nag_superlu_refine_lu (f11mhc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_superlu_refine_lu (f11mhc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_superlu_refine_lu (f11mhc) returns error bounds for the solution of a real sparse system of linear equations with multiple right-hand sides, AX=B or ATX=B. It improves the solution by iterative refinement in standard precision, in order to reduce the backward error as much as possible.

2  Specification

#include <nag.h>
#include <nagf11.h>
void  nag_superlu_refine_lu (Nag_OrderType order, Nag_TransType trans, Integer n, const Integer icolzp[], const Integer irowix[], const double a[], const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], Integer nrhs, const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3  Description

nag_superlu_refine_lu (f11mhc) returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides AX=B or ATX=B. The function handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of nag_superlu_refine_lu (f11mhc) in terms of a single right-hand side b and solution x.
Given a computed solution x, the function computes the component-wise backward error β. This is the size of the smallest relative perturbation in each element of A and b such that if x is the exact solution of a perturbed system:
A+δA x = b + δ b then   δaij β aij   and   δbi β bi .
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
maxi xi - x^i / maxi xi
where x^ is the true solution.
The function uses the LU  factorization Pr A Pc = LU  computed by nag_superlu_lu_factorize (f11mec) and the solution computed by nag_superlu_solve_lu (f11mfc).

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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 whether AX=B or ATX=B is solved.
trans=Nag_NoTrans
AX=B is solved.
trans=Nag_Trans
ATX=B is solved.
Constraint: trans=Nag_NoTrans or Nag_Trans.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     icolzp[dim]const IntegerInput
Note: the dimension, dim, of the array icolzp must be at least n+1.
On entry: icolzp[i-1] contains the index in A of the start of a new column. See Section 2.1.3 in the f11 Chapter Introduction.
5:     irowix[dim]const IntegerInput
Note: the dimension, dim, of the array irowix must be at least icolzp[n]-1, the number of nonzeros of the sparse matrix A.
On entry: the row index array of the sparse matrix A.
6:     a[dim]const doubleInput
Note: the dimension, dim, of the array a must be at least icolzp[n]-1, the number of nonzeros of the sparse matrix A.
On entry: the array of nonzero values in the sparse matrix A.
7:     iprm[7×n]const IntegerInput
On entry: the column permutation which defines Pc, the row permutation which defines Pr, plus associated data structures as computed by nag_superlu_lu_factorize (f11mec).
8:     il[dim]const IntegerInput
Note: the dimension, dim, of the array il must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records the sparsity pattern of matrix L as computed by nag_superlu_lu_factorize (f11mec).
9:     lval[dim]const doubleInput
Note: the dimension, dim, of the array lval must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records the nonzero values of matrix L and some nonzero values of matrix U as computed by nag_superlu_lu_factorize (f11mec).
10:   iu[dim]const IntegerInput
Note: the dimension, dim, of the array iu must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records the sparsity pattern of matrix U as computed by nag_superlu_lu_factorize (f11mec).
11:   uval[dim]const doubleInput
Note: the dimension, dim, of the array uval must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records some nonzero values of matrix U as computed by nag_superlu_lu_factorize (f11mec).
12:   nrhsIntegerInput
On entry: nrhs, the number of right-hand sides in B.
Constraint: nrhs0.
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 nrhs right-hand side matrix 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 nrhs solution matrix X, as returned by nag_superlu_solve_lu (f11mfc).
On exit: the n by nrhs improved 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: ferr[j-1] contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,nrhs.
18:   berr[nrhs]doubleOutput
On exit: berr[j-1] contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,,nrhs.
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 number value had an illegal value.
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.
NE_INVALID_PERM_COL
Incorrect Column Permutations in array iprm.
NE_INVALID_PERM_ROW
Incorrect Row Permutations in array iprm.

7  Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

8  Further Comments

At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax=b or ATx=b;

9  Example

This example solves the system of equations AX=B using iterative refinement and to compute the forward and backward error bounds, where
A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00   and  B= 1.56 3.12 -0.25 -0.50 3.60 7.20 1.33 2.66 0.52 1.04 .
Here A is nonsymmetric and must first be factorized by nag_superlu_lu_factorize (f11mec).

9.1  Program Text

Program Text (f11mhce.c)

9.2  Program Data

Program Data (f11mhce.d)

9.3  Program Results

Program Results (f11mhce.r)


nag_superlu_refine_lu (f11mhc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012