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:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     n IntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3:     nrhs IntegerInput
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 constraints 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:     pdb IntegerInput
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:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
Element value of the diagonal 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
Parallelism and Performance

nag_zgtsv (f07cnc) 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 x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
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).

10
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 .  

10.1
Program Text

Program Text (f07cnce.c)

10.2
Program Data

Program Data (f07cnce.d)

10.3
Program Results

Program Results (f07cnce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017