that has been stored in band-symmetric storage. nag_lapack_dpbtrf (f07hd) must be called first to supply the Cholesky factorized form. The storage (upper or lower triangular) used by nag_lapack_dpbtrf (f07hd) is relevant as this determines which elements of the stored factorized form are referenced.

Syntax

[d, id, ifail] = f03bh(uplo, kd, ab, 'n', n)

[d, id, ifail] = nag_det_real_band_sym(uplo, kd, ab, 'n', n)

Description

The determinant of

A

is calculated using the Cholesky factorization

A = U^{T} U

, where

U

is an upper triangular band matrix, or

A = L L^{T}

, where

L

is a lower triangular band matrix. The determinant of

A

is the product of the squares of the diagonal elements of

U

L

References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Indicates whether the upper or lower triangular part of

A

was stored and how it was factorized. This should not be altered following a call to nag_lapack_dpbtrf (f07hd).

$uplo ='U'$: The upper triangular part of $A$ was originally stored and $A$ was factorized as $U^{T} U$ where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ was originally stored and $A$ was factorized as $L L^{T}$ where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

3: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

kd + 1

The second dimension of the array ab must be at least

\max (1, n)

The Cholesky factor of

A

, as returned by nag_lapack_dpbtrf (f07hd).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the array ab.
$n$ , the order of the matrix $A$ .

Constraint: $n > 0$ .

Output Parameters

1: $d$ – double scalar
2: $id$ – int64int32nag_int scalar: The determinant of $A$ is given by $d \times {2.0}^{id}$ . It is given in this form to avoid overflow or underflow.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $uplo ='L'$ or $'U'$ .

$ifail = 2$: Constraint: $n > 0$ .

$ifail = 3$: Constraint: $kd \geq 0$ .

$ifail = 5$: Constraint: $ldab \geq kd + 1$ .

$ifail = 6$: The matrix $A$ is not positive definite.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy of the determinant depends on the conditioning of the original matrix. For a detailed error analysis see page 54 of Wilkinson and Reinsch (1971).

Further Comments

The time taken by nag_det_real_band_sym (f03bh) is approximately proportional to

n

This function should only be used when

m ≪ n

since as

m

approaches

n

, it becomes less efficient to take advantage of the band form.

Example

This example calculates the determinant of the real symmetric positive definite band matrix

(\begin{array}{r} 5 & - 4 & 1 \\ - 4 & 6 & - 4 & 1 \\ 1 & - 4 & 6 & - 4 & 1 \\ 1 & - 4 & 6 & - 4 & 1 \\ 1 & - 4 & 6 & - 4 & 1 \\ 1 & - 4 & 6 & - 4 \\ 1 & - 4 & 5 \end{array}) .

Open in the MATLAB editor: f03bh_example

function f03bh_example


fprintf('f03bh example results\n\n');

uplo = 'l';
kd   = int64(2);
n    = int64(7);
ab = [ 5,  6,  6,  6,  6,  6,  5;
      -4, -4, -4, -4, -4, -4,  0;
       1,  1,  1,  1,  1,  0,  0];
% Factorize a
[ab, info] = f07hd(uplo, kd, ab);

if info == 0
  fprintf('\n');
  [ifail] = x04ce(n, n, kd, int64(0), ab, 'Array ab after factorization');

  [d, id, ifail] = f03bh(uplo, kd, ab);

  fprintf('d = %13.5f id = %d\n', d, id);
  fprintf('Value of determinant = %13.5e\n', d*2^id);
else
  fprintf('\n** Factorization routine returned error flag info = %d\n', info);
end

f03bh example results


 Array ab after factorization
             1          2          3          4          5          6          7
 1      2.2361
 2     -1.7889     1.6733
 3      0.4472    -1.9124     1.4639
 4                 0.5976    -1.9518     1.3540
 5                            0.6831    -1.9695     1.2863
 6                                       0.7385    -1.9789     1.2403
 7                                                  0.7774    -1.9846     0.6761
d =       0.25000 id = 8
Value of determinant =   6.40000e+01

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