nag_zgtsv (f07cnc) (PDF version)
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f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgtsv (f07cnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgtsv (f07cnc) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n tridiagonal matrix and X and B are n by r matrices.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgtsv (Nag_OrderType order, Integer n, Integer nrhs, Complex dl[], Complex d[], Complex du[], Complex b[], Integer pdb, NagError *fail)

3  Description

nag_zgtsv (f07cnc) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations AX=B . The matrix A  is factorized as A=PLU , where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element per column, and U  is an upper triangular band matrix, with two superdiagonals.
Note that the equations ATX=B may be solved by interchanging the order of the arguments du and dl.

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

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:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4:     dl[dim]ComplexInput/Output
Note: the dimension, dim, of the array dl must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the matrix A.
On exit: if no constrains are violated, dl is overwritten by the (n-2) elements of the second superdiagonal of the upper triangular matrix U from the LU factorization of A, in dl[0],dl[1],,dl[n-3].
5:     d[dim]ComplexInput/Output
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the matrix A.
On exit: if no constraints are violated, d is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
6:     du[dim]ComplexInput/Output
Note: the dimension, dim, of the array du must be at least max1,n-1.
On entry: must contain the n-1 superdiagonal elements of the matrix A.
On exit: if no constraints are violated, du is overwritten by the n-1 elements of the first superdiagonal of U.
7:     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.
8:     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.
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, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
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_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, and the solution has not been computed. The factorization has not been completed unless n=value.

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.
Alternatives to nag_zgtsv (f07cnc), which return condition and error estimates are nag_complex_tridiag_lin_solve (f04ccc) and nag_zgtsvx (f07cpc).

8  Further Comments

The total number of floating point operations required to solve the equations AX=B  is proportional to nr .
The real analogue of this function is nag_dgtsv (f07cac).

9  Example

This example solves the equations
Ax=b ,
where A  is the tridiagonal matrix
A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i
and
b = 2.4-05.0i 3.4+18.2i -14.7+09.7i 31.9-07.7i -1.0+01.6i .

9.1  Program Text

Program Text (f07cnce.c)

9.2  Program Data

Program Data (f07cnce.d)

9.3  Program Results

Program Results (f07cnce.r)


nag_zgtsv (f07cnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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