nag_zgbsv (f07bnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG 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
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 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 9 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.
  • 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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
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.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
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.
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 ,
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  Parallelism and Performance

nag_zgbsv (f07bnc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgbsv (f07bnc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  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).

10  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
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.

10.1  Program Text

Program Text (f07bnce.c)

10.2  Program Data

Program Data (f07bnce.d)

10.3  Program Results

Program Results (f07bnce.r)

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014