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<a name="Functions-for-numerical-solution-of-equations"></a>
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<p>
Next: <a href="maxima_88.html#Introduction-to-numerical-solution-of-differential-equations" accesskey="n" rel="next">Introduction to numerical solution of differential equations</a>, Previous: <a href="maxima_86.html#Functions-and-Variables-for-FFTPACK5" accesskey="p" rel="previous">Functions and Variables for FFTPACK5</a>, Up: <a href="maxima_83.html#Numerical" accesskey="u" rel="up">Numerical</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-for-numerical-solution-of-equations-1"></a>
<h3 class="section">21.4 Functions for numerical solution of equations</h3>
<a name="horner"></a><a name="Item_003a-Numerical_002fdeffn_002fhorner"></a><dl>
<dt><a name="index-horner"></a>Function: <strong>horner</strong> <em><br> <tt>horner</tt> (<var>expr</var>, <var>x</var>) <br> <tt>horner</tt> (<var>expr</var>)</em></dt>
<dd>
<p>Returns a rearranged representation of <var>expr</var> as in Horner’s rule, using
<var>x</var> as the main variable if it is specified. <code>x</code> may be omitted in
which case the main variable of the canonical rational expression form of
<var>expr</var> is used.
</p>
<p><code>horner</code> sometimes improves stability if <code>expr</code> is
to be numerically evaluated. It is also useful if Maxima is used to
generate programs to be run in Fortran. See also <code><a href="maxima_41.html#stringout">stringout</a></code>.
</p>
<div class="example">
<pre class="example">(%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
2
(%o1) 1.e-155 x - 5.5 x + 5.2e+155
(%i2) expr2: horner (%, x), keepfloat: true;
(%o2) 1.0 ((1.e-155 x - 5.5) x + 5.2e+155)
(%i3) ev (expr, x=1e155);
Maxima encountered a Lisp error:
arithmetic error FLOATING-POINT-OVERFLOW signalled
Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.
(%i4) ev (expr2, x=1e155);
(%o4) 7.00000000000001e+154
</pre></div>
</dd></dl>
<a name="find_005froot"></a><a name="bf_005ffind_005froot"></a><a name="find_005froot_005ferror"></a><a name="find_005froot_005fabs"></a><a name="find_005froot_005frel"></a><a name="Item_003a-Numerical_002fdeffn_002ffind_005froot"></a><dl>
<dt><a name="index-find_005froot"></a>Function: <strong>find_root</strong> <em>(<var>expr</var>, <var>x</var>, <var>a</var>, <var>b</var>, [<var>abserr</var>, <var>relerr</var>])</em></dt>
<dt><a name="index-find_005froot-1"></a>Function: <strong>find_root</strong> <em>(<var>f</var>, <var>a</var>, <var>b</var>, [<var>abserr</var>, <var>relerr</var>])</em></dt>
<dd><a name="Item_003a-Numerical_002fdeffn_002fbf_005ffind_005froot"></a></dd><dt><a name="index-bf_005ffind_005froot"></a>Function: <strong>bf_find_root</strong> <em>(<var>expr</var>, <var>x</var>, <var>a</var>, <var>b</var>, [<var>abserr</var>, <var>relerr</var>])</em></dt>
<dt><a name="index-bf_005ffind_005froot-1"></a>Function: <strong>bf_find_root</strong> <em>(<var>f</var>, <var>a</var>, <var>b</var>, [<var>abserr</var>, <var>relerr</var>])</em></dt>
<dd><a name="Item_003a-Numerical_002fdeffn_002ffind_005froot_005ferror"></a></dd><dt><a name="index-find_005froot_005ferror"></a>Option variable: <strong>find_root_error</strong></dt>
<dd><a name="Item_003a-Numerical_002fdeffn_002ffind_005froot_005fabs"></a></dd><dt><a name="index-find_005froot_005fabs"></a>Option variable: <strong>find_root_abs</strong></dt>
<dd><a name="Item_003a-Numerical_002fdeffn_002ffind_005froot_005frel"></a></dd><dt><a name="index-find_005froot_005frel"></a>Option variable: <strong>find_root_rel</strong></dt>
<dd>
<p>Finds a root of the expression <var>expr</var> or the function <var>f</var> over the
closed interval <em>[<var>a</var>, <var>b</var>]</em>. The expression <var>expr</var> may be an
equation, in which case <code><a href="#find_005froot">find_root</a></code> seeks a root of
<code>lhs(<var>expr</var>) - rhs(<var>expr</var>)</code>.
</p>
<p>Given that Maxima can evaluate <var>expr</var> or <var>f</var> over
<em>[<var>a</var>, <var>b</var>]</em> and that <var>expr</var> or <var>f</var> is continuous,
<code>find_root</code> is guaranteed to find the root,
or one of the roots if there is more than one.
</p>
<p><code>find_root</code> initially applies binary search.
If the function in question appears to be smooth enough,
<code>find_root</code> applies linear interpolation instead.
</p>
<p><code>bf_find_root</code> is a bigfloat version of <code>find_root</code>. The
function is computed using bigfloat arithmetic and a bigfloat result
is returned. Otherwise, <code>bf_find_root</code> is identical to
<code>find_root</code>, and the following description is equally applicable
to <code>bf_find_root</code>.
</p>
<p>The accuracy of <code>find_root</code> is governed by <code>abserr</code> and
<code>relerr</code>, which are optional keyword arguments to
<code>find_root</code>. These keyword arguments take the form
<code>key=val</code>. The keyword arguments are
</p>
<dl compact="compact">
<dt><code>abserr</code></dt>
<dd><p>Desired absolute error of function value at root. Default is
<code>find_root_abs</code>.
</p></dd>
<dt><code>relerr</code></dt>
<dd><p>Desired relative error of root. Default is <code>find_root_rel</code>.
</p></dd>
</dl>
<p><code>find_root</code> stops when the function in question evaluates to
something less than or equal to <code>abserr</code>, or if successive
approximants <var>x_0</var>, <var>x_1</var> differ by no more than <code>relerr
* max(abs(x_0), abs(x_1))</code>. The default values of
<code>find_root_abs</code> and <code>find_root_rel</code> are both zero.
</p>
<p><code>find_root</code> expects the function in question to have a different sign at
the endpoints of the search interval.
When the function evaluates to a number at both endpoints
and these numbers have the same sign,
the behavior of <code>find_root</code> is governed by <code>find_root_error</code>.
When <code>find_root_error</code> is <code>true</code>,
<code>find_root</code> prints an error message.
Otherwise <code>find_root</code> returns the value of <code>find_root_error</code>.
The default value of <code>find_root_error</code> is <code>true</code>.
</p>
<p>If <var>f</var> evaluates to something other than a number at any step in the search
algorithm, <code>find_root</code> returns a partially-evaluated <code>find_root</code>
expression.
</p>
<p>The order of <var>a</var> and <var>b</var> is ignored; the region in which a root is
sought is <em>[min(<var>a</var>, <var>b</var>), max(<var>a</var>, <var>b</var>)]</em>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) f(x) := sin(x) - x/2;
x
(%o1) f(x) := sin(x) - -
2
(%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
(%o2) 1.895494267033981
(%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
(%o3) 1.895494267033981
(%i4) find_root (f(x), x, 0.1, %pi);
(%o4) 1.895494267033981
(%i5) find_root (f, 0.1, %pi);
(%o5) 1.895494267033981
(%i6) find_root (exp(x) = y, x, 0, 100);
x
(%o6) find_root(%e = y, x, 0.0, 100.0)
(%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
(%o7) 2.302585092994046
(%i8) log (10.0);
(%o8) 2.302585092994046
(%i9) fpprec:32;
(%o9) 32
(%i10) bf_find_root (exp(x) = y, x, 0, 100), y = 10;
(%o10) 2.3025850929940456840179914546844b0
(%i11) log(10b0);
(%o11) 2.3025850929940456840179914546844b0
</pre></div>
</dd></dl>
<a name="newton"></a><a name="Item_003a-Numerical_002fdeffn_002fnewton"></a><dl>
<dt><a name="index-newton"></a>Function: <strong>newton</strong> <em>(<var>expr</var>, <var>x</var>, <var>x_0</var>, <var>eps</var>)</em></dt>
<dd>
<p>Returns an approximate solution of <code><var>expr</var> = 0</code> by Newton’s method,
considering <var>expr</var> to be a function of one variable, <var>x</var>.
The search begins with <code><var>x</var> = <var>x_0</var></code>
and proceeds until <code>abs(<var>expr</var>) < <var>eps</var></code>
(with <var>expr</var> evaluated at the current value of <var>x</var>).
</p>
<p><code>newton</code> allows undefined variables to appear in <var>expr</var>,
so long as the termination test <code>abs(<var>expr</var>) < <var>eps</var></code> evaluates
to <code>true</code> or <code>false</code>.
Thus it is not necessary that <var>expr</var> evaluate to a number.
</p>
<p><code>load("newton1")</code> loads this function.
</p>
<p>See also <code><a href="maxima_79.html#realroots">realroots</a></code>, <code><a href="maxima_79.html#allroots">allroots</a></code>, <code><a href="#find_005froot">find_root</a></code> and
<code><a href="maxima_219.html#mnewton">mnewton</a></code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) load ("newton1");
(%o1) /maxima/share/numeric/newton1.mac
(%i2) newton (cos (u), u, 1, 1/100);
(%o2) 1.570675277161251
(%i3) ev (cos (u), u = %);
(%o3) 1.2104963335033529e-4
(%i4) assume (a > 0);
(%o4) [a > 0]
(%i5) newton (x^2 - a^2, x, a/2, a^2/100);
(%o5) 1.00030487804878 a
(%i6) ev (x^2 - a^2, x = %);
2
(%o6) 6.098490481853958e-4 a
</pre></div>
</dd></dl>
<a name="Item_003a-Numerical_002fnode_002fIntroduction-to-numerical-solution-of-differential-equations"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_88.html#Introduction-to-numerical-solution-of-differential-equations" accesskey="n" rel="next">Introduction to numerical solution of differential equations</a>, Previous: <a href="maxima_86.html#Functions-and-Variables-for-FFTPACK5" accesskey="p" rel="previous">Functions and Variables for FFTPACK5</a>, Up: <a href="maxima_83.html#Numerical" accesskey="u" rel="up">Numerical</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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