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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sparse_real_gen_sort (f11za)

## Purpose

nag_sparse_real_gen_sort (f11za) sorts the nonzero elements of a real sparse nonsymmetric matrix, represented in coordinate storage format.

## Syntax

[nz, a, irow, icol, istr, ifail] = f11za(n, nz, a, irow, icol, dup, zer)
[nz, a, irow, icol, istr, ifail] = nag_sparse_real_gen_sort(n, nz, a, irow, icol, dup, zer)

## Description

nag_sparse_real_gen_sort (f11za) takes a coordinate storage (CS) representation (see Coordinate storage (CS) format in the F11 Chapter Introduction) of a real $n$ by $n$ sparse nonsymmetric matrix $A$, and reorders the nonzero elements by increasing row index and increasing column index within each row. Entries with duplicate row and column indices may be removed, or the values may be summed. Any entries with zero values may optionally be removed.
nag_sparse_real_gen_sort (f11za) also returns a pointer istr to the starting address of each row in $A$. This can be used to construct a compressed column storage (CCS) representation of the matrix (see Further Comments).

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathrm{nz}$int64int32nag_int scalar
The number of nonzero elements in the matrix $A$.
Constraint: ${\mathbf{nz}}\ge 0$.
3:     $\mathrm{a}\left(:\right)$ – double array
The dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nz}}\right)$
The nonzero elements of the matrix $A$. These may be in any order and there may be multiple nonzero elements with the same row and column indices.
4:     $\mathrm{irow}\left(:\right)$int64int32nag_int array
The dimension of the array irow must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nz}}\right)$
The row indices corresponding to the nonzero elements supplied in the array a.
Constraint: $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$.
5:     $\mathrm{icol}\left(:\right)$int64int32nag_int array
The dimension of the array icol must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nz}}\right)$
The column indices corresponding to the nonzero elements supplied in the array a.
Constraint: $1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$.
6:     $\mathrm{dup}$ – string (length ≥ 1)
Indicates how any nonzero elements with duplicate row and column indices are to be treated.
${\mathbf{dup}}=\text{'R'}$
The entries are removed.
${\mathbf{dup}}=\text{'S'}$
The relevant values in a are summed.
${\mathbf{dup}}=\text{'F'}$
The function fails on detecting a duplicate, with ${\mathbf{ifail}}={\mathbf{3}}$.
Constraint: ${\mathbf{dup}}=\text{'R'}$, $\text{'S'}$ or $\text{'F'}$.
7:     $\mathrm{zer}$ – string (length ≥ 1)
Indicates how any elements with zero values in a are to be treated.
${\mathbf{zer}}=\text{'R'}$
The entries are removed.
${\mathbf{zer}}=\text{'K'}$
The entries are kept.
${\mathbf{zer}}=\text{'F'}$
The function fails on detecting a zero, with ${\mathbf{ifail}}={\mathbf{4}}$.
Constraint: ${\mathbf{zer}}=\text{'R'}$, $\text{'K'}$ or $\text{'F'}$.

None.

### Output Parameters

1:     $\mathrm{nz}$int64int32nag_int scalar
The number of nonzero elements with unique row and column indices.
2:     $\mathrm{a}\left(:\right)$ – double array
The dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nz}}\right)$
The nonzero elements ordered by increasing row index, and by increasing column index within each row. Each nonzero element has a unique row and column index.
3:     $\mathrm{irow}\left(:\right)$int64int32nag_int array
The dimension of the array irow will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nz}}\right)$
The first nz elements contain the row indices corresponding to the nonzero elements returned in the array a.
4:     $\mathrm{icol}\left(:\right)$int64int32nag_int array
The dimension of the array icol will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nz}}\right)$
The first nz elements contain the row indices corresponding to the nonzero elements returned in the array a.
5:     $\mathrm{istr}\left({\mathbf{n}}+1\right)$int64int32nag_int array
${\mathbf{istr}}\left(\mathit{i}\right)$, 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 $A$. ${\mathbf{istr}}\left({\mathbf{n}}+1\right)$ is the address of the last nonzero element in $A$ plus one.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{nz}}<0$, or ${\mathbf{dup}}\ne \text{'R'},\text{'S'}$, or 'F', or ${\mathbf{zer}}\ne \text{'R'},\text{'K'}$, or 'F'.
${\mathbf{ifail}}=2$
On entry, a nonzero element has been supplied which does not lie within the matrix $A$, i.e., one or more of the following constraints has been violated:
• $1\le {\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$,
• $1\le {\mathbf{icol}}\left(i\right)\le {\mathbf{n}}$,
for $i=1,2,\dots ,{\mathbf{nz}}$
${\mathbf{ifail}}=3$
On entry, ${\mathbf{dup}}=\text{'F'}$ and nonzero elements have been supplied which have duplicate row and column indices.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{zer}}=\text{'F'}$ and at least one matrix element has been supplied with a zero coefficient value.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

