NAG FL Interface
f07bdf (dgbtrf)

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1 Purpose

f07bdf computes the LU factorization of a real m×n band matrix.

2 Specification

Fortran Interface
Subroutine f07bdf ( m, n, kl, ku, ab, ldab, ipiv, info)
Integer, Intent (In) :: m, n, kl, ku, ldab
Integer, Intent (Out) :: ipiv(min(m,n)), info
Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*)
C Header Interface
#include <nag.h>
void  f07bdf_ (const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, double ab[], const Integer *ldab, Integer ipiv[], Integer *info)
The routine may be called by the names f07bdf, nagf_lapacklin_dgbtrf or its LAPACK name dgbtrf.

3 Description

f07bdf forms the LU factorization of a real m×n band matrix A using partial pivoting, with row interchanges. Usually m=n, and then, if A has kl nonzero subdiagonals and ku nonzero superdiagonals, the factorization has the form A=PLU, where P is a permutation matrix, L is a lower triangular matrix with unit diagonal elements and at most kl nonzero elements in each column, and U is an upper triangular band matrix with kl+ku superdiagonals.
Note that L is not a band matrix, but the nonzero elements of L can be stored in the same space as the subdiagonal elements of A. U is a band matrix but with kl additional superdiagonals compared with A. These additional superdiagonals are created by the row interchanges.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3: kl Integer Input
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
4: ku Integer Input
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
5: ab(ldab,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the m×n matrix A.
The matrix is stored in rows kl+1 to 2kl+ku+1; the first kl rows need not be set, more precisely, the element Aij must be stored in
ab(kl+ku+1+i-j,j)=Aij  for ​max(1,j-ku)imin(m,j+kl).  
See Section 9 in f07baf for further details.
On exit: if info0, ab is overwritten by details of the factorization.
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
6: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bdf is called.
Constraint: ldab2×kl+ku+1.
7: ipiv(min(m,n)) Integer array Output
On exit: the pivot indices that define the permutation matrix. At the ith step, if ipiv(i)>i then row i of the matrix A was interchanged with row ipiv(i), for i=1,2,,min(m,n). ipiv(i)i indicates that, at the ith step, a row interchange was not required.
8: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value.
If info=−999, 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 value of the diagonal is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7 Accuracy

The computed factors L and U are the exact factors of a perturbed matrix A+E, where
|E|c(k)εP|L||U| ,  
c(k) is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kmin(m,n).

8 Parallelism 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.

9 Further Comments

The total number of floating-point operations varies between approximately 2nkl(ku+1) and 2nkl(kl+ku+1), depending on the interchanges, assuming m=nkl and nku.
A call to f07bdf may be followed by calls to the routines:
The complex analogue of this routine is f07brf.

10 Example

This example computes the LU factorization of the matrix A, where
A= ( -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 ) .  
Here A is treated as a band matrix with one subdiagonal and two superdiagonals.

10.1 Program Text

Program Text (f07bdfe.f90)

10.2 Program Data

Program Data (f07bdfe.d)

10.3 Program Results

Program Results (f07bdfe.r)