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# 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(nz, 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(nz, 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 $n$ by $n$ matrix $A$. 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
 $A=M+R$
where
 $M=PLDLTPT$
and $P$ is a permutation matrix, $L$ is lower triangular with unit diagonal elements, $D$ is diagonal and $R$ 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 Further Comments.
The sparse matrix $A$ is represented in symmetric coordinate storage (SCS) format (see Symmetric coordinate storage (SCS) format in the F11 Chapter Introduction). The array a stores all the nonzero elements of the lower triangular part of $A$, 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 $M$ is returned in terms of the SCS representation of the lower triangular matrix
 $C=L+D-1-I.$

## 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:     $\mathrm{nz}$int64int32nag_int scalar
The number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
2:     $\mathrm{a}\left({\mathbf{la}}\right)$ – double array
The nonzero elements in the lower triangular part of the matrix $A$, 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:     $\mathrm{irow}\left({\mathbf{la}}\right)$int64int32nag_int array
4:     $\mathrm{icol}\left({\mathbf{la}}\right)$int64int32nag_int array
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)):
• $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{irow}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$;
• ${\mathbf{irow}}\left(\mathit{i}-1\right)<{\mathbf{irow}}\left(\mathit{i}\right)$ or ${\mathbf{irow}}\left(\mathit{i}-1\right)={\mathbf{irow}}\left(\mathit{i}\right)$ and ${\mathbf{icol}}\left(\mathit{i}-1\right)<{\mathbf{icol}}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,\dots ,{\mathbf{nz}}$.
5:     $\mathrm{lfill}$int64int32nag_int scalar
If ${\mathbf{lfill}}\ge 0$ its value is the maximum level of fill allowed in the decomposition (see Control of Fill-in). A negative value of lfill indicates that dtol will be used to control the fill instead.
6:     $\mathrm{dtol}$ – double scalar
If ${\mathbf{lfill}}<0$, dtol is used as a drop tolerance to control the fill-in (see Control of Fill-in); otherwise dtol is not referenced.
Constraint: if ${\mathbf{lfill}}<0$, ${\mathbf{dtol}}\ge 0.0$.
7:     $\mathrm{mic}$ – string (length ≥ 1)
Indicates whether or not the factorization should be modified to preserve row sums (see Choice of s).
${\mathbf{mic}}=\text{'M'}$
The factorization is modified.
${\mathbf{mic}}=\text{'N'}$
The factorization is not modified.
Constraint: ${\mathbf{mic}}=\text{'M'}$ or $\text{'N'}$.
8:     $\mathrm{dscale}$ – double scalar
The diagonal scaling parameter. All diagonal elements are multiplied by the factor ($1+{\mathbf{dscale}}$) at the start of the factorization. This can be used to ensure that the preconditioner is positive definite. See Choice of s.
9:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
If ${\mathbf{pstrat}}=\text{'U'}$, then ${\mathbf{ipiv}}\left(i\right)$ must specify the row index of the diagonal element used as a pivot at elimination stage $i$. Otherwise ipiv need not be initialized.
Constraint: if ${\mathbf{pstrat}}=\text{'U'}$, ipiv must contain a valid permutation of the integers on [1,n].

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array ipiv.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathrm{la}$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. these arrays must be of sufficient size to store both $A$ (nz elements) and $C$ (nnzc elements).
Constraint: ${\mathbf{la}}\ge 2×{\mathbf{nz}}$.
3:     $\mathrm{pstrat}$ – string (length ≥ 1)
Default: $\text{'M'}$
Specifies the pivoting strategy to be adopted.
${\mathbf{pstrat}}=\text{'N'}$
No pivoting is carried out.
${\mathbf{pstrat}}=\text{'M'}$
Diagonal pivoting aimed at minimizing fill-in is carried out, using the Markowitz strategy.
${\mathbf{pstrat}}=\text{'U'}$
Diagonal pivoting is carried out according to the user-defined input value of ipiv.
Constraint: ${\mathbf{pstrat}}=\text{'N'}$, $\text{'M'}$ or $\text{'U'}$.

