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<a name="Sparse-Linear-Algebra"></a>
<div class="header">
<p>
Next: <a href="Iterative-Techniques.html#Iterative-Techniques" accesskey="n" rel="next">Iterative Techniques</a>, Previous: <a href="Basics.html#Basics" accesskey="p" rel="prev">Basics</a>, Up: <a href="Sparse-Matrices.html#Sparse-Matrices" accesskey="u" rel="up">Sparse Matrices</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Linear-Algebra-on-Sparse-Matrices"></a>
<h3 class="section">22.2 Linear Algebra on Sparse Matrices</h3>
<p>Octave includes a polymorphic solver for sparse matrices, where
the exact solver used to factorize the matrix, depends on the properties
of the sparse matrix itself. Generally, the cost of determining the matrix
type is small relative to the cost of factorizing the matrix itself, but in
any case the matrix type is cached once it is calculated, so that it is not
re-determined each time it is used in a linear equation.
</p>
<p>The selection tree for how the linear equation is solve is
</p>
<ol>
<li> If the matrix is diagonal, solve directly and goto 8
</li><li> If the matrix is a permuted diagonal, solve directly taking into
account the permutations. Goto 8
</li><li> If the matrix is square, banded and if the band density is less
than that given by <code>spparms ("bandden")</code> continue, else goto 4.
<ol type="a" start="1">
<li> If the matrix is tridiagonal and the right-hand side is not sparse
continue, else goto 3b.
<ol>
<li> If the matrix is Hermitian, with a positive real diagonal, attempt
Cholesky factorization using <small>LAPACK</small> xPTSV.
</li><li> If the above failed or the matrix is not Hermitian with a positive
real diagonal use Gaussian elimination with pivoting using
<small>LAPACK</small> xGTSV, and goto 8.
</li></ol>
</li><li> If the matrix is Hermitian with a positive real diagonal, attempt
Cholesky factorization using <small>LAPACK</small> xPBTRF.
</li><li> if the above failed or the matrix is not Hermitian with a positive
real diagonal use Gaussian elimination with pivoting using
<small>LAPACK</small> xGBTRF, and goto 8.
</li></ol>
</li><li> If the matrix is upper or lower triangular perform a sparse forward
or backward substitution, and goto 8
</li><li> If the matrix is an upper triangular matrix with column permutations
or lower triangular matrix with row permutations, perform a sparse forward
or backward substitution, and goto 8
</li><li> If the matrix is square, Hermitian with a real positive diagonal, attempt
sparse Cholesky factorization using <small>CHOLMOD</small>.
</li><li> If the sparse Cholesky factorization failed or the matrix is not
Hermitian with a real positive diagonal, and the matrix is square, factorize,
solve, and perform one refinement iteration using <small>UMFPACK</small>.
</li><li> If the matrix is not square, or any of the previous solvers flags
a singular or near singular matrix, find a minimum norm solution using
<small>CXSPARSE</small><a name="DOCF10" href="#FOOT10"><sup>10</sup></a>.
</li></ol>
<p>The band density is defined as the number of nonzero values in the band
divided by the total number of values in the full band. The banded
matrix solvers can be entirely disabled by using <em>spparms</em> to set
<code>bandden</code> to 1 (i.e., <code>spparms ("bandden", 1)</code>).
</p>
<p>The QR solver factorizes the problem with a Dulmage-Mendelsohn
decomposition, to separate the problem into blocks that can be treated
as over-determined, multiple well determined blocks, and a final
over-determined block. For matrices with blocks of strongly connected
nodes this is a big win as LU decomposition can be used for many
blocks. It also significantly improves the chance of finding a solution
to over-determined problems rather than just returning a vector of
<em>NaN</em>’s.
</p>
<p>All of the solvers above, can calculate an estimate of the condition
number. This can be used to detect numerical stability problems in the
solution and force a minimum norm solution to be used. However, for
narrow banded, triangular or diagonal matrices, the cost of
calculating the condition number is significant, and can in fact
exceed the cost of factoring the matrix. Therefore the condition
number is not calculated in these cases, and Octave relies on simpler
techniques to detect singular matrices or the underlying <small>LAPACK</small> code in
the case of banded matrices.
</p>
<p>The user can force the type of the matrix with the <code>matrix_type</code>
function. This overcomes the cost of discovering the type of the matrix.
However, it should be noted that identifying the type of the matrix incorrectly
will lead to unpredictable results, and so <code>matrix_type</code> should be
used with care.
</p>
<a name="XREFnormest"></a><dl>
<dt><a name="index-normest"></a><em><var>nest</var> =</em> <strong>normest</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-normest-1"></a><em><var>nest</var> =</em> <strong>normest</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dt><a name="index-normest-2"></a><em>[<var>nest</var>, <var>iter</var>] =</em> <strong>normest</strong> <em>(…)</em></dt>
<dd><p>Estimate the 2-norm of the matrix <var>A</var> using a power series analysis.
</p>
<p>This is typically used for large matrices, where the cost of calculating
<code>norm (<var>A</var>)</code> is prohibitive and an approximation to the 2-norm is
acceptable.
