f11dac computes an incomplete $LU$ factorization of a real sparse nonsymmetric matrix, represented in coordinate storage format. This factorization may be used as a preconditioner in combination with f11dcc.
and $L$ is lower triangular with unit diagonal elements, $D$ is diagonal, $U$ is upper triangular with unit diagonals, $P$ and $Q$ are permutation matrices, 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 argument pstrat defines the pivoting strategy to be used. The options currently available are no pivoting, user-defined pivoting, partial pivoting by columns for stability, and complete pivoting by rows for sparsity and by columns for stability. The factorization may optionally be modified to preserve the row-sums of the original matrix.
The sparse matrix $A$ is represented in coordinate storage (CS) format (see Section 2.1.2 in the F11 Chapter Introduction). The array a stores all the nonzero elements of the matrix $A$, 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 CS representation of the matrix
$$C=L+{D}^{\mathrm{-1}}+U-2I\text{.}$$
Further algorithmic details are given in Section 9.3.
4References
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
Meijerink J and Van der Vorst H (1981) Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems J. Comput. Phys.44 134–155
Salvini S A and Shaw G J (1996) An evaluation of new NAG Library solvers for large sparse unsymmetric linear systems NAG Technical Report TR2/96
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 1$.
2: $\mathbf{nnz}$ – IntegerInput
On entry: the number of nonzero elements in the matrix $A$.
On entry: the nonzero elements in 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 f11zac may be used to order the elements in this way.
On exit: the first nnz entries of a contain the nonzero elements of $A$ and the next nnzc entries contain the elements of the matrix $C$. Matrix elements are ordered by increasing row index, and by increasing column index within each row.
4: $\mathbf{la}$ – Integer *Input/Output
On entry: the
second
dimension of the arrays a, irow and icol.
These arrays must be of sufficient size to store both $A$ (nnz elements) and $C$ (nnzc elements); for this reason the length of the arrays may be changed internally by calls to realloc. It is, therefore, imperative that these arrays are allocated using NAG_ALLOC and not declared as automatic arrays
On exit: if internal allocation has taken place then la is set to ${\mathbf{nnz}}+{\mathbf{nnzc}}$, otherwise it remains unchanged.
On entry: the row and column indices of the nonzero elements supplied in a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to f11zac):;
$1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{n}}$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$;
${\mathbf{irow}}\left[\mathit{i}-1\right]<{\mathbf{irow}}\left[\mathit{i}\right]$ 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 $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
On exit: the row and column indices of the nonzero elements returned in a.
7: $\mathbf{lfill}$ – IntegerInput
On entry: if ${\mathbf{lfill}}\ge 0$ its value is the maximum level of fill allowed in the decomposition (see Section 9.2). A negative value of lfill indicates that dtol will be used to control the fill instead.
8: $\mathbf{dtol}$ – doubleInput
On entry: if ${\mathbf{lfill}}<0$ then dtol is used as a drop tolerance to control the fill-in (see Section 9.2); otherwise dtol is not referenced.
Constraint:
if ${\mathbf{lfill}}<0$, ${\mathbf{dtol}}\ge 0.0$.
9: $\mathbf{pstrat}$ – Nag_SparseNsym_PivInput
On entry: specifies the pivoting strategy to be adopted as follows:
if ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_NoPiv}$, no pivoting is carried out;
if ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_UserPiv}$, pivoting is carried out according to the user-defined input value of ipivp and ipivq;
if ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_PartialPiv}$, partial pivoting by columns for stability is carried out;
if ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_CompletePiv}$, complete pivoting by rows for sparsity, and by columns for stability, is carried out.
On entry: if ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_UserPiv}$, ${\mathbf{ipivp}}\left[k-1\right]$ and ${\mathbf{ipivq}}\left[k-1\right]$ must specify the row and column indices of the element used as a pivot at elimination stage $k$. Otherwise ipivp and ipivq need not be initialized.
Constraint:
if ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_UserPiv}$, ipivp and ipivq must both hold valid permutations of the integers on $[1,{\mathbf{n}}]$.
On exit: the pivot indices. If ${\mathbf{ipivp}}\left[k-1\right]=i$ and ${\mathbf{ipivq}}\left[k-1\right]=j$ then the element in row $i$ and column $j$ was used as the pivot at elimination stage $k$.
On exit: ${\mathbf{istr}}\left[\mathit{i}-1\right]-1$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ is the index of arrays a, irow and icol where row $i$ of the matrix $C$ starts. ${\mathbf{istr}}\left[{\mathbf{n}}\right]-1$ is the address of the last nonzero element in $C$ plus one.
On exit: ${\mathbf{idiag}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ holds the index in the arrays a, irow and icol which holds the diagonal element in row $\mathit{i}$ of the matrix $C$.
15: $\mathbf{nnzc}$ – Integer *Output
On exit: the number of nonzero elements in the matrix $C$.
16: $\mathbf{npivm}$ – Integer *Output
On exit: if ${\mathbf{npivm}}>0$ it gives the number of pivots which were modified during the factorization to ensure that $M$ exists.
If ${\mathbf{npivm}}=\mathrm{-1}$ no pivot modifications were required, but a local restart occurred (Section 9.4). 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 f11dac again with an increased value of lfill, a reduced value of dtol, or ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_CompletePiv}$.
17: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_2_INT_ARG_LT
On entry, ${\mathbf{la}}=\u27e8\mathit{\text{value}}\u27e9$ while ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$. These arguments must satisfy ${\mathbf{la}}\ge 2\times {\mathbf{nnz}}$.
On entry, ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}\text{.}$.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
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.
