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<hr>
<h4 class="subsubsection" id="Mathematical-Considerations-1"><span>22.1.4.3 Mathematical Considerations<a class="copiable-link" href="#Mathematical-Considerations-1"> ¶</a></span></h4>
<p>The attempt has been made to make sparse matrices behave in exactly the
same manner as there full counterparts. However, there are certain differences
and especially differences with other products sparse implementations.
</p>
<p>First, the <code class="code">"./"</code> and <code class="code">".^"</code> operators must be used with care.
Consider what the examples
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> s = speye (4);
a1 = s .^ 2;
a2 = s .^ s;
a3 = s .^ -2;
a4 = s ./ 2;
a5 = 2 ./ s;
a6 = s ./ s;
</pre></div></div>
<p>will give. The first example of <var class="var">s</var> raised to the power of 2 causes
no problems. However <var class="var">s</var> raised element-wise to itself involves a
large number of terms <code class="code">0 .^ 0</code> which is 1. There <code class="code"><var class="var">s</var> .^
<var class="var">s</var></code> is a full matrix.
</p>
<p>Likewise <code class="code"><var class="var">s</var> .^ -2</code> involves terms like <code class="code">0 .^ -2</code> which
is infinity, and so <code class="code"><var class="var">s</var> .^ -2</code> is equally a full matrix.
</p>
<p>For the "./" operator <code class="code"><var class="var">s</var> ./ 2</code> has no problems, but
<code class="code">2 ./ <var class="var">s</var></code> involves a large number of infinity terms as well
and is equally a full matrix. The case of <code class="code"><var class="var">s</var> ./ <var class="var">s</var></code>
involves terms like <code class="code">0 ./ 0</code> which is a <code class="code">NaN</code> and so this
is equally a full matrix with the zero elements of <var class="var">s</var> filled with
<code class="code">NaN</code> values.
</p>
<p>The above behavior is consistent with full matrices, but is not
consistent with sparse implementations in other products.
</p>
<p>A particular problem of sparse matrices comes about due to the fact that
as the zeros are not stored, the sign-bit of these zeros is equally not
stored. In certain cases the sign-bit of zero is important. For example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> a = 0 ./ [-1, 1; 1, -1];
b = 1 ./ a
⇒ -Inf Inf
Inf -Inf
c = 1 ./ sparse (a)
⇒ Inf Inf
Inf Inf
</pre></div></div>
<p>To correct this behavior would mean that zero elements with a negative
sign-bit would need to be stored in the matrix to ensure that their
sign-bit was respected. This is not done at this time, for reasons of
efficiency, and so the user is warned that calculations where the sign-bit
of zero is important must not be done using sparse matrices.
</p>
<p>In general any function or operator used on a sparse matrix will
result in a sparse matrix with the same or a larger number of nonzero
elements than the original matrix. This is particularly true for the
important case of sparse matrix factorizations. The usual way to
address this is to reorder the matrix, such that its factorization is
sparser than the factorization of the original matrix. That is the
factorization of <code class="code">L * U = P * S * Q</code> has sparser terms <code class="code">L</code>
and <code class="code">U</code> than the equivalent factorization <code class="code">L * U = S</code>.
</p>
<p>Several functions are available to reorder depending on the type of the
matrix to be factorized. If the matrix is symmetric positive-definite,
then <em class="dfn">symamd</em> or <em class="dfn">csymamd</em> should be used. Otherwise
<em class="dfn">amd</em>, <em class="dfn">colamd</em> or <em class="dfn">ccolamd</em> should be used. For completeness
the reordering functions <em class="dfn">colperm</em> and <em class="dfn">randperm</em> are
also available.
</p>
<p>See <a class="xref" href="#fig_003asimplematrix">Figure 22.3</a>, for an example of the structure of a simple
positive definite matrix.
</p>
<div class="float" id="fig_003asimplematrix">
<div class="center"><img class="image" src="spmatrix.png" alt="spmatrix">
</div><div class="caption"><p><strong class="strong">Figure 22.3: </strong>Structure of simple sparse matrix.</p></div></div>
<p>The standard Cholesky factorization of this matrix can be
obtained by the same command that would be used for a full
matrix. This can be visualized with the command
<code class="code">r = chol (A); spy (r);</code>.
See <a class="xref" href="#fig_003asimplechol">Figure 22.4</a>.
The original matrix had
598
nonzero terms, while this Cholesky factorization has
10200,
with only half of the symmetric matrix being stored. This
is a significant level of fill in, and although not an issue
for such a small test case, can represents a large overhead
in working with other sparse matrices.