</p>
<p><var>tol</var> is the tolerance to which the 2-norm is calculated. By default
<var>tol</var> is 1e-6.
</p>
<p>The optional output <var>iter</var> returns the number of iterations that were
required for <code>normest</code> to converge.
</p>
<p><strong>See also:</strong> <a href="#XREFnormest1">normest1</a>, <a href="Basic-Matrix-Functions.html#XREFnorm">norm</a>, <a href="Basic-Matrix-Functions.html#XREFcond">cond</a>, <a href="#XREFcondest">condest</a>.
</p></dd></dl>
<a name="XREFnormest1"></a><dl>
<dt><a name="index-normest1"></a><em><var>nest</var> =</em> <strong>normest1</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-normest1-1"></a><em><var>nest</var> =</em> <strong>normest1</strong> <em>(<var>A</var>, <var>t</var>)</em></dt>
<dt><a name="index-normest1-2"></a><em><var>nest</var> =</em> <strong>normest1</strong> <em>(<var>A</var>, <var>t</var>, <var>x0</var>)</em></dt>
<dt><a name="index-normest1-3"></a><em><var>nest</var> =</em> <strong>normest1</strong> <em>(<var>Afun</var>, <var>t</var>, <var>x0</var>, <var>p1</var>, <var>p2</var>, …)</em></dt>
<dt><a name="index-normest1-4"></a><em>[<var>nest</var>, <var>v</var>] =</em> <strong>normest1</strong> <em>(<var>A</var>, …)</em></dt>
<dt><a name="index-normest1-5"></a><em>[<var>nest</var>, <var>v</var>, <var>w</var>] =</em> <strong>normest1</strong> <em>(<var>A</var>, …)</em></dt>
<dt><a name="index-normest1-6"></a><em>[<var>nest</var>, <var>v</var>, <var>w</var>, <var>iter</var>] =</em> <strong>normest1</strong> <em>(<var>A</var>, …)</em></dt>
<dd><p>Estimate the 1-norm of the matrix <var>A</var> using a block algorithm.
</p>
<p><code>normest1</code> is best for large sparse matrices where only an estimate of
the norm is required. For small to medium sized matrices, consider using
<code>norm (<var>A</var>, 1)</code>. In addition, <code>normest1</code> can be used for the
estimate of the 1-norm of a linear operator <var>A</var> when matrix-vector
products <code><var>A</var> * <var>x</var></code> and <code><var>A</var>' * <var>x</var></code> can be
cheaply computed. In this case, instead of the matrix <var>A</var>, a function
<code><var>Afun</var> (<var>flag</var>, <var>x</var>)</code> is used; it must return:
</p>
<ul>
<li> the dimension <var>n</var> of <var>A</var>, if <var>flag</var> is <code>"dim"</code>
</li><li> true if <var>A</var> is a real operator, if <var>flag</var> is <code>"real"</code>
</li><li> the result <code><var>A</var> * <var>x</var></code>, if <var>flag</var> is <code>"notransp"</code>
</li><li> the result <code><var>A</var>' * <var>x</var></code>, if <var>flag</var> is <code>"transp"</code>
</li></ul>
<p>A typical case is <var>A</var> defined by <code><var>b</var> ^ <var>m</var></code>, in which the
result <code><var>A</var> * <var>x</var></code> can be computed without even forming
explicitly <code><var>b</var> ^ <var>m</var></code> by:
</p>
<div class="example">
<pre class="example"><var>y</var> = <var>x</var>;
for <var>i</var> = 1:<var>m</var>
<var>y</var> = <var>b</var> * <var>y</var>;
endfor
</pre></div>
<p>The parameters <var>p1</var>, <var>p2</var>, … are arguments of
<code><var>Afun</var> (<var>flag</var>, <var>x</var>, <var>p1</var>, <var>p2</var>, …)</code>.
</p>
<p>The default value for <var>t</var> is 2. The algorithm requires matrix-matrix
products with sizes <var>n</var> x <var>n</var> and <var>n</var> x <var>t</var>.
</p>
<p>The initial matrix <var>x0</var> should have columns of unit 1-norm. The default
initial matrix <var>x0</var> has the first column
<code>ones (<var>n</var>, 1) / <var>n</var></code> and, if <var>t</var> > 1, the remaining
columns with random elements <code>-1 / <var>n</var></code>, <code>1 / <var>n</var></code>,
divided by <var>n</var>.
</p>
<p>On output, <var>nest</var> is the desired estimate, <var>v</var> and <var>w</var>
are vectors such that <code><var>w</var> = <var>A</var> * <var>v</var></code>, with
<code>norm (<var>w</var>, 1)</code> = <code><var>c</var> * norm (<var>v</var>, 1)</code>. <var>iter</var>
contains in <code><var>iter</var>(1)</code> the number of iterations (the maximum is
hardcoded to 5) and in <code><var>iter</var>(2)</code> the total number of products
<code><var>A</var> * <var>x</var></code> or <code><var>A</var>' * <var>x</var></code> performed by the
algorithm.
</p>
<p>Algorithm Note: <code>normest1</code> uses random numbers during evaluation.
Therefore, if consistent results are required, the <code>"state"</code> of the
random generator should be fixed before invoking <code>normest1</code>.
</p>
<p>Reference: N. J. Higham and F. Tisseur,
<cite>A block algorithm for matrix 1-norm estimation, with and
application to 1-norm pseudospectra</cite>,
SIAM J. Matrix Anal. Appl.,
pp. 1185–1201, Vol 21, No. 4, 2000.
</p>
<p><strong>See also:</strong> <a href="#XREFnormest">normest</a>, <a href="Basic-Matrix-Functions.html#XREFnorm">norm</a>, <a href="Basic-Matrix-Functions.html#XREFcond">cond</a>, <a href="#XREFcondest">condest</a>.
</p></dd></dl>
<a name="XREFcondest"></a><dl>
<dt><a name="index-condest"></a><em><var>cest</var> =</em> <strong>condest</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-condest-1"></a><em><var>cest</var> =</em> <strong>condest</strong> <em>(<var>A</var>, <var>t</var>)</em></dt>
<dt><a name="index-condest-2"></a><em><var>cest</var> =</em> <strong>condest</strong> <em>(<var>A</var>, <var>solvefun</var>, <var>t</var>, <var>p1</var>, <var>p2</var>, …)</em></dt>
<dt><a name="index-condest-3"></a><em><var>cest</var> =</em> <strong>condest</strong> <em>(<var>Afcn</var>, <var>solvefun</var>, <var>t</var>, <var>p1</var>, <var>p2</var>, …)</em></dt>
<dt><a name="index-condest-4"></a><em>[<var>cest</var>, <var>v</var>] =</em> <strong>condest</strong> <em>(…)</em></dt>
<dd>
<p>Estimate the 1-norm condition number of a square matrix <var>A</var> using
<var>t</var> test vectors and a randomized 1-norm estimator.
</p>
<p>The optional input <var>t</var> specifies the number of test vectors (default 5).
</p>
<p>If the matrix is not explicit, e.g., when estimating the condition number of
<var>A</var> given an LU factorization, <code>condest</code> uses the following
functions:
</p>
<ul class="no-bullet">
<li>- <var>Afcn</var> which must return
<ul>
<li> the dimension <var>n</var> of <var>a</var>, if <var>flag</var> is <code>"dim"</code>
</li><li> true if <var>a</var> is a real operator, if <var>flag</var> is <code>"real"</code>
</li><li> the result <code><var>a</var> * <var>x</var></code>, if <var>flag</var> is "notransp"
</li><li> the result <code><var>a</var>' * <var>x</var></code>, if <var>flag</var> is "transp"
</li></ul>
</li><li>- <var>solvefun</var> which must return
<ul>
<li> the dimension <var>n</var> of <var>a</var>, if <var>flag</var> is <code>"dim"</code>
</li><li> true if <var>a</var> is a real operator, if <var>flag</var> is <code>"real"</code>
</li><li> the result <code><var>a</var> \ <var>x</var></code>, if <var>flag</var> is "notransp"
</li><li> the result <code><var>a</var>' \ <var>x</var></code>, if <var>flag</var> is "transp"
</li></ul>
</li></ul>
<p>The parameters <var>p1</var>, <var>p2</var>, … are arguments of
<code><var>Afcn</var> (<var>flag</var>, <var>x</var>, <var>p1</var>, <var>p2</var>, …)</code>
and <code><var>solvefcn</var> (<var>flag</var>, <var>x</var>, <var>p1</var>, <var>p2</var>,
…)</code>.
</p>
<p>The principal output is the 1-norm condition number estimate <var>cest</var>.
</p>
<p>The optional second output is an approximate null vector when <var>cest</var> is
large; it satisfies the equation
<code>norm (A*v, 1) == norm (A, 1) * norm (<var>v</var>, 1) / <var>est</var></code>.
</p>
<p>Algorithm Note: <code>condest</code> uses a randomized algorithm to approximate
the 1-norms. Therefore, if consistent results are required, the
<code>"state"</code> of the random generator should be fixed before invoking
<code>condest</code>.
</p>
<p>References:
</p>
<ul>
<li> N.J. Higham and F. Tisseur, <cite>A Block Algorithm
for Matrix 1-Norm Estimation, with an Application to 1-Norm
Pseudospectra</cite>. SIMAX vol 21, no 4, pp 1185-1201.
<a href="https://dx.doi.org/10.1137/S0895479899356080">https://dx.doi.org/10.1137/S0895479899356080</a>
</li><li> N.J. Higham and F. Tisseur, <cite>A Block Algorithm
for Matrix 1-Norm Estimation, with an Application to 1-Norm
Pseudospectra</cite>. <a href="https://citeseer.ist.psu.edu/223007.html">https://citeseer.ist.psu.edu/223007.html</a>
</li></ul>
<p><strong>See also:</strong> <a href="Basic-Matrix-Functions.html#XREFcond">cond</a>, <a href="Basic-Matrix-Functions.html#XREFnorm">norm</a>, <a href="#XREFnormest1">normest1</a>, <a href="#XREFnormest">normest</a>.
</p></dd></dl>
<a name="XREFspparms"></a><dl>
<dt><a name="index-spparms"></a><em></em> <strong>spparms</strong> <em>()</em></dt>
<dt><a name="index-spparms-1"></a><em><var>vals</var> =</em> <strong>spparms</strong> <em>()</em></dt>
<dt><a name="index-spparms-2"></a><em>[<var>keys</var>, <var>vals</var>] =</em> <strong>spparms</strong> <em>()</em></dt>
<dt><a name="index-spparms-3"></a><em><var>val</var> =</em> <strong>spparms</strong> <em>(<var>key</var>)</em></dt>
<dt><a name="index-spparms-4"></a><em></em> <strong>spparms</strong> <em>(<var>vals</var>)</em></dt>
<dt><a name="index-spparms-5"></a><em></em> <strong>spparms</strong> <em>("default")</em></dt>
<dt><a name="index-spparms-6"></a><em></em> <strong>spparms</strong> <em>("tight")</em></dt>
<dt><a name="index-spparms-7"></a><em></em> <strong>spparms</strong> <em>(<var>key</var>, <var>val</var>)</em></dt>
<dd><p>Query or set the parameters used by the sparse solvers and factorization
functions.
</p>
<p>The first four calls above get information about the current settings, while
the others change the current settings. The parameters are stored as pairs
of keys and values, where the values are all floats and the keys are one of
the following strings:
</p>
<dl compact="compact">
<dt>‘<samp>spumoni</samp>’</dt>
<dd><p>Printing level of debugging information of the solvers (default 0)
</p>
</dd>
<dt>‘<samp>ths_rel</samp>’</dt>
<dd><p>Included for compatibility. Not used. (default 1)
</p>
</dd>
<dt>‘<samp>ths_abs</samp>’</dt>
<dd><p>Included for compatibility. Not used. (default 1)
</p>
</dd>
<dt>‘<samp>exact_d</samp>’</dt>
<dd><p>Included for compatibility. Not used. (default 0)
</p>
</dd>
<dt>‘<samp>supernd</samp>’</dt>
<dd><p>Included for compatibility. Not used. (default 3)
</p>
</dd>
<dt>‘<samp>rreduce</samp>’</dt>
<dd><p>Included for compatibility. Not used. (default 3)
</p>
</dd>
<dt>‘<samp>wh_frac</samp>’</dt>
<dd><p>Included for compatibility. Not used. (default 0.5)
</p>
</dd>
<dt>‘<samp>autommd</samp>’</dt>
<dd><p>Flag whether the LU/QR and the ’\’ and ’/’ operators will automatically
use the sparsity preserving mmd functions (default 1)
</p>
</dd>
<dt>‘<samp>autoamd</samp>’</dt>
<dd><p>Flag whether the LU and the ’\’ and ’/’ operators will automatically
use the sparsity preserving amd functions (default 1)
</p>
</dd>
<dt>‘<samp>piv_tol</samp>’</dt>
<dd><p>The pivot tolerance of the <small>UMFPACK</small> solvers (default 0.1)
</p>
</dd>
<dt>‘<samp>sym_tol</samp>’</dt>
<dd><p>The pivot tolerance of the <small>UMFPACK</small> symmetric solvers (default 0.001)
</p>
</dd>
<dt>‘<samp>bandden</samp>’</dt>
<dd><p>The density of nonzero elements in a banded matrix before it is treated
by the <small>LAPACK</small> banded solvers (default 0.5)
</p>
</dd>
<dt>‘<samp>umfpack</samp>’</dt>
<dd><p>Flag whether the <small>UMFPACK</small> or mmd solvers are used for the LU, ’\’ and
’/’ operations (default 1)
</p></dd>
</dl>
<p>The value of individual keys can be set with
<code>spparms (<var>key</var>, <var>val</var>)</code>.
The default values can be restored with the special keyword
<code>"default"</code>. The special keyword <code>"tight"</code> can be used to
set the mmd solvers to attempt a sparser solution at the potential cost of
longer running time.
</p>
<p><strong>See also:</strong> <a href="Matrix-Factorizations.html#XREFchol">chol</a>, <a href="Mathematical-Considerations.html#XREFcolamd">colamd</a>, <a href="Matrix-Factorizations.html#XREFlu">lu</a>, <a href="Matrix-Factorizations.html#XREFqr">qr</a>, <a href="Mathematical-Considerations.html#XREFsymamd">symamd</a>.
</p></dd></dl>
<a name="XREFsprank"></a><dl>
<dt><a name="index-sprank"></a><em><var>p</var> =</em> <strong>sprank</strong> <em>(<var>S</var>)</em></dt>
<dd><a name="index-structural-rank"></a>
<p>Calculate the structural rank of the sparse matrix <var>S</var>.
</p>
<p>Note that only the structure of the matrix is used in this calculation based
on a Dulmage-Mendelsohn permutation to block triangular form. As
such the numerical rank of the matrix <var>S</var> is bounded by
<code>sprank (<var>S</var>) >= rank (<var>S</var>)</code>. Ignoring floating point errors
<code>sprank (<var>S</var>) == rank (<var>S</var>)</code>.
