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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sparse_real_symm_precon_ichol (f11ja)

Purpose

nag_sparse_real_symm_precon_ichol (f11ja) computes an incomplete Cholesky factorization of a real sparse symmetric matrix, represented in symmetric coordinate storage format. This factorization may be used as a preconditioner in combination with nag_sparse_real_symm_basic_solver (f11ge) or nag_sparse_real_symm_solve_ichol (f11jc).

Syntax

[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = f11ja(nnz, a, irow, icol, lfill, dtol, mic, dscale, ipiv, 'n', n, 'la', la, 'pstrat', pstrat)
[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = nag_sparse_real_symm_precon_ichol(nnz, a, irow, icol, lfill, dtol, mic, dscale, ipiv, 'n', n, 'la', la, 'pstrat', pstrat)

Description

nag_sparse_real_symm_precon_ichol (f11ja) computes an incomplete Cholesky factorization (see Meijerink and Van der Vorst (1977)) of a real sparse symmetric nn by nn matrix AA. It is designed specifically for positive definite matrices, but may also work for some mildly indefinite cases. The factorization is intended primarily for use as a preconditioner with one of the symmetric iterative solvers nag_sparse_real_symm_basic_solver (f11ge) or nag_sparse_real_symm_solve_ichol (f11jc).
The decomposition is written in the form
A = M + R
A=M+R
where
M = PLDLTPT
M=PLDLTPT
and PP is a permutation matrix, LL is lower triangular with unit diagonal elements, DD is diagonal and RR is a remainder matrix.
The amount of fill-in occurring in the factorization can vary from zero to complete fill, and can be controlled by specifying either the maximum level of fill lfill, or the drop tolerance dtol. The factorization may be modified in order to preserve row sums, and the diagonal elements may be perturbed to ensure that the preconditioner is positive definite. Diagonal pivoting may optionally be employed, either with a user-defined ordering, or using the Markowitz strategy (see Markowitz (1957)), which aims to minimize fill-in. For further details see Section [Further Comments].
The sparse matrix AA is represented in symmetric coordinate storage (SCS) format (see Section [Symmetric coordinate storage (SCS) format] in the F11 Chapter Introduction). The array a stores all the nonzero elements of the lower triangular part of AA, while arrays irow and icol store the corresponding row and column indices respectively. Multiple nonzero elements may not be specified for the same row and column index.
The preconditioning matrix MM is returned in terms of the SCS representation of the lower triangular matrix
C = L + D1I.
C=L+D-1-I.

References

Chan T F (1991) Fourier analysis of relaxed incomplete factorization preconditioners SIAM J. Sci. Statist. Comput. 12(2) 668–680
Markowitz H M (1957) The elimination form of the inverse and its application to linear programming Management Sci. 3 255–269
Meijerink J and Van der Vorst H (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix Math. Comput. 31 148–162
Salvini S A and Shaw G J (1995) An evaluation of new NAG Library solvers for large sparse symmetric linear systems NAG Technical Report TR1/95
Van der Vorst H A (1990) The convergence behaviour of preconditioned CG and CG-S in the presence of rounding errors Lecture Notes in Mathematics (eds O Axelsson and L Y Kolotilina) 1457 Springer–Verlag

Parameters

Compulsory Input Parameters

1:     nnz – int64int32nag_int scalar
The number of nonzero elements in the lower triangular part of the matrix AA.
Constraint: 1nnzn × (n + 1) / 21nnzn×(n+1)/2.
2:     a(la) – double array
la, the dimension of the array, must satisfy the constraint la2 × nnzla2×nnz.
The nonzero elements in the lower triangular part of the matrix AA, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The function nag_sparse_real_symm_sort (f11zb) may be used to order the elements in this way.
3:     irow(la) – int64int32nag_int array
4:     icol(la) – int64int32nag_int array
la, the dimension of the array, must satisfy the constraint la2 × nnzla2×nnz.
The row and column indices of the nonzero elements supplied in a.
Constraints:
irow and icol must satisfy these constraints (which may be imposed by a call to nag_sparse_real_symm_sort (f11zb)):
  • 1irow(i)n1irowin and 1icol(i)irow(i)1icoliirowi, for i = 1,2,,nnzi=1,2,,nnz;
  • irow(i1) < irow(i)irowi-1<irowi or irow(i1) = irow(i)irowi-1=irowi and icol(i1) < icol(i)icoli-1<icoli, for i = 2,3,,nnzi=2,3,,nnz.
5:     lfill – int64int32nag_int scalar
If lfill0lfill0 its value is the maximum level of fill allowed in the decomposition (see Section [Control of Fill-in]). A negative value of lfill indicates that dtol will be used to control the fill instead.
6:     dtol – double scalar
If lfill < 0lfill<0, dtol is used as a drop tolerance to control the fill-in (see Section [Control of Fill-in]); otherwise dtol is not referenced.
Constraint: if lfill < 0lfill<0, dtol0.0dtol0.0.
7:     mic – string (length ≥ 1)
Indicates whether or not the factorization should be modified to preserve row sums (see Section [Choice of s]).
mic = 'M'mic='M'
The factorization is modified.
mic = 'N'mic='N'
The factorization is not modified.
Constraint: mic = 'M'mic='M' or 'N''N'.
8:     dscale – double scalar
The diagonal scaling parameter. All diagonal elements are multiplied by the factor (1 + dscale1+dscale) at the start of the factorization. This can be used to ensure that the preconditioner is positive definite. See Section [Choice of s].
9:     ipiv(n) – int64int32nag_int array
n, the dimension of the array, must satisfy the constraint n1n1.
If pstrat = 'U'pstrat='U', then ipiv(i)ipivi must specify the row index of the diagonal element used as a pivot at elimination stage ii. Otherwise ipiv need not be initialized.
Constraint: if pstrat = 'U'pstrat='U', ipiv must contain a valid permutation of the integers on [1,n].

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array ipiv.
nn, the order of the matrix AA.
Constraint: n1n1.
2:     la – int64int32nag_int scalar
Default: The dimension of the arrays a, irow, icol. (An error is raised if these dimensions are not equal.)
The dimension of the arrays a, irow and icol as declared in the (sub)program from which nag_sparse_real_symm_precon_ichol (f11ja) is called. These arrays must be of sufficient size to store both AA (nnz elements) and CC (nnzc elements).
Constraint: la2 × nnzla2×nnz.
3:     pstrat – string (length ≥ 1)
Specifies the pivoting strategy to be adopted.
pstrat = 'N'pstrat='N'
No pivoting is carried out.
pstrat = 'M'pstrat='M'
Diagonal pivoting aimed at minimizing fill-in is carried out, using the Markowitz strategy.
pstrat = 'U'pstrat='U'
Diagonal pivoting is carried out according to the user-defined input value of ipiv.
Default: 'M''M'
Constraint: pstrat = 'N'pstrat='N', 'M''M' or 'U''U'.

