NAG Library Routine Document

f07chf  (dgtrfs)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f07chf (dgtrfs) computes error bounds and refines the solution to a real system of linear equations AX=B  or ATX=B , where A  is an n  by n  tridiagonal matrix and X  and B  are n  by r  matrices, using the LU  factorization returned by f07cdf (dgttrf) and an initial solution returned by f07cef (dgttrs). Iterative refinement is used to reduce the backward error as much as possible.

2
Specification

Fortran Interface
Subroutine f07chf ( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
Integer, Intent (In):: n, nrhs, ipiv(*), ldb, ldx
Integer, Intent (Out):: iwork(n), info
Real (Kind=nag_wp), Intent (In):: dl(*), d(*), du(*), dlf(*), df(*), duf(*), du2(*), b(ldb,*)
Real (Kind=nag_wp), Intent (Inout):: x(ldx,*)
Real (Kind=nag_wp), Intent (Out):: ferr(nrhs), berr(nrhs), work(3*n)
Character (1), Intent (In):: trans
C Header Interface
#include nagmk26.h
void  f07chf_ ( const char *trans, const Integer *n, const Integer *nrhs, const double dl[], const double d[], const double du[], const double dlf[], const double df[], const double duf[], const double du2[], const Integer ipiv[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name dgtrfs.

3
Description

f07chf (dgtrfs) should normally be preceded by calls to f07cdf (dgttrf) and f07cef (dgttrs). f07cdf (dgttrf) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A  as
A=PLU ,  
where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element in each column, and U  is an upper triangular band matrix, with two superdiagonals. f07cef (dgttrs) then utilizes the factorization to compute a solution, X^ , to the required equations. Letting x^  denote a column of X^ , f07chf (dgtrfs) 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 .

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

5
Arguments

1:     trans – Character(1)Input
On entry: specifies the equations to be solved as follows:
trans='N'
Solve AX=B for X.
trans='T' or 'C'
Solve ATX=B for X.
Constraint: trans='N', 'T' or 'C'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     nrhs – IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4:     dl* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array dl must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the matrix A.
5:     d* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the matrix A.
6:     du* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array du must be at least max1,n-1.
On entry: must contain the n-1 superdiagonal elements of the matrix A.
7:     dlf* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array dlf must be at least max1,n-1.
On entry: must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
8:     df* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array df must be at least max1,n.
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
9:     duf* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array duf must be at least max1,n-1.
On entry: must contain the n-1 elements of the first superdiagonal of U.
10:   du2* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array du2 must be at least max1,n-2.
On entry: must contain the n-2 elements of the second superdiagonal of U.
11:   ipiv* – Integer arrayInput
Note: the dimension of the array ipiv must be at least max1,n.
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipivi, and ipivi must always be either i or i+1, ipivi=i indicating that a row interchange was not performed.
12:   bldb* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r matrix of right-hand sides B.
13:   ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07chf (dgtrfs) is called.
Constraint: ldbmax1,n.
14:   xldx* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array x must be at least max1,nrhs.
On entry: the n by r initial solution matrix X.
On exit: the n by r refined solution matrix X.
15:   ldx – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07chf (dgtrfs) is called.
Constraint: ldxmax1,n.
16:   ferrnrhs – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the forward error bound for each computed solution vector, such that x^j-xj/x^jferrj, 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.
17:   berrnrhs – Real (Kind=nag_wp) arrayOutput
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).
18:   work3×n – Real (Kind=nag_wp) arrayWorkspace
19:   iworkn – Integer arrayWorkspace
20:   info – IntegerOutput
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=OεA  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x x κA E A ,  
where κA=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 f07cgf (dgtcon) can be used to estimate the condition number of A .

8
Parallelism and Performance

f07chf (dgtrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07chf (dgtrfs) 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  or ATX=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 f07cvf (zgtrfs).

10
Example

This example solves the equations
AX=B ,  
where A  is the tridiagonal matrix
A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1   and   B = 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 .  
Estimates for the backward errors and forward errors are also output.

10.1
Program Text

Program Text (f07chfe.f90)

10.2
Program Data

Program Data (f07chfe.d)

10.3
Program Results

Program Results (f07chfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017