</p>
<p><strong>See also:</strong> <a href="Mathematical-Considerations.html#XREFdmperm">dmperm</a>.
</p></dd></dl>
<a name="XREFsymbfact"></a><dl>
<dt><a name="index-symbfact"></a><em>[<var>count</var>, <var>h</var>, <var>parent</var>, <var>post</var>, <var>R</var>] =</em> <strong>symbfact</strong> <em>(<var>S</var>)</em></dt>
<dt><a name="index-symbfact-1"></a><em>[…] =</em> <strong>symbfact</strong> <em>(<var>S</var>, <var>typ</var>)</em></dt>
<dt><a name="index-symbfact-2"></a><em>[…] =</em> <strong>symbfact</strong> <em>(<var>S</var>, <var>typ</var>, <var>mode</var>)</em></dt>
<dd>
<p>Perform a symbolic factorization analysis of the sparse matrix <var>S</var>.
</p>
<p>The input variables are
</p>
<dl compact="compact">
<dt><var>S</var></dt>
<dd><p><var>S</var> is a real or complex sparse matrix.
</p>
</dd>
<dt><var>typ</var></dt>
<dd><p>Is the type of the factorization and can be one of
</p>
<dl compact="compact">
<dt><code>"sym"</code> (default)</dt>
<dd><p>Factorize <var>S</var>. Assumes <var>S</var> is symmetric and uses the upper
triangular portion of the matrix.
</p>
</dd>
<dt><code>"col"</code></dt>
<dd><p>Factorize <code><var>S</var>' * <var>S</var></code>.
</p>
</dd>
<dt><code>"row"</code></dt>
<dd><p>Factorize <code><var>S</var> * <var>S</var>'</code>.
</p>
</dd>
<dt><code>"lo"</code></dt>
<dd><p>Factorize <code><var>S</var>'</code>. Assumes <var>S</var> is symmetric and uses the lower
triangular portion of the matrix.
</p></dd>
</dl>
</dd>
<dt><var>mode</var></dt>
<dd><p>When <var>mode</var> is unspecified return the Cholesky factorization for
<var>R</var>. If <var>mode</var> is <code>"lower"</code> or <code>"L"</code> then return
the conjugate transpose <code><var>R</var>'</code> which is a lower triangular factor.
The conjugate transpose version is faster and uses less memory, but still
returns the same values for all other outputs: <var>count</var>, <var>h</var>,
<var>parent</var>, and <var>post</var>.
</p></dd>
</dl>
<p>The output variables are:
</p>
<dl compact="compact">
<dt><var>count</var></dt>
<dd><p>The row counts of the Cholesky factorization as determined by
<var>typ</var>. The computational difficulty of performing the true
factorization using <code>chol</code> is <code>sum (<var>count</var> .^ 2)</code>.
</p>
</dd>
<dt><var>h</var></dt>
<dd><p>The height of the elimination tree.
</p>
</dd>
<dt><var>parent</var></dt>
<dd><p>The elimination tree itself.
</p>
</dd>
<dt><var>post</var></dt>
<dd><p>A sparse boolean matrix whose structure is that of the
Cholesky factorization as determined by <var>typ</var>.
</p></dd>
</dl>
<p><strong>See also:</strong> <a href="Matrix-Factorizations.html#XREFchol">chol</a>, <a href="Information.html#XREFetree">etree</a>, <a href="Information.html#XREFtreelayout">treelayout</a>.
</p></dd></dl>
<p>For non square matrices, the user can also utilize the <code>spaugment</code>
function to find a least squares solution to a linear equation.
</p>
<a name="XREFspaugment"></a><dl>
<dt><a name="index-spaugment"></a><em><var>s</var> =</em> <strong>spaugment</strong> <em>(<var>A</var>, <var>c</var>)</em></dt>
<dd><p>Create the augmented matrix of <var>A</var>.
</p>
<p>This is given by
</p>
<div class="example">
<pre class="example">[<var>c</var> * eye(<var>m</var>, <var>m</var>), <var>A</var>;
<var>A</var>', zeros(<var>n</var>, <var>n</var>)]
</pre></div>
<p>This is related to the least squares solution of
<code><var>A</var> \ <var>b</var></code>, by
</p>
<div class="example">
<pre class="example"><var>s</var> * [ <var>r</var> / <var>c</var>; x] = [ <var>b</var>, zeros(<var>n</var>, columns(<var>b</var>)) ]
</pre></div>
<p>where <var>r</var> is the residual error
</p>
<div class="example">
<pre class="example"><var>r</var> = <var>b</var> - <var>A</var> * <var>x</var>
</pre></div>
<p>As the matrix <var>s</var> is symmetric indefinite it can be factorized with
<code>lu</code>, and the minimum norm solution can therefore be found without the
need for a <code>qr</code> factorization. As the residual error will be
<code>zeros (<var>m</var>, <var>m</var>)</code> for underdetermined problems, and example
can be
</p>
<div class="example">
<pre class="example">m = 11; n = 10; mn = max (m, n);
A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],
[-1, 0, 1], m, n);
x0 = A \ ones (m,1);
s = spaugment (A);
[L, U, P, Q] = lu (s);
x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)])));
x1 = x1(end - n + 1 : end);
</pre></div>
<p>To find the solution of an overdetermined problem needs an estimate of the
residual error <var>r</var> and so it is more complex to formulate a minimum norm
solution using the <code>spaugment</code> function.
