The routine may be called by the names f07bdf, nagf_lapacklin_dgbtrf or its LAPACK name dgbtrf.
f07bdf 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: – IntegerInput
On entry: , the number of rows of the matrix .
2: – IntegerInput
On entry: , the number of columns of the matrix .
3: – IntegerInput
On entry: , the number of subdiagonals within the band of the matrix .
4: – IntegerInput
On entry: , the number of superdiagonals within the band of the matrix .
5: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
On entry: the matrix .
The matrix is stored in rows to ; the first rows need not be set, more precisely, the element must be stored in
On exit: if , ab is overwritten by details of the factorization.
The upper triangular band matrix , with superdiagonals, is stored in rows to of the array, and the multipliers used to form the matrix are stored in rows to .
6: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bdf is called.
7: – Integer arrayOutput
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.
8: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value.
If , dynamic memory allocation failed. See Section 9 in the Introduction to the NAG Library FL Interface for further information. An explanatory message is output, and execution of the program is terminated.
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
Background information to multithreading can be found in the Multithreading documentation.
f07bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bdf 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 routine. 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 f07bdf may be followed by calls to the routines: