nag_dgbtrf (f07bdc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dgbtrf (f07bdc)

+ Contents

    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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     klIntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5:     kuIntegerInput
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 8 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:     pdabIntegerInput
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
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.
NE_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+1.
NE_INTERNAL_ERROR
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.
NE_SINGULAR
Uvalue,value 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 nag_dgbtrf (f07bdc) may be followed by calls to the functions:
The complex analogue of this function is nag_zgbtrf (f07brc).

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 (f07bdce.c)

9.2  Program Data

Program Data (f07bdce.d)

9.3  Program Results

Program Results (f07bdce.r)


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

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