</p>
<p>In general the left division operator is more stable and faster than using
the <code>spaugment</code> function.
</p>
<p><strong>See also:</strong> <a href="Arithmetic-Ops.html#XREFmldivide">mldivide</a>.
</p></dd></dl>
<p>Finally, the function <code>eigs</code> can be used to calculate a limited
number of eigenvalues and eigenvectors based on a selection criteria
and likewise for <code>svds</code> which calculates a limited number of
singular values and vectors.
</p>
<a name="XREFeigs"></a><dl>
<dt><a name="index-eigs"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-eigs-1"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>k</var>)</em></dt>
<dt><a name="index-eigs-2"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>k</var>, <var>sigma</var>)</em></dt>
<dt><a name="index-eigs-3"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>k</var>, <var>sigma</var>, <var>opts</var>)</em></dt>
<dt><a name="index-eigs-4"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt><a name="index-eigs-5"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>B</var>, <var>k</var>)</em></dt>
<dt><a name="index-eigs-6"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>B</var>, <var>k</var>, <var>sigma</var>)</em></dt>
<dt><a name="index-eigs-7"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>A</var>, <var>B</var>, <var>k</var>, <var>sigma</var>, <var>opts</var>)</em></dt>
<dt><a name="index-eigs-8"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>)</em></dt>
<dt><a name="index-eigs-9"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>B</var>)</em></dt>
<dt><a name="index-eigs-10"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>k</var>)</em></dt>
<dt><a name="index-eigs-11"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>B</var>, <var>k</var>)</em></dt>
<dt><a name="index-eigs-12"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>k</var>, <var>sigma</var>)</em></dt>
<dt><a name="index-eigs-13"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>B</var>, <var>k</var>, <var>sigma</var>)</em></dt>
<dt><a name="index-eigs-14"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>k</var>, <var>sigma</var>, <var>opts</var>)</em></dt>
<dt><a name="index-eigs-15"></a><em><var>d</var> =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, <var>B</var>, <var>k</var>, <var>sigma</var>, <var>opts</var>)</em></dt>
<dt><a name="index-eigs-16"></a><em>[<var>V</var>, <var>d</var>] =</em> <strong>eigs</strong> <em>(<var>A</var>, …)</em></dt>
<dt><a name="index-eigs-17"></a><em>[<var>V</var>, <var>d</var>] =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, …)</em></dt>
<dt><a name="index-eigs-18"></a><em>[<var>V</var>, <var>d</var>, <var>flag</var>] =</em> <strong>eigs</strong> <em>(<var>A</var>, …)</em></dt>
<dt><a name="index-eigs-19"></a><em>[<var>V</var>, <var>d</var>, <var>flag</var>] =</em> <strong>eigs</strong> <em>(<var>af</var>, <var>n</var>, …)</em></dt>
<dd><p>Calculate a limited number of eigenvalues and eigenvectors of <var>A</var>,
based on a selection criteria.
</p>
<p>The number of eigenvalues and eigenvectors to calculate is given by
<var>k</var> and defaults to 6.
</p>
<p>By default, <code>eigs</code> solve the equation
where
is the corresponding eigenvector. If given the positive definite matrix
<var>B</var> then <code>eigs</code> solves the general eigenvalue equation
</p>
<p>The argument <var>sigma</var> determines which eigenvalues are returned.
<var>sigma</var> can be either a scalar or a string. When <var>sigma</var> is a
scalar, the <var>k</var> eigenvalues closest to <var>sigma</var> are returned. If
<var>sigma</var> is a string, it must have one of the following values.
</p>
<dl compact="compact">
<dt><code>"lm"</code></dt>
<dd><p>Largest Magnitude (default).
</p>
</dd>
<dt><code>"sm"</code></dt>
<dd><p>Smallest Magnitude.
</p>
</dd>
<dt><code>"la"</code></dt>
<dd><p>Largest Algebraic (valid only for real symmetric problems).
</p>
</dd>
<dt><code>"sa"</code></dt>
<dd><p>Smallest Algebraic (valid only for real symmetric problems).
</p>
</dd>
<dt><code>"be"</code></dt>
<dd><p>Both Ends, with one more from the high-end if <var>k</var> is odd (valid only for
real symmetric problems).
</p>
</dd>
<dt><code>"lr"</code></dt>
<dd><p>Largest Real part (valid only for complex or unsymmetric problems).
</p>
</dd>
<dt><code>"sr"</code></dt>
<dd><p>Smallest Real part (valid only for complex or unsymmetric problems).