</p>
<p>The appropriate sparsity preserving permutation of the original
matrix is given by <em class="dfn">symamd</em> and the factorization using this
reordering can be visualized using the command <code class="code">q = symamd (A);
r = chol (A(q,q)); spy (r)</code>. This gives
399
nonzero terms which is a significant improvement.
</p>
<p>The Cholesky factorization itself can be used to determine the
appropriate sparsity preserving reordering of the matrix during the
factorization, In that case this might be obtained with three return
arguments as <code class="code">[r, p, q] = chol (A); spy (r)</code>.
</p>
<div class="float" id="fig_003asimplechol">
<div class="center"><img class="image" src="spchol.png" alt="spchol">
</div><div class="caption"><p><strong class="strong">Figure 22.4: </strong>Structure of the unpermuted Cholesky factorization of the above matrix.</p></div></div>
<div class="float" id="fig_003asimplecholperm">
<div class="center"><img class="image" src="spcholperm.png" alt="spcholperm">
</div><div class="caption"><p><strong class="strong">Figure 22.5: </strong>Structure of the permuted Cholesky factorization of the above matrix.</p></div></div>
<p>In the case of an asymmetric matrix, the appropriate sparsity
preserving permutation is <em class="dfn">colamd</em> and the factorization using
this reordering can be visualized using the command
<code class="code">q = colamd (A); [l, u, p] = lu (A(:,q)); spy (l+u)</code>.
</p>
<p>Finally, Octave implicitly reorders the matrix when using the div (/)
and ldiv (\) operators, and so no the user does not need to explicitly
reorder the matrix to maximize performance.
</p>
<a class="anchor" id="XREFamd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-amd"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">amd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-amd"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-amd-1"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">amd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">opts</var>)</code><a class="copiable-link" href="#index-amd-1"> ¶</a></span></dt>
<dd>
<p>Return the approximate minimum degree permutation of a matrix.
</p>
<p>This is a permutation such that the Cholesky factorization of
<code class="code"><var class="var">S</var> (<var class="var">p</var>, <var class="var">p</var>)</code> tends to be sparser than the
Cholesky factorization of <var class="var">S</var> itself. <code class="code">amd</code> is typically
faster than <code class="code">symamd</code> but serves a similar purpose.
</p>
<p>The optional parameter <var class="var">opts</var> is a structure that controls the behavior
of <code class="code">amd</code>. The fields of the structure are
</p>
<dl class="table">
<dt><var class="var">opts</var>.dense</dt>
<dd><p>Determines what <code class="code">amd</code> considers to be a dense row or column of the
input matrix. Rows or columns with more than <code class="code">max (16, (dense *
sqrt (<var class="var">n</var>)))</code> entries, where <var class="var">n</var> is the order of the matrix <var class="var">S</var>,
are ignored by <code class="code">amd</code> during the calculation of the permutation.
The value of dense must be a positive scalar and the default value is 10.0
</p>
</dd>
<dt><var class="var">opts</var>.aggressive</dt>
<dd><p>If this value is a nonzero scalar, then <code class="code">amd</code> performs aggressive
absorption. The default is not to perform aggressive absorption.
</p></dd>
</dl>
<p>The author of the code itself is Timothy A. Davis
(see <a class="url" href="http://faculty.cse.tamu.edu/davis/suitesparse.html">http://faculty.cse.tamu.edu/davis/suitesparse.html</a>).
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFsymamd">symamd</a>, <a class="ref" href="#XREFcolamd">colamd</a>.
</p></dd></dl>
<a class="anchor" id="XREFccolamd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-ccolamd"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">ccolamd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-ccolamd"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ccolamd-1"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">ccolamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>)</code><a class="copiable-link" href="#index-ccolamd-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ccolamd-2"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">ccolamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>, <var class="var">cmember</var>)</code><a class="copiable-link" href="#index-ccolamd-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ccolamd-3"><span><code class="def-type">[<var class="var">p</var>, <var class="var">stats</var>] =</code> <strong class="def-name">ccolamd</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-ccolamd-3"> ¶</a></span></dt>
<dd>
<p>Constrained column approximate minimum degree permutation.
</p>
<p><code class="code"><var class="var">p</var> = ccolamd (<var class="var">S</var>)</code> returns the column approximate minimum
degree permutation vector for the sparse matrix <var class="var">S</var>. For a
non-symmetric matrix <var class="var">S</var>, <code class="code"><var class="var">S</var>(:, <var class="var">p</var>)</code> tends to have
sparser LU factors than <var class="var">S</var>.