NE_INVALID_ROWCOL_PIVOT
On entry, ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_UserPiv}$, but one or both of the arrays ipivp and ipivq does not represent a valid permutation of the integers in $[1,{\mathbf{n}}]$. An input value of ipivp or ipivq is either out of range or repeated.
NE_NONSYMM_MATRIX_DUP
A nonzero matrix element has been supplied which does not lie within the matrix $A$, is out of order or has duplicate row and column indices, i.e., one or more of the following constraints has been violated:
$1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$, $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{n}}$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$.
${\mathbf{irow}}\left[\mathit{i}-1\right]<{\mathbf{irow}}\left[\mathit{i}\right]$, 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 $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
Call f11zac to reorder and sum or remove duplicates.
NE_REAL_INT_ARG_CONS
On entry, ${\mathbf{dtol}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{lfill}}=\u27e8\mathit{\text{value}}\u27e9$. These arguments must satisfy ${\mathbf{dtol}}\ge 0.0$ if ${\mathbf{lfill}}<0$.
7Accuracy
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 pivot 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 f11dac is used in combination with f11dcc, the more accurate the factorization the fewer iterations will be required. However, the cost of the decomposition will also generally increase.
8Parallelism and Performance
f11dac is not threaded in any implementation.
9Further Comments
9.1Timing
The time taken for a call to f11dac is roughly proportional to ${{\mathbf{nnzc}}}^{2}/{\mathbf{n}}$.
9.2Control 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:
where ${k}_{e}$ is the level of fill of the element being eliminated, and ${k}_{c}$ 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 tolerancedtol. A potential fill-in element ${a}_{ij}$ occurring in row $i$ and column $j$ will not be included if:
where $\alpha $ is the maximum absolute value element in the matrix $A$.
For either method of control, any elements which are not included are discarded unless ${\mathbf{milu}}=\mathrm{Nag\_SparseNsym\_ModFact}$, in which case their contributions are subtracted from the pivot element in the relevant elimination row, to preserve the row-sums of the original matrix.
Should the factorization process break down a local restart process is implemented as described in Section 9.4. This will affect the amount of fill present in the final factorization.
9.3Algorithmic Details
The factorization is constructed row by row. At each elimination stage a row index is chosen. In the case of complete pivoting this index is chosen in order to reduce fill-in. Otherwise the rows are treated in the order given, or some user-defined order.
The chosen row is copied from the original matrix $A$ and modified according to those previous elimination stages which affect it. During this process any fill-in elements are either dropped or kept according to the values of lfill or dtol. In the case of a modified factorization (${\mathbf{milu}}=\mathrm{Nag\_SparseNsym\_ModFact}$) the sum of the dropped terms for the given row is stored.
Finally the pivot element for the row is chosen and the multipliers are computed for this elimination stage. For partial or complete pivoting the pivot element is chosen in the interests of stability as the element of largest absolute value in the row. Otherwise the pivot element is chosen in the order given, or some user-defined order.
If the factorization breaks down because the chosen pivot element is zero, or there is no nonzero pivot available, a local restart recovery process is implemented. The modification of the given pivot row according to previous elimination stages is repeated, but this time keeping all fill. Note that in this case the final factorization will include more fill than originally specified by the user-supplied value of lfill or dtol. The local restart usually results in a suitable nonzero pivot arising. The original criteria for dropping fill-in elements is then resumed for the next elimination stage (hence the local nature of the restart process). Should this restart process also fail to produce a nonzero pivot element an arbitrary unit pivot is introduced in an arbitrarily chosen column. f11dac returns an integer argument npivm which gives the number of these arbitrary unit pivots introduced. If no pivots were modified but local restarts occurred ${\mathbf{npivm}}=\mathrm{-1}$ is returned.
9.4Choice of Parameters
There is unfortunately no choice of the various algorithmic arguments which is optimal for all types of 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 ${\mathbf{lfill}}=0$ and ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_CompletePiv}$. If the value returned for npivm is significantly larger than zero, i.e., a large number of pivot modifications were required to ensure that $M$ existed, the preconditioner is not likely to be satisfactory. In this case increase lfill until npivm falls to a value close to zero.
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 $LU$ factorization of an M-matrix (Meijerink and Van der Vorst (1977)). In this case a good preconditioner can generally be expected by setting ${\mathbf{lfill}}=0$, ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_NoPiv}$ and ${\mathbf{milu}}=\mathrm{Nag\_SparseNsym\_ModFact}$.
Some illustrations of the application of f11dac to linear systems arising from the discretization of two-dimensional elliptic partial differential equations, and to random-valued randomly structured linear systems, can be found in Salvini and Shaw (1996).
9.5Direct Solution of Sparse Linear Systems
Although it is not their primary purpose f11dac and f11dbc may be used together to obtain a direct solution to a nonsingular sparse linear system. To achieve this the call to f11dbc should be preceded by a complete$LU$ factorization
$$A=PLDUQ=M\text{.}$$
A complete factorization is obtained from a call to f11dac with ${\mathbf{lfill}}<0$ and ${\mathbf{dtol}}=0.0$, provided ${\mathbf{npivm}}\le 0$ on exit. A positive value of npivm indicates that $A$ is singular, or ill-conditioned. A factorization with positive 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 f11dac and f11dbc as a direct method is illustrated in Section 10 in f11dbc.
10Example
This example program reads in a sparse matrix $A$ and calls f11dac to compute an incomplete $LU$ factorization. It then outputs the nonzero elements of both $A$ and $C=L+{D}^{\mathrm{-1}}+U-2I$.
The call to f11dac has ${\mathbf{lfill}}=0$, and ${\mathbf{pstrat}}=\mathrm{Nag\_SparseNsym\_CompletePiv}$, giving an unmodified zero-fill $LU$ factorization, with row pivoting for sparsity and column pivoting for stability.