### Output Parameters

1:     $\mathrm{a}\left({\mathbf{la}}\right)$ – double array
The first nz elements of a contain the nonzero elements of $A$ and the next nnzc elements contain the elements of the lower triangular matrix $C$. Matrix elements are ordered by increasing row index, and by increasing column index within each row.
2:     $\mathrm{irow}\left({\mathbf{la}}\right)$int64int32nag_int array
3:     $\mathrm{icol}\left({\mathbf{la}}\right)$int64int32nag_int array
The row and column indices of the nonzero elements returned in a.
4:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
The pivot indices. If ${\mathbf{ipiv}}\left(i\right)=j$ then the diagonal element in row $j$ was used as the pivot at elimination stage $i$.
5:     $\mathrm{istr}\left({\mathbf{n}}+1\right)$int64int32nag_int array
${\mathbf{istr}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$, is the starting address in the arrays a, irow and icol of row $i$ of the matrix $C$. ${\mathbf{istr}}\left({\mathbf{n}}+1\right)$ is the address of the last nonzero element in $C$ plus one.
6:     $\mathrm{nnzc}$int64int32nag_int scalar
The number of nonzero elements in the lower triangular matrix $C$.
7:     $\mathrm{npivm}$int64int32nag_int scalar
The number of pivots which were modified during the factorization to ensure that $M$ 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 Direct Solution of Systems.
8:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{nz}}<1$, or ${\mathbf{nz}}>{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$, or ${\mathbf{la}}<2×{\mathbf{nz}}$, or ${\mathbf{dtol}}<0.0$, or ${\mathbf{mic}}\ne \text{'M'}$ or $\text{'N'}$, or ${\mathbf{pstrat}}\ne \text{'N'}$, $\text{'M'}$ or $\text{'U'}$, or $\mathit{liwork}<2×{\mathbf{la}}-3×{\mathbf{nz}}+7×{\mathbf{n}}+1$, and ${\mathbf{lfill}}\ge 0$, or $\mathit{liwork}<{\mathbf{la}}-{\mathbf{nz}}+7×{\mathbf{n}}+1$, and ${\mathbf{lfill}}<0$.
${\mathbf{ifail}}=2$
On entry, the arrays irow and icol fail to satisfy the following constraints:
• $1\le {\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(i\right)\le {\mathbf{irow}}\left(i\right)$, for $i=1,2,\dots ,{\mathbf{nz}}$;
• ${\mathbf{irow}}\left(i-1\right)<{\mathbf{irow}}\left(i\right)$, or ${\mathbf{irow}}\left(i-1\right)={\mathbf{irow}}\left(i\right)$ and ${\mathbf{icol}}\left(i-1\right)<{\mathbf{icol}}\left(i\right)$, for $i=2,3,\dots ,{\mathbf{nz}}$.
Therefore a nonzero element has been supplied which does not lie in the lower triangular part of $A$, 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.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{pstrat}}=\text{'U'}$, but ipiv does not represent a valid permutation of the integers in $\left[1,{\mathbf{n}}\right]$. An input value of ipiv is either out of range or repeated.
${\mathbf{ifail}}=4$
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 ${\mathbf{pstrat}}=\text{'M'}$, reducing lfill, or increasing dtol.
${\mathbf{ifail}}=5$ (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.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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 $A$. The factorization can generally be made more accurate by increasing lfill, or by reducing dtol with ${\mathbf{lfill}}<0$.
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 ${\left({\mathbf{nnzc}}\right)}^{2}/{\mathbf{n}}$.

### Control of Fill-in

If ${\mathbf{lfill}}\ge 0$ 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 $A$ are defined to be of level $0$. The fill level of a new nonzero location occurring during the factorization is defined as
 $k=maxke,kc+1,$
where ${k}_{\mathrm{e}}$ is the level of fill of the element being eliminated, and ${k}_{\mathrm{c}}$ is the level of fill of the element causing the fill-in.
If ${\mathbf{lfill}}<0$ the fill-in is controlled by means of the drop tolerance dtol. A potential fill-in element ${a}_{ij}$ occurring in row $i$ and column $j$ will not be included if
 $aij
For either method of control, any elements which are not included are discarded if ${\mathbf{mic}}=\text{'N'}$, or subtracted from the diagonal element in the elimination row if ${\mathbf{mic}}=\text{'M'}$.

