f03bhc computes the determinant of an symmetric positive definite banded matrix that has been stored in band-symmetric storage. f07hdc must be called first to supply the Cholesky factorized form. The storage (upper or lower triangular) used by f07hdc is relevant as this determines which elements of the stored factorized form are referenced.
The function may be called by the names: f03bhc or nag_det_real_band_sym.
The determinant of is calculated using the Cholesky factorization , where is an upper triangular band matrix, or , where is a lower triangular band matrix. The determinant of is the product of the squares of the diagonal elements of or .
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of was stored and how it was factorized. This should not be altered following a call to f07hdc.
The upper triangular part of was originally stored and was factorized as where is upper triangular.
The lower triangular part of was originally stored and was factorized as where is lower triangular.
3: – IntegerInput
On entry: , the order of the matrix .
4: – IntegerInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
5: – const doubleInput
Note: the dimension, dim, of the array ab
must be at least
On entry: the Cholesky factor of , as returned by f07hdc.
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
7: – double *Output
8: – Integer *Output
On exit: the determinant of is given by . It is given in this form to avoid overflow or underflow.
9: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, .
On entry, and .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
The matrix is not positive definite.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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).
8Parallelism and Performance
f03bhc is not threaded in any implementation.
The time taken by f03bhc is approximately proportional to .
This function should only be used when since as approaches , it becomes less efficient to take advantage of the band form.
This example calculates the determinant of the real symmetric positive definite band matrix