Input Parameters Omitted from the MATLAB Interface

iwork liwork

Output Parameters

1:     a(la) – double array
The first nnz elements of a contain the nonzero elements of AA and the next nnzc elements contain the elements of the lower triangular matrix CC. Matrix elements are ordered by increasing row index, and by increasing column index within each row.
2:     irow(la) – int64int32nag_int array
3:     icol(la) – int64int32nag_int array
The row and column indices of the nonzero elements returned in a.
4:     ipiv(n) – int64int32nag_int array
The pivot indices. If ipiv(i) = jipivi=j then the diagonal element in row jj was used as the pivot at elimination stage ii.
5:     istr(n + 1n+1) – int64int32nag_int array
istr(i)istri, for i = 1,2,,ni=1,2,,n, is the starting address in the arrays a, irow and icol of row ii of the matrix CC. istr(n + 1)istrn+1 is the address of the last nonzero element in CC plus one.
6:     nnzc – int64int32nag_int scalar
The number of nonzero elements in the lower triangular matrix CC.
7:     npivm – int64int32nag_int scalar
The number of pivots which were modified during the factorization to ensure that MM was positive definite. The quality of the preconditioner will generally depend on the returned value of npivm. If npivm is large the preconditioner may not be satisfactory. In this case it may be advantageous to call nag_sparse_real_symm_precon_ichol (f11ja) again with an increased value of either lfill or dscale. See also Section [Direct Solution of Systems].
8:     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 < 1n<1,
ornnz < 1nnz<1,
ornnz > n × (n + 1) / 2nnz>n×(n+1)/2,
orla < 2 × nnzla<2×nnz,
ordtol < 0.0dtol<0.0,
ormic'M'mic'M' or 'N''N',
orpstrat'N'pstrat'N', 'M''M' or 'U''U',
orliwork < 2 × la3 × nnz + 7 × n + 1liwork<2×la-3×nnz+7×n+1, and lfill0lfill0,
orliwork < lannz + 7 × n + 1liwork<la-nnz+7×n+1, and lfill < 0lfill<0.
  ifail = 2ifail=2
On entry, the arrays irow and icol fail to satisfy the following constraints:
  • 1irow(i)n1irowin and 1icol(i)irow(i)1icoliirowi, for i = 1,2,,nnzi=1,2,,nnz;
  • irow(i1) < irow(i)irowi-1<irowi, or irow(i1) = irow(i)irowi-1=irowi and icol(i1) < icol(i)icoli-1<icoli, for i = 2,3,,nnzi=2,3,,nnz.
Therefore a nonzero element has been supplied which does not lie in the lower triangular part of AA, is out of order, or has duplicate row and column indices. Call nag_sparse_real_symm_sort (f11zb) to reorder and sum or remove duplicates.
  ifail = 3ifail=3
On entry, pstrat = 'U'pstrat='U', but ipiv does not represent a valid permutation of the integers in [1,n][1,n]. An input value of ipiv is either out of range or repeated.
  ifail = 4ifail=4
la is too small, resulting in insufficient storage space for fill-in elements. The decomposition has been terminated before completion. Either increase la or reduce the amount of fill by setting pstrat = 'M'pstrat='M', reducing lfill, or increasing dtol.
  ifail = 5ifail=5 (nag_sparse_real_symm_sort (f11zb))
A serious error has occurred in an internal call to the specified function. Check all function calls and array sizes. Seek expert help.

Accuracy

The accuracy of the factorization will be determined by the size of the elements that are dropped and the size of any modifications made to the diagonal elements. If these sizes are small then the computed factors will correspond to a matrix close to AA. The factorization can generally be made more accurate by increasing lfill, or by reducing dtol with lfill < 0lfill<0.
If nag_sparse_real_symm_precon_ichol (f11ja) is used in combination with nag_sparse_real_symm_basic_solver (f11ge) or nag_sparse_real_symm_solve_ichol (f11jc), the more accurate the factorization the fewer iterations will be required. However, the cost of the decomposition will also generally increase.

Further Comments

Timing

The time taken for a call to nag_sparse_real_symm_precon_ichol (f11ja) is roughly proportional to (nnzc)2 / n (nnzc) 2/n.

Control of Fill-in

If lfill0lfill0 the amount of fill-in occurring in the incomplete factorization is controlled by limiting the maximum level of fill-in to lfill. The original nonzero elements of AA are defined to be of level 00. The fill level of a new nonzero location occurring during the factorization is defined as
k = max (ke,kc) + 1,
k=max(ke,kc)+1,
where keke is the level of fill of the element being eliminated, and kckc is the level of fill of the element causing the fill-in.
If lfill < 0lfill<0 the fill-in is controlled by means of the drop tolerance dtol. A potential fill-in element aijaij occurring in row ii and column jj will not be included if
|aij| < dtol × sqrt(|aiiajj|).
|aij|<dtol×|aiiajj|.
For either method of control, any elements which are not included are discarded if mic = 'N'mic='N', or subtracted from the diagonal element in the elimination row if mic = 'M'mic='M'.

