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NAG Toolbox

NAG Toolbox: nag_sparse_direct_real_gen_lu (f11me)

Purpose

nag_sparse_direct_real_gen_lu (f11me) computes the LU 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 AA, nag_sparse_direct_real_gen_lu (f11me) computes an LU LU  factorization of AA with partial pivoting, Pr A Pc = LU Pr A Pc = LU , where PrPr is a row permutation matrix (computed by nag_sparse_direct_real_gen_lu (f11me)), PcPc is a (supplied) column permutation matrix, LL is unit lower triangular and UU is upper triangular. The column permutation matrix, PcPc, must be computed by a prior call to nag_sparse_direct_real_gen_setup (f11md). The matrix AA must be presented in the column permuted, compressed column (Harwell–Boeing) format.
The LU 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 LL and UU matrices. The minimum required dimensions of these arrays cannot be given as a simple function of the size parameters (order and number of nonzero values) of the matrix AA. 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
nn, the order of the matrix AA.
Constraint: n0n0.
2:     irowix( : :) – int64int32nag_int array
Note: the dimension of the array irowix must be at least nnznnz, the number of nonzeros of the sparse matrix AA.
The row index array of sparse matrix AA.
3:     a( : :) – double array
Note: the dimension of the array a must be at least nnznnz, the number of nonzeros of the sparse matrix AA.
The array of nonzero values in the sparse matrix AA.
4:     iprm(7 × n7×n) – int64int32nag_int array
Contains the column permutation which defines the permutation PcPc and associated data structures as computed by function nag_sparse_direct_real_gen_setup (f11md).
5:     thresh – double scalar
The diagonal pivoting threshold, tt. At step jj of the Gaussian elimination, if |Ajj|t(maxij |Aij|)|Ajj|t(maxij|Aij|), use AjjAjj as a pivot, otherwise use maxij |Aij|maxij|Aij|. A value of t = 1t=1 corresponds to partial pivoting, a value of t = 0t=0 corresponds to always choosing the pivot on the diagonal (unless it is zero).
Constraint: 0.0thresh1.00.0thresh1.0.
6:     nzlmx – int64int32nag_int scalar
Indicates the available size of array il. The dimension of il should be at least 7 × n + nzlmx + 47×n+nzlmx+4. A good range for nzlmx that works for many problems is nnznnz to 8 × nnz8×nnz, where nnznnz is the number of nonzeros in the sparse matrix AA. If, on exit, ifail = 2ifail=2, the given nzlmx was too small and you should attempt to provide more storage and call the function again.
Constraint: nzlmx1nzlmx1.
7:     nzlumx – int64int32nag_int scalar
Indicates the available size of array lval. The dimension of lval should be at least nzlumx.
Constraint: nzlumx1nzlumx1.
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 + 12×n+nzumx+1 and the dimension of uval should be at least nzumx. A good range for nzumx that works for many problems is nnznnz to 8 × nnz8×nnz, where nnznnz is the number of nonzeros in the sparse matrix AA. If, on exit, ifail = 3ifail=3, the given nzumx was too small and you should attempt to provide more storage and call the function again.
Constraint: nzumx1nzumx1.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     iprm(7 × n7×n) – int64int32nag_int array
Part of the array is modified to record the row permutation PrPr determined by pivoting.
2:     nzlumx – int64int32nag_int scalar
If ifail = 4ifail=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:     il(7 × n + nzlmx + 47×n+nzlmx+4 ) – int64int32nag_int array
Encapsulates the sparsity pattern of matrix LL.
4:     lval(nzlumx) – double array
Records the nonzero values of matrix LL and some of the nonzero values of matrix UU.
5:     iu(2 × n + nzumx + 12×n+nzumx+1) – int64int32nag_int array
Encapsulates the sparsity pattern of matrix UU.
6:     uval(nzumxnzumx) – double array
Records some of the nonzero values of matrix UU.
7:     nnzl – int64int32nag_int scalar
The number of nonzero values in the matrix LL.
8:     nnzu – int64int32nag_int scalar
The number of nonzero values in the matrix UU.
9:     flop – double scalar
The number of floating point operations performed.
10:   ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,n < 0n<0,
ornzlmx < 1nzlmx<1,
ornzlumx < 1nzlumx<1,
ornzumx < 1nzumx<1,
orthresh < 0.0thresh<0.0,
orthresh > 1.0thresh>1.0.
  ifail = 2ifail=2
nzlmxnzlmx was not large enough. You should repeat the call with a larger value of nzlmxnzlmx, providing more storage for the output array ilil.
  ifail = 3ifail=3
nzumxnzumx was not large enough. You should repeat the call with a larger value of nzumxnzumx, providing more storage for the output arrays iuiu and uvaluval.
  ifail = 4ifail=4
nzlumxnzlumx was not large enough. You should repeat the call with the value of nzlumxnzlumx returned on exit, providing more storage for the output array lvallval.
  ifail = 5ifail=5
The matrix AA is singular and no factorization will be attempted.
  ifail = 301ifail=301
Unable to allocate required internal workspace.

