naginterfaces.library.sparse.direct_​real_​gen_​setup

naginterfaces.library.sparse.direct_real_gen_setup(spec, n, icolzp, irowix, iprm)

direct_real_gen_setup computes a column permutation suitable for $LU$ factorization (by direct_real_gen_lu()) of a real sparse matrix in compressed column (Harwell–Boeing) format and applies it to the matrix. This function must be called prior to direct_real_gen_lu().

For full information please refer to the NAG Library document for f11md

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f11/f11mdf.html

Parameters

specstr, length 1

Indicates the permutation to be applied.

$spec=\text{'N'}$

The identity permutation is used (i.e., the columns are not permuted).

$spec=\text{'U'}$

The permutation in the $iprm$ array is used, as supplied by you.

$spec=\text{'M'}$

The permutation computed by the COLAMD algorithm is used

nint

$n$, the order of the matrix $A$.

icolzpint, array-like, shape $(max(1,(n+1)))$

The new column index array of sparse matrix $A$. See the F11 Introduction.

irowixint, array-like, shape $(max(1,\left(icolzp\right[max(1,(n+1))-1]-1)))$

$irowix[i-1]$ contains the row index in $A$ for element $A\left(i\right)$. See the F11 Introduction.

iprmint, array-like, shape $(max(1,(7\times n)))$

The first $n$ entries contain the column permutation if supplied by the user. This will be used if $spec=\text{'U'}$, and ignored otherwise. If used, it must consist of a permutation of all the integers in the range $[0,(n-1)]$, the leftmost column of the matrix $A$ denoted by $0$ and the rightmost by $n-1$. Labelling columns in this way, $iprm\left[i\right]=j$ means that column $i-1$ of $A$ is in position $j$ in $A{P}_c$, where $P_{r}A{P}_c=LU$ expresses the factorization to be performed.

Returns

iprmint, ndarray, shape $(max(1,(7\times n)))$

The column permutation given or computed is returned in the second $n$ entries. The rest of the array contains data structures that will be used by other functions in the suite. The function computes the column elimination tree for $A$ and a post-order permutation on the tree. It then compounds the $iprm$ permutation given or computed by the COLAMD algorthm with the post-order permutation and this permutation is returned in the first $n$ entries. This whole array is needed by the $LU$ factorization function direct_real_gen_lu() and associated functions direct_real_gen_solve(), direct_real_gen_refine() and direct_real_gen_diag() and should be passed to them unchanged.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

On entry, $spec=⟨value⟩$.

Constraint: $spec=\text{'N'}$, $\text{'U'}$ or $\text{'M'}$.

(errno $2$)

Incorrect column permutations in array $iprm$.

(errno $3$)

COLAMD algorithm failed.

(errno $4$)

Incorrect specification of argument $icolzp$.

(errno $5$)

Incorrect specification of argument $irowix$.

Notes

Given a sparse matrix in compressed column (Harwell–Boeing) format $A$ and a choice of column permutation schemes, the function computes those data structures that will be needed by the $LU$ factorization function direct_real_gen_lu() and associated functions direct_real_gen_diag(), direct_real_gen_solve() and direct_real_gen_refine(). The column permutation choices are:

original order (that is, no permutation);

user-supplied permutation;

a permutation, computed by the function, designed to minimize fill-in during the $LU$ factorization.

The algorithm for this computed permutation is based on the approximate minimum degree column ordering algorithm COLAMD. The computed permutation is not sensitive to the magnitude of the nonzero values of $A$.

References

Amestoy, P R, Davis, T A and Duff, I S, 1996, An approximate minimum degree ordering algorithm, SIAM J. Matrix Anal. Appl. (17), 886–905

Gilbert, J R and Larimore, S I, 2004, A column approximate minimum degree ordering algorithm, ACM Trans. Math. Software (30,3), 353–376

Gilbert, J R, Larimore, S I and Ng, E G, 2004, Algorithm 836: COLAMD, an approximate minimum degree ordering algorithm, ACM Trans. Math. Software (30, 3), 377–380