Choice of Parameters

There is unfortunately no choice of the various algorithmic parameters which is optimal for all types of symmetric matrix, and some experimentation will generally be required for each new type of matrix encountered.
If the matrix AA is not known to have any particular special properties the following strategy is recommended. Start with lfill = 0lfill=0, mic = 'N'mic='N' and dscale = 0.0dscale=0.0. If the value returned for npivm is significantly larger than zero, i.e., a large number of pivot modifications were required to ensure that MM was positive definite, the preconditioner is not likely to be satisfactory. In this case increase either lfill or dscale until npivm falls to a value close to zero. Once suitable values of lfill and dscale have been found try setting mic = 'M'mic='M' to see if any improvement can be obtained by using modified incomplete Cholesky.
nag_sparse_real_symm_precon_ichol (f11ja) is primarily designed for positive definite matrices, but may work for some mildly indefinite problems. If npivm cannot be satisfactorily reduced by increasing lfill or dscale then AA is probably too indefinite for this function.
If AA has non-positive off-diagonal elements, is nonsingular, and has only non-negative elements in its inverse, it is called an ‘M-matrix’. It can be shown that no pivot modifications are required in the incomplete Cholesky factorization of an M-matrix (see Meijerink and Van der Vorst (1977)). In this case a good preconditioner can generally be expected by setting lfill = 0lfill=0, mic = 'M'mic='M' and dscale = 0.0dscale=0.0.
For certain mesh-based problems involving M-matrices it can be shown in theory that setting mic = 'M'mic='M', and choosing dscale appropriately can reduce the order of magnitude of the condition number of the preconditioned matrix as a function of the mesh steplength (see Chan (1991)). In practise this property often holds even with dscale = 0.0dscale=0.0, although an improvement in condition can result from increasing dscale slightly (see Van der Vorst (1990)).
Some illustrations of the application of nag_sparse_real_symm_precon_ichol (f11ja) to linear systems arising from the discretization of two-dimensional elliptic partial differential equations, and to random-valued randomly structured symmetric positive definite linear systems, can be found in Salvini and Shaw (1995).

Direct Solution of positive definite Systems

Although it is not their primary purpose, nag_sparse_real_symm_precon_ichol (f11ja) and nag_sparse_real_symm_precon_ichol_solve (f11jb) may be used together to obtain a direct solution to a symmetric positive definite linear system. To achieve this the call to nag_sparse_real_symm_precon_ichol_solve (f11jb) should be preceded by a complete Cholesky factorization
A = PLDLTPT = M.
A=PLDLTPT=M.
A complete factorization is obtained from a call to nag_sparse_real_symm_precon_ichol (f11ja) with lfill < 0lfill<0 and dtol = 0.0dtol=0.0, provided npivm = 0npivm=0 on exit. A nonzero value of npivm indicates that a is not positive definite, or is ill-conditioned. A factorization with nonzero npivm may serve as a preconditioner, but will not result in a direct solution. It is therefore essential to check the output value of npivm if a direct solution is required.
The use of nag_sparse_real_symm_precon_ichol (f11ja) and nag_sparse_real_symm_precon_ichol_solve (f11jb) as a direct method is illustrated in Section [Example] in (f11jb).

Example

function nag_sparse_real_symm_precon_ichol_example
nz = int64(16);
a = zeros(2*nz, 1);
a(1:nz) = [4; 1; 5; 2; 2; 3; -1; 1; 4; 1; -2; 3; 2; -1; -2; 5];
irow = zeros(2*nz, 1, 'int64');
irow(1:nz) = [int64(1); 2; 2; 3; 4; 4; 5; 5; 5; 6; 6; 6; 7; 7; 7; 7];
icol = zeros(2*nz, 1, 'int64');
icol(1:nz) = [int64(1); 1; 2; 3; 2; 4; 1; 4; 5; 2; 5; 6; 1; 2; 3; 7];
lfill = int64(0);
dtol = 0;
mic = 'N';
dscale = 0;
ipiv = [int64(0);0;0;0;0;0;0];
[aOut, irowOut, icolOut, ipivOut, istr, nnzc, npivm, ifail] = ...
    nag_sparse_real_symm_precon_ichol(nz, a, irow, icol, lfill, dtol, mic, ...
                                      dscale, ipiv)
 