</p>
</dd>
<dt><code>"li"</code></dt>
<dd><p>Largest Imaginary part (valid only for complex or unsymmetric problems).
</p>
</dd>
<dt><code>"si"</code></dt>
<dd><p>Smallest Imaginary part (valid only for complex or unsymmetric problems).
</p></dd>
</dl>
<p>If <var>opts</var> is given, it is a structure defining possible options that
<code>eigs</code> should use. The fields of the <var>opts</var> structure are:
</p>
<dl compact="compact">
<dt><code>issym</code></dt>
<dd><p>If <var>af</var> is given, then flags whether the function <var>af</var> defines a
symmetric problem. It is ignored if <var>A</var> is given. The default is
false.
</p>
</dd>
<dt><code>isreal</code></dt>
<dd><p>If <var>af</var> is given, then flags whether the function <var>af</var> defines a
real problem. It is ignored if <var>A</var> is given. The default is true.
</p>
</dd>
<dt><code>tol</code></dt>
<dd><p>Defines the required convergence tolerance, calculated as
<code>tol * norm (A)</code>. The default is <code>eps</code>.
</p>
</dd>
<dt><code>maxit</code></dt>
<dd><p>The maximum number of iterations. The default is 300.
</p>
</dd>
<dt><code>p</code></dt>
<dd><p>The number of Lanzcos basis vectors to use. More vectors will result in
faster convergence, but a greater use of memory. The optimal value of
<code>p</code> is problem dependent and should be in the range <code><var>k</var> + 1</code>
to <var>n</var>. The default value is <code>2 * <var>k</var></code>.
</p>
</dd>
<dt><code>v0</code></dt>
<dd><p>The starting vector for the algorithm. An initial vector close to the
final vector will speed up convergence. The default is for <small>ARPACK</small>
to randomly generate a starting vector. If specified, <code>v0</code> must be
an <var>n</var>-by-1 vector where <code><var>n</var> = rows (<var>A</var>)</code>
</p>
</dd>
<dt><code>disp</code></dt>
<dd><p>The level of diagnostic printout (0|1|2). If <code>disp</code> is 0 then
diagnostics are disabled. The default value is 0.
</p>
</dd>
<dt><code>cholB</code></dt>
<dd><p>Flag if <code>chol (<var>B</var>)</code> is passed rather than <var>B</var>. The default is
false.
</p>
</dd>
<dt><code>permB</code></dt>
<dd><p>The permutation vector of the Cholesky factorization of <var>B</var> if
<code>cholB</code> is true. It is obtained by <code>[R, ~, permB] =
chol (<var>B</var>, <code>"vector"</code>)</code>. The default is <code>1:<var>n</var></code>.
</p>
</dd>
</dl>
<p>It is also possible to represent <var>A</var> by a function denoted <var>af</var>.
<var>af</var> must be followed by a scalar argument <var>n</var> defining the length
of the vector argument accepted by <var>af</var>. <var>af</var> can be a function
handle, an inline function, or a string. When <var>af</var> is a string it
holds the name of the function to use.
</p>
<p><var>af</var> is a function of the form <code>y = af (x)</code> where the required
return value of <var>af</var> is determined by the value of <var>sigma</var>. The
four possible forms are
</p>
<dl compact="compact">
<dt><code>A * x</code></dt>
<dd><p>if <var>sigma</var> is not given or is a string other than "sm".
</p>
</dd>
<dt><code>A \ x</code></dt>
<dd><p>if <var>sigma</var> is 0 or "sm".
</p>
</dd>
<dt><code>(A - sigma * I) \ x</code></dt>
<dd><p>for the standard eigenvalue problem, where <code>I</code> is the identity matrix
of the same size as <var>A</var>.
</p>
</dd>
<dt><code>(A - sigma * B) \ x</code></dt>
<dd><p>for the general eigenvalue problem.
</p></dd>
</dl>
<p>The return arguments of <code>eigs</code> depend on the number of return arguments
requested. With a single return argument, a vector <var>d</var> of length
<var>k</var> is returned containing the <var>k</var> eigenvalues that have been
found. With two return arguments, <var>V</var> is a <var>n</var>-by-<var>k</var> matrix
whose columns are the <var>k</var> eigenvectors corresponding to the returned
eigenvalues. The eigenvalues themselves are returned in <var>d</var> in the
form of a <var>n</var>-by-<var>k</var> matrix, where the elements on the diagonal
are the eigenvalues.
</p>
<p>Given a third return argument <var>flag</var>, <code>eigs</code> returns the status
of the convergence. If <var>flag</var> is 0 then all eigenvalues have converged.
Any other value indicates a failure to converge.
</p>
<p>This function is based on the <small>ARPACK</small> package, written by
R. Lehoucq, K. Maschhoff, D. Sorensen, and C. Yang. For more
information see <a href="http://www.caam.rice.edu/software/ARPACK/">http://www.caam.rice.edu/software/ARPACK/</a>.
</p>
<p><strong>See also:</strong> <a href="Basic-Matrix-Functions.html#XREFeig">eig</a>, <a href="#XREFsvds">svds</a>.
