NAG FL Interface
f07jhf (dptrfs)

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1 Purpose

f07jhf computes error bounds and refines 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 modified Cholesky factorization returned by f07jdf and an initial solution returned by f07jef. Iterative refinement is used to reduce the backward error as much as possible.

2 Specification

Fortran Interface
Subroutine f07jhf ( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
Integer, Intent (In) :: n, nrhs, ldb, ldx
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: d(*), e(*), df(*), ef(*), b(ldb,*)
Real (Kind=nag_wp), Intent (Inout) :: x(ldx,*)
Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), work(2*n)
C Header Interface
#include <nag.h>
void  f07jhf_ (const Integer *n, const Integer *nrhs, const double d[], const double e[], const double df[], const double ef[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer *info)
The routine may be called by the names f07jhf, nagf_lapacklin_dptrfs or its LAPACK name dptrfs.

3 Description

f07jhf should normally be preceded by calls to f07jdf and f07jef. f07jdf 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 compute a solution, X^ , to the required equations. Letting x^ denote a column of X^ , f07jhf computes a component-wise backward error, β , the smallest relative perturbation in each element of A and b such that x^ is the exact solution of a perturbed system
(A+E) x^ = b + f , with  |eij| β |aij| , and  |fj| β |bj| .  
The routine also estimates a bound for the component-wise forward error in the computed solution defined by max |xi-xi^| / max |xi^| , where x is the corresponding column of the exact solution, X .
Note that the modified Cholesky factorization of A can also be expressed as
A=UTDU ,  
where U is unit upper bidiagonal.

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 https://www.netlib.org/lapack/lug

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 matrix 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 matrix A.
5: df(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array df 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.
6: ef(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array ef must be at least max(1,n).
On entry: must contain the (n-1) subdiagonal elements of the unit bidiagonal matrix L from the LDLT factorization of A.
7: b(ldb,*) Real (Kind=nag_wp) array Input
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.
8: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07jhf is called.
Constraint: ldbmax(1,n).
9: x(ldx,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array x must be at least max(1,nrhs).
On entry: the n×r initial solution matrix X.
On exit: the n×r refined solution matrix X.
10: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07jhf is called.
Constraint: ldxmax(1,n).
11: ferr(nrhs) Real (Kind=nag_wp) array Output
On exit: estimate of the forward error bound for each computed solution vector, such that x^j-xj/x^jferr(j), where x^j is the jth column of the computed solution returned in the array x and xj is the corresponding column of the exact solution X. The estimate is almost always a slight overestimate of the true error.
12: berr(nrhs) Real (Kind=nag_wp) array Output
On exit: estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
13: work(2×n) Real (Kind=nag_wp) array Workspace
14: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
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 ,  
where
E=O(ε)A  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x x κ(A) E A ,  
where κ(A)=A-1 A , 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.
Routine f07jgf can be used to compute the condition number of A .

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07jhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07jhf 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 . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this routine is f07jvf.

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 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 ) .  
Estimates for the backward errors and forward errors are also output.

10.1 Program Text

Program Text (f07jhfe.f90)

10.2 Program Data

Program Data (f07jhfe.d)

10.3 Program Results

Program Results (f07jhfe.r)