\text{'I'}$, m = iuil + 1${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
4:     w(n) – double array
The first m elements contain the selected eigenvalues in ascending order.
5:     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,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first m columns of Z$Z$ contain the orthonormal eigenvectors of the matrix A$A$ corresponding to the selected eigenvalues, with the i$i$th column of Z$Z$ holding the eigenvector associated with w(i)${\mathbf{w}}\left(i\right)$. 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 an eigenvector fails to converge (INFO = 1ton${\mathbf{INFO}}={\mathbf{1}} \text{to} {\mathbf{n}}$), then that column of Z$Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
6:     jfail( : $:$) – int64int32nag_int array
Note: the dimension of the array jfail must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first m elements of jfail are zero;
• if INFO = 1ton${\mathbf{INFO}}={\mathbf{1}} \text{to} {\mathbf{n}}$, jfail contains the indices of the eigenvectors that failed to converge.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
7:     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: range, 4: uplo, 5: n, 6: a, 7: lda, 8: b, 9: ldb, 10: vl, 11: vu, 12: il, 13: iu, 14: abstol, 15: m, 16: w, 17: z, 18: ldz, 19: work, 20: lwork, 21: iwork, 22: jfail, 23: 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 = 1ton${\mathbf{INFO}}=1 \text{to} {\mathbf{n}}$
If info = i${\mathbf{info}}=i$, nag_lapack_dsyevx (f08fb) failed to converge; i$i$ eigenvectors failed to converge. Their indices are stored in array jfail.
${\mathbf{INFO}}>{\mathbf{N}}$
nag_lapack_dpotrf (f07fd) returned an error code; i.e., if info = n + i${\mathbf{info}}={\mathbf{n}}+i$, for 1in$1\le i\le {\mathbf{n}}$, 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 total number of floating point operations is proportional to n3${n}^{3}$.
The complex analogue of this function is nag_lapack_zhegvx (f08sp).

## Example

```function nag_lapack_dsygvx_example
itype = int64(1);
jobz = 'Vectors';
range = 'Values in range';
uplo = 'Upper';
a = [0.24, 0.39, 0.42, -0.16;
0, -0.11, 0.79, 0.63;
0, 0, -0.25, 0.48;
0, 0 0, -0.03];
b = [4.16, -3.12, 0.56, -0.1;
0, 5.03, -0.83, 1.09;
0, 0, 0.76, 0.34;
0, 0, 0, 1.18];
vl = -1;
vu = 1;
il = int64(0);
iu = int64(8185080);
abstol = 0;
[aOut, bOut, m, w, z, jfail, info] = ...
nag_lapack_dsygvx(itype, jobz, range, uplo, a, b, vl, vu, il, iu, abstol)
```
```

aOut =

0.0875    0.4683    0.4892   -0.4476
0    0.5244   -0.6812   -0.4576
0         0   -0.3775   -0.9487
0         0         0   -1.6875

bOut =

2.0396   -1.5297    0.2746   -0.0490
0    1.6401   -0.2500    0.6189
0         0    0.7887    0.6443
0         0         0    0.6161

m =

2

w =

-0.4548
0.1001
0
0

z =

0.3080    0.4469
0.5329    0.0371
-0.3496   -0.0505
-0.6211   -0.4743

jfail =

0
0
0
0

info =

0

```
```function f08sb_example
itype = int64(1);
jobz = 'Vectors';
range = 'Values in range';
uplo = 'Upper';
a = [0.24, 0.39, 0.42, -0.16;
0, -0.11, 0.79, 0.63;
0, 0, -0.25, 0.48;
0, 0 0, -0.03];
b = [4.16, -3.12, 0.56, -0.1;
0, 5.03, -0.83, 1.09;
0, 0, 0.76, 0.34;
0, 0, 0, 1.18];
vl = -1;
vu = 1;
il = int64(0);
iu = int64(8185080);
abstol = 0;
[aOut, bOut, m, w, z, jfail, info] = ...
f08sb(itype, jobz, range, uplo, a, b, vl, vu, il, iu, abstol)
```
```

aOut =

0.0875    0.4683    0.4892   -0.4476
0    0.5244   -0.6812   -0.4576
0         0   -0.3775   -0.9487
0         0         0   -1.6875

bOut =

2.0396   -1.5297    0.2746   -0.0490
0    1.6401   -0.2500    0.6189
0         0    0.7887    0.6443
0         0         0    0.6161

m =

2

w =

-0.4548
0.1001
0
0

z =

0.3080    0.4469
0.5329    0.0371
-0.3496   -0.0505
-0.6211   -0.4743

jfail =

0
0
0
0

info =

0

```