<code class="code">chol (<var class="var">S</var>(:, <var class="var">p</var>)' * <var class="var">S</var>(:, <var class="var">p</var>))</code> also tends to be
sparser than <code class="code">chol (<var class="var">S</var>' * <var class="var">S</var>)</code>.
<code class="code"><var class="var">p</var> = ccolamd (<var class="var">S</var>, 1)</code> optimizes the ordering for
<code class="code">lu (<var class="var">S</var>(:, <var class="var">p</var>))</code>. The ordering is followed by a column
elimination tree post-ordering.
</p>
<p><var class="var">knobs</var> is an optional 1-element to 5-element input vector, with a
default value of <code class="code">[0 10 10 1 0]</code> if not present or empty. Entries not
present are set to their defaults.
</p>
<dl class="table">
<dt><code class="code"><var class="var">knobs</var>(1)</code></dt>
<dd><p>if nonzero, the ordering is optimized for <code class="code">lu (S(:, p))</code>. It will be a
poor ordering for <code class="code">chol (<var class="var">S</var>(:, <var class="var">p</var>)' * <var class="var">S</var>(:, <var class="var">p</var>))</code>.
This is the most important knob for ccolamd.
</p>
</dd>
<dt><code class="code"><var class="var">knobs</var>(2)</code></dt>
<dd><p>if <var class="var">S</var> is m-by-n, rows with more than
<code class="code">max (16, <var class="var">knobs</var>(2) * sqrt (n))</code> entries are ignored.
</p>
</dd>
<dt><code class="code"><var class="var">knobs</var>(3)</code></dt>
<dd><p>columns with more than
<code class="code">max (16, <var class="var">knobs</var>(3) * sqrt (min (<var class="var">m</var>, <var class="var">n</var>)))</code> entries are
ignored and ordered last in the output permutation
(subject to the cmember constraints).
</p>
</dd>
<dt><code class="code"><var class="var">knobs</var>(4)</code></dt>
<dd><p>if nonzero, aggressive absorption is performed.
</p>
</dd>
<dt><code class="code"><var class="var">knobs</var>(5)</code></dt>
<dd><p>if nonzero, statistics and knobs are printed.
</p>
</dd>
</dl>
<p><var class="var">cmember</var> is an optional vector of length <em class="math">n</em>. It defines the
constraints on the column ordering. If <code class="code"><var class="var">cmember</var>(j) = <var class="var">c</var></code>,
then column <var class="var">j</var> is in constraint set <var class="var">c</var> (<var class="var">c</var> must be in the
range 1 to n). In the output permutation <var class="var">p</var>, all columns in set 1
appear first, followed by all columns in set 2, and so on.
<code class="code"><var class="var">cmember</var> = ones (1,n)</code> if not present or empty.
<code class="code">ccolamd (<var class="var">S</var>, [], 1 : n)</code> returns <code class="code">1 : n</code>
</p>
<p><code class="code"><var class="var">p</var> = ccolamd (<var class="var">S</var>)</code> is about the same as
<code class="code"><var class="var">p</var> = colamd (<var class="var">S</var>)</code>. <var class="var">knobs</var> and its default values
differ. <code class="code">colamd</code> always does aggressive absorption, and it finds an
ordering suitable for both <code class="code">lu (<var class="var">S</var>(:, <var class="var">p</var>))</code> and <code class="code">chol
(<var class="var">S</var>(:, <var class="var">p</var>)' * <var class="var">S</var>(:, <var class="var">p</var>))</code>; it cannot optimize its
ordering for <code class="code">lu (<var class="var">S</var>(:, <var class="var">p</var>))</code> to the extent that
<code class="code">ccolamd (<var class="var">S</var>, 1)</code> can.
</p>
<p><var class="var">stats</var> is an optional 20-element output vector that provides data
about the ordering and the validity of the input matrix <var class="var">S</var>. Ordering
statistics are in <code class="code"><var class="var">stats</var>(1 : 3)</code>. <code class="code"><var class="var">stats</var>(1)</code> and
<code class="code"><var class="var">stats</var>(2)</code> are the number of dense or empty rows and columns
ignored by <small class="sc">CCOLAMD</small> and <code class="code"><var class="var">stats</var>(3)</code> is the number of garbage
collections performed on the internal data structure used by <small class="sc">CCOLAMD</small>
(roughly of size <code class="code">2.2 * nnz (<var class="var">S</var>) + 4 * <var class="var">m</var> + 7 * <var class="var">n</var></code>
integers).
</p>
<p><code class="code"><var class="var">stats</var>(4 : 7)</code> provide information if CCOLAMD was able to
continue. The matrix is OK if <code class="code"><var class="var">stats</var>(4)</code> is zero, or 1 if
invalid. <code class="code"><var class="var">stats</var>(5)</code> is the rightmost column index that is
unsorted or contains duplicate entries, or zero if no such column exists.
<code class="code"><var class="var">stats</var>(6)</code> is the last seen duplicate or out-of-order row
index in the column index given by <code class="code"><var class="var">stats</var>(5)</code>, or zero if no
such row index exists. <code class="code"><var class="var">stats</var>(7)</code> is the number of duplicate
or out-of-order row indices. <code class="code"><var class="var">stats</var>(8 : 20)</code> is always zero in
the current version of <small class="sc">CCOLAMD</small> (reserved for future use).
</p>
<p>The authors of the code itself are S. Larimore, T. Davis and
S. Rajamanickam in collaboration with J. Bilbert and E. Ng.
Supported by the National Science Foundation
(DMS-9504974, DMS-9803599, CCR-0203270), and a grant from
Sandia National Lab.
See <a class="url" href="http://faculty.cse.tamu.edu/davis/suitesparse.html">http://faculty.cse.tamu.edu/davis/suitesparse.html</a> for ccolamd,
csymamd, amd, colamd, symamd, and other related orderings.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcolamd">colamd</a>, <a class="ref" href="#XREFcsymamd">csymamd</a>.
</p></dd></dl>
<a class="anchor" id="XREFcolamd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-colamd"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">colamd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-colamd"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-colamd-1"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">colamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>)</code><a class="copiable-link" href="#index-colamd-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-colamd-2"><span><code class="def-type">[<var class="var">p</var>, <var class="var">stats</var>] =</code> <strong class="def-name">colamd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-colamd-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-colamd-3"><span><code class="def-type">[<var class="var">p</var>, <var class="var">stats</var>] =</code> <strong class="def-name">colamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>)</code><a class="copiable-link" href="#index-colamd-3"> ¶</a></span></dt>
<dd>
<p>Compute the column approximate minimum degree permutation.
</p>
<p><code class="code"><var class="var">p</var> = colamd (<var class="var">S</var>)</code> returns the column approximate minimum
degree permutation vector for the sparse matrix <var class="var">S</var>. For a
non-symmetric matrix <var class="var">S</var>, <code class="code"><var class="var">S</var>(:,<var class="var">p</var>)</code> tends to have
sparser LU factors than <var class="var">S</var>. The Cholesky factorization of
<code class="code"><var class="var">S</var>(:,<var class="var">p</var>)' * <var class="var">S</var>(:,<var class="var">p</var>)</code> also tends to be sparser
than that of <code class="code"><var class="var">S</var>' * <var class="var">S</var></code>.
</p>
<p><var class="var">knobs</var> is an optional one- to three-element input vector. If <var class="var">S</var>
is m-by-n, then rows with more than <code class="code">max(16,<var class="var">knobs</var>(1)*sqrt(n))</code>
entries are ignored. Columns with more than
<code class="code">max (16,<var class="var">knobs</var>(2)*sqrt(min(m,n)))</code> entries are removed prior to
ordering, and ordered last in the output permutation <var class="var">p</var>. Only
completely dense rows or columns are removed if <code class="code"><var class="var">knobs</var>(1)</code> and
<code class="code"><var class="var">knobs</var>(2)</code> are < 0, respectively. If <code class="code"><var class="var">knobs</var>(3)</code> is
nonzero, <var class="var">stats</var> and <var class="var">knobs</var> are printed. The default is
<code class="code"><var class="var">knobs</var> = [10 10 0]</code>. Note that <var class="var">knobs</var> differs from earlier
versions of colamd.
</p>
<p><var class="var">stats</var> is an optional 20-element output vector that provides data
about the ordering and the validity of the input matrix <var class="var">S</var>. Ordering
statistics are in <code class="code"><var class="var">stats</var>(1:3)</code>. <code class="code"><var class="var">stats</var>(1)</code> and
<code class="code"><var class="var">stats</var>(2)</code> are the number of dense or empty rows and columns
ignored by <small class="sc">COLAMD</small> and <code class="code"><var class="var">stats</var>(3)</code> is the number of garbage
collections performed on the internal data structure used by <small class="sc">COLAMD</small>
(roughly of size <code class="code">2.2 * nnz(<var class="var">S</var>) + 4 * <var class="var">m</var> + 7 * <var class="var">n</var></code>
integers).
</p>
<p>Octave built-in functions are intended to generate valid sparse matrices,
with no duplicate entries, with ascending row indices of the nonzeros
in each column, with a non-negative number of entries in each column (!)
and so on. If a matrix is invalid, then <small class="sc">COLAMD</small> may or may not be able
to continue. If there are duplicate entries (a row index appears two or
more times in the same column) or if the row indices in a column are out
of order, then <small class="sc">COLAMD</small> can correct these errors by ignoring the
duplicate entries and sorting each column of its internal copy of the
matrix <var class="var">S</var> (the input matrix <var class="var">S</var> is not repaired, however). If a
matrix is invalid in other ways then <small class="sc">COLAMD</small> cannot continue, an error
message is printed, and no output arguments (<var class="var">p</var> or <var class="var">stats</var>) are
returned.
<small class="sc">COLAMD</small> is thus a simple way to check a sparse matrix to see if it’s
valid.
</p>
<p><code class="code"><var class="var">stats</var>(4:7)</code> provide information if <small class="sc">COLAMD</small> was able to
continue. The matrix is OK if <code class="code"><var class="var">stats</var>(4)</code> is zero, or 1 if
invalid. <code class="code"><var class="var">stats</var>(5)</code> is the rightmost column index that is
unsorted or contains duplicate entries, or zero if no such column exists.
<code class="code"><var class="var">stats</var>(6)</code> is the last seen duplicate or out-of-order row
index in the column index given by <code class="code"><var class="var">stats</var>(5)</code>, or zero if no
such row index exists. <code class="code"><var class="var">stats</var>(7)</code> is the number of duplicate
or out-of-order row indices. <code class="code"><var class="var">stats</var>(8:20)</code> is always zero in
the current version of <small class="sc">COLAMD</small> (reserved for future use).
</p>
<p>The ordering is followed by a column elimination tree post-ordering.
</p>
<p>The authors of the code itself are Stefan I. Larimore and
Timothy A. Davis. The algorithm was developed in collaboration with
John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National
Laboratory. (see <a class="url" href="http://faculty.cse.tamu.edu/davis/suitesparse.html">http://faculty.cse.tamu.edu/davis/suitesparse.html</a>)
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcolperm">colperm</a>, <a class="ref" href="#XREFsymamd">symamd</a>, <a class="ref" href="#XREFccolamd">ccolamd</a>.
</p></dd></dl>
<a class="anchor" id="XREFcolperm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-colperm"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">colperm</strong> <code class="def-code-arguments">(<var class="var">s</var>)</code><a class="copiable-link" href="#index-colperm"> ¶</a></span></dt>
<dd><p>Return the column permutations such that the columns of
<code class="code"><var class="var">s</var>(:, <var class="var">p</var>)</code> are ordered in terms of increasing number of
nonzero elements.
</p>
<p>If <var class="var">s</var> is symmetric, then <var class="var">p</var> is chosen such that
<code class="code"><var class="var">s</var>(<var class="var">p</var>, <var class="var">p</var>)</code> orders the rows and columns with
increasing number of nonzero elements.
</p></dd></dl>
<a class="anchor" id="XREFcsymamd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-csymamd"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">csymamd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-csymamd"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-csymamd-1"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">csymamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>)</code><a class="copiable-link" href="#index-csymamd-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-csymamd-2"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">csymamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>, <var class="var">cmember</var>)</code><a class="copiable-link" href="#index-csymamd-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-csymamd-3"><span><code class="def-type">[<var class="var">p</var>, <var class="var">stats</var>] =</code> <strong class="def-name">csymamd</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-csymamd-3"> ¶</a></span></dt>
<dd>
<p>For a symmetric positive definite matrix <var class="var">S</var>, return the permutation
vector <var class="var">p</var> such that <code class="code"><var class="var">S</var>(<var class="var">p</var>,<var class="var">p</var>)</code> tends to have a
sparser Cholesky factor than <var class="var">S</var>.
</p>
<p>Sometimes <code class="code">csymamd</code> works well for symmetric indefinite matrices too.
The matrix <var class="var">S</var> is assumed to be symmetric; only the strictly lower
triangular part is referenced. <var class="var">S</var> must be square. The ordering is
followed by an elimination tree post-ordering.
</p>
<p><var class="var">knobs</var> is an optional 1-element to 3-element input vector, with a
default value of <code class="code">[10 1 0]</code>. Entries not present are set to their
defaults.
