nag_zpttrs (f07jsc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zpttrs (f07jsc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpttrs (f07jsc) computes the solution to a complex system of linear equations AX=B , where A  is an n  by n  Hermitian positive definite tridiagonal matrix and X  and B  are n  by r  matrices, using the LDLH  factorization returned by nag_zpttrf (f07jrc).

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zpttrs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, const double d[], const Complex e[], Complex b[], Integer pdb, NagError *fail)

3  Description

nag_zpttrs (f07jsc) should be preceded by a call to nag_zpttrf (f07jrc), which computes a modified Cholesky factorization of the matrix A  as
A=LDLH ,
where L  is a unit lower bidiagonal matrix and D  is a diagonal matrix, with positive diagonal elements. nag_zpttrs (f07jsc) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UHDU , where U  is a unit upper bidiagonal matrix.

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:     uploNag_UploTypeInput
On entry: specifies the form of the factorization as follows:
uplo=Nag_Upper
A=UHDU.
uplo=Nag_Lower
A=LDLH.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     d[dim]const doubleInput
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the diagonal matrix D from the LDLH or UHDU factorization of A.
6:     e[dim]const ComplexInput
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: if uplo=Nag_Upper, e must contain the n-1 superdiagonal elements of the unit upper bidiagonal matrix U from the UHDU factorization of A.
If uplo=Nag_Lower, e must contain the n-1 subdiagonal elements of the unit lower bidiagonal matrix L from the LDLH factorization of A.
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 matrix of right-hand sides B.
On exit: 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.

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 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 this function nag_zptcon (f07juc) can be used to estimate the condition number of A  and nag_zptrfs (f07jvc) can be used to obtain approximate error bounds.

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_dpttrs (f07jec).

9  Example

This example solves the equations
AX=B ,
where A  is the Hermitian positive definite tridiagonal matrix
A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0
and
B = 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i .

9.1  Program Text

Program Text (f07jsce.c)

9.2  Program Data

Program Data (f07jsce.d)

9.3  Program Results

Program Results (f07jsce.r)


nag_zpttrs (f07jsc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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