NAG FL Interface
f07brf computes the factorization of a complex by band matrix.
|Integer, Intent (In)
||m, n, kl, ku, ldab
|Integer, Intent (Out)
|Complex (Kind=nag_wp), Intent (Inout)
|C Header Interface
f07brf_ (const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, Complex ab, const Integer *ldab, Integer ipiv, Integer *info)
The routine may be called by the names f07brf, nagf_lapacklin_zgbtrf or its LAPACK name zgbtrf.
f07brf forms the factorization of a complex by 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
On entry: , the number of rows of the matrix .
On entry: , the number of columns of the matrix .
On entry: , the number of subdiagonals within the band of the matrix .
On entry: , the number of superdiagonals within the band of the matrix .
– Complex (Kind=nag_wp) array
the second dimension of the array ab
must be at least
The matrix is stored in rows
; the first
rows need not be set, more precisely, the element
must be stored in
See Section 9
for further details.
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 .
: the first dimension of the array ab
as declared in the (sub)program from which f07brf
– Integer array
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.
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value.
, 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
are the exact factors of a perturbed matrix
is a modest linear function of
is the machine precision
. This assumes
Parallelism and Performance
f07brf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07brf 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 real floating-point operations varies between approximately and , depending on the interchanges, assuming and .
A call to f07brf
may be followed by calls to the routines:
- f07bsf to solve , or ;
- f07buf to estimate the condition number of .
The real analogue of this routine is f07bdf
This example computes the
factorization of the matrix
is treated as a band matrix with one subdiagonal and two superdiagonals.