nag_dptsv (f07jac) (PDF version)
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f07 Chapter Introduction
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 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:     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]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.
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.
NE_MAT_NOT_POS_DEF
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 ,
where
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