nag_complex_band_lin_solve (f04cbc) (PDF version)
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f04 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_complex_band_lin_solve (f04cbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_complex_band_lin_solve (f04cbc) 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. An estimate of the condition number of A and an error bound for the computed solution are also returned.

2  Specification

#include <nag.h>
#include <nagf04.h>
void  nag_complex_band_lin_solve (Nag_OrderType order, Integer n, Integer kl, Integer ku, Integer nrhs, Complex ab[], Integer pdab, Integer ipiv[], Complex b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)

3  Description

The LU decomposition with partial pivoting and row interchanges is used to factor A as A=PLU, where P is a permutation matrix, L is the product of permutation matrices 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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: the number of linear equations n, i.e., the order of the matrix A.
Constraint: n0.
3:     klIntegerInput
On entry: the number of subdiagonals kl, within the band of A.
Constraint: kl0.
4:     kuIntegerInput
On entry: the number of superdiagonals ku, within the band of A.
Constraint: ku0.
5:     nrhsIntegerInput
On entry: the number of right-hand sides r, 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 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 fail.code= NE_NOERROR, 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 matrix of right-hand sides B.
On exit: if fail.code= NE_NOERROR or NE_RCOND, 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:   rconddouble *Output
On exit: if fail.code= NE_NOERROR, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=A1A-11.
12:   errbnddouble *Output
On exit: if fail.code= NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and x is the corresponding column of the exact solution X. If rcond is less than machine precision, then errbnd is returned as unity.
13:   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, pdab =value, kl =value and ku=value.
Constraint: pdab2×kl+ku+1.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
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.
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
Diagonal element value of the upper triangular factor is zero. The factorization has been completed, but 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^-x1 x1 κA E1 A1 ,
where κA=A-11A1, the condition number of A with respect to the solution of the linear equations. nag_complex_band_lin_solve (f04cbc) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

nag_complex_band_lin_solve (f04cbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_complex_band_lin_solve (f04cbc) 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=5, kl=2, and ku=1. Storage of the band matrix A in the array ab:
Band matrix A Band storage in array ab
order=Nag_ColMajor order=Nag_RowMajor
a11 a12 a21 a22 a23 a31 a32 a33 a34 a42 a43 a44 a45 a53 a54 a55 * * * + + * * + + + * a12 a23 a34 a45 a11 a22 a33 a44 a55 a21 a32 a43 a54 * a31 a42 a53 * * * * a11 a12 + + * a21 a22 a23 + + a31 a32 a33 a34 + * a42 a43 a44 a45 * * a53 a54 a55 * * *
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 u13, u14, u24, u25 and u35. In this example when order=Nag_ColMajor the first referenced element of ab is ab[3]=a11; while for order=Nag_RowMajor the first referenced element is ab[2]=a11.
In general, elements aij are stored as follows: where max1,i-kl j minn,i+ku .
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 n kl kl + ku  for the factorization and O n 2 kl + ku ,r  for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of nag_complex_band_lin_solve (f04cbc) is nag_real_band_lin_solve (f04bbc).

10  Example

This example solves the equations
where A is the band matrix
A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 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 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.73-01.23i -34.56+16.73i 26.01+31.97i .
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.

10.1  Program Text

Program Text (f04cbce.c)

10.2  Program Data

Program Data (f04cbce.d)

10.3  Program Results

Program Results (f04cbce.r)

nag_complex_band_lin_solve (f04cbc) (PDF version)
f04 Chapter Contents
f04 Chapter Introduction
NAG Library Manual

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