</p>
<dl class="table">
<dt><code class="code"><var class="var">knobs</var>(1)</code></dt>
<dd><p>If <var class="var">S</var> is n-by-n, then rows and columns with more than
<code class="code">max(16,<var class="var">knobs</var>(1)*sqrt(n))</code> entries are ignored, and ordered
last in the output permutation (subject to the cmember constraints).
</p>
</dd>
<dt><code class="code"><var class="var">knobs</var>(2)</code></dt>
<dd><p>If nonzero, aggressive absorption is performed.
</p>
</dd>
<dt><code class="code"><var class="var">knobs</var>(3)</code></dt>
<dd><p>If nonzero, statistics and knobs are printed.
</p>
</dd>
</dl>
<p><var class="var">cmember</var> is an optional vector of length n. It defines the constraints
on the ordering. If <code class="code"><var class="var">cmember</var>(j) = <var class="var">S</var></code>, then row/column j is
in constraint set <var class="var">c</var> (<var class="var">c</var> must be in the range 1 to n). In the
output permutation <var class="var">p</var>, rows/columns in set 1 appear first, followed
by all rows/columns in set 2, and so on. <code class="code"><var class="var">cmember</var> = ones (1,n)</code>
if not present or empty. <code class="code">csymamd (<var class="var">S</var>,[],1:n)</code> returns
<code class="code">1:n</code>.
</p>
<p><code class="code"><var class="var">p</var> = csymamd (<var class="var">S</var>)</code> is about the same as
<code class="code"><var class="var">p</var> = symamd (<var class="var">S</var>)</code>. <var class="var">knobs</var> and its default values
differ.
</p>
<p><code class="code"><var class="var">stats</var>(4:7)</code> provide information if CCOLAMD was able to
continue. The matrix is OK if <code class="code"><var class="var">stats</var>(4)</code> is zero, or 1 if
invalid. <code class="code"><var class="var">stats</var>(5)</code> is the rightmost column index that is
unsorted or contains duplicate entries, or zero if no such column exists.
<code class="code"><var class="var">stats</var>(6)</code> is the last seen duplicate or out-of-order row
index in the column index given by <code class="code"><var class="var">stats</var>(5)</code>, or zero if no
such row index exists. <code class="code"><var class="var">stats</var>(7)</code> is the number of duplicate
or out-of-order row indices. <code class="code"><var class="var">stats</var>(8:20)</code> is always zero in
the current version of <small class="sc">CCOLAMD</small> (reserved for future use).
</p>
<p>The authors of the code itself are S. Larimore, T. Davis and
S. Rajamanickam in collaboration with J. Bilbert and E. Ng.
Supported by the National Science Foundation
(DMS-9504974, DMS-9803599, CCR-0203270), and a grant from
Sandia National Lab.
See <a class="url" href="http://faculty.cse.tamu.edu/davis/suitesparse.html">http://faculty.cse.tamu.edu/davis/suitesparse.html</a> for ccolamd,
colamd, csymamd, amd, colamd, symamd, and other related orderings.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFsymamd">symamd</a>, <a class="ref" href="#XREFccolamd">ccolamd</a>.
</p></dd></dl>
<a class="anchor" id="XREFdmperm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-dmperm"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">dmperm</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-dmperm"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-dmperm-1"><span><code class="def-type">[<var class="var">p</var>, <var class="var">q</var>, <var class="var">r</var>, <var class="var">s</var>, <var class="var">cc</var>, <var class="var">rr</var>] =</code> <strong class="def-name">dmperm</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-dmperm-1"> ¶</a></span></dt>
<dd>
<a class="index-entry-id" id="index-Dulmage_002dMendelsohn-decomposition"></a>
<p>Perform a Dulmage-Mendelsohn permutation of the sparse matrix
<var class="var">A</var>.
</p>
<p>With a single output argument <code class="code">dmperm</code>, return a maximum matching <var class="var">p</var>
such that <code class="code">p(j) = i</code> if column <var class="var">j</var> is matched to row <var class="var">i</var>, or 0 if
column <var class="var">j</var> is unmatched. If <var class="var">A</var> is square and full structural rank,
<var class="var">p</var> is a row permutation and <code class="code">A(p,:)</code> has a zero-free diagonal. The
structural rank of <var class="var">A</var> is <code class="code">sprank(A) = sum(p>0)</code>.
</p>
<p>Called with two or more output arguments, return the
Dulmage-Mendelsohn decomposition of <var class="var">A</var>. <var class="var">p</var> and <var class="var">q</var> are
permutation vectors. <var class="var">cc</var> and <var class="var">rr</var> are vectors of length 5.