The time taken for a call to nag_sparse_real_gen_sort (f11za) is proportional to nz.
Note that the resulting matrix may have either rows or columns with no entries. If row $i$ has no entries then ${\mathbf{istr}}\left(i\right)={\mathbf{istr}}\left(i+1\right)$.
It is also possible to use this function to convert between coordinate storage (CS) and compressed column storage (CCS) formats. To achieve this the CS storage format arrays irow and icol must be interchanged in the call to nag_sparse_real_gen_sort (f11za). On exit from nag_sparse_real_gen_sort (f11za), the CCS representation of the matrix is then defined by arrays a, irow and istr. This is illustrated in Example.

## Example

This example reads the CS representation of a real sparse matrix $A$, calls nag_sparse_real_gen_sort (f11za) to reorder the nonzero elements, and outputs the original and the reordered representations. It then calls nag_sparse_real_gen_sort (f11za) again with the alternative ordering, creating a CCS representation which is then passed to a function that computes a matrix norm for that representation.
 $A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 .$
```function f11za_example

fprintf('f11za example results\n\n');

% Sparse 5-by-5 matrix in coordinate storage form to be reordered
n = int64(5);
nz = int64(15);
irow = zeros(nz,1,'int64');
icol = zeros(nz,1,'int64');
a(1:nz)     = [4; -2;  1; -2; -3;  1;  0;  1; -1;  6;  2;  2;  1;  1;  2];
irow(1:nz) = [3;  5;  4;  4;  5;  1;  1;  3;  2;  5;  1;  4;  2;  3;  4];
icol(1:nz) = [1;  2;  4;  2;  5;  2;  5;  5;  4;  5;  1;  2;  3;  3;  5];

fprintf('Number of elements in original sparse matrix: %5d\n',nz);
disp('Original matrix ordering:');
fprintf('   k   a(k)  i_k  j_k\n');
for j = 1:nz
fprintf('%4d%6.1f%5d%5d\n',j,a(j),irow(j),icol(j));
end

% Sum duplicates and remove zeros
dup = 'S';
zero = 'R';

% Reorder along rows first (swap row, col to reorder by columns)
[nz, a, irow, icol, istr, ifail] = ...
f11za( ...
n, nz, a, irow, icol, dup, zero);

fprintf('\nNumber of elements in reordered sparse matrix: %5d\n',nz);
disp('New ordering (by row first):');
fprintf('   k   a(k)  i_k  j_k\n');
for j = 1:nz
fprintf('%4d%6.1f%5d%5d\n',j,a(j),irow(j),icol(j));
end

```
```f11za example results

Number of elements in original sparse matrix:    15
Original matrix ordering:
k   a(k)  i_k  j_k
1   4.0    3    1
2  -2.0    5    2
3   1.0    4    4
4  -2.0    4    2
5  -3.0    5    5
6   1.0    1    2
7   0.0    1    5
8   1.0    3    5
9  -1.0    2    4
10   6.0    5    5
11   2.0    1    1
12   2.0    4    2
13   1.0    2    3
14   1.0    3    3
15   2.0    4    5

Number of elements in reordered sparse matrix:    11
New ordering (by row first):
k   a(k)  i_k  j_k
1   2.0    1    1
2   1.0    1    2
3   1.0    2    3
4  -1.0    2    4
5   4.0    3    1
6   1.0    3    3
7   1.0    3    5
8   1.0    4    4
9   2.0    4    5
10  -2.0    5    2
11   3.0    5    5
```