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<a name="Basic-Matrix-Functions"></a>
<p>
Next: <a rel="next" accesskey="n" href="Matrix-Factorizations.html#Matrix-Factorizations">Matrix Factorizations</a>,
Previous: <a rel="previous" accesskey="p" href="Techniques-used-for-Linear-Algebra.html#Techniques-used-for-Linear-Algebra">Techniques used for Linear Algebra</a>,
Up: <a rel="up" accesskey="u" href="Linear-Algebra.html#Linear-Algebra">Linear Algebra</a>
<hr>
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<h3 class="section">18.2 Basic Matrix Functions</h3>
<!-- ./DLD-FUNCTIONS/balance.cc -->
<p><a name="doc_002dbalance"></a>
<div class="defun">
— Loadable Function: <var>aa</var> = <b>balance</b> (<var>a, opt</var>)<var><a name="index-balance-1549"></a></var><br>
— Loadable Function: [<var>dd</var>, <var>aa</var>] = <b>balance</b> (<var>a, opt</var>)<var><a name="index-balance-1550"></a></var><br>
— Loadable Function: [<var>d</var>, <var>p</var>, <var>aa</var>] = <b>balance</b> (<var>a, opt</var>)<var><a name="index-balance-1551"></a></var><br>
— Loadable Function: [<var>cc</var>, <var>dd</var>, <var>aa</var>, <var>bb</var>] = <b>balance</b> (<var>a, b, opt</var>)<var><a name="index-balance-1552"></a></var><br>
<blockquote>
<p>Compute <code>aa = dd \ a * dd</code> in which <code>aa</code> is a matrix whose
row and column norms are roughly equal in magnitude, and
<code>dd</code> = <code>p * d</code>, in which <code>p</code> is a permutation
matrix and <code>d</code> is a diagonal matrix of powers of two. This allows
the equilibration to be computed without roundoff. Results of
eigenvalue calculation are typically improved by balancing first.
<p>If two output values are requested, <code>balance</code> returns
the diagonal <code>d</code> and the permutation <code>p</code> separately as vectors.
In this case, <code>dd = eye(n)(p,:) * diag (d)</code>, where <code>n</code> is the matrix
size.
<p>If four output values are requested, compute <code>aa = cc*a*dd</code> and
<code>bb = cc*b*dd)</code>, in which <code>aa</code> and <code>bb</code> have non-zero
elements of approximately the same magnitude and <code>cc</code> and <code>dd</code>
are permuted diagonal matrices as in <code>dd</code> for the algebraic
eigenvalue problem.
<p>The eigenvalue balancing option <code>opt</code> may be one of:
<dl>
<dt><code>"noperm"</code>, <code>"S"</code><dd>Scale only; do not permute.
<br><dt><code>"noscal"</code>, <code>"P"</code><dd>Permute only; do not scale.
</dl>
<p>Algebraic eigenvalue balancing uses standard <span class="sc">lapack</span> routines.
<p>Generalized eigenvalue problem balancing uses Ward's algorithm
(SIAM Journal on Scientific and Statistical Computing, 1981).
</p></blockquote></div>
<!-- ./linear-algebra/cond.m -->
<p><a name="doc_002dcond"></a>
<div class="defun">
— Function File: <b>cond</b> (<var>a,p</var>)<var><a name="index-cond-1553"></a></var><br>
<blockquote><p>Compute the <var>p</var>-norm condition number of a matrix. <code>cond (</code><var>a</var><code>)</code> is
defined as <code>norm (</code><var>a</var><code>, </code><var>p</var><code>) * norm (inv (</code><var>a</var><code>), </code><var>p</var><code>)</code>.
By default <var>p</var><code>=2</code> is used which implies a (relatively slow)
singular value decomposition. Other possible selections are
<var>p</var><code>= 1, Inf, inf, 'Inf', 'fro'</code> which are generally faster.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcondest.html#doc_002dcondest">condest</a>, <a href="doc_002drcond.html#doc_002drcond">rcond</a>, <a href="doc_002dnorm.html#doc_002dnorm">norm</a>, <a href="doc_002dsvd.html#doc_002dsvd">svd</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/det.cc -->
<p><a name="doc_002ddet"></a>
<div class="defun">
— Loadable Function: [<var>d</var>, <var>rcond</var>] = <b>det</b> (<var>a</var>)<var><a name="index-det-1554"></a></var><br>
<blockquote><p>Compute the determinant of <var>a</var> using <span class="sc">lapack</span> for full and UMFPACK
for sparse matrices. Return an estimate of the reciprocal condition number
if requested.
</p></blockquote></div>
<!-- ./deprecated/dmult.m -->
<p><a name="doc_002ddmult"></a>
<div class="defun">
— Function File: <b>dmult</b> (<var>a, b</var>)<var><a name="index-dmult-1555"></a></var><br>
<blockquote><p>This function has been deprecated. Use the direct syntax <code>diag(A)*B</code>
which is more readable and now also more efficient.
</p></blockquote></div>
<!-- ./linear-algebra/dot.m -->
<p><a name="doc_002ddot"></a>
<div class="defun">
— Function File: <b>dot</b> (<var>x, y, dim</var>)<var><a name="index-dot-1556"></a></var><br>
<blockquote><p>Computes the dot product of two vectors. If <var>x</var> and <var>y</var>
are matrices, calculate the dot-product along the first
non-singleton dimension. If the optional argument <var>dim</var> is
given, calculate the dot-product along this dimension.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/eig.cc -->
<p><a name="doc_002deig"></a>
<div class="defun">
— Loadable Function: <var>lambda</var> = <b>eig</b> (<var>a</var>)<var><a name="index-eig-1557"></a></var><br>
— Loadable Function: <var>lambda</var> = <b>eig</b> (<var>a, b</var>)<var><a name="index-eig-1558"></a></var><br>
— Loadable Function: [<var>v</var>, <var>lambda</var>] = <b>eig</b> (<var>a</var>)<var><a name="index-eig-1559"></a></var><br>
— Loadable Function: [<var>v</var>, <var>lambda</var>] = <b>eig</b> (<var>a, b</var>)<var><a name="index-eig-1560"></a></var><br>
<blockquote><p>The eigenvalues (and eigenvectors) of a matrix are computed in a several
step process which begins with a Hessenberg decomposition, followed by a
Schur decomposition, from which the eigenvalues are apparent. The
eigenvectors, when desired, are computed by further manipulations of the
Schur decomposition.
<p>The eigenvalues returned by <code>eig</code> are not ordered.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002deigs.html#doc_002deigs">eigs</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/givens.cc -->
<p><a name="doc_002dgivens"></a>
<div class="defun">
— Loadable Function: <var>g</var> = <b>givens</b> (<var>x, y</var>)<var><a name="index-givens-1561"></a></var><br>
— Loadable Function: [<var>c</var>, <var>s</var>] = <b>givens</b> (<var>x, y</var>)<var><a name="index-givens-1562"></a></var><br>
<blockquote><p>Return a 2 by 2 orthogonal matrix
<var>g</var><code> = [</code><var>c</var> <var>s</var><code>; -</code><var>s</var><code>' </code><var>c</var><code>]</code> such that
<var>g</var><code> [</code><var>x</var><code>; </code><var>y</var><code>] = [*; 0]</code> with <var>x</var> and <var>y</var> scalars.
<p>For example,
<pre class="example"> givens (1, 1)
⇒ 0.70711 0.70711
-0.70711 0.70711
</pre>
</blockquote></div>
<!-- ./linear-algebra/planerot.m -->
<p><a name="doc_002dplanerot"></a>
<div class="defun">
— Function File: [<var>g</var>, <var>y</var>] = <b>planerot</b> (<var>x</var>)<var><a name="index-planerot-1563"></a></var><br>
<blockquote><p>Given a two-element column vector, returns the
2 by 2 orthogonal matrix
<var>G</var> such that
<var>y</var><code> = </code><var>g</var><code> * </code><var>x</var> and <var>y</var><code>(2) = 0</code>.
