NAG Toolbox: nag_lapack_zhegvd (f08sq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{itype}$ – int64int32nag_int scalar

Specifies the problem type to be solved.

$\mathbf{itype}=1$

$Az=\lambda Bz$.

$\mathbf{itype}=2$

$ABz=\lambda z$.

$\mathbf{itype}=3$

$BAz=\lambda z$.

Constraint: $\mathbf{itype}=1$, $2$ or $3$.

2:     $\mathrm{jobz}$ – string (length ≥ 1)

Indicates whether eigenvectors are computed.

$\mathbf{jobz}=\text{'N'}$

Only eigenvalues are computed.

$\mathbf{jobz}=\text{'V'}$

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{jobz}=\text{'N'}$ or $\text{'V'}$.

3:     $\mathrm{uplo}$ – string (length ≥ 1)

If $\mathbf{uplo}=\text{'U'}$, the upper triangles of $A$ and $B$ are stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

4:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ Hermitian 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.

5:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

The first dimension of the array b must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array b must be at least $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ Hermitian matrix $B$.

• If $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $b$ 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 $b$ 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 arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)

$n$, the order of the matrices $A$ and $B$.

Constraint: $\mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

If $\mathbf{jobz}=\text{'V'}$, a contains the matrix $Z$ of eigenvectors. The eigenvectors are normalized as follows:

• if $\mathbf{itype}=1$ or $2$, $Z^{\mathrm{H}}BZ=I$;
• if $\mathbf{itype}=3$, $Z^{\mathrm{H}}{B}^{-1}Z=I$.

If $\mathbf{jobz}=\text{'N'}$, the upper triangle (if $\mathbf{uplo}=\text{'U'}$) or the lower triangle (if $\mathbf{uplo}=\text{'L'}$) of a, including the diagonal, is overwritten.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

The first dimension of the array b will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array b will be $\max \left(1,\mathbf{n}\right)$.

The triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$.

3:     $\mathrm{w}\left(\mathbf{n}\right)$ – double array

The eigenvalues in ascending order.

4:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: itype, 2: jobz, 3: uplo, 4: n, 5: a, 6: lda, 7: b, 8: ldb, 9: w, 10: work, 11: lwork, 12: rwork, 13: lrwork, 14: iwork, 15: liwork, 16: 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.

   $\mathbf{info}=1\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$

If $\mathbf{info}=i$, nag_lapack_zheevd (f08fq) failed to converge; $i$ $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

   $\mathbf{info}>\mathbf{n}$

nag_lapack_zpotrf (f07fr) returned an error code; i.e., if $\mathbf{info}=\mathbf{n}+i$, for $1\le i\le \mathbf{n}$, then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08sq_example

function f08sq_example


fprintf('f08sq example results\n\n');

% Upper triangular parts of Hermitian matrix A and positive definite matrix B
uplo = 'Upper';
n = int64(4);
a = [-7.36 + 0i,  0.77 - 0.43i, -0.64 - 0.92i,  3.01 - 6.97i;
      0    + 0i,  3.49 + 0i,     2.19 + 4.45i,  1.90 + 3.73i;
      0    + 0i,  0    + 0i,     0.12 + 0i,     2.88 - 3.17i;
      0    + 0i,  0    + 0i,     0    + 0i,    -2.54 + 0i];
b = [ 3.23 + 0i,  1.51 - 1.92i,  1.90 + 0.84i,  0.42 + 2.50i;
      0    + 0i,  3.58 + 0i,    -0.23 + 1.11i, -1.18 + 1.37i;
      0    + 0i,  0    + 0i,     4.09 + 0i,     2.33 - 0.14i;
      0    + 0i,  0    + 0i,     0    + 0i,     4.29 + 0i];

anorm = norm(a+a'-diag(diag(a)),1);
bnorm = norm(b+b'-diag(diag(b)),1);

% Generalized eigenvalues and eigenvectors for problem ABz = lambda z.
itype = int64(2);
jobz = 'Vectors';
[Z, L, w, info] = f08sq( ...
                         itype, jobz, uplo, a, b);

% Normalize: largest elements are real (preserving Z^HBZ = I)
for i = 1:n
  [~,k] = max(abs(real(Z(:,i)))+abs(imag(Z(:,i))));
  Z(:,i) = Z(:,i)*conj(Z(k,i))/abs(Z(k,i));
end

disp('Eigenvalues');
disp(w');
disp('Eigenvectors');
disp(Z);

% Get reciprocal condition number
[rcond, info] = f07tu( ...
                       '1', uplo, 'N', L);

rcondb = rcond^2;
fprintf('Estimate of reciprocal condition number for B\n%12.1e\n', rcondb);

% Eigenvector condition numbers
[rcondz, info] = f08fl( ...
		        'Eigenvectors', n, n, w);

% Error bounds
t1 = 1/rcond;
t2 = x02aj*t1;
t3 = anorm*bnorm;
errbnd(1:n) = x02aj*t3 + (x02aj/rcondb).*abs(w);
zerrbd(1:n) = t1*t2 + (t2*t3)./rcondz;

fprintf('\n');
disp('Error estimate for the eigenvalues');
fprintf('%12.1e',errbnd);
fprintf('\n\n');
disp('Error estimates for the eigenvectors');
fprintf('%12.1e',zerrbd);
fprintf('\n');

f08sq example results

Eigenvalues
  -61.7321   -6.6195    0.0725   43.1883

Eigenvectors
   0.3903 + 0.0000i  -0.1560 - 0.0404i   2.2909 + 0.0000i  -0.1943 - 0.0690i
  -0.1814 + 0.0114i  -0.1552 - 0.3651i  -0.5042 - 0.7120i   0.3884 + 0.0000i
   0.0438 + 0.0338i   0.5364 + 0.0000i  -1.2701 - 0.4547i   0.0657 - 0.2095i
  -0.2221 - 0.2272i  -0.1298 - 0.1880i   0.5706 + 1.3132i   0.2924 - 0.0675i

Estimate of reciprocal condition number for B
     2.5e-03

Error estimate for the eigenvalues
     2.7e-12     3.1e-13     2.6e-14     1.9e-12

Error estimates for the eigenvectors
     5.2e-14     1.1e-13     1.1e-13     5.4e-14