nag_dpttrs (f07jec) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dpttrs (f07jec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpttrs (f07jec) computes the solution to a real system of linear equations AX=B , where A  is an n  by n  symmetric positive definite tridiagonal matrix and X  and B  are n  by r  matrices, using the LDLT  factorization returned by nag_dpttrf (f07jdc).

2  Specification

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

3  Description

nag_dpttrs (f07jec) should be preceded by a call to nag_dpttrf (f07jdc), which computes a modified Cholesky factorization of the matrix A  as
A=LDLT ,
where L  is a unit lower bidiagonal matrix and D  is a diagonal matrix, with positive diagonal elements. nag_dpttrs (f07jec) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UTDU , 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:     nIntegerInput
On entry: n, 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:     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 LDLT factorization of A.
5:     e[dim]const doubleInput
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the unit lower bidiagonal matrix L. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix U from the UTDU factorization of A.)
6:     b[dim]doubleInput/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.
7:     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.
8:     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_dptcon (f07jgc) can be used to estimate the condition number of A  and nag_dptrfs (f07jhc) 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 complex analogue of this function is nag_zpttrs (f07jsc).

9  Example

This example solves the equations
AX=B ,
where A  is the symmetric positive definite tridiagonal matrix
A = 4.0 -2.0 0.0 0.0 0.0 -2.0 10.0 -6.0 0.0 0.0 0.0 -6.0 29.0 15.0 0.0 0.0 0.0 15.0 25.0 8.0 0.0 0.0 0.0 8.0 5.0   and   B = 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 .

9.1  Program Text

Program Text (f07jece.c)

9.2  Program Data

Program Data (f07jece.d)

9.3  Program Results

Program Results (f07jece.r)


nag_dpttrs (f07jec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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