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<!-- @sp 1 -->
<p class="noindent"><strong>See also:</strong> <a href="doc_002dgivens.html#doc_002dgivens">givens</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/inv.cc -->
<p><a name="doc_002dinv"></a>
<div class="defun">
— Loadable Function: [<var>x</var>, <var>rcond</var>] = <b>inv</b> (<var>a</var>)<var><a name="index-inv-1564"></a></var><br>
— Loadable Function: [<var>x</var>, <var>rcond</var>] = <b>inverse</b> (<var>a</var>)<var><a name="index-inverse-1565"></a></var><br>
<blockquote><p>Compute the inverse of the square matrix <var>a</var>. Return an estimate
of the reciprocal condition number if requested, otherwise warn of an
ill-conditioned matrix if the reciprocal condition number is small.
<p>If called with a sparse matrix, then in general <var>x</var> will be a full
matrix, and so if possible forming the inverse of a sparse matrix should
be avoided. It is significantly more accurate and faster to do
<var>y</var><code> = </code><var>a</var><code> \ </code><var>b</var>, rather than
<var>y</var><code> = inv (</code><var>a</var><code>) * </code><var>b</var>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/matrix_type.cc -->
<p><a name="doc_002dmatrix_005ftype"></a>
<div class="defun">
— Loadable Function: <var>type</var> = <b>matrix_type</b> (<var>a</var>)<var><a name="index-matrix_005ftype-1566"></a></var><br>
— Loadable Function: <var>a</var> = <b>matrix_type</b> (<var>a, type</var>)<var><a name="index-matrix_005ftype-1567"></a></var><br>
— Loadable Function: <var>a</var> = <b>matrix_type</b> (<var>a, 'upper', perm</var>)<var><a name="index-matrix_005ftype-1568"></a></var><br>
— Loadable Function: <var>a</var> = <b>matrix_type</b> (<var>a, 'lower', perm</var>)<var><a name="index-matrix_005ftype-1569"></a></var><br>
— Loadable Function: <var>a</var> = <b>matrix_type</b> (<var>a, 'banded', nl, nu</var>)<var><a name="index-matrix_005ftype-1570"></a></var><br>
<blockquote><p>Identify the matrix type or mark a matrix as a particular type. This allows rapid
for solutions of linear equations involving <var>a</var> to be performed. Called with a
single argument, <code>matrix_type</code> returns the type of the matrix and caches it for
future use. Called with more than one argument, <code>matrix_type</code> allows the type
of the matrix to be defined.
<p>The possible matrix types depend on whether the matrix is full or sparse, and can be
one of the following
<dl>
<dt>'unknown'<dd>Remove any previously cached matrix type, and mark type as unknown
<br><dt>'full'<dd>Mark the matrix as full.
<br><dt>'positive definite'<dd>Probable full positive definite matrix.
<br><dt>'diagonal'<dd>Diagonal Matrix. (Sparse matrices only)
<br><dt>'permuted diagonal'<dd>Permuted Diagonal matrix. The permutation does not need to be specifically
indicated, as the structure of the matrix explicitly gives this. (Sparse matrices
only)
<br><dt>'upper'<dd>Upper triangular. If the optional third argument <var>perm</var> is given, the matrix is
assumed to be a permuted upper triangular with the permutations defined by the
vector <var>perm</var>.
<br><dt>'lower'<dd>Lower triangular. If the optional third argument <var>perm</var> is given, the matrix is
assumed to be a permuted lower triangular with the permutations defined by the
vector <var>perm</var>.
<br><dt>'banded'<dt>'banded positive definite'<dd>Banded matrix with the band size of <var>nl</var> below the diagonal and <var>nu</var> above
it. If <var>nl</var> and <var>nu</var> are 1, then the matrix is tridiagonal and treated
with specialized code. In addition the matrix can be marked as probably a
positive definite (Sparse matrices only)
<br><dt>'singular'<dd>The matrix is assumed to be singular and will be treated with a minimum norm solution
</dl>
<p>Note that the matrix type will be discovered automatically on the first attempt to
solve a linear equation involving <var>a</var>. Therefore <code>matrix_type</code> is only
useful to give Octave hints of the matrix type. Incorrectly defining the
matrix type will result in incorrect results from solutions of linear equations,
and so it is entirely the responsibility of the user to correctly identify the
matrix type.
