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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).

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$n$ by n$n$ matrix A$A$. 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

where

and P$P$ is a permutation matrix, L$L$ is lower triangular with unit diagonal elements, D$D$ is diagonal and R$R$ is a remainder matrix.

A = M + R
$$A=M+R$$ |

M = PLDL ^{T}P^{T}
$$M=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$$ |

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 A$A$ 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 A$A$, 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$M$ is returned in terms of the SCS representation of the lower triangular matrix

C = L + D ^{ − 1} − I.
$$C=L+{D}^{-1}-I\text{.}$$ |

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

- 1: nnz – int64int32nag_int scalar
- The number of nonzero elements in the lower triangular part of the matrix A$A$.
- 2: a(la) – double array
- la, the dimension of the array, must satisfy the constraint la ≥ 2 × nnz${\mathbf{la}}\ge 2\times {\mathbf{nnz}}$.The nonzero elements in the lower triangular part of the matrix A$A$, 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: irow(la) – int64int32nag_int array
- 4: icol(la) – int64int32nag_int array
- la, the dimension of the array, must satisfy the constraint la ≥ 2 × nnz${\mathbf{la}}\ge 2\times {\mathbf{nnz}}$.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)):- 1 ≤ irow(i) ≤ n$1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ and 1 ≤ icol(i) ≤ irow(i)$1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{irow}}\left(\mathit{i}\right)$, for i = 1,2, … ,nnz$\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$;
- irow(i − 1) < irow(i)${\mathbf{irow}}\left(\mathit{i}-1\right)<{\mathbf{irow}}\left(\mathit{i}\right)$ or irow(i − 1) = irow(i)${\mathbf{irow}}\left(\mathit{i}-1\right)={\mathbf{irow}}\left(\mathit{i}\right)$ and icol(i − 1) < icol(i)${\mathbf{icol}}\left(\mathit{i}-1\right)<{\mathbf{icol}}\left(\mathit{i}\right)$, for i = 2,3, … ,nnz$\mathit{i}=2,3,\dots ,{\mathbf{nnz}}$.

- 5: lfill – int64int32nag_int scalar
- If lfill ≥ 0${\mathbf{lfill}}\ge 0$ 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 < 0${\mathbf{lfill}}<0$, dtol is used as a drop tolerance to control the fill-in (see Section [Control of Fill-in]); otherwise dtol is not referenced.
- 7: mic – string (length ≥ 1)
- Indicates whether or not the factorization should be modified to preserve row sums (see Section [Choice of s]).
- 8: dscale – double scalar
- The diagonal scaling parameter. All diagonal elements are multiplied by the factor (1 + dscale$1+{\mathbf{dscale}}$) 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

- 1: n – int64int32nag_int scalar
*Default*: The dimension of the array ipiv.n$n$, the order of the matrix A$A$.- 2: la – int64int32nag_int scalar
*Default*: The dimension of the arrays a, irow, icol. (An error is raised if these dimensions are not equal.)- 3: pstrat – string (length ≥ 1)
- Specifies the pivoting strategy to be adopted.
- pstrat = 'N'${\mathbf{pstrat}}=\text{'N'}$
- No pivoting is carried out.
- pstrat = 'M'${\mathbf{pstrat}}=\text{'M'}$
- Diagonal pivoting aimed at minimizing fill-in is carried out, using the Markowitz strategy.
- pstrat = 'U'${\mathbf{pstrat}}=\text{'U'}$
- Diagonal pivoting is carried out according to the user-defined input value of ipiv.

*Default*: 'M'$\text{'M'}$

- iwork liwork

- 1: a(la) – double array
- 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) = j${\mathbf{ipiv}}\left(i\right)=j$ then the diagonal element in row j$j$ was used as the pivot at elimination stage i$i$.
- 5: istr(n + 1${\mathbf{n}}+1$) – int64int32nag_int array
- istr(i)${\mathbf{istr}}\left(\mathit{i}\right)$, for i = 1,2, … ,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$, is the starting address in the arrays a, irow and icol of row i$i$ of the matrix C$C$. istr(n + 1)${\mathbf{istr}}\left({\mathbf{n}}+1\right)$ is the address of the last nonzero element in C$C$ plus one.
- 6: nnzc – int64int32nag_int scalar
- The number of nonzero elements in the lower triangular matrix C$C$.
- 7: npivm – int64int32nag_int scalar
- The number of pivots which were modified during the factorization to ensure that M$M$ 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 Section [Direct Solution of Systems].
- 8: ifail – int64int32nag_int scalar
- ifail = 0${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Errors or warnings detected by the function:

On entry, n < 1${\mathbf{n}}<1$, or nnz < 1${\mathbf{nnz}}<1$, or nnz > n × (n + 1) / 2${\mathbf{nnz}}>{\mathbf{n}}\times ({\mathbf{n}}+1)/2$, or la < 2 × nnz${\mathbf{la}}<2\times {\mathbf{nnz}}$, or dtol < 0.0${\mathbf{dtol}}<0.0$, or mic ≠ 'M'${\mathbf{mic}}\ne \text{'M'}$ or 'N'$\text{'N'}$, or pstrat ≠ 'N'${\mathbf{pstrat}}\ne \text{'N'}$, 'M'$\text{'M'}$ or 'U'$\text{'U'}$, or liwork < 2 × la − 3 × nnz + 7 × n + 1$\mathit{liwork}<2\times {\mathbf{la}}-3\times {\mathbf{nnz}}+7\times {\mathbf{n}}+1$, and lfill ≥ 0${\mathbf{lfill}}\ge 0$, or liwork < la − nnz + 7 × n + 1$\mathit{liwork}<{\mathbf{la}}-{\mathbf{nnz}}+7\times {\mathbf{n}}+1$, and lfill < 0${\mathbf{lfill}}<0$.

- On entry, the arrays irow and icol fail to satisfy the following constraints:
- 1 ≤ irow(i) ≤ n$1\le {\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$ and 1 ≤ icol(i) ≤ irow(i)$1\le {\mathbf{icol}}\left(i\right)\le {\mathbf{irow}}\left(i\right)$, for i = 1,2, … ,nnz$i=1,2,\dots ,{\mathbf{nnz}}$;
- irow(i − 1) < irow(i)${\mathbf{irow}}\left(i-1\right)<{\mathbf{irow}}\left(i\right)$, or irow(i − 1) = irow(i)${\mathbf{irow}}\left(i-1\right)={\mathbf{irow}}\left(i\right)$ and icol(i − 1) < icol(i)${\mathbf{icol}}\left(i-1\right)<{\mathbf{icol}}\left(i\right)$, for i = 2,3, … ,nnz$i=2,3,\dots ,{\mathbf{nnz}}$.

Therefore a nonzero element has been supplied which does not lie in the lower triangular part of A$A$, 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.

- A serious error has occurred in an internal call to the specified function. Check all function calls and array sizes. Seek expert help.

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$A$. The factorization can generally be made more accurate by increasing lfill, or by reducing dtol with lfill < 0${\mathbf{lfill}}<0$.

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.

The time taken for a call to nag_sparse_real_symm_precon_ichol (f11ja) is roughly proportional to (nnzc)^{2} / n${\left({\mathbf{nnzc}}\right)}^{2}/{\mathbf{n}}$.

If lfill ≥ 0${\mathbf{lfill}}\ge 0$ 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$A$ are defined to be of level 0$0$. The fill level of a new nonzero location occurring during the factorization is defined as

where k_{e}${k}_{\mathrm{e}}$ is the level of fill of the element being eliminated, and k_{c}${k}_{\mathrm{c}}$ is the level of fill of the element causing the fill-in.

k = max (k _{e},k_{c}) + 1,
$$k=\mathrm{max}\phantom{\rule{0.125em}{0ex}}({k}_{\mathrm{e}},{k}_{\mathrm{c}})+1\text{,}$$ |

If lfill < 0${\mathbf{lfill}}<0$ the fill-in is controlled by means of the **drop tolerance**
dtol. A potential fill-in element a_{ij}${a}_{ij}$ occurring in row i$i$ and column j$j$ will not be included if

$$\left|{a}_{ij}\right|<{\mathbf{dtol}}\times \sqrt{\left|{a}_{ii}{a}_{jj}\right|}\text{.}$$ |

For either method of control, any elements which are not included are discarded if mic = 'N'${\mathbf{mic}}=\text{'N'}$, or subtracted from the diagonal element in the elimination row if mic = 'M'${\mathbf{mic}}=\text{'M'}$.

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 A$A$ is not known to have any particular special properties the following strategy is recommended. Start with lfill = 0${\mathbf{lfill}}=0$, mic = 'N'${\mathbf{mic}}=\text{'N'}$ and dscale = 0.0${\mathbf{dscale}}=0.0$. 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$M$ 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 mic = 'M'${\mathbf{mic}}=\text{'M'}$ 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$A$ is probably too indefinite for this function.

If A$A$ 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 lfill = 0${\mathbf{lfill}}=0$, mic = 'M'${\mathbf{mic}}=\text{'M'}$ and dscale = 0.0${\mathbf{dscale}}=0.0$.

For certain mesh-based problems involving M-matrices it can be shown in theory that setting mic = 'M'${\mathbf{mic}}=\text{'M'}$, 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 dscale = 0.0${\mathbf{dscale}}=0.0$, 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).

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 complete factorization is obtained from a call to nag_sparse_real_symm_precon_ichol (f11ja) with lfill < 0${\mathbf{lfill}}<0$ and dtol = 0.0${\mathbf{dtol}}=0.0$, provided npivm = 0${\mathbf{npivm}}=0$ 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.

A = PLDL ^{T}P^{T} = M.
$$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}=M\text{.}$$ |

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).

Open in the MATLAB editor: nag_sparse_real_symm_precon_ichol_example

function nag_sparse_real_symm_precon_ichol_examplenz = 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

Open in the MATLAB editor: f11ja_example

function f11ja_examplenz = 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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