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<a name="Arithmetic-Ops"></a>
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Next: <a rel="next" accesskey="n" href="Comparison-Ops.html#Comparison-Ops">Comparison Ops</a>,
Previous: <a rel="previous" accesskey="p" href="Calling-Functions.html#Calling-Functions">Calling Functions</a>,
Up: <a rel="up" accesskey="u" href="Expressions.html#Expressions">Expressions</a>
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<h3 class="section">8.3 Arithmetic Operators</h3>
<p><a name="index-arithmetic-operators-531"></a><a name="index-operators_002c-arithmetic-532"></a><a name="index-addition-533"></a><a name="index-subtraction-534"></a><a name="index-multiplication-535"></a><a name="index-matrix-multiplication-536"></a><a name="index-division-537"></a><a name="index-quotient-538"></a><a name="index-negation-539"></a><a name="index-unary-minus-540"></a><a name="index-exponentiation-541"></a><a name="index-transpose-542"></a><a name="index-Hermitian-operator-543"></a><a name="index-transpose_002c-complex_002dconjugate-544"></a><a name="index-complex_002dconjugate-transpose-545"></a>
The following arithmetic operators are available, and work on scalars
and matrices. The element-by-element operators and functions broadcast
(see <a href="Broadcasting.html#Broadcasting">Broadcasting</a>).
<dl>
<dt><var>x</var> + <var>y</var><dd><a name="index-g_t_002b-546"></a>Addition. If both operands are matrices, the number of rows and columns
must both agree, or they must be broadcastable to the same shape.
<br><dt><var>x</var> .+ <var>y</var><dd><a name="index-g_t_002e_002b-547"></a>Element-by-element addition. This operator is equivalent to <code>+</code>.
<br><dt><var>x</var> - <var>y</var><dd><a name="index-g_t_002d-548"></a>Subtraction. If both operands are matrices, the number of rows and
columns of both must agree, or they must be broadcastable to the same
shape.
<br><dt><var>x</var> .- <var>y</var><dd>Element-by-element subtraction. This operator is equivalent to <code>-</code>.
<br><dt><var>x</var> * <var>y</var><dd><a name="index-g_t_002a-549"></a>Matrix multiplication. The number of columns of <var>x</var> must agree with
the number of rows of <var>y</var>, or they must be broadcastable to the same
shape.
<br><dt><var>x</var> .* <var>y</var><dd><a name="index-g_t_002e_002a-550"></a>Element-by-element multiplication. If both operands are matrices, the
number of rows and columns must both agree, or they must be
broadcastable to the same shape.
<br><dt><var>x</var> / <var>y</var><dd><a name="index-g_t_002f-551"></a>Right division. This is conceptually equivalent to the expression
<pre class="example"> (inverse (y') * x')'
</pre>
<p class="noindent">but it is computed without forming the inverse of <var>y'</var>.
<p>If the system is not square, or if the coefficient matrix is singular,
a minimum norm solution is computed.
<br><dt><var>x</var> ./ <var>y</var><dd><a name="index-g_t_002e_002f-552"></a>Element-by-element right division.
<br><dt><var>x</var> \ <var>y</var><dd><a name="index-g_t_005c-553"></a>Left division. This is conceptually equivalent to the expression
<pre class="example"> inverse (x) * y
</pre>
<p class="noindent">but it is computed without forming the inverse of <var>x</var>.
<p>If the system is not square, or if the coefficient matrix is singular,
a minimum norm solution is computed.
<br><dt><var>x</var> .\ <var>y</var><dd><a name="index-g_t_002e_005c-554"></a>Element-by-element left division. Each element of <var>y</var> is divided
by each corresponding element of <var>x</var>.
<br><dt><var>x</var> ^ <var>y</var><dt><var>x</var> ** <var>y</var><dd><a name="index-g_t_002a_002a-555"></a><a name="index-g_t_005e-556"></a>Power operator. If <var>x</var> and <var>y</var> are both scalars, this operator
returns <var>x</var> raised to the power <var>y</var>. If <var>x</var> is a scalar and
<var>y</var> is a square matrix, the result is computed using an eigenvalue
expansion. If <var>x</var> is a square matrix, the result is computed by
repeated multiplication if <var>y</var> is an integer, and by an eigenvalue
expansion if <var>y</var> is not an integer. An error results if both
<var>x</var> and <var>y</var> are matrices.
<p>The implementation of this operator needs to be improved.
<br><dt><var>x</var> .^ <var>y</var><dt><var>x</var> .** <var>y</var><dd><a name="index-g_t_002e_002a_002a-557"></a><a name="index-g_t_002e_005e-558"></a>Element-by-element power operator. If both operands are matrices, the
number of rows and columns must both agree, or they must be
broadcastable to the same shape. If several complex results are
possible, the one with smallest non-negative argument (angle) is taken.
This rule may return a complex root even when a real root is also possible.
Use <code>realpow</code>, <code>realsqrt</code>, <code>cbrt</code>, or <code>nthroot</code> if a
real result is preferred.
<br><dt>-<var>x</var><dd><a name="index-g_t_002d-559"></a>Negation.
<br><dt>+<var>x</var><dd><a name="index-g_t_002b-560"></a>Unary plus. This operator has no effect on the operand.
<br><dt><var>x</var>'<dd><a name="index-g_t_0027-561"></a>Complex conjugate transpose. For real arguments, this operator is the
same as the transpose operator. For complex arguments, this operator is
equivalent to the expression
<pre class="example"> conj (x.')
</pre>
<br><dt><var>x</var>.'<dd><a name="index-g_t_002e_0027-562"></a>Transpose.
</dl>
<p>Note that because Octave's element-by-element operators begin with a
‘<samp><span class="samp">.</span></samp>’, there is a possible ambiguity for statements like
<pre class="example"> 1./m
</pre>
<p class="noindent">because the period could be interpreted either as part of the constant
or as part of the operator. To resolve this conflict, Octave treats the
expression as if you had typed
<pre class="example"> (1) ./ m
</pre>
<p class="noindent">and not
<pre class="example"> (1.) / m
</pre>
<p class="noindent">Although this is inconsistent with the normal behavior of Octave's
lexer, which usually prefers to break the input into tokens by
preferring the longest possible match at any given point, it is more
useful in this case.
<p><a name="index-g_t_0027-563"></a><!-- ctranspose src/data.cc -->
<a name="doc_002dctranspose"></a>
<div class="defun">
— Built-in Function: <b>ctranspose</b> (<var>x</var>)<var><a name="index-ctranspose-564"></a></var><br>
<blockquote><p>Return the complex conjugate transpose of <var>x</var>.
This function and <code>x'</code> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dtranspose.html#doc_002dtranspose">transpose</a>.
</p></blockquote></div>
<p><a name="index-g_t_002e_005c-565"></a><!-- ldivide src/data.cc -->
<a name="doc_002dldivide"></a>
<div class="defun">
— Built-in Function: <b>ldivide</b> (<var>x, y</var>)<var><a name="index-ldivide-566"></a></var><br>
<blockquote><p>Return the element-by-element left division of <var>x</var> and <var>y</var>.
This function and <code>x . y</code><!-- /@w --> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002drdivide.html#doc_002drdivide">rdivide</a>, <a href="doc_002dmldivide.html#doc_002dmldivide">mldivide</a>.
</p></blockquote></div>
<p><a name="index-g_t_002d-567"></a><!-- minus src/data.cc -->
<a name="doc_002dminus"></a>
<div class="defun">
— Built-in Function: <b>minus</b> (<var>x, y</var>)<var><a name="index-minus-568"></a></var><br>
<blockquote><p>This function and <code>x - y</code><!-- /@w --> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dplus.html#doc_002dplus">plus</a>.
</p></blockquote></div>
<p><a name="index-g_t_005c-569"></a><!-- mldivide src/data.cc -->
<a name="doc_002dmldivide"></a>
<div class="defun">
— Built-in Function: <b>mldivide</b> (<var>x, y</var>)<var><a name="index-mldivide-570"></a></var><br>
<blockquote><p>Return the matrix left division of <var>x</var> and <var>y</var>.
This function and <code>x y</code><!-- /@w --> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmrdivide.html#doc_002dmrdivide">mrdivide</a>, <a href="doc_002dldivide.html#doc_002dldivide">ldivide</a>.
</p></blockquote></div>
<p><a name="index-g_t_002a_002a-571"></a><a name="index-g_t_005e-572"></a><!-- mpower src/data.cc -->
<a name="doc_002dmpower"></a>
<div class="defun">
— Built-in Function: <b>mpower</b> (<var>x, y</var>)<var><a name="index-mpower-573"></a></var><br>
<blockquote><p>Return the matrix power operation of <var>x</var> raised to the <var>y</var> power.
This function and <code>x ^ y</code><!-- /@w --> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpower.html#doc_002dpower">power</a>.
</p></blockquote></div>
<p><a name="index-g_t_002f-574"></a><!-- mrdivide src/data.cc -->
<a name="doc_002dmrdivide"></a>
<div class="defun">
— Built-in Function: <b>mrdivide</b> (<var>x, y</var>)<var><a name="index-mrdivide-575"></a></var><br>
<blockquote><p>Return the matrix right division of <var>x</var> and <var>y</var>.
This function and <code>x / y</code><!-- /@w --> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmldivide.html#doc_002dmldivide">mldivide</a>, <a href="doc_002drdivide.html#doc_002drdivide">rdivide</a>.
</p></blockquote></div>
<p><a name="index-g_t_002a-576"></a><!-- mtimes src/data.cc -->
<a name="doc_002dmtimes"></a>
<div class="defun">
— Built-in Function: <b>mtimes</b> (<var>x, y</var>)<var><a name="index-mtimes-577"></a></var><br>
— Built-in Function: <b>mtimes</b> (<var>x1, x2, <small class="dots">...</small></var>)<var><a name="index-mtimes-578"></a></var><br>
<blockquote><p>Return the matrix multiplication product of inputs.
This function and <code>x * y</code><!-- /@w --> are equivalent.
If more arguments are given, the multiplication is applied
cumulatively from left to right:
<pre class="example"> (...((x1 * x2) * x3) * ...)
</pre>
<p>At least one argument is required.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dtimes.html#doc_002dtimes">times</a>.
</p></blockquote></div>
<p><a name="index-g_t_002b-579"></a><!-- plus src/data.cc -->
<a name="doc_002dplus"></a>
<div class="defun">
— Built-in Function: <b>plus</b> (<var>x, y</var>)<var><a name="index-plus-580"></a></var><br>
— Built-in Function: <b>plus</b> (<var>x1, x2, <small class="dots">...</small></var>)<var><a name="index-plus-581"></a></var><br>
<blockquote><p>This function and <code>x + y</code><!-- /@w --> are equivalent.
If more arguments are given, the summation is applied
cumulatively from left to right:
<pre class="example"> (...((x1 + x2) + x3) + ...)
</pre>
<p>At least one argument is required.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dminus.html#doc_002dminus">minus</a>.
</p></blockquote></div>
<p><a name="index-g_t_002e_002a_002a-582"></a><a name="index-g_t_002e_005e-583"></a><!-- power src/data.cc -->
<a name="doc_002dpower"></a>
<div class="defun">
— Built-in Function: <b>power</b> (<var>x, y</var>)<var><a name="index-power-584"></a></var><br>
<blockquote><p>Return the element-by-element operation of <var>x</var> raised to the
<var>y</var> power. If several complex results are possible,
returns the one with smallest non-negative argument (angle). Use
<code>realpow</code>, <code>realsqrt</code>, <code>cbrt</code>, or <code>nthroot</code> if a
real result is preferred.
<p>This function and <code>x .^ y</code><!-- /@w --> are equivalent.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmpower.html#doc_002dmpower">mpower</a>, <a href="doc_002drealpow.html#doc_002drealpow">realpow</a>, <a href="doc_002drealsqrt.html#doc_002drealsqrt">realsqrt</a>, <a href="doc_002dcbrt.html#doc_002dcbrt">cbrt</a>, <a href="doc_002dnthroot.html#doc_002dnthroot">nthroot</a>.
</p></blockquote></div>
<p><a name="index-g_t_002e_002f-585"></a><!-- rdivide src/data.cc -->
<a name="doc_002drdivide"></a>
<div class="defun">
— Built-in Function: <b>rdivide</b> (<var>x, y</var>)<var><a name="index-rdivide-586"></a></var><br>
<blockquote><p>Return the element-by-element right division of <var>x</var> and <var>y</var>.
This function and <code>x ./ y</code><!-- /@w --> are equivalent.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dldivide.html#doc_002dldivide">ldivide</a>, <a href="doc_002dmrdivide.html#doc_002dmrdivide">mrdivide</a>.
</p></blockquote></div>
<p><a name="index-g_t_002e_002a-587"></a><!-- times src/data.cc -->
<a name="doc_002dtimes"></a>
<div class="defun">
— Built-in Function: <b>times</b> (<var>x, y</var>)<var><a name="index-times-588"></a></var><br>
— Built-in Function: <b>times</b> (<var>x1, x2, <small class="dots">...</small></var>)<var><a name="index-times-589"></a></var><br>
<blockquote><p>Return the element-by-element multiplication product of inputs.
This function and <code>x .* y</code><!-- /@w --> are equivalent.
If more arguments are given, the multiplication is applied
cumulatively from left to right:
<pre class="example"> (...((x1 .* x2) .* x3) .* ...)
</pre>
<p>At least one argument is required.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dmtimes.html#doc_002dmtimes">mtimes</a>.
</p></blockquote></div>
<p><a name="index-g_t_002e_0027-590"></a><!-- transpose src/data.cc -->
<a name="doc_002dtranspose"></a>
<div class="defun">
— Built-in Function: <b>transpose</b> (<var>x</var>)<var><a name="index-transpose-591"></a></var><br>
<blockquote><p>Return the transpose of <var>x</var>.
This function and <code>x.'</code> are equivalent.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dctranspose.html#doc_002dctranspose">ctranspose</a>.
</p></blockquote></div>
<p><a name="index-g_t_002d-592"></a><!-- uminus src/data.cc -->
<a name="doc_002duminus"></a>
<div class="defun">
— Built-in Function: <b>uminus</b> (<var>x</var>)<var><a name="index-uminus-593"></a></var><br>
<blockquote><p>This function and <code>- x</code><!-- /@w --> are equivalent.
</p></blockquote></div>
<p><a name="index-g_t_002b-594"></a><!-- uplus src/data.cc -->
<a name="doc_002duplus"></a>
<div class="defun">
— Built-in Function: <b>uplus</b> (<var>x</var>)<var><a name="index-uplus-595"></a></var><br>
<blockquote><p>This function and <code>+ x</code><!-- /@w --> are equivalent.
</p></blockquote></div>
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