aOut =

    4.0000
    1.0000
    5.0000
    2.0000
    2.0000
    3.0000
   -1.0000
    1.0000
    4.0000
    1.0000
   -2.0000
    3.0000
    2.0000
   -1.0000
   -2.0000
    5.0000
    0.5000
    0.3333
    0.3333
    0.2727
   -0.5455
    0.5238
   -0.2727
    0.2683
    0.6667
    0.5238
    0.2683
    0.3479
   -1.0000
    0.5366
   -0.5345
    0.9046


irowOut =

                    1
                    2
                    2
                    3
                    4
                    4
                    5
                    5
                    5
                    6
                    6
                    6
                    7
                    7
                    7
                    7
                    1
                    2
                    3
                    3
                    4
                    4
                    5
                    5
                    6
                    6
                    6
                    6
                    7
                    7
                    7
                    7


icolOut =

                    1
                    1
                    2
                    3
                    2
                    4
                    1
                    4
                    5
                    2
                    5
                    6
                    1
                    2
                    3
                    7
                    1
                    2
                    2
                    3
                    3
                    4
                    3
                    5
                    2
                    4
                    5
                    6
                    1
                    5
                    6
                    7


ipivOut =

                    3
                    4
                    5
                    6
                    1
                    2
                    7


istr =

                   17
                   18
                   19
                   21
                   23
                   25
                   29
                   33


nnzc =

                   16


npivm =

                    0


ifail =

                    0


function f11ja_example
nz = int64(16);
a = zeros(2*nz, 1);
a(1:nz) = [4; 1; 5; 2; 2; 3; -1; 1; 4; 1; -2; 3; 2; -1; -2; 5];
irow = zeros(2*nz, 1, 'int64');
irow(1:nz) = [int64(1); 2; 2; 3; 4; 4; 5; 5; 5; 6; 6; 6; 7; 7; 7; 7];
icol = zeros(2*nz, 1, 'int64');
icol(1:nz) = [int64(1); 1; 2; 3; 2; 4; 1; 4; 5; 2; 5; 6; 1; 2; 3; 7];
lfill = int64(0);
dtol = 0;
mic = 'N';
dscale = 0;
ipiv = [int64(0);0;0;0;0;0;0];
[aOut, irowOut, icolOut, ipivOut, istr, nnzc, npivm, ifail] = ...
    f11ja(nz, a, irow, icol, lfill, dtol, mic, dscale, ipiv)
 

aOut =

    4.0000
    1.0000
    5.0000
    2.0000
    2.0000
    3.0000
   -1.0000
    1.0000
    4.0000
    1.0000
   -2.0000
    3.0000
    2.0000
   -1.0000
   -2.0000
    5.0000
    0.5000
    0.3333
    0.3333
    0.2727
   -0.5455
    0.5238
   -0.2727
    0.2683
    0.6667
    0.5238
    0.2683
    0.3479
   -1.0000
    0.5366
   -0.5345
    0.9046


irowOut =

                    1
                    2
                    2
                    3
                    4
                    4
                    5
                    5
                    5
                    6
                    6
                    6
                    7
                    7
                    7
                    7
                    1
                    2
                    3
                    3
                    4
                    4
                    5
                    5
                    6
                    6
                    6
                    6
                    7
                    7
                    7
                    7


icolOut =

                    1
                    1
                    2
                    3
                    2
                    4
                    1
                    4
                    5
                    2
                    5
                    6
                    1
                    2
                    3
                    7
                    1
                    2
                    2
                    3
                    3
                    4
                    3
                    5
                    2
                    4
                    5
                    6
                    1
                    5
                    6
                    7


ipivOut =

                    3
                    4
                    5
                    6
                    1
                    2
                    7


istr =

                   17
                   18
                   19
                   21
                   23
                   25
                   29
                   33


nnzc =

                   16


npivm =

                    0


ifail =

                    0



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