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<a name="Functions-for-Numbers"></a>
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<p>
Next: <a href="maxima_49.html#Functions-for-Complex-Numbers" accesskey="n" rel="next">Functions for Complex Numbers</a>, Up: <a href="maxima_47.html#Elementary-Functions" accesskey="u" rel="up">Elementary Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-for-Numbers-1"></a>
<h3 class="section">10.1 Functions for Numbers</h3>
<a name="abs"></a><a name="Item_003a-MathFunctions_002fdeffn_002fabs"></a><dl>
<dt><a name="index-abs"></a>Function: <strong>abs</strong> <em>(<var>z</var>)</em></dt>
<dd>
<p>The <code>abs</code> function represents the mathematical absolute value function and
works for both numerical and symbolic values. If the argument, <var>z</var>, is a
real or complex number, <code>abs</code> returns the absolute value of <var>z</var>. If
possible, symbolic expressions using the absolute value function are
also simplified.
</p>
<p>Maxima can differentiate, integrate and calculate limits for expressions
containing <code>abs</code>. The <code>abs_integrate</code> package further extends
Maxima’s ability to calculate integrals involving the abs function. See
(%i12) in the examples below.
</p>
<p>When applied to a list or matrix, <code>abs</code> automatically distributes over
the terms. Similarly, it distributes over both sides of an
equation. To alter this behaviour, see the variable <code><a href="maxima_46.html#distribute_005fover">distribute_over</a></code>.
</p>
<p>See also <code><a href="maxima_49.html#cabs">cabs</a></code>.
</p>
<p>Examples:
</p>
<p>Calculation of <code>abs</code> for real and complex numbers, including numerical
constants and various infinities. The first example shows how <code>abs</code>
distributes over the elements of a list.
</p>
<div class="example">
<pre class="example">(%i1) abs([-4, 0, 1, 1+%i]);
(%o1) [4, 0, 1, sqrt(2)]
(%i2) abs((1+%i)*(1-%i));
(%o2) 2
(%i3) abs(%e+%i);
2
(%o3) sqrt(%e + 1)
(%i4) abs([inf, infinity, minf]);
(%o4) [inf, inf, inf]
</pre></div>
<p>Simplification of expressions containing <code>abs</code>:
</p>
<div class="example">
<pre class="example">(%i5) abs(x^2);
2
(%o5) x
(%i6) abs(x^3);
2
(%o6) x abs(x)
(%i7) abs(abs(x));
(%o7) abs(x)
(%i8) abs(conjugate(x));
(%o8) abs(x)
</pre></div>
<p>Integrating and differentiating with the <code>abs</code> function. Note that more
integrals involving the <code>abs</code> function can be performed, if the
<code>abs_integrate</code> package is loaded. The last example shows the Laplace
transform of <code>abs</code>: see <code><a href="maxima_104.html#laplace">laplace</a></code>.
</p>
<div class="example">
<pre class="example">(%i9) diff(x*abs(x),x),expand;
(%o9) 2 abs(x)
(%i10) integrate(abs(x),x);
x abs(x)
(%o10) --------
2
(%i11) integrate(x*abs(x),x);
/
[
(%o11) I x abs(x) dx
]
/
(%i12) load("abs_integrate")$
(%i13) integrate(x*abs(x),x);
2 3
x abs(x) x signum(x)
(%o13) --------- - ------------
2 6
(%i14) integrate(abs(x),x,-2,%pi);
2
%pi
(%o14) ---- + 2
2
(%i15) laplace(abs(x),x,s);
1
(%o15) --
2
s
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="ceiling"></a><a name="Item_003a-MathFunctions_002fdeffn_002fceiling"></a><dl>
<dt><a name="index-ceiling"></a>Function: <strong>ceiling</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>When <var>x</var> is a real number, return the least integer that
is greater than or equal to <var>x</var>.
</p>
<p>If <var>x</var> is a constant expression (<code>10 * %pi</code>, for example),
<code>ceiling</code> evaluates <var>x</var> using big floating point numbers, and
applies <code>ceiling</code> to the resulting big float. Because <code>ceiling</code> uses
floating point evaluation, it’s possible, although unlikely, that <code>ceiling</code>
could return an erroneous value for constant inputs. To guard against errors,
the floating point evaluation is done using three values for <code><a href="maxima_13.html#fpprec">fpprec</a></code>.
</p>
<p>For non-constant inputs, <code>ceiling</code> tries to return a simplified value.
Here are examples of the simplifications that <code>ceiling</code> knows about:
</p>
<div class="example">
<pre class="example">(%i1) ceiling (ceiling (x));
(%o1) ceiling(x)
</pre><pre class="example">(%i2) ceiling (floor (x));
(%o2) floor(x)
</pre><pre class="example">(%i3) declare (n, integer)$
</pre><pre class="example">(%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))];
(%o4) [n, abs(n), max(6, n)]
</pre><pre class="example">(%i5) assume (x > 0, x < 1)$
</pre><pre class="example">(%i6) ceiling (x);
(%o6) 1
</pre><pre class="example">(%i7) tex (ceiling (a));
$$\left \lceil a \right \rceil$$
(%o7) false
</pre></div>
<p>The <code>ceiling</code> function distributes over lists, matrices and equations.
See <code><a href="maxima_46.html#distribute_005fover">distribute_over</a></code>.
</p>
<p>Finally, for all inputs that are manifestly complex, <code>ceiling</code> returns
a noun form.
</p>
<p>If the range of a function is a subset of the integers, it can be declared to
be <code>integervalued</code>. Both the <code>ceiling</code> and <code><a href="#floor">floor</a></code> functions
can use this information; for example:
</p>
<div class="example">
<pre class="example">(%i1) declare (f, integervalued)$
</pre><pre class="example">(%i2) floor (f(x));
(%o2) f(x)
</pre><pre class="example">(%i3) ceiling (f(x) - 1);
(%o3) f(x) - 1
</pre></div>
<p>Example use:
</p>
<div class="example">
<pre class="example">(%i1) unitfrac(r) := block([uf : [], q],
if not(ratnump(r)) then
error("unitfrac: argument must be a rational number"),
while r # 0 do (
uf : cons(q : 1/ceiling(1/r), uf),
r : r - q),
reverse(uf));
(%o1) unitfrac(r) := block([uf : [], q],
if not ratnump(r) then
error("unitfrac: argument must be a rational number"),
1
while r # 0 do (uf : cons(q : ----------, uf), r : r - q),
1
ceiling(-)
r
reverse(uf))
</pre><pre class="example">(%i2) unitfrac (9/10);
1 1 1
(%o2) [-, -, --]
2 3 15
</pre><pre class="example">(%i3) apply ("+", %);
9
(%o3) --
10
</pre><pre class="example">(%i4) unitfrac (-9/10);
1
(%o4) [- 1, --]
10
</pre><pre class="example">(%i5) apply ("+", %);
9
(%o5) - --
10
</pre><pre class="example">(%i6) unitfrac (36/37);
1 1 1 1 1
(%o6) [-, -, -, --, ----]
2 3 8 69 6808
</pre><pre class="example">(%i7) apply ("+", %);
36
(%o7) --
37
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="entier"></a><a name="Item_003a-MathFunctions_002fdeffn_002fentier"></a><dl>
<dt><a name="index-entier"></a>Function: <strong>entier</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>Returns the largest integer less than or equal to <var>x</var> where <var>x</var> is
numeric. <code><a href="#fix">fix</a></code> (as in <code>fixnum</code>) is a synonym for this, so
<code>fix(<var>x</var>)</code> is precisely the same.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="floor"></a><a name="Item_003a-MathFunctions_002fdeffn_002ffloor"></a><dl>
<dt><a name="index-floor"></a>Function: <strong>floor</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>When <var>x</var> is a real number, return the largest integer that is less than or
equal to <var>x</var>.
</p>
<p>If <var>x</var> is a constant expression (<code>10 * %pi</code>, for example), <code>floor</code>
evaluates <var>x</var> using big floating point numbers, and applies <code>floor</code> to
the resulting big float. Because <code>floor</code> uses floating point evaluation,
it’s possible, although unlikely, that <code>floor</code> could return an erroneous
value for constant inputs. To guard against errors, the floating point
evaluation is done using three values for <code><a href="maxima_13.html#fpprec">fpprec</a></code>.
</p>
<p>For non-constant inputs, <code>floor</code> tries to return a simplified value. Here
are examples of the simplifications that <code>floor</code> knows about:
</p>
<div class="example">
<pre class="example">(%i1) floor (ceiling (x));
(%o1) ceiling(x)
</pre><pre class="example">(%i2) floor (floor (x));
(%o2) floor(x)
</pre><pre class="example">(%i3) declare (n, integer)$
</pre><pre class="example">(%i4) [floor (n), floor (abs (n)), floor (min (n, 6))];
(%o4) [n, abs(n), min(6, n)]
</pre><pre class="example">(%i5) assume (x > 0, x < 1)$
</pre><pre class="example">(%i6) floor (x);
(%o6) 0
</pre><pre class="example">(%i7) tex (floor (a));
$$\left \lfloor a \right \rfloor$$
(%o7) false
</pre></div>
<p>The <code>floor</code> function distributes over lists, matrices and equations.
See <code><a href="maxima_46.html#distribute_005fover">distribute_over</a></code>.
</p>
<p>Finally, for all inputs that are manifestly complex, <code>floor</code> returns
a noun form.
</p>
<p>If the range of a function is a subset of the integers, it can be declared to
be <code>integervalued</code>. Both the <code><a href="#ceiling">ceiling</a></code> and <code>floor</code> functions
can use this information; for example:
</p>
<div class="example">
<pre class="example">(%i1) declare (f, integervalued)$
</pre><pre class="example">(%i2) floor (f(x));
(%o2) f(x)
</pre><pre class="example">(%i3) ceiling (f(x) - 1);
(%o3) f(x) - 1
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="fix"></a><a name="Item_003a-MathFunctions_002fdeffn_002ffix"></a><dl>
<dt><a name="index-fix"></a>Function: <strong>fix</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>A synonym for <code>entier (<var>x</var>)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="hstep"></a><a name="Item_003a-MathFunctions_002fdeffn_002fhstep"></a><dl>
<dt><a name="index-hstep"></a>Function: <strong>hstep</strong> <em>(<var>x</var>)</em></dt>
<dd><p>The Heaviside unit step function, equal to 0 if <var>x</var> is negative,
equal to 1 if <var>x</var> is positive and equal to 1/2 if <var>x</var> is equal
to zero.
</p>
<p>If you want a unit step function that takes on the value of 0 at <var>x</var>
equal to zero, use <code><a href="maxima_338.html#unit_005fstep">unit_step</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Laplace-transform">Laplace transform</a>
·<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div>
</dd></dl>
<a name="lmax"></a><a name="Item_003a-MathFunctions_002fdeffn_002flmax"></a><dl>
<dt><a name="index-lmax"></a>Function: <strong>lmax</strong> <em>(<var>L</var>)</em></dt>
<dd>
<p>When <var>L</var> is a list or a set, return <code>apply ('max, args (<var>L</var>))</code>.
When <var>L</var> is not a list or a set, signal an error.
See also <code><a href="#lmin">lmin</a></code> and <code><a href="#max">max</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·<a href="maxima_424.html#Category_003a-Lists">Lists</a>
·<a href="maxima_424.html#Category_003a-Sets">Sets</a>
·</div></dd></dl>
<a name="lmin"></a><a name="Item_003a-MathFunctions_002fdeffn_002flmin"></a><dl>
<dt><a name="index-lmin"></a>Function: <strong>lmin</strong> <em>(<var>L</var>)</em></dt>
<dd>
<p>When <var>L</var> is a list or a set, return <code>apply ('min, args (<var>L</var>))</code>.
When <var>L</var> is not a list or a set, signal an error.
See also <code><a href="#lmax">lmax</a></code> and <code><a href="#min">min</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·<a href="maxima_424.html#Category_003a-Lists">Lists</a>
·<a href="maxima_424.html#Category_003a-Sets">Sets</a>
·</div></dd></dl>
<a name="max"></a><a name="Item_003a-MathFunctions_002fdeffn_002fmax"></a><dl>
<dt><a name="index-max"></a>Function: <strong>max</strong> <em>(<var>x_1</var>, …, <var>x_n</var>)</em></dt>
<dd>
<p>Return a simplified value for the numerical maximum of the expressions <var>x_1</var>
through <var>x_n</var>. For an empty argument list, <code>max</code> yields <code>minf</code>.
</p>
<p>The option variable <code>maxmin_effort</code> controls which simplification methods are
applied. Using the default value of <em>twelve</em> for <code>maxmin_effort</code>,
<code>max</code> uses <em>all</em> available simplification methods. To to inhibit all
simplifications, set <code>maxmin_effort</code> to zero.
</p>
<p>When <code>maxmin_effort</code> is one or more, for an explicit list of real numbers,
<code>max</code> returns a number.
</p>
<p>Unless <code>max</code> needs to simplify a lengthy list of expressions, we suggest using
the default value of <code>maxmin_effort</code>. Setting <code>maxmin_effort</code> to zero
(no simplifications), will cause problems for some Maxima functions; accordingly,
generally <code>maxmin_effort</code> should be nonzero.
</p>
<p>See also <code><a href="#min">min</a></code>, <code><a href="#lmax">lmax</a></code>., and <code><a href="#lmin">lmin</a></code>..
</p>
<p><b>Examples:</b>
</p>
<p>In the first example, setting <code>maxmin_effort</code> to zero suppresses simplifications.
</p><div class="example">
<pre class="example">(%i1) block([maxmin_effort : 0], max(1,2,x,x, max(a,b)));
(%o1) max(1,2,max(a,b),x,x)
(%i2) block([maxmin_effort : 1], max(1,2,x,x, max(a,b)));
(%o2) max(2,a,b,x)
</pre></div>
<p>When <code>maxmin_effort</code> is two or more, <code>max</code> compares pairs of members:
</p><div class="example">
<pre class="example">(%i1) block([maxmin_effort : 1], max(x,x+1,x+3));
(%o1) max(x,x+1,x+3)
(%i2) block([maxmin_effort : 2], max(x,x+1,x+3));
(%o2) x+3
</pre></div>
<p>Finally, when <code>maxmin_effort</code> is three or more, <code>max</code> compares triples
members and excludes those that are in between; for example
</p><div class="example">
<pre class="example">(%i1) block([maxmin_effort : 4], max(x, 2*x, 3*x, 4*x));
(%o1) max(x,4*x)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="min"></a><a name="Item_003a-MathFunctions_002fdeffn_002fmin"></a><dl>
<dt><a name="index-min"></a>Function: <strong>min</strong> <em>(<var>x_1</var>, …, <var>x_n</var>)</em></dt>
<dd>
<p>Return a simplified value for the numerical minimum of the expressions <var>x_1</var>
through <var>x_n</var>. For an empty argument list, <code>minf</code> yields <code>inf</code>.
</p>
<p>The option variable <code>maxmin_effort</code> controls which simplification methods are
applied. Using the default value of <em>twelve</em> for <code>maxmin_effort</code>,
<code>max</code> uses <em>all</em> available simplification methods. To to inhibit all
simplifications, set <code>maxmin_effort</code> to zero.
</p>
<p>When <code>maxmin_effort</code> is one or more, for an explicit list of real numbers,
<code>min</code> returns a number.
</p>
<p>Unless <code>min</code> needs to simplify a lengthy list of expressions, we suggest using
the default value of <code>maxmin_effort</code>. Setting <code>maxmin_effort</code> to zero
(no simplifications), will cause problems for some Maxima functions; accordingly,
generally <code>maxmin_effort</code> should be nonzero.
</p>
<p>See also <code><a href="#max">max</a></code>, <code><a href="#lmax">lmax</a></code>., and <code><a href="#lmin">lmin</a></code>..
</p>
<p><b>Examples:</b>
</p>
<p>In the first example, setting <code>maxmin_effort</code> to zero suppresses simplifications.
</p><div class="example">
<pre class="example">(%i1) block([maxmin_effort : 0], min(1,2,x,x, min(a,b)));
(%o1) min(1,2,a,b,x,x)
(%i2) block([maxmin_effort : 1], min(1,2,x,x, min(a,b)));
(%o2) min(1,a,b,x)
</pre></div>
<p>When <code>maxmin_effort</code> is two or more, <code>min</code> compares pairs of members:
</p><div class="example">
<pre class="example">(%i1) block([maxmin_effort : 1], min(x,x+1,x+3));
(%o1) min(x,x+1,x+3)
(%i2) block([maxmin_effort : 2], min(x,x+1,x+3));
(%o2) x
</pre></div>
<p>Finally, when <code>maxmin_effort</code> is three or more, <code>min</code> compares triples
members and excludes those that are in between; for example
</p><div class="example">
<pre class="example">(%i1) block([maxmin_effort : 4], min(x, 2*x, 3*x, 4*x));
(%o1) max(x,4*x)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="round"></a><a name="Item_003a-MathFunctions_002fdeffn_002fround"></a><dl>
<dt><a name="index-round"></a>Function: <strong>round</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>When <var>x</var> is a real number, returns the closest integer to <var>x</var>.
Multiples of 1/2 are rounded to the nearest even integer. Evaluation of
<var>x</var> is similar to <code><a href="#floor">floor</a></code> and <code><a href="#ceiling">ceiling</a></code>.
</p>
<p>The <code>round</code> function distributes over lists, matrices and equations.
See <code><a href="maxima_46.html#distribute_005fover">distribute_over</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="signum"></a><a name="Item_003a-MathFunctions_002fdeffn_002fsignum"></a><dl>
<dt><a name="index-signum"></a>Function: <strong>signum</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>For either real or complex numbers <var>x</var>, the signum function returns
0 if <var>x</var> is zero; for a nonzero numeric input <var>x</var>, the signum function
returns <code>x/abs(x)</code>.
</p>
<p>For non-numeric inputs, Maxima attempts to determine the sign of the input.
When the sign is negative, zero, or positive, <code>signum</code> returns -1,0, 1,
respectively. For all other values for the sign, <code>signum</code> a simplified but
equivalent form. The simplifications include reflection (<code>signum(-x)</code>
gives <code>-signum(x)</code>) and multiplicative identity (<code>signum(x*y)</code> gives
<code>signum(x) * signum(y)</code>).
</p>
<p>The <code>signum</code> function distributes over a list, a matrix, or an
equation. See <code><a href="maxima_63.html#sign">sign</a></code> and <code><a href="maxima_46.html#distribute_005fover">distribute_over</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
<a name="truncate"></a><a name="Item_003a-MathFunctions_002fdeffn_002ftruncate"></a><dl>
<dt><a name="index-truncate"></a>Function: <strong>truncate</strong> <em>(<var>x</var>)</em></dt>
<dd>
<p>When <var>x</var> is a real number, return the closest integer to <var>x</var> not
greater in absolute value than <var>x</var>. Evaluation of <var>x</var> is similar
to <code><a href="#floor">floor</a></code> and <code><a href="#ceiling">ceiling</a></code>.
</p>
<p>The <code>truncate</code> function distributes over lists, matrices and equations.
See <code><a href="maxima_46.html#distribute_005fover">distribute_over</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Mathematical-functions">Mathematical functions</a>
·</div></dd></dl>
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<div class="header">
<p>
Next: <a href="maxima_49.html#Functions-for-Complex-Numbers" accesskey="n" rel="next">Functions for Complex Numbers</a>, Up: <a href="maxima_47.html#Elementary-Functions" accesskey="u" rel="up">Elementary Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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