nag_superlu_solve_lu (f11mfc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_superlu_solve_lu (f11mfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_superlu_solve_lu (f11mfc) solves a real sparse system of linear equations with multiple right-hand sides given an LU  factorization of the sparse matrix computed by nag_superlu_lu_factorize (f11mec).

2  Specification

#include <nag.h>
#include <nagf11.h>
void  nag_superlu_solve_lu (Nag_OrderType order, Nag_TransType trans, Integer n, const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], Integer nrhs, double b[], Integer pdb, NagError *fail)

3  Description

nag_superlu_solve_lu (f11mfc) solves a real system of linear equations with multiple right-hand sides AX=B or ATX=B, according to the value of the argument trans, where the matrix factorization Pr A Pc = LU  corresponds to an LU  decomposition of a sparse matrix stored in compressed column (Harwell–Boeing) format, as computed by nag_superlu_lu_factorize (f11mec).
In the above decomposition L is a lower triangular sparse matrix with unit diagonal elements and U is an upper triangular sparse matrix; Pr and Pc are permutation matrices.

4  References

None.

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:     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).
5:     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).
6:     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).
7:     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).
8:     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).
9:     nrhsIntegerInput
On entry: nrhs, the number of right-hand sides in B.
Constraint: nrhs0.
10:   b[dim]doubleInput/Output
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.
On exit: the n by nrhs solution matrix X.
11:   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.
12:   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.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,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

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EcnεLU,
cn is a modest linear function of n, and ε is the machine precision, when partial pivoting is used.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε
where condA,x= A-1 A x / xcondA= A-1 A κ A. Note that condA,x can be much smaller than condA, and condAT can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_superlu_refine_lu (f11mhc), and an estimate for κA can be obtained by calling nag_superlu_condition_number_lu (f11mgc).

8  Further Comments

nag_superlu_solve_lu (f11mfc) may be followed by a call to nag_superlu_refine_lu (f11mhc) to refine the solution and return an error estimate.

9  Example

This example solves the system of equations AX=B, 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 (f11mfce.c)

9.2  Program Data

Program Data (f11mfce.d)

9.3  Program Results

Program Results (f11mfce.r)


nag_superlu_solve_lu (f11mfc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

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