NAG Toolbox: nag_lapack_dsbgst (f08ue)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates whether $X$ is to be returned.

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

$X$ is not returned.

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

$X$ is returned.

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

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

Indicates whether the upper or lower triangular part of $A$ is stored.

$\mathbf{uplo}=\text{'U'}$

The upper triangular part of $A$ is stored.

$\mathbf{uplo}=\text{'L'}$

The lower triangular part of $A$ is stored.

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

3:     $\mathrm{ka}$ – 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: $\mathbf{ka}\ge 0$.

4:     $\mathrm{kb}$ – 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: $\mathbf{ka}\ge \mathbf{kb}\ge 0$.

5:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double 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 $n$ symmetric 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 $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_a\right)\text{.}$

6:     $\mathrm{bb}\left(\mathit{ldbb},:\right)$ – double 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 banded split Cholesky factor of $B$ as specified by uplo, n and kb and returned by nag_lapack_dpbstf (f08uf).

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:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double 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 upper or lower triangle of ab stores the corresponding upper or lower triangle of $C$ as specified by uplo.

2:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array

The first dimension, $\mathit{ldx}$, of the array x will be

• if $\mathbf{vect}=\text{'V'}$, $\mathit{ldx}=\max \left(1,\mathbf{n}\right)$;
• if $\mathbf{vect}=\text{'N'}$, $\mathit{ldx}=1$.

The second dimension of the array x will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{vect}=\text{'V'}$ and at least $1$ if $\mathbf{vect}=\text{'N'}$.

The $n$ by $n$ matrix $X=S^{-1}Q$, if $\mathbf{vect}=\text{'V'}$.

If $\mathbf{vect}=\text{'N'}$, x is not referenced.

3:     $\mathrm{info}$ – 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: vect, 2: uplo, 3: n, 4: ka, 5: kb, 6: ab, 7: ldab, 8: bb, 9: ldbb, 10: x, 11: ldx, 12: work, 13: 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.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08ue_example

function f08ue_example


fprintf('f08ue example results\n\n');

% Sove Az = lambda Bz
% A and B are the symmetric banded positive definite matrices:
n = 4;
% A has 2 off-diagonals
ka = int64(2); 
a = [ 0.24,  0.39,  0.42,  0.00;
      0.39, -0.11,  0.79,  0.63;
      0.42,  0.79, -0.25,  0.48;
      0.00,  0.63,  0.48, -0.03];
% B has 1 off-diagonal
kb = int64(1); 
b = [ 2.07   0.95   0.00   0.00;
      0.95   1.69  -0.29   0.00;
      0.00  -0.29   0.65  -0.33;
      0.00   0.00  -0.33   1.17];

% Convert to general banded format ...
[~, ab, ifail] = f01zc( ...
			'P', ka, ka, a, zeros(ka+ka+1,n));
[~, bb, ifail] = f01zc( ...
			'P', kb, kb, b, zeros(kb+kb+1,n));
% ... and chop to give 'Upper' symmetric banded format
ab = ab(1:ka+1,1:n);
bb = bb(1:kb+1,1:n);

% Factorize B
uplo = 'Upper';
[ub, info] = f08uf( ...
		    uplo, kb, bb);

% Reduce problem to standard form Cy = lambda*y
vect = 'N';
[cb, x, info] = f08ue( ...
		       vect, uplo, ka, kb, ab, ub);

% Find eigenvalues lambda
jobz = 'No Vectors';
[~, w, ~, info] = f08ha( ...
			 jobz, uplo, ka, cb);

disp('Eigenvalues:');
disp(w');

f08ue example results

Eigenvalues:
   -0.8305   -0.6401    0.0992    1.8525