<p>Also the test for positive definiteness is a low-cost test for a hermitian
matrix with a real positive diagonal. This does not guarantee that the matrix
is positive definite, but only that it is a probable candidate. When such a
matrix is factorized, a Cholesky factorization is first attempted, and if
that fails the matrix is then treated with an LU factorization. Once the
matrix has been factorized, <code>matrix_type</code> will return the correct
classification of the matrix.
</p></blockquote></div>
<!-- data.cc -->
<p><a name="doc_002dnorm"></a>
<div class="defun">
— Built-in Function: <b>norm</b> (<var>a, p, opt</var>)<var><a name="index-norm-1571"></a></var><br>
<blockquote><p>Compute the p-norm of the matrix <var>a</var>. If the second argument is
missing, <code>p = 2</code> is assumed.
<p>If <var>a</var> is a matrix (or sparse matrix):
<dl>
<dt><var>p</var> = <code>1</code><dd>1-norm, the largest column sum of the absolute values of <var>a</var>.
<br><dt><var>p</var> = <code>2</code><dd>Largest singular value of <var>a</var>.
<br><dt><var>p</var> = <code>Inf</code> or <code>"inf"</code><dd><a name="index-infinity-norm-1572"></a>Infinity norm, the largest row sum of the absolute values of <var>a</var>.
<br><dt><var>p</var> = <code>"fro"</code><dd><a name="index-Frobenius-norm-1573"></a>Frobenius norm of <var>a</var>, <code>sqrt (sum (diag (</code><var>a</var><code>' * </code><var>a</var><code>)))</code>.
<br><dt>other <var>p</var>, <var>p</var><code> > 1</code><dd><a name="index-general-p_002dnorm-1574"></a>maximum <code>norm (A*x, p)</code> such that <code>norm (x, p) == 1</code>
</dl>
<p>If <var>a</var> is a vector or a scalar:
<dl>
<dt><var>p</var> = <code>Inf</code> or <code>"inf"</code><dd><code>max (abs (</code><var>a</var><code>))</code>.
<br><dt><var>p</var> = <code>-Inf</code><dd><code>min (abs (</code><var>a</var><code>))</code>.
<br><dt><var>p</var> = <code>"fro"</code><dd>Frobenius norm of <var>a</var>, <code>sqrt (sumsq (abs (a)))</code>.
<br><dt><var>p</var> = 0<dd>Hamming norm - the number of nonzero elements.
<br><dt>other <var>p</var>, <var>p</var><code> > 1</code><dd>p-norm of <var>a</var>, <code>(sum (abs (</code><var>a</var><code>) .^ </code><var>p</var><code>)) ^ (1/</code><var>p</var><code>)</code>.
<br><dt>other <var>p</var> <var>p</var><code> < 1</code><dd>the p-pseudonorm defined as above.
</dl>
<p>If <code>"rows"</code> is given as <var>opt</var>, the norms of all rows of the matrix <var>a</var> are
returned as a column vector. Similarly, if <code>"columns"</code> or <code>"cols"</code> is passed
column norms are computed.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dcond.html#doc_002dcond">cond</a>, <a href="doc_002dsvd.html#doc_002dsvd">svd</a>.
</p></blockquote></div>
<!-- ./linear-algebra/null.m -->
<p><a name="doc_002dnull"></a>
<div class="defun">
— Function File: <b>null</b> (<var>a, tol</var>)<var><a name="index-null-1575"></a></var><br>
<blockquote><p>Return an orthonormal basis of the null space of <var>a</var>.