<code class="code">c = A(p,q)</code> is split into a 4-by-4 set of coarse blocks:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> A11 A12 A13 A14
0 0 A23 A24
0 0 0 A34
0 0 0 A44
</pre></div></div>
<p>where <code class="code">A12</code>, <code class="code">A23</code>, and <code class="code">A34</code> are square with zero-free
diagonals. The columns of <code class="code">A11</code> are the unmatched columns, and the rows
of <code class="code">A44</code> are the unmatched rows. Any of these blocks can be empty. In
the "coarse" decomposition, the (i,j)-th block is
<code class="code">C(rr(i):rr(i+1)-1,cc(j):cc(j+1)-1)</code>. In terms of a linear system,
<code class="code">[A11 A12]</code> is the underdetermined part of the system (it is always
rectangular and with more columns and rows, or 0-by-0), <code class="code">A23</code> is the
well-determined part of the system (it is always square), and
<code class="code">[A34 ; A44]</code> is the over-determined part of the system (it is always
rectangular with more rows than columns, or 0-by-0).
</p>
<p>The structural rank of <var class="var">A</var> is <code class="code">sprank (A) = rr(4)-1</code>, which is an
upper bound on the numerical rank of <var class="var">A</var>.
<code class="code">sprank(A) = rank(full(sprand(A)))</code> with probability 1 in exact
arithmetic.
</p>
<p>The <code class="code">A23</code> submatrix is further subdivided into block upper triangular form
via the "fine" decomposition (the strongly-connected components of <code class="code">A23</code>).
If <var class="var">A</var> is square and structurally non-singular, <code class="code">A23</code> is the entire
matrix.
</p>
<p><code class="code">C(r(i):r(i+1)-1,s(j):s(j+1)-1)</code> is the (i,j)-th block of the fine
decomposition. The (1,1) block is the rectangular block <code class="code">[A11 A12]</code>,
unless this block is 0-by-0. The (b,b) block is the rectangular block
<code class="code">[A34 ; A44]</code>, unless this block is 0-by-0, where <code class="code">b = length(r)-1</code>.
All other blocks of the form <code class="code">C(r(i):r(i+1)-1,s(i):s(i+1)-1)</code> are diagonal
blocks of <code class="code">A23</code>, and are square with a zero-free diagonal.
</p>
<p>The method used is described in: A. Pothen & C.-J. Fan.
<cite class="cite">Computing the Block Triangular Form of a Sparse Matrix</cite>.
ACM Trans. Math. Software, 16(4):303–324, 1990.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcolamd">colamd</a>, <a class="ref" href="#XREFccolamd">ccolamd</a>.
</p></dd></dl>
<a class="anchor" id="XREFsymamd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-symamd"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">symamd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-symamd"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-symamd-1"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">symamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>)</code><a class="copiable-link" href="#index-symamd-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-symamd-2"><span><code class="def-type">[<var class="var">p</var>, <var class="var">stats</var>] =</code> <strong class="def-name">symamd</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-symamd-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-symamd-3"><span><code class="def-type">[<var class="var">p</var>, <var class="var">stats</var>] =</code> <strong class="def-name">symamd</strong> <code class="def-code-arguments">(<var class="var">S</var>, <var class="var">knobs</var>)</code><a class="copiable-link" href="#index-symamd-3"> ¶</a></span></dt>
<dd>
<p>For a symmetric positive definite matrix <var class="var">S</var>, returns the permutation
vector p such that <code class="code"><var class="var">S</var>(<var class="var">p</var>, <var class="var">p</var>)</code> tends to have a
sparser Cholesky factor than <var class="var">S</var>.
</p>
<p>Sometimes <code class="code">symamd</code> works well for symmetric indefinite matrices too.
The matrix <var class="var">S</var> is assumed to be symmetric; only the strictly lower
triangular part is referenced. <var class="var">S</var> must be square.
</p>
<p><var class="var">knobs</var> is an optional one- to two-element input vector. If <var class="var">S</var> is
n-by-n, then rows and columns with more than
<code class="code">max (16,<var class="var">knobs</var>(1)*sqrt(n))</code> entries are removed prior to
ordering, and ordered last in the output permutation <var class="var">p</var>. No
rows/columns are removed if <code class="code"><var class="var">knobs</var>(1) < 0</code>. If
<code class="code"><var class="var">knobs</var>(2)</code> is nonzero, <var class="var">stats</var> and <var class="var">knobs</var> are
printed. The default is <code class="code"><var class="var">knobs</var> = [10 0]</code>. Note that
<var class="var">knobs</var> differs from earlier versions of <code class="code">symamd</code>.
