NAG Toolbox: nag_lapack_zpbstf (f08ut)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

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

The upper triangular part of $B$ is stored.

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

The lower triangular part of $B$ is stored.

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

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

3:     $\mathrm{bb}\left(\mathit{ldbb},:\right)$ – 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 $n$ 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 $\mathbf{bb}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_b\right)\text{.}$

Optional Input Parameters

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

Default: the first dimension of the array bb and the second dimension of the array bb. (An error is raised if these dimensions are not equal.)

$n$, the order of the matrix $B$.

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

Output Parameters

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

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

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

$B$ stores the elements of its split Cholesky factor $S$.

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: uplo, 2: n, 3: kb, 4: bb, 5: ldbb, 6: 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.

   $\mathbf{info}>0$

If $\mathbf{info}=i$, the factorization could not be completed, because the updated element $b\left(i,i\right)$ would be the square root of a negative number. Hence $B$ is not positive definite. This may indicate an error in forming the matrix $B$.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08ut_example

function f08ut_example


fprintf('f08ut example results\n\n');

% Sove Az = lambda Bz
% A and B are the Hermitian banded positive definite matrices:
n = 4;
ka = int64(2); 
a = [ -1.13+0.00i  1.94-2.10i -1.40+0.25i  0.00+0.00i;
       1.94+2.10i -1.91+0.00i -0.82-0.89i -0.67+0.34i;
      -1.40-0.25i -0.82+0.89i -1.87+0.00i -1.10-0.16i;
       0.00+0.00i -0.67-0.34i -1.10+0.16i  0.50+0.00i];

kb = int64(1); 
b = [  9.89+0.00i  1.08-1.73i  0.00+0.00i  0.00+0.00i;
       1.08+1.73i  1.69+0.00i -0.04+0.29i  0.00+0.00i;
       0.00+0.00i -0.04-0.29i  2.65+0.00i -0.33+2.24i;
       0.00+0.00i  0.00+0.00i -0.33-2.24i  2.17+0.00i];

% Convert to general banded format ...
[~, ab, ifail] = f01zd( ...
			'P', ka, ka, a, complex(zeros(ka+ka+1,n)));
[~, bb, ifail] = f01zd( ...
			'P', kb, kb, b, complex(zeros(kb+kb+1,n)));
% ... and chop to give 'Upper' Hermitian banded format
ab = ab(1:ka+1,1:n);
bb = bb(1:kb+1,1:n);

% Factorize B
uplo = 'Upper';
[bbfac, info] = f08ut( ...
		       uplo, kb, bb);

% Reduce problem to standard form Cy = lambda*y
vect = 'N';
[cb, x, info] = f08us( ...
		       vect, uplo, ka, kb, ab, bbfac);

% Find eigenvalues lambda
jobz = 'No Vectors';
[~, w, ~, info] = f08hn( ...
			 jobz, uplo, ka, cb);

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

f08ut example results

Eigenvalues:
   -6.6089   -2.0416    0.1603    1.7712