nag_zgbsv (f07bnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgbsv (f07bnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgbsv (f07bnc) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n band matrix, with kl subdiagonals and ku superdiagonals, and X and B are n by r matrices.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgbsv (Nag_OrderType order, Integer n, Integer kl, Integer ku, Integer nrhs, Complex ab[], Integer pdab, Integer ipiv[], Complex b[], Integer pdb, NagError *fail)

3  Description

nag_zgbsv (f07bnc) uses the LU decomposition with partial pivoting and row interchanges to factor A as A=PLU, where P is a permutation matrix, L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. The factored form of A is then used to solve the system of equations AX=B.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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:     nIntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3:     klIntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
4:     kuIntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
5:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6:     ab[dim]ComplexInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n coefficient 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,,n 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 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,,n and j=i,,minn,i+kl+ku, and lij, for i=1,,n 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[n]IntegerOutput
On exit: if no constraints are violated, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipiv[i-1]. ipiv[i-1]=i indicates a row interchange was not required.
9:     b[dim]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by r right-hand side matrix B.
On exit: if fail.code= NE_NOERROR, the n by r solution matrix X.
10:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
11:   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, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
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, so the solution could not be computed.

7  Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^ = b ,
where
E1 = Oε A1
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x 1 x1 κA E1 A1 ,
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of nag_zgbsv (f07bnc), nag_zgbcon (f07buc) can be used to estimate the condition number of A  and nag_zgbrfs (f07bvc) can be used to obtain approximate error bounds. Alternatives to nag_zgbsv (f07bnc), which return condition and error estimates directly are nag_complex_band_lin_solve (f04cbc) and nag_zgbsvx (f07bpc).

8  Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n=6 , kl=1 , and ku=2 . Storage of the band matrix A  in the array ab:
order=Nag_ColMajor * * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * order=Nag_RowMajor * a11 a12 a13 + a21 a22 a23 a24 + a32 a33 a34 a35 + a43 a44 a45 a46 * a54 a55 a56 * * a65 a66 * * *
Array elements marked * need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements u14 , u25  and u36 .
The total number of floating point operations required to solve the equations AX=B  depends upon the pivoting required, but if nkl + ku  then it is approximately bounded by O nkl kl + ku  for the factorization and O n 2 kl + ku r  for the solution following the factorization.
The real analogue of this function is nag_dgbsv (f07bac).

9  Example

This example solves the equations
Ax=b ,
where A  is the band matrix
A = -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i   and   b = -1.06+21.50i -22.72-53.90i 28.24-38.60i -34.56+16.73i .
Details of the LU  factorization of A  are also output.

9.1  Program Text

Program Text (f07bnce.c)

9.2  Program Data

Program Data (f07bnce.d)

9.3  Program Results

Program Results (f07bnce.r)


nag_zgbsv (f07bnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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