F07BDF (DGBTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07BDF (DGBTRF) computes the LU factorization of a real m by n band matrix.

2  Specification

REAL (KIND=nag_wp)  AB(LDAB,*)
The routine may be called by its LAPACK name dgbtrf.

3  Description

F07BDF (DGBTRF) forms the LU factorization of a real m by 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  Parameters

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
3:     KL – INTEGERInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: KL0.
4:     KU – INTEGERInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: KU0.
5:     AB(LDAB,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least max1,N.
On entry: the m by 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
ABkl+ku+1+i-jj=Aij  for ​max1,j-kuiminm,j+kl.
See Section 8 in F07BAF (DGBSV) 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 – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07BDF (DGBTRF) is called.
Constraint: LDAB2×KL+KU+1.
7:     IPIV(minM,N) – INTEGER arrayOutput
On exit: the pivot indices that define the permutation matrix. At the ith step, if IPIVi>i then row i of the matrix A was interchanged with row IPIVi, for i=1,2,,minm,n. IPIVii indicates that, at the ith step, a row interchange was not required.
8:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, Ui,i 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
EckεPLU ,
ck is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kminm,n.

8  Further Comments

The total number of floating point operations varies between approximately 2nklku+1 and 2nklkl+ku+1, depending on the interchanges, assuming m=nkl and nku.
A call to F07BDF (DGBTRF) may be followed by calls to the routines:
The complex analogue of this routine is F07BRF (ZGBTRF).

9  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.

9.1  Program Text

Program Text (f07bdfe.f90)

9.2  Program Data

Program Data (f07bdfe.d)

9.3  Program Results

Program Results (f07bdfe.r)

F07BDF (DGBTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012