1: $n$ – int64int32nag_int scalar: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $icolzp (:)$ – int64int32nag_int array: The dimension of the array icolzp must be at least $n + 1$

$icolzp (i)$ contains the index in $A$ of the start of a new column. See Compressed column storage (CCS) format in the F11 Chapter Introduction.
3: $a (:)$ – double array: The dimension of the array a must be at least $icolzp (n + 1) - 1$ , the number of nonzeros of the sparse matrix $A$

The array of nonzero values in the sparse matrix $A$ .
4: $iprm (7 \times n)$ – int64int32nag_int array: The column permutation which defines $P_{c}$ , the row permutation which defines $P_{r}$ , plus associated data structures as computed by nag_sparse_direct_real_gen_lu (f11me).
5: $il (:)$ – int64int32nag_int array: The dimension of the array il must be at least as large as the dimension of the array of the same name in nag_sparse_direct_real_gen_lu (f11me)

Records the sparsity pattern of matrix $L$ as computed by nag_sparse_direct_real_gen_lu (f11me).
6: $lval (:)$ – double array: The dimension of the array lval must be at least as large as the dimension of the array of the same name in nag_sparse_direct_real_gen_lu (f11me)

Records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by nag_sparse_direct_real_gen_lu (f11me).
7: $iu (:)$ – int64int32nag_int array: The dimension of the array iu must be at least as large as the dimension of the array of the same name in nag_sparse_direct_real_gen_lu (f11me)

Records the sparsity pattern of matrix $U$ as computed by nag_sparse_direct_real_gen_lu (f11me).
8: $uval (:)$ – double array: The dimension of the array uval must be at least as large as the dimension of the array of the same name in nag_sparse_direct_real_gen_lu (f11me)

Records some nonzero values of matrix $U$ as computed by nag_sparse_direct_real_gen_lu (f11me).

Optional Input Parameters

None.

Output Parameters

1: $rpg$ – double scalar: The reciprocal pivot growth factor $\max_{j} ({‖A_{j}‖}_{\infty} / {‖U_{j}‖}_{\infty})$ . If the reciprocal pivot growth factor is much less than $1$ , the stability of the $L U$ factorization may be poor.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n \geq 0$ .

$ifail = 2$: Incorrect column permutations in array iprm.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

If the reciprocal pivot growth factor, rpg, is much less than

1

, then the factorization of the matrix

A

could be poor. This means that using the factorization to obtain solutions to a linear system, forward error bounds and estimates of the condition number could be unreliable. Consider increasing the thresh argument in the call to nag_sparse_direct_real_gen_lu (f11me).

Example

To compute the reciprocal pivot growth for the factorization of the matrix

A

, where

A = (\begin{matrix} 2.00 & 1.00 & 0 & 0 & 0 \\ 0 & 0 & 1.00 & - 1.00 & 0 \\ 4.00 & 0 & 1.00 & 0 & 1.00 \\ 0 & 0 & 0 & 1.00 & 2.00 \\ 0 & - 2.00 & 0 & 0 & 3.00 \end{matrix}) .

In this case, it should be equal to

1.0

Open in the MATLAB editor: f11mm_example

function f11mm_example


fprintf('f11mm example results\n\n');

% Calculate reciprocal pivot growth from factorization of sparse A 
n      = int64(5);
nz     = int64(11);
icolzp = [int64(1); 3; 5;  7; 9; 12];
irowix = [int64(1); 3; 1;  5; 2;  3;  2; 4; 3; 4; 5];
a      = [         2; 4; 1; -2; 1;  1; -1; 1; 1; 2; 3];

% Calculate COLAMD permutation
spec = 'M';
iprm = zeros(1, 7*n, 'int64');
[iprm, ifail] = f11md( ...
                       spec, n, icolzp, irowix, iprm);

% Factorise
thresh = 1;
nzlmx = int64(8*nz);
nzlumx = int64(8*nz);
nzumx = int64(8*nz);
[iprm, nzlumx, il, lval, iu, uval, nnzl, nnzu, flop, ifail] = ...
  f11me( ...
         n, irowix, a, iprm, thresh, nzlmx, nzlumx, nzumx);

% Calculate reciprocal pivot growth
[rpg, ifail] = f11mm( ...
                      n, icolzp, a, iprm, il, lval, iu, uval);

disp('Reciprocal pivot growth');
disp(rpg);

f11mm example results

Reciprocal pivot growth
     1

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox