naginterfaces.library.det.real_​band_​sym

naginterfaces.library.det.real_band_sym(uplo, kd, ab)

real_band_sym computes the determinant of an $n\times n$ symmetric positive definite banded matrix $A$ that has been stored in band-symmetric storage. lapacklin.dpbtrf must be called first to supply the Cholesky factorized form. The storage (upper or lower triangular) used by lapacklin.dpbtrf is relevant as this determines which elements of the stored factorized form are referenced.

For full information please refer to the NAG Library document for f03bh

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f03/f03bhf.html

Parameters

uplostr, 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 lapacklin.dpbtrf.

$uplo=\text{'U'}$

The upper triangular part of $A$ was originally stored and $A$ was factorized as $U^{T}U$ where $U$ is upper triangular.

$uplo=\text{'L'}$

The lower triangular part of $A$ was originally stored and $A$ was factorized as $L{L}^T$ where $L$ is lower triangular.

kdint

$k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

abfloat, array-like, shape $(kd+1,n)$

The Cholesky factor of $A$, as returned by lapacklin.dpbtrf.

Returns

dfloat

The determinant of $A$ is given by $d\times {2.0}^{dexp}$. It is given in this form to avoid overflow or underflow.

dexpint

The determinant of $A$ is given by $d\times {2.0}^{dexp}$. It is given in this form to avoid overflow or underflow.

Raises

NagValueError

(errno $1$)

On entry, $uplo=⟨value⟩$.

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

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $3$)

On entry, $kd=⟨value⟩$.

Constraint: $kd\ge 0$.

(errno $6$)

The matrix $A$ is not positive definite.

Notes

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 $U$ or $L$.

References

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