nag_dptsv (f07jac) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dptsv (f07jac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dptsv (f07jac) 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.

2  Specification

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

3  Description

nag_dptsv (f07jac) factors A as A=LDLT. The factored form of A is then used to solve the system of 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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 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]doubleInput/Output
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: the n diagonal elements of the tridiagonal matrix A.
On exit: the n diagonal elements of the diagonal matrix D from the factorization A=LDLT.
5:     e[dim]doubleInput/Output
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: the n-1 subdiagonal elements of the tridiagonal matrix A.
On exit: the n-1 subdiagonal elements of the unit bidiagonal factor L from the LDLT factorization of A. (e can also be regarded as the superdiagonal of the unit bidiagonal factor 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 right-hand side matrix B.
On exit: if fail.code= NE_NOERROR, 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.
  • 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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
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.
The leading minor of order value is not positive definite, 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 ,
E1 = Oε A1
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,
where κ A = 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.
nag_dptsvx (f07jbc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_real_sym_posdef_tridiag_lin_solve (f04bgc) solves Ax=b  and returns a forward error bound and condition estimate. nag_real_sym_posdef_tridiag_lin_solve (f04bgc) calls nag_dptsv (f07jac) to solve the equations.

8  Further Comments

The number of floating point operations required for the factorization of A  is proportional to n , and the number of floating point operations required for the solution of the equations is proportional to nr , where r  is the number of right-hand sides.
The complex analogue of this function is nag_zptsv (f07jnc).

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 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0   and   b = 6.0 9.0 2.0 14.0 7.0 .
Details of the LDLT  factorization of A  are also output.

9.1  Program Text

Program Text (f07jace.c)

9.2  Program Data

Program Data (f07jace.d)

9.3  Program Results

Program Results (f07jace.r)

nag_dptsv (f07jac) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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