NAG Library Function Document

nag_zgttrs (f07csc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zgttrs (f07csc) computes the solution to a complex system of linear equations AX=B  or ATX=B  or AHX=B , where A  is an n  by n  tridiagonal matrix and X  and B  are n  by r  matrices, using the LU  factorization returned by nag_zgttrf (f07crc).

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgttrs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer nrhs, const Complex dl[], const Complex d[], const Complex du[], const Complex du2[], const Integer ipiv[], Complex b[], Integer pdb, NagError *fail)

3
Description

nag_zgttrs (f07csc) should be preceded by a call to nag_zgttrf (f07crc), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A  as
A=PLU ,  
where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element in each column, and U  is an upper triangular band matrix, with two superdiagonals. nag_zgttrs (f07csc) then utilizes the factorization to solve the required equations.

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:     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:     trans Nag_TransTypeInput
On entry: specifies the equations to be solved as follows:
trans=Nag_NoTrans
Solve AX=B for X.
trans=Nag_Trans
Solve ATX=B for X.
trans=Nag_ConjTrans
Solve AHX=B for X.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     nrhs IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     dl[dim] const ComplexInput
Note: the dimension, dim, of the array dl must be at least max1,n-1.
On entry: must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
6:     d[dim] const ComplexInput
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
7:     du[dim] const ComplexInput
Note: the dimension, dim, of the array du must be at least max1,n-1.
On entry: must contain the n-1 elements of the first superdiagonal of U.
8:     du2[dim] const ComplexInput
Note: the dimension, dim, of the array du2 must be at least max1,n-2.
On entry: must contain the n-2 elements of the second superdiagonal of U.
9:     ipiv[dim] const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipiv[i-1], and ipiv[i-1] must always be either i or i+1, ipiv[i-1]=i indicating that a row interchange was not performed.
10:   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: the n by r solution matrix X.
11:   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.
12:   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.

7
Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
where
E1 =OεA1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x 1 x 1 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 this function nag_zgtcon (f07cuc) can be used to estimate the condition number of A  and nag_zgtrfs (f07cvc) can be used to obtain approximate error bounds.

8
Parallelism and Performance

nag_zgttrs (f07csc) is not threaded in any implementation.

9
Further Comments

The total number of floating-point operations required to solve the equations AX=B  or ATX=B  or AHX=B  is proportional to nr .
The real analogue of this function is nag_dgttrs (f07cec).

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 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i .  

10.1
Program Text

Program Text (f07csce.c)

10.2
Program Data

Program Data (f07csce.d)

10.3
Program Results

Program Results (f07csce.r)

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