The function may be called by the names: f07bdc, nag_lapacklin_dgbtrf or nag_dgbtrf.
f07bdc forms the factorization of a real band matrix using partial pivoting, with row interchanges. Usually , and then, if has nonzero subdiagonals and nonzero superdiagonals, the factorization has the form , where is a permutation matrix, is a lower triangular matrix with unit diagonal elements and at most nonzero elements in each column, and is an upper triangular band matrix with superdiagonals.
Note that is not a band matrix, but the nonzero elements of can be stored in the same space as the subdiagonal elements of . is a band matrix but with additional superdiagonals compared with . These additional superdiagonals are created by the row interchanges.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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: – IntegerInput
On entry: , the number of rows of the matrix .
3: – IntegerInput
On entry: , the number of columns of the matrix .
4: – IntegerInput
On entry: , the number of subdiagonals within the band of the matrix .
5: – IntegerInput
On entry: , the number of superdiagonals within the band of the matrix .
6: – doubleInput/Output
Note: the dimension, dim, of the array ab
must be at least
On entry: the matrix .
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements , for row and column , depends on the order argument as follows:
On exit: ab is overwritten by details of the factorization.
The elements, , of the upper triangular band factor with super-diagonals, and the multipliers, , used to form the lower triangular factor are stored. The elements , for and , and , for and , are stored where is stored on entry.
7: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
8: – IntegerOutput
On exit: the pivot indices that define the permutation matrix. At the
th step, if then row of the matrix was interchanged with row , for . indicates that, at the th step, a row interchange was not required.
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, .
On entry, .
On entry, . Constraint: .
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.
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.
Element of the diagonal is exactly zero.
The factorization has been completed, but the factor
is exactly singular, and division by zero will occur if it is used to solve
a system of equations.
The computed factors and are the exact factors of a perturbed matrix , where
is a modest linear function of , and is the machine precision. This assumes .
8Parallelism and Performance
f07bdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bdc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations varies between approximately and , depending on the interchanges, assuming and .
A call to f07bdc may be followed by calls to the functions: