nag_dgtrfs (f07chc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dgtrfs (f07chc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgtrfs (f07chc) 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 nag_dgttrf (f07cdc) and an initial solution returned by nag_dgttrs (f07cec). Iterative refinement is used to reduce the backward error as much as possible.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dgtrfs (Nag_OrderType order, Nag_TransType trans, Integer n, 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[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3  Description

nag_dgtrfs (f07chc) should normally be preceded by calls to nag_dgttrf (f07cdc) and nag_dgttrs (f07cec). nag_dgttrf (f07cdc) 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. nag_dgttrs (f07cec) then utilizes the factorization to compute a solution, X^ , to the required equations. Letting x^  denote a column of X^ , nag_dgtrfs (f07chc) 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 .

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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     transNag_TransTypeInput
On entry: specifies the equations to be solved as follows:
trans=Nag_NoTrans
Solve AX=B for X.
trans=Nag_Trans or Nag_ConjTrans
Solve ATX=B for X.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     dl[dim]const doubleInput
Note: the dimension, dim, of the array dl must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the matrix A.
6:     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 A.
7:     du[dim]const doubleInput
Note: the dimension, dim, of the array du must be at least max1,n-1.
On entry: must contain the n-1 superdiagonal elements of the matrix A.
8:     dlf[dim]const doubleInput
Note: the dimension, dim, 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.
9:     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 upper triangular matrix U from the LU factorization of A.
10:   duf[dim]const doubleInput
Note: the dimension, dim, 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.
11:   du2[dim]const doubleInput
Note: the dimension, dim, 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.
12:   ipiv[dim]const IntegerInput
Note: the dimension, dim, 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 ipiv[i-1], and ipiv[i-1] must always be either i or i+1, ipiv[i-1]=i indicating that a row interchange was not performed.
13:   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.
14:   pdbIntegerInput
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.
15:   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.
16:   pdxIntegerInput
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.
17:   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.
18:   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).
19:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.

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_dgtcon (f07cgc) can be used to estimate the condition number of A .

8  Parallelism and Performance

nag_dgtrfs (f07chc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgtrfs (f07chc) 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 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 function is nag_zgtrfs (f07cvc).

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 (f07chce.c)

10.2  Program Data

Program Data (f07chce.d)

10.3  Program Results

Program Results (f07chce.r)


nag_dgtrfs (f07chc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014