Accuracy

The computed factors LL and UU are the exact factors of a perturbed matrix A + EA+E, where
|E| c (n) ε |L| |U| ,
| E | c ( n ) ε |L| |U| ,
c(n)c(n) is a modest linear function of nn, and εε 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 AA.
A call to nag_sparse_direct_real_gen_lu (f11me) may be followed by calls to the functions:

Example

function nag_sparse_direct_real_gen_lu_example
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];
iprm = zeros(1, 7*n, 'int64');
spec = 'M';
thresh = 1;
nzlmx = int64(8*nz);
nzlumx = int64(8*nz);
nzumx = int64(8*nz);

% Calculate COLAMD permutation
[iprm, ifail] = ...
    nag_sparse_direct_real_gen_setup(spec, n, icolzp, irowix, iprm);

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

fprintf('\nNumber of nonzeros in factors (excluding unit diagonal)\n%10d\n', nnzl+nnzu-n);
fprintf('\nFactor elements in lval\n');
fprintf('%6.2f',lval(1:10));
fprintf('\n');
fprintf('\nFactor elements in uval\n');
fprintf('%6.2f',uval(1:4));
fprintf('\n');
 

iprm =

  Columns 1 through 4

                    1                    0                    4                    3

  Columns 5 through 8

                    2                    4                    3                    1

  Columns 9 through 12

                    2                    0                    2                    0

  Columns 13 through 16

                    8                    6                    4                    4

  Columns 17 through 20

                    2                   11                    8                    6

  Columns 21 through 24

                    1                    2                    3                    4

  Columns 25 through 28

                    5                    2                    2                    2

  Columns 29 through 32

                    2                    1                    1                    1

  Columns 33 through 35

                    1                    2                    0


nzlumx =

                   88


nnzl =

                    9


nnzu =

                   10


flop =

    19


ifail =

                    0


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

function f11me_example
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];
iprm = zeros(1, 7*n, 'int64');
spec = 'M';
thresh = 1;
nzlmx = int64(8*nz);
nzlumx = int64(8*nz);
nzumx = int64(8*nz);

% Calculate COLAMD permutation
[iprm, ifail] = f11md(spec, n, icolzp, irowix, iprm);

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

fprintf('\nNumber of nonzeros in factors (excluding unit diagonal)\n%10d\n', nnzl+nnzu-n);
fprintf('\nFactor elements in lval\n');
fprintf('%6.2f',lval(1:10));
fprintf('\n');
fprintf('\nFactor elements in uval\n');
fprintf('%6.2f',uval(1:4));
fprintf('\n');
 

iprm =

  Columns 1 through 4

                    1                    0                    4                    3

  Columns 5 through 8

                    2                    4                    3                    1

  Columns 9 through 12

                    2                    0                    2                    0

  Columns 13 through 16

                    8                    6                    4                    4

  Columns 17 through 20

                    2                   11                    8                    6

  Columns 21 through 24

                    1                    2                    3                    4

  Columns 25 through 28

                    5                    2                    2                    2

  Columns 29 through 32

                    2                    1                    1                    1

  Columns 33 through 35

                    1                    2                    0


nzlumx =

                   88


nnzl =

                    9


nnzu =

                   10


flop =

    19


ifail =

                    0


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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