NAG Toolbox: nag_lapack_dspgst (f08te)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates how the standard form is computed.

$\mathbf{itype}=1$

• if $\mathbf{uplo}=\text{'U'}$, $C=U^{-\mathrm{T}}A{U}^{-1}$;
• if $\mathbf{uplo}=\text{'L'}$, $C=L^{-1}A{L}^{-\mathrm{T}}$.

$\mathbf{itype}=2$ or $3$

• if $\mathbf{uplo}=\text{'U'}$, $C=UA{U}^{\mathrm{T}}$;
• if $\mathbf{uplo}=\text{'L'}$, $C=L^{\mathrm{T}}AL$.

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

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

Indicates whether the upper or lower triangular part of $A$ is stored and how $B$ has been factorized.

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

The upper triangular part of $A$ is stored and $B=U^{\mathrm{T}}U$.

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

The lower triangular part of $A$ is stored and $B=L{L}^{\mathrm{T}}$.

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

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

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

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

4:     $\mathrm{ap}\left(:\right)$ – 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 $i\ge j$.

5:     $\mathrm{bp}\left(:\right)$ – 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 Cholesky factor of $B$ as specified by uplo and returned by nag_lapack_dpptrf (f07gd).

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 upper or lower triangle of ap stores the corresponding upper or lower triangle of $C$ as specified by itype and uplo, using the same packed storage format as described above.

2:     $\mathrm{info}$ – 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: uplo, 3: n, 4: ap, 5: bp, 6: info.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08te_example

function f08te_example


fprintf('f08te example results\n\n');

% Symmetric matrices A and B stored in packed (Lower) format
n = int64(4);
uplo = 'L';
ap = [0.24;  0.39;  0.42; -0.16;
            -0.11;  0.79;  0.63;
                   -0.25;  0.48;
                          -0.03];
bp = [4.16; -3.12;  0.56; -0.10;
             5.03; -0.83;  1.09;
                    0.76;  0.34;
                           1.18];

% Cholesky factorize B = LL^T
[lp, info] = f07gd( ...
                   uplo, n, bp);

% Reduce Generalized eigenproblem Az = lambda Bz to Cy = lambda y
% where C = L^-1 A L^-T and y = L^T z
itype = int64(1);
[cp, info] = f08te( ...
                    itype, uplo, n, ap, lp);

% Reduce C to tridiagonal form T = Q'CQ
[cpf, d, e, tau, info] = f08ge( ...
                                uplo, n, cp);

% Calculate the eigenvalues of T (same as C)
[w, ~, info] = f08jf( ...
                      d, e);

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

f08te example results

Eigenvalues
   -2.2254   -0.4548    0.1001    1.1270