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:     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:     trans Nag_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:     n IntegerInput
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:     nrhs IntegerInput
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:   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.
12:   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.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_PERM_COL
Incorrect column permutations in array iprm.
NE_INVALID_PERM_ROW
Incorrect row permutations in array iprm.
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

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
Parallelism and Performance

nag_superlu_solve_lu (f11mfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_superlu_solve_lu (f11mfc) 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

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.

10
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).

10.1
Program Text

Program Text (f11mfce.c)

10.2
Program Data

Program Data (f11mfce.d)

10.3
Program Results

Program Results (f11mfce.r)

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