NAG Toolbox: nag_lapack_dspgvx (f08tb)

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:     $\mathrm{range}$ – 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 $\text{'I'}$.

4:     $\mathrm{uplo}$ – 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 $\text{'L'}$.

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

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

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

6:     $\mathrm{ap}\left(:\right)$ – double array

The dimension of the array ap must be at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The upper or lower triangle of the $n$ by $n$ symmetric matrix $A$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

7:     $\mathrm{bp}\left(:\right)$ – double array

The dimension of the array bp must be at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The upper or lower triangle of the $n$ by $n$ symmetric matrix $B$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $B$ must be stored with element $B_{ij}$ in $\mathbf{bp}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $B$ must be stored with element $B_{ij}$ in $\mathbf{bp}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

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

None.

Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – double array

The dimension of the array ap will be $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The contents of ap are destroyed.

2:     $\mathrm{bp}\left(:\right)$ – double array

The dimension of the array bp will be $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{T}}U$ or $B=L{L}^{\mathrm{T}}$, in the same storage format as $B$.

3:     $\mathrm{m}$ – 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 $\mathbf{range}=\text{'I'}$, $\mathbf{m}=\mathbf{iu}-\mathbf{il}+1$.

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

The first m elements contain the selected eigenvalues in ascending order.

5:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double 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 $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobz}=\text{'V'}$ and $1$ otherwise.

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

• if $\mathbf{info}=\mathbf{0}$, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left(i\right)$. The eigenvectors are normalized as follows:
  • if $\mathbf{itype}=1$ or $2$, $Z^{\mathrm{T}}BZ=I$;
  • if $\mathbf{itype}=3$, $Z^{\mathrm{T}}{B}^{-1}Z=I$;
• if an eigenvector fails to converge ($\mathbf{info}=\mathbf{1}\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

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

6:     $\mathrm{jfail}\left(:\right)$ – 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.

7:     $\mathrm{info}$ – 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: itype, 2: jobz, 3: range, 4: uplo, 5: n, 6: ap, 7: bp, 8: vl, 9: vu, 10: il, 11: iu, 12: abstol, 13: m, 14: w, 15: z, 16: ldz, 17: work, 18: iwork, 19: jfail, 20: 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 $\mathbf{info}=i$, nag_lapack_dspevx (f08gb) failed to converge; $i$ eigenvectors failed to converge. Their indices are stored in array jfail.

   $\mathbf{info}>\mathbf{n}$

nag_lapack_dpptrf (f07gd) 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: f08tb_example

function f08tb_example


fprintf('f08tb example results\n\n');

% Symmetric matrices A and B stored in packed (Upper) format
n = int64(4);
uplo = 'U';
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;
      -0.10;  1.09;  0.34;  1.18];

% Eigenvalues in the range [-1,1] and corresponding eigenvectors
itype = int64(1);
jobz  = 'Vectors';
range = 'Values in range';
vl = -1;         vu = 1;
il = int64(0); iu = int64(0);
abstol = 0;
[~, ~, m, w, z, jfail, info] = ...
f08tb( ...
       itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol);

disp('Selected Eigenvalues');
disp(w(1:m)');
% Normalize eigenvectors (wrt B): largest element positive
for j = 1:m
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);
  end
end                            
disp('Corresponding Eigenvectors');
disp(z);

f08tb example results

Selected Eigenvalues
   -0.4548    0.1001

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