NAG FL Interface
f07jaf (dptsv)

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1 Purpose

f07jaf computes the solution to a real system of linear equations
AX=B ,  
where A is an n×n symmetric positive definite tridiagonal matrix, and X and B are n×r matrices.

2 Specification

Fortran Interface
Subroutine f07jaf ( n, nrhs, d, e, b, ldb, info)
Integer, Intent (In) :: n, nrhs, ldb
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), b(ldb,*)
C Header Interface
#include <nag.h>
void  f07jaf_ (const Integer *n, const Integer *nrhs, double d[], double e[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07jaf, nagf_lapacklin_dptsv or its LAPACK name dptsv.

3 Description

f07jaf 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: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
3: d(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array d must be at least max(1,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.
4: e(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array e must be at least max(1,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.)
5: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,nrhs).
On entry: the n×r right-hand side matrix B.
On exit: if info=0, the n×r solution matrix X.
6: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07jaf is called.
Constraint: ldbmax(1,n).
7: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
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 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) 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.
f07jbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bgf solves Ax=b and returns a forward error bound and condition estimate. f04bgf calls f07jaf to solve the equations.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07jaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 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 routine is f07jnf.

10 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.

10.1 Program Text

Program Text (f07jafe.f90)

10.2 Program Data

Program Data (f07jafe.d)

10.3 Program Results

Program Results (f07jafe.r)