NAG Toolbox: nag_lapack_zhbgvx (f08up)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{jobz}$ – 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 $\text{'V'}$.

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

If $\mathbf{range}=\text{'A'}$, all eigenvalues will be found.

If $\mathbf{range}=\text{'V'}$, all eigenvalues in the half-open interval $\left(\mathbf{vl},\mathbf{vu}\right]$ will be found.

If $\mathbf{range}=\text{'I'}$, the ilth to iuth eigenvalues will be found.

Constraint: $\mathbf{range}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

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:     $\mathrm{ka}$ – int64int32nag_int scalar

If $\mathbf{uplo}=\text{'U'}$, the number of superdiagonals, $k_a$, of the matrix $A$.

If $\mathbf{uplo}=\text{'L'}$, the number of subdiagonals, $k_a$, of the matrix $A$.

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

5:     $\mathrm{kb}$ – int64int32nag_int scalar

If $\mathbf{uplo}=\text{'U'}$, the number of superdiagonals, $k_b$, of the matrix $B$.

If $\mathbf{uplo}=\text{'L'}$, the number of subdiagonals, $k_b$, of the matrix $B$.

Constraint: $\mathbf{ka}\ge \mathbf{kb}\ge 0$.

6:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array

The first dimension of the array ab must be at least $\mathbf{ka}+1$.

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

The upper or lower triangle of the $n$ by $n$ Hermitian band matrix $A$.

The matrix is stored in rows $1$ to $k_a+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_a+1+i-j,j\right)\text{ for }\max \left(1,j-k_a\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_a\right)\text{.}$

7:     $\mathrm{bb}\left(\mathit{ldbb},:\right)$ – complex array

The first dimension of the array bb must be at least $\mathbf{kb}+1$.

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

The upper or lower triangle of the $n$ by $n$ Hermitian positive definite band matrix $B$.

The matrix is stored in rows $1$ to $k_b+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $B$ within the band must be stored with element $B_{ij}$ in $\mathbf{bb}\left(k_b+1+i-j,j\right)\text{ for }\max \left(1,j-k_b\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $B$ within the band must be stored with element $B_{ij}$ in $\mathbf{bb}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_b\right)\text{.}$

8:     $\mathrm{vl}$ – double scalar

9:     $\mathrm{vu}$ – double scalar

If $\mathbf{range}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $\mathbf{range}=\text{'A'}$ or $\text{'I'}$, vl and vu are not referenced.

Constraint: if $\mathbf{range}=\text{'V'}$, $\mathbf{vl}<\mathbf{vu}$.

10:   $\mathrm{il}$ – int64int32nag_int scalar

11:   $\mathrm{iu}$ – int64int32nag_int scalar

If $\mathbf{range}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

If $\mathbf{range}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.

Constraints:

• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

12:   $\mathrm{abstol}$ – 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 $\left[a,b\right]$ of width less than or equal to

$$\mathbf{abstol}+\epsilon \max \left(\left|a\right|,\left|b\right|\right)\text{,}$$

where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_1$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold $2\times \mathbf{x02am}\left( \right)$, not zero. If this function returns with $\mathbf{info}=\mathbf{1}\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$, indicating that some eigenvectors did not converge, try setting abstol to $2\times \mathbf{x02am}\left( \right)$. See Demmel and Kahan (1990).

Optional Input Parameters

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

Default: the second dimension of the arrays ab, bb.

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

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

Output Parameters

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

The first dimension of the array ab will be $\mathbf{ka}+1$.

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

The contents of ab are overwritten.

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

The first dimension of the array bb will be $\mathbf{kb}+1$.

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

The factor $S$ from the split Cholesky factorization $B=S^{\mathrm{H}}S$, as returned by nag_lapack_zpbstf (f08ut).

3:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – complex array

The first dimension, $\mathit{ldq}$, of the array q will be

• if $\mathbf{jobz}=\text{'V'}$, $\mathit{ldq}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldq}=1$.

The second dimension of the array q will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobz}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'V'}$, the $n$ by $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form.

If $\mathbf{jobz}=\text{'N'}$, q is not referenced.

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

The total number of eigenvalues found. $0\le \mathbf{m}\le \mathbf{n}$.

If $\mathbf{range}=\text{'A'}$, $\mathbf{m}=\mathbf{n}$.

If $\mathbf{range}=\text{'I'}$, $\mathbf{m}=\mathbf{iu}-\mathbf{il}+1$.

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

The eigenvalues in ascending order.

6:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – complex array

The first dimension, $\mathit{ldz}$, of the array z will be

• if $\mathbf{jobz}=\text{'V'}$, $\mathit{ldz}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldz}=1$.

The second dimension of the array z will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobz}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'V'}$, z contains the matrix $Z$ of eigenvectors, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left(i\right)$. The eigenvectors are normalized so that $Z^{\mathrm{H}}BZ=I$.

If $\mathbf{jobz}=\text{'N'}$, z is not referenced.

7:     $\mathrm{jfail}\left(:\right)$ – int64int32nag_int array

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

If $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m elements of jfail are zero;
• if $\mathbf{info}=\mathbf{1}\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$, jfail contains the indices of the eigenvectors that failed to converge.

If $\mathbf{jobz}=\text{'N'}$, jfail is not referenced.

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

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

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{info}=-i$

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

1: jobz, 2: range, 3: uplo, 4: n, 5: ka, 6: kb, 7: ab, 8: ldab, 9: bb, 10: ldbb, 11: q, 12: ldq, 13: vl, 14: vu, 15: il, 16: iu, 17: abstol, 18: m, 19: w, 20: z, 21: ldz, 22: work, 23: rwork, 24: iwork, 25: jfail, 26: 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.

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

If $\mathbf{info}=i$, then $i$ eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.

   $\mathbf{info}>\mathbf{n}$

nag_lapack_dpbstf (f08uf) 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: f08up_example

function f08up_example


fprintf('f08up example results\n\n');

% Hermitian banded matrices A and B stored in symmetric banded format
uplo = 'U';
ka = int64(2);
ab = [ 0,          0    + 0i,    -1.40 + 0.25i, -0.67 + 0.34i;
       0    + 0i,  1.94 - 2.10i, -0.82 - 0.89i, -1.10 - 0.16i;
      -1.13 + 0i, -1.91 + 0i,    -1.87 + 0i,     0.50 + 0i];
kb = int64(1);
bb = [ 0,          1.08 - 1.73i, -0.04 + 0.29i, -0.33 + 2.24i;
       9.89 + 0i,  1.69 + 0i,     2.65 + 0i,     2.17 + 0i];

% Eigenvalues in range [0,2] and corresponding eigenvectors of Ax = lambda Bx
jobz  = 'Vectors';
range = 'Values in range';
vl = 0;          vu = 2;
il = int64(0); iu = il;
abstol = 0;
[~, ~, ~, m, w, z, jfail, info] = ...
  f08up( ...
	 jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol);

disp('Selected eigenvalues');
disp(w(1:m)');

% Normalize vectors, largest element is real and positive.
for i = 1:m
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

disp('Corresponding eigenvectors');
disp(z(:,1:m));

f08up example results

Selected eigenvalues
    0.1603    1.7712

Corresponding eigenvectors
  -0.1026 - 0.1614i   0.0068 + 0.0492i
  -0.0028 - 0.1738i  -0.4307 + 0.2705i
  -0.0466 + 0.0890i  -0.1123 - 0.9664i
   0.3586 + 0.0000i   1.3241 + 0.0000i