### Choice of Arguments

There is unfortunately no choice of the various algorithmic arguments 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 $A$ is not known to have any particular special properties the following strategy is recommended. Start with ${\mathbf{lfill}}=0$, ${\mathbf{mic}}=\text{'N'}$ and ${\mathbf{dscale}}=0.0$. If the value returned for npivm is significantly larger than zero, i.e., a large number of pivot modifications were required to ensure that $M$ 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 ${\mathbf{mic}}=\text{'M'}$ 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 $A$ is probably too indefinite for this function.
If $A$ 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 ${\mathbf{lfill}}=0$, ${\mathbf{mic}}=\text{'M'}$ and ${\mathbf{dscale}}=0.0$.
For certain mesh-based problems involving M-matrices it can be shown in theory that setting ${\mathbf{mic}}=\text{'M'}$, 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 ${\mathbf{dscale}}=0.0$, 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 complete factorization is obtained from a call to nag_sparse_real_symm_precon_ichol (f11ja) with ${\mathbf{lfill}}<0$ and ${\mathbf{dtol}}=0.0$, provided ${\mathbf{npivm}}=0$ 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 Example in nag_sparse_real_symm_precon_ichol_solve (f11jb).

## Example

This example reads in a symmetric sparse matrix $A$ and calls nag_sparse_real_symm_precon_ichol (f11ja) to compute an incomplete Cholesky factorization. It then outputs the nonzero elements of both $A$ and $C=L+{D}^{-1}-I$.
The call to nag_sparse_real_symm_precon_ichol (f11ja) has ${\mathbf{lfill}}=0$, ${\mathbf{mic}}=\text{'N'}$, ${\mathbf{dscale}}=0.0$ and ${\mathbf{pstrat}}=\text{'M'}$, giving an unmodified zero-fill factorization of an unperturbed matrix, with Markowitz diagonal pivoting.
```function f11ja_example

fprintf('f11ja example results\n\n');

% Sparse matrix A
n    = int64(7);
nz   = int64(16);
a    = zeros(1000,1);
irow = zeros(1000,1,'int64');
icol = irow;

a(1:16)    = [4  1  5  2  2  3 -1  1  4  1 -2  3  2 -1 -2  5];
irow(1:16) = [1  2  2  3  4  4  5  5  5  6  6  6  7  7  7  7];
icol(1:16) = [1  1  2  3  2  4  1  4  5  2  5  6  1  2  3  7];

% Incomplete Cholesky factorization, zero fill
lfill  = int64(0);
dtol   = 0;
mic    = 'N';
dscale = 0;
ipiv   = zeros(n, 1, 'int64');

[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = ...
f11ja( ...
nz, a, irow, icol, lfill, dtol, mic, dscale, ipiv);

% Display details
fprintf(' Original Matrix\n');
inda = 1:nz;
amat = [inda' a(inda) irow(inda) icol(inda)];
fprintf('n     = %4d\n', n);
fprintf('nz    = %4d\n', nz);
fprintf('\n            a        irow    icol\n');
fprintf('%4d %11.4f%8d%8d\n',amat');

fprintf('\n Factorization\n');
inda = nz+1:nz+nnzc;
amat = [inda' a(inda) irow(inda) icol(inda)];
fprintf('n     = %4d\n', n);
fprintf('nz    = %4d\n', nnzc);
fprintf('npivm = %4d\n', npivm);
fprintf('\n            a        irow    icol\n');
fprintf('%4d %11.4f%8d%8d\n',amat');
fprintf('\n   i      ipiv(i)\n');
fprintf('%4d %8d\n',[[1:n]'; ipiv]);

```
```f11ja example results

Original Matrix
n     =    7
nz    =   16

a        irow    icol
1      4.0000       1       1
2      1.0000       2       1
3      5.0000       2       2
4      2.0000       3       3
5      2.0000       4       2
6      3.0000       4       4
7     -1.0000       5       1
8      1.0000       5       4
9      4.0000       5       5
10      1.0000       6       2
11     -2.0000       6       5
12      3.0000       6       6
13      2.0000       7       1
14     -1.0000       7       2
15     -2.0000       7       3
16      5.0000       7       7

Factorization
n     =    7
nz    =   16
npivm =    0

a        irow    icol
17      1.0000       1       1
18      0.0000       2       2
19      0.0000       3       2
20      0.0000       3       3
21     -1.0000       4       3
22      1.0000       4       4
23      0.0000       5       3
24      0.0000       5       5
25      1.0000       6       2
26      1.0000       6       4
27      0.0000       6       5
28      0.0000       6       6
29     -1.0000       7       1
30      1.0000       7       5
31     -1.0000       7       6
32      1.0000       7       7

i      ipiv(i)
1        2
3        4
5        6
7        3
4        5
6        1
2        7
```

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