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NAG Toolbox: nag_sparse_direct_real_gen_lu (f11me)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_sparse_direct_real_gen_lu (f11me) computes the LU  factorization of a real sparse matrix in compressed column (Harwell–Boeing), column-permuted format.

Syntax

[iprm, nzlumx, il, lval, iu, uval, nnzl, nnzu, flop, ifail] = f11me(n, irowix, a, iprm, thresh, nzlmx, nzlumx, nzumx)
[iprm, nzlumx, il, lval, iu, uval, nnzl, nnzu, flop, ifail] = nag_sparse_direct_real_gen_lu(n, irowix, a, iprm, thresh, nzlmx, nzlumx, nzumx)

Description

Given a real sparse matrix A, nag_sparse_direct_real_gen_lu (f11me) computes an LU  factorization of A with partial pivoting, Pr A Pc = LU , where Pr is a row permutation matrix (computed by nag_sparse_direct_real_gen_lu (f11me)), Pc is a (supplied) column permutation matrix, L is unit lower triangular and U is upper triangular. The column permutation matrix, Pc, must be computed by a prior call to nag_sparse_direct_real_gen_setup (f11md). The matrix A must be presented in the column permuted, compressed column (Harwell–Boeing) format.
The LU  factorization is output in the form of four one-dimensional arrays: integer arrays il and iu and real-valued arrays lval and uval. These describe the sparsity pattern and numerical values in the L and U matrices. The minimum required dimensions of these arrays cannot be given as a simple function of the size arguments (order and number of nonzero values) of the matrix A. This is due to unpredictable fill-in created by partial pivoting. nag_sparse_direct_real_gen_lu (f11me) will, on return, indicate which dimensions of these arrays were not adequate for the computation or (in the case of one of them) give a firm bound. You should then allocate more storage and try again.

References

Demmel J W, Eisenstat S C, Gilbert J R, Li X S and Li J W H (1999) A supernodal approach to sparse partial pivoting SIAM J. Matrix Anal. Appl. 20 720–755
Demmel J W, Gilbert J R and Li X S (1999) An asynchronous parallel supernodal algorithm for sparse gaussian elimination SIAM J. Matrix Anal. Appl. 20 915–952

Parameters

Compulsory Input Parameters

1:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
2:     irowix: int64int32nag_int array
The dimension of the array irowix must be at least nnz, the number of nonzeros of the sparse matrix A
The row index array of sparse matrix A.
3:     a: – double array
The dimension of the array a must be at least nnz, the number of nonzeros of the sparse matrix A
The array of nonzero values in the sparse matrix A.
4:     iprm7×n int64int32nag_int array
Contains the column permutation which defines the permutation Pc and associated data structures as computed by function nag_sparse_direct_real_gen_setup (f11md).
5:     thresh – double scalar
Suggested value: thresh=1.0. Smaller values may result in a faster factorization, but the benefits are likely to be small in most cases. It might be possible to use thresh=0.0 if you are confident about the stability of the factorization, for example, if A is diagonally dominant.
The diagonal pivoting threshold, t. At step j of the Gaussian elimination, if AjjtmaxijAij, use Ajj as a pivot, otherwise use maxijAij. A value of t=1 corresponds to partial pivoting, a value of t=0 corresponds to always choosing the pivot on the diagonal (unless it is zero).
Constraint: 0.0thresh1.0.
6:     nzlmx int64int32nag_int scalar
Indicates the available size of array il. The dimension of il should be at least 7×n+nzlmx+4. A good range for nzlmx that works for many problems is nnz to 8×nnz, where nnz is the number of nonzeros in the sparse matrix A. If, on exit, ifail=2, the given nzlmx was too small and you should attempt to provide more storage and call the function again.
Constraint: nzlmx1.
7:     nzlumx int64int32nag_int scalar
Indicates the available size of array lval. The dimension of lval should be at least nzlumx.
Constraint: nzlumx1.
8:     nzumx int64int32nag_int scalar
Indicates the available sizes of arrays iu and uval. The dimension of iu should be at least 2×n+nzumx+1 and the dimension of uval should be at least nzumx. A good range for nzumx that works for many problems is nnz to 8×nnz, where nnz is the number of nonzeros in the sparse matrix A. If, on exit, ifail=3, the given nzumx was too small and you should attempt to provide more storage and call the function again.
Constraint: nzumx1.