</p></dd></dl>
<a name="XREFsvds"></a><dl>
<dt><a name="index-svds"></a><em><var>s</var> =</em> <strong>svds</strong> <em>(<var>A</var>)</em></dt>
<dt><a name="index-svds-1"></a><em><var>s</var> =</em> <strong>svds</strong> <em>(<var>A</var>, <var>k</var>)</em></dt>
<dt><a name="index-svds-2"></a><em><var>s</var> =</em> <strong>svds</strong> <em>(<var>A</var>, <var>k</var>, <var>sigma</var>)</em></dt>
<dt><a name="index-svds-3"></a><em><var>s</var> =</em> <strong>svds</strong> <em>(<var>A</var>, <var>k</var>, <var>sigma</var>, <var>opts</var>)</em></dt>
<dt><a name="index-svds-4"></a><em>[<var>u</var>, <var>s</var>, <var>v</var>] =</em> <strong>svds</strong> <em>(…)</em></dt>
<dt><a name="index-svds-5"></a><em>[<var>u</var>, <var>s</var>, <var>v</var>, <var>flag</var>] =</em> <strong>svds</strong> <em>(…)</em></dt>
<dd>
<p>Find a few singular values of the matrix <var>A</var>.
</p>
<p>The singular values are calculated using
</p>
<div class="example">
<pre class="example">[<var>m</var>, <var>n</var>] = size (<var>A</var>);
<var>s</var> = eigs ([sparse(<var>m</var>, <var>m</var>), <var>A</var>;
<var>A</var>', sparse(<var>n</var>, <var>n</var>)])
</pre></div>
<p>The eigenvalues returned by <code>eigs</code> correspond to the singular values
of <var>A</var>. The number of singular values to calculate is given by <var>k</var>
and defaults to 6.
</p>
<p>The argument <var>sigma</var> specifies which singular values to find. When
<var>sigma</var> is the string <code>'L'</code>, the default, the largest singular
values of <var>A</var> are found. Otherwise, <var>sigma</var> must be a real scalar
and the singular values closest to <var>sigma</var> are found. As a corollary,
<code><var>sigma</var> = 0</code> finds the smallest singular values. Note that for
relatively small values of <var>sigma</var>, there is a chance that the
requested number of singular values will not be found. In that case
<var>sigma</var> should be increased.
</p>
<p><var>opts</var> is a structure defining options that <code>svds</code> will pass
to <code>eigs</code>. The possible fields of this structure are documented in
<code>eigs</code>. By default, <code>svds</code> sets the following three fields:
</p>
<dl compact="compact">
<dt><code>tol</code></dt>
<dd><p>The required convergence tolerance for the singular values. The default
value is 1e-10. <code>eigs</code> is passed <code><var>tol</var> / sqrt(2)</code>.
</p>
</dd>
<dt><code>maxit</code></dt>
<dd><p>The maximum number of iterations. The default is 300.
</p>
</dd>
<dt><code>disp</code></dt>
<dd><p>The level of diagnostic printout (0|1|2). If <code>disp</code> is 0 then
diagnostics are disabled. The default value is 0.
</p></dd>
</dl>
<p>If more than one output is requested then <code>svds</code> will return an
approximation of the singular value decomposition of <var>A</var>
</p>
<div class="example">
<pre class="example"><var>A</var>_approx = <var>u</var>*<var>s</var>*<var>v</var>'
</pre></div>
<p>where <var>A</var>_approx is a matrix of size <var>A</var> but only rank <var>k</var>.
</p>
<p><var>flag</var> returns 0 if the algorithm has succesfully converged, and 1
otherwise. The test for convergence is
</p>
<div class="example">
<pre class="example">norm (<var>A</var>*<var>v</var> - <var>u</var>*<var>s</var>, 1) <= <var>tol</var> * norm (<var>A</var>, 1)
</pre></div>
<p><code>svds</code> is best for finding only a few singular values from a large
sparse matrix. Otherwise, <code>svd (full (<var>A</var>))</code> will likely be more
efficient.
</p>
<p><strong>See also:</strong> <a href="Matrix-Factorizations.html#XREFsvd">svd</a>, <a href="#XREFeigs">eigs</a>.
</p></dd></dl>
<div class="footnote">
<hr>
<h4 class="footnotes-heading">Footnotes</h4>
<h3><a name="FOOT10" href="#DOCF10">(10)</a></h3>
<p>The <small>CHOLMOD</small>, <small>UMFPACK</small> and <small>CXSPARSE</small>
packages were written by Tim Davis and are available at
<a href="http://faculty.cse.tamu.edu/davis/suitesparse.html">http://faculty.cse.tamu.edu/davis/suitesparse.html</a></p>
</div>
<hr>
<div class="header">
<p>
Next: <a href="Iterative-Techniques.html#Iterative-Techniques" accesskey="n" rel="next">Iterative Techniques</a>, Previous: <a href="Basics.html#Basics" accesskey="p" rel="prev">Basics</a>, Up: <a href="Sparse-Matrices.html#Sparse-Matrices" accesskey="u" rel="up">Sparse Matrices</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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