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NAG Toolbox

NAG Toolbox: nag_det_real_band_sym (f03bh)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_det_real_band_sym (f03bh) computes the determinant of a n by n symmetric positive definite banded matrix A 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.


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


The determinant of A is calculated using the Cholesky factorization A=UTU, where U is an upper triangular band matrix, or A=LLT, where L is a lower triangular band matrix. The determinant of A is the product of the squares of the diagonal elements of U or L.


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


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).
The upper triangular part of A was originally stored and A was factorized as UTU where U is upper triangular.
The lower triangular part of A was originally stored and A was factorized as LLT where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     kd int64int32nag_int scalar
kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
3:     abldab: – 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 max1,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×2.0id. 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:
Constraint: uplo='L' or 'U'.
Constraint: n>0.
Constraint: kd0.
Constraint: ldabkd+1.
The matrix A is not positive definite.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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 mn since as m approaches n, it becomes less efficient to take advantage of the band form.


This example calculates the determinant of the real symmetric positive definite band matrix
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 .  
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
  [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);
  fprintf('\n** Factorization routine returned error flag info = %d\n', info);

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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Chapter Introduction
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