Optional Input Parameters

None.

Output Parameters

1:     iprm7×n int64int32nag_int array
Part of the array is modified to record the row permutation Pr determined by pivoting.
2:     nzlumx int64int32nag_int scalar
If ifail=4, the given nzlumx was too small and is reset to a value that will be sufficient. You should then provide the indicated storage and call the function again.
3:     il7×n+nzlmx+4 int64int32nag_int array
Encapsulates the sparsity pattern of matrix L.
4:     lvalnzlumx – double array
Records the nonzero values of matrix L and some of the nonzero values of matrix U.
5:     iu2×n+nzumx+1 int64int32nag_int array
Encapsulates the sparsity pattern of matrix U.
6:     uvalnzumx – double array
Records some of the nonzero values of matrix U.
7:     nnzl int64int32nag_int scalar
The number of nonzero values in the matrix L.
8:     nnzu int64int32nag_int scalar
The number of nonzero values in the matrix U.
9:     flop – double scalar
The number of floating-point operations performed.
10:   ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
Constraint: 0.0thresh1.0.
Constraint: n0.
Constraint: nzlmx1.
Constraint: nzlumx1.
Constraint: nzumx1.
   ifail=2
Insufficient nzlmx.
   ifail=3
Insufficient nzumx.
   ifail=4
Insufficient nzlumx.
   ifail=5
The matrix is singular – no factorization possible.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

The computed factors L and U are the exact factors of a perturbed matrix A+E, where
E c n ε L U ,  
cn is a modest linear function of n, and ε is the machine precision, when partial pivoting is used. If no partial pivoting is used, the factorization accuracy can be considerably worse. A call to nag_sparse_direct_real_gen_diag (f11mm) after nag_sparse_direct_real_gen_lu (f11me) can help determine the quality of the factorization.

Further Comments

The total number of floating-point operations depends on the sparsity pattern of the matrix A.
A call to nag_sparse_direct_real_gen_lu (f11me) may be followed by calls to the functions:

Example

This example computes the LU factorization of the matrix A, 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 .  
function f11me_example


fprintf('f11me example results\n\n');

% LU factorize sparse A and solve Ax = b
n  = int64(5);
nz = int64(11);
icolzp = int64([1; 3; 5; 7; 9; 12]);
irowix = int64([1; 3;
                  1; 5;
                  2; 3;
                  2; 4;
                  3; 4; 5]);
a = [ 2;  4;
      1; -2;
      1;  1;
     -1;  1;
      1;  2; 3];

% Permute matrix
iprm   = zeros(1, 7*n, 'int64');
spec   = 'M';
% Calculate COLAMD permutation
[iprm, ifail] = f11md( ...
                       spec, n, icolzp, irowix, iprm);

% Factorize
thresh = 1;
nzlmx  = int64(8*nz);
nzlumx = nzlmx;
nzumx  = nzlmx;
[iprm, nzlumx, il, lval, iu, uval, nnzl, nnzu, flop, ifail] = ...
    f11me(...
          n, irowix, a, iprm, thresh, nzlmx, nzlumx, nzumx);

fprintf('\nNumber of nonzeros in factors (excluding unit diagonal)');
fprintf('%8d\n',nnzl + nnzu - n);

disp('Factor elements in lval');
fprintf('%7.2f%7.2f%7.2f%7.2f%7.2f\n',lval(1:10)');
disp('Factor elements in uval');
fprintf('%7.2f%7.2f%7.2f%7.2f\n',uval(1:4)');


f11me example results


Number of nonzeros in factors (excluding unit diagonal)      14
Factor elements in lval
  -2.00  -0.50   4.00   0.50   2.00
   0.50  -1.00   0.50   1.00  -1.00
Factor elements in uval
   1.00   3.00   1.00   1.00

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