nag_dgbtrf (f07bdc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_dgbtrf (f07bdc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_dgbtrf (f07bdc) computes the LU factorization of a real m by n band matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dgbtrf (Nag_OrderType order, Integer m, Integer n, Integer kl, Integer ku, double ab[], Integer pdab, Integer ipiv[], NagError *fail)

3  Description

nag_dgbtrf (f07bdc) 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  Arguments

1:     order 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 order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     m IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     n IntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     kl IntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5:     ku IntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6:     ab[dim] doubleInput/Output
Note: the dimension, dim, of the array ab must be at least
  • max1,pdab×n when order=Nag_ColMajor;
  • max1,m×pdab when order=Nag_RowMajor.
On entry: the m by n matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements Aij, for row i=1,,m and column j=max1,i-kl,,minn,i+ku, depends on the order argument as follows:
  • if order=Nag_ColMajor, Aij is stored as ab[j-1×pdab+kl+ku+i-j];
  • if order=Nag_RowMajor, Aij is stored as ab[i-1×pdab+kl+j-i].
See Section 9 in nag_dgbsv (f07bac) for further details.
On exit: ab is overwritten by details of the factorization.
The elements, uij, of the upper triangular band factor U with kl+ku super-diagonals, and the multipliers, lij, used to form the lower triangular factor L are stored. The elements uij, for i=1,,m and j=i,,minn,i+kl+ku, and lij, for i=1,,m and j=max1,i-kl,,i, are stored where Aij is stored on entry.
7:     pdab IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdab2×kl+ku+1.
8:     ipiv[minm,n] IntegerOutput
On exit: the pivot indices that define the permutation matrix. At the ith step, if ipiv[i-1]>i then row i of the matrix A was interchanged with row ipiv[i-1], for i=1,2,,minm,n. ipiv[i-1]i indicates that, at the ith step, a row interchange was not required.
9:     fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in the Essential Introduction for further information.
On entry, argument value had an illegal value.
On entry, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+1.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
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
EckεPLU ,  
ck is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kminm,n.

8  Parallelism and Performance

nag_dgbtrf (f07bdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgbtrf (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.

9  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 nag_dgbtrf (f07bdc) may be followed by calls to the functions:
The complex analogue of this function is nag_zgbtrf (f07brc).

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 (f07bdce.c)

10.2  Program Data

Program Data (f07bdce.d)

10.3  Program Results

Program Results (f07bdce.r)

nag_dgbtrf (f07bdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015