NAG C Library Function Document

nag_sparse_herm_chol_fac (f11jnc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 1$.

2:    $\mathbf{nnz}$ – IntegerInput

On entry: the number of nonzero elements in the lower triangular part of the matrix $A$.

Constraint: $1\le \mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$.

3:    $\mathbf{a}\left[\mathbf{la}\right]$ – ComplexInput/Output

On entry: 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_herm_sort (f11zpc) may be used to order the elements in this way.

On exit: the first nnz 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.

4:    $\mathbf{la}$ – IntegerInput

On entry: the 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).

Constraint: $\mathbf{la}\ge 2\times \mathbf{nnz}$.

5:    $\mathbf{irow}\left[\mathbf{la}\right]$ – IntegerInput/Output

6:    $\mathbf{icol}\left[\mathbf{la}\right]$ – IntegerInput/Output

On entry: 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_herm_sort (f11zpc)):

• $1\le \mathbf{irow}\left[\mathit{i}\right]\le \mathbf{n}$ and $1\le \mathbf{icol}\left[\mathit{i}\right]\le \mathbf{irow}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{nnz}-1$;
• $\mathbf{irow}\left[\mathit{i}-1\right]<\mathbf{irow}\left[\mathit{i}\right]$ 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 $\mathbf{lfill}<0$, 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:    $\mathbf{mic}$ – Nag_SparseSym_FactInput

On entry: indicates whether or not the factorization should be modified to preserve row sums (see Section 9.3).

$\mathbf{mic}=\mathrm{Nag\_SparseSym\_ModFact}$

The factorization is modified.

$\mathbf{mic}=\mathrm{Nag\_SparseSym\_UnModFact}$

The factorization is not modified.

Constraint: $\mathbf{mic}=\mathrm{Nag\_SparseSym\_ModFact}$ or $\mathrm{Nag\_SparseSym\_UnModFact}$.

10:  $\mathbf{dscale}$ – doubleInput

On entry: the diagonal scaling parameter. All diagonal elements are multiplied by the factor ($1.0+\mathbf{dscale}$) at the start of the factorization. This can be used to ensure that the preconditioner is positive definite. See also Section 9.3.

11:  $\mathbf{pstrat}$ – Nag_SparseSym_PivInput

On entry: specifies the pivoting strategy to be adopted.

$\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_NoPiv}$

No pivoting is carried out.

$\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_MarkPiv}$

Diagonal pivoting aimed at minimizing fill-in is carried out, using the Markowitz strategy (see Markowitz (1957)).

$\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_UserPiv}$

Diagonal pivoting is carried out according to the user-defined input array ipiv.

Suggested value: $\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_MarkPiv}$.

Constraint: $\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_NoPiv}$, $\mathrm{Nag\_SparseSym\_MarkPiv}$ or $\mathrm{Nag\_SparseSym\_UserPiv}$.

12:  $\mathbf{ipiv}\left[\mathbf{n}\right]$ – IntegerInput/Output

On entry: if $\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_UserPiv}$, $\mathbf{ipiv}\left[i-1\right]$ must specify the row index of the diagonal element to be used as a pivot at elimination stage $i$. Otherwise ipiv need not be initialized.

Constraint: if $\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_UserPiv}$, ipiv must contain a valid permutation of the integers on $\left[1,\mathbf{n}\right]$.

On exit: the pivot indices. If $\mathbf{ipiv}\left[i-1\right]=j$, the diagonal element in row $j$ was used as the pivot at elimination stage $i$.

13:  $\mathbf{istr}\left[\mathbf{n}+1\right]$ – IntegerOutput

On exit: $\mathbf{istr}\left[\mathit{i}-1\right]-1$, 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 $C$. $\mathbf{istr}\left[\mathbf{n}\right]-1$ is the address of the last nonzero element in $C$ plus one.

14:  $\mathbf{nnzc}$ – Integer *Output

On exit: the number of nonzero elements in the lower triangular matrix $C$.

15:  $\mathbf{npivm}$ – Integer *Output

On exit: 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_herm_chol_fac (f11jnc) again with an increased value of either lfill or dscale. See also Sections 9.3 and 9.4.

16:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

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. Consider calling nag_sparse_herm_sort (f11zpc) to reorder and sum or remove duplicates.

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}\ge 1$.

NE_INT_2

On entry, $\mathbf{la}=〈\mathit{\text{value}}〉$ and $\mathbf{nnz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{la}\ge 2\times \mathbf{nnz}$.

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$ 

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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

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

NE_INVALID_ROWCOL_PIVOT

On entry, a user-supplied value of ipiv is repeated.

On entry, a user-supplied value of ipiv lies outside the range $\left[1,\mathbf{n}\right]$.

NE_INVALID_SCS

On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathbf{icol}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$ and $\mathbf{irow}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{icol}\left[\mathit{I}-1\right]\ge 1$ and $\mathbf{icol}\left[\mathit{I}-1\right]\le \mathbf{irow}\left[\mathit{I}-1\right]$.

On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathbf{irow}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{irow}\left[\mathit{I}-1\right]\ge 1$ and $\mathbf{irow}\left[\mathit{I}-1\right]\le \mathbf{n}$.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_NOT_STRICTLY_INCREASING

On entry, $\mathbf{a}\left[i-1\right]$ is out of order: $i=〈\mathit{\text{value}}〉$.

On entry, the location ($\mathbf{irow}\left[\mathit{I}-1\right],\mathbf{icol}\left[\mathit{I}-1\right]$) is a duplicate: $\mathit{I}=〈\mathit{\text{value}}〉$.

NE_REAL

On entry, $\mathbf{dtol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{dtol}\ge 0.0$ 

NE_TOO_SMALL

The number of nonzero entries in the decomposition is too large. The decomposition has been terminated before completion. Either increase la, or reduce the fill by setting $\mathbf{pstrat}=\mathrm{Nag\_SparseSym\_MarkPiv}$, reducing lfill, or increasing dtol.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

Timing

9.2

Control of Fill-in

9.3

Choice of Arguments

9.4

Direct Solution of positive definite Systems

10

Example

10.1

Program Text

Program Text (f11jnce.c)

10.2

Program Data

Program Data (f11jnce.d)

10.3

Program Results

Program Results (f11jnce.r)