</p>
<p><var class="var">stats</var> is an optional 20-element output vector that provides data
about the ordering and the validity of the input matrix <var class="var">S</var>. Ordering
statistics are in <code class="code"><var class="var">stats</var>(1:3)</code>.
<code class="code"><var class="var">stats</var>(1) = <var class="var">stats</var>(2)</code> is the number of dense or empty rows
and columns ignored by SYMAMD and <code class="code"><var class="var">stats</var>(3)</code> is the number of
garbage collections performed on the internal data structure used by SYMAMD
(roughly of size <code class="code">8.4 * nnz (tril (<var class="var">S</var>, -1)) + 9 * <var class="var">n</var></code>
integers).
</p>
<p>Octave built-in functions are intended to generate valid sparse matrices,
with no duplicate entries, with ascending row indices of the nonzeros
in each column, with a non-negative number of entries in each column (!)
and so on. If a matrix is invalid, then SYMAMD may or may not be able
to continue. If there are duplicate entries (a row index appears two or
more times in the same column) or if the row indices in a column are out
of order, then SYMAMD can correct these errors by ignoring the duplicate
entries and sorting each column of its internal copy of the matrix S (the
input matrix S is not repaired, however). If a matrix is invalid in
other ways then SYMAMD cannot continue, an error message is printed, and
no output arguments (<var class="var">p</var> or <var class="var">stats</var>) are returned. SYMAMD is
thus a simple way to check a sparse matrix to see if it’s valid.
</p>
<p><code class="code"><var class="var">stats</var>(4:7)</code> provide information if SYMAMD was able to
continue. The matrix is OK if <code class="code"><var class="var">stats</var> (4)</code> is zero, or 1
if invalid. <code class="code"><var class="var">stats</var>(5)</code> is the rightmost column index that
is unsorted or contains duplicate entries, or zero if no such column
exists. <code class="code"><var class="var">stats</var>(6)</code> is the last seen duplicate or out-of-order
row index in the column index given by <code class="code"><var class="var">stats</var>(5)</code>, or zero
if no such row index exists. <code class="code"><var class="var">stats</var>(7)</code> is the number of
duplicate or out-of-order row indices. <code class="code"><var class="var">stats</var>(8:20)</code> is
always zero in the current version of SYMAMD (reserved for future use).
</p>
<p>The ordering is followed by a column elimination tree post-ordering.
</p>
<p>The authors of the code itself are Stefan I. Larimore and
Timothy A. Davis. The algorithm was developed in collaboration with
John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National
Laboratory. (see <a class="url" href="http://faculty.cse.tamu.edu/davis/suitesparse.html">http://faculty.cse.tamu.edu/davis/suitesparse.html</a>)
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcolperm">colperm</a>, <a class="ref" href="#XREFcolamd">colamd</a>.
</p></dd></dl>
<a class="anchor" id="XREFsymrcm"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-symrcm"><span><code class="def-type"><var class="var">p</var> =</code> <strong class="def-name">symrcm</strong> <code class="def-code-arguments">(<var class="var">S</var>)</code><a class="copiable-link" href="#index-symrcm"> ¶</a></span></dt>
<dd><p>Return the symmetric reverse Cuthill-McKee permutation of <var class="var">S</var>.
</p>
<p><var class="var">p</var> is a permutation vector such that
<code class="code"><var class="var">S</var>(<var class="var">p</var>, <var class="var">p</var>)</code> tends to have its diagonal elements closer
to the diagonal than <var class="var">S</var>. This is a good preordering for LU or
Cholesky factorization of matrices that come from “long, skinny”
problems. It works for both symmetric and asymmetric <var class="var">S</var>.
</p>
<p>The algorithm represents a heuristic approach to the NP-complete bandwidth
minimization problem. The implementation is based in the descriptions found
in
</p>
<p>E. Cuthill, J. McKee.
<cite class="cite">Reducing the Bandwidth of Sparse Symmetric Matrices</cite>.
Proceedings of the 24th ACM National Conference,
157–172 1969, Brandon Press, New Jersey.
</p>
<p>A. George, J.W.H. Liu. <cite class="cite">Computer Solution of Large Sparse
Positive Definite Systems</cite>, Prentice Hall Series in Computational
Mathematics, ISBN 0-13-165274-5, 1981.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcolperm">colperm</a>, <a class="ref" href="#XREFcolamd">colamd</a>, <a class="ref" href="#XREFsymamd">symamd</a>.
</p></dd></dl>
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