<p>The dimension of the null space is taken as the number of singular
values of <var>a</var> not greater than <var>tol</var>. If the argument <var>tol</var>
is missing, it is computed as
<pre class="example"> max (size (<var>a</var>)) * max (svd (<var>a</var>)) * eps
</pre>
</blockquote></div>
<!-- ./linear-algebra/orth.m -->
<p><a name="doc_002dorth"></a>
<div class="defun">
— Function File: <b>orth</b> (<var>a, tol</var>)<var><a name="index-orth-1576"></a></var><br>
<blockquote><p>Return an orthonormal basis of the range space of <var>a</var>.
<p>The dimension of the range space is taken as the number of singular
values of <var>a</var> greater than <var>tol</var>. If the argument <var>tol</var> is
missing, it is computed as
<pre class="example"> max (size (<var>a</var>)) * max (svd (<var>a</var>)) * eps
</pre>
</blockquote></div>
<!-- ./DLD-FUNCTIONS/pinv.cc -->
<p><a name="doc_002dpinv"></a>
<div class="defun">
— Loadable Function: <b>pinv</b> (<var>x, tol</var>)<var><a name="index-pinv-1577"></a></var><br>
<blockquote><p>Return the pseudoinverse of <var>x</var>. Singular values less than
<var>tol</var> are ignored.
<p>If the second argument is omitted, it is assumed that
<pre class="example"> tol = max (size (<var>x</var>)) * sigma_max (<var>x</var>) * eps,
</pre>
<p class="noindent">where <code>sigma_max (</code><var>x</var><code>)</code> is the maximal singular value of <var>x</var>.
</p></blockquote></div>
<!-- ./linear-algebra/rank.m -->
<p><a name="doc_002drank"></a>
<div class="defun">
— Function File: <b>rank</b> (<var>a, tol</var>)<var><a name="index-rank-1578"></a></var><br>
<blockquote><p>Compute the rank of <var>a</var>, using the singular value decomposition.
The rank is taken to be the number of singular values of <var>a</var> that
are greater than the specified tolerance <var>tol</var>. If the second
argument is omitted, it is taken to be
<pre class="example"> tol = max (size (<var>a</var>)) * sigma(1) * eps;
</pre>
<p class="noindent">where <code>eps</code> is machine precision and <code>sigma(1)</code> is the largest
singular value of <var>a</var>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/rcond.cc -->
<p><a name="doc_002drcond"></a>
<div class="defun">
— Loadable Function: <var>c</var> = <b>rcond</b> (<var>a</var>)<var><a name="index-rcond-1579"></a></var><br>
<blockquote><p>Compute the 1-norm estimate of the reciprocal condition as returned
by <span class="sc">lapack</span>. If the matrix is well-conditioned then <var>c</var> will be near
1 and if the matrix is poorly conditioned it will be close to zero.
<p>The matrix <var>a</var> must not be sparse. If the matrix is sparse then
<code>condest (</code><var>a</var><code>)</code> or <code>rcond (full (</code><var>a</var><code>))</code> should be used
instead.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinv.html#doc_002dinv">inv</a>.
</p></blockquote></div>
<!-- ./linear-algebra/trace.m -->
<p><a name="doc_002dtrace"></a>
<div class="defun">
— Function File: <b>trace</b> (<var>a</var>)<var><a name="index-trace-1580"></a></var><br>
<blockquote><p>Compute the trace of <var>a</var>, <code>sum (diag (</code><var>a</var><code>))</code>.
</p></blockquote></div>
<!-- ./linear-algebra/rref.m -->
<p><a name="doc_002drref"></a>
<div class="defun">
— Function File: [<var>r</var>, <var>k</var>] = <b>rref</b> (<var>a, tol</var>)<var><a name="index-rref-1581"></a></var><br>
<blockquote>
<p>Returns the reduced row echelon form of <var>a</var>. <var>tol</var> defaults
to <code>eps * max (size (</code><var>a</var><code>)) * norm (</code><var>a</var><code>, inf)</code>.
<p>Called with two return arguments, <var>k</var> returns the vector of
"bound variables", which are those columns on which elimination
has been performed.
</blockquote></div>
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