NAG Library Function Document

nag_dptrfs (f07jhc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_dptrfs (f07jhc) computes error bounds and refines 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, using the modified Cholesky factorization returned by nag_dpttrf (f07jdc) and an initial solution returned by nag_dpttrs (f07jec). Iterative refinement is used to reduce the backward error as much as possible.

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dptrfs (Nag_OrderType order, Integer n, Integer nrhs, const double d[], const double e[], const double df[], const double ef[], const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3
Description

nag_dptrfs (f07jhc) should normally be preceded by calls to nag_dpttrf (f07jdc) and nag_dpttrs (f07jec). nag_dpttrf (f07jdc) 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. nag_dpttrs (f07jec) then utilizes the factorization to compute a solution, X^ , to the required equations. Letting x^  denote a column of X^ , nag_dptrfs (f07jhc) 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 function 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 http://www.netlib.org/lapack/lug

5
Arguments

1:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
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:     d[dim] const doubleInput
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the matrix of A.
5:     e[dim] const doubleInput
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the matrix A.
6:     df[dim] const doubleInput
Note: the dimension, dim, of the array df must be at least max1,n.
On entry: must contain the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
7:     ef[dim] const doubleInput
Note: the dimension, dim, of the array ef must be at least max1,n.
On entry: must contain the n-1 subdiagonal elements of the unit bidiagonal matrix L from the LDLT factorization of A.
8:     b[dim] const doubleInput
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 matrix of right-hand sides B.
9:     pdb IntegerInput
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.
10:   x[dim] doubleInput/Output
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×nrhs when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On entry: the n by r initial solution matrix X.
On exit: the n by r refined solution matrix X.
11:   pdx IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax1,n;
  • if order=Nag_RowMajor, pdxmax1,nrhs.
12:   ferr[nrhs] doubleOutput
On exit: estimate of the forward error bound for each computed solution vector, such that x^j-xj/x^jferr[j-1], 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.
13:   berr[nrhs] doubleOutput
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).
14:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdx=value and n=value.
Constraint: pdxmax1,n.
On entry, pdx=value and nrhs=value.
Constraint: pdxmax1,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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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.
Function nag_dptcon (f07jgc) can be used to compute the condition number of A .

8
Parallelism and Performance

nag_dptrfs (f07jhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dptrfs (f07jhc) 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 function. 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 function is nag_zptrfs (f07jvc).

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 (f07jhce.c)

10.2
Program Data

Program Data (f07jhce.d)

10.3
Program Results

Program Results (f07jhce.r)

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