NAG FL Interface
f07jef (dpttrs)

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1 Purpose

f07jef 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, using the LDLT factorization returned by f07jdf.

2 Specification

Fortran Interface
Subroutine f07jef ( n, nrhs, d, e, b, ldb, info)
Integer, Intent (In) :: n, nrhs, ldb
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: d(*), e(*)
Real (Kind=nag_wp), Intent (Inout) :: b(ldb,*)
C Header Interface
#include <nag.h>
void  f07jef_ (const Integer *n, const Integer *nrhs, const double d[], const double e[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07jef, nagf_lapacklin_dpttrs or its LAPACK name dpttrs.

3 Description

f07jef should be preceded by a call to f07jdf, 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. f07jef 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

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
Note: the dimension of the array d must be at least max(1,n).
On entry: must contain the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
4: e(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array e must be at least max(1,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.)
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 matrix of right-hand sides B.
On exit: 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 f07jef 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.

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.
Following the use of this routine f07jgf can be used to estimate the condition number of A and f07jhf can be used to obtain approximate error bounds.

8 Parallelism and Performance

f07jef 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 total number of floating-point operations required to solve the equations AX=B is proportional to nr .
The complex analogue of this routine is f07jsf.

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 0.0 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 ) .  

10.1 Program Text

Program Text (f07jefe.f90)

10.2 Program Data

Program Data (f07jefe.d)

10.3 Program Results

Program Results (f07jefe.r)