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The Maxima on-line user's manual

Algebra Calculator

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Find_root Calculator

Find_root

Function: find_root (<expr>, <x>, <a>, <b>)

f(x) := x^3 - x^2 + 5;
define(Df(x), diff(f(x), x));
Df(5);
plot2d(f(x), [x, -4, 4]);
find_root(f(x), x, -2, -1);

Function: find_root (<f>, <a>, <b>) -- Option variable: find_root_error -- Option variable: find_root_abs -- Option variable: find_root_rel Finds a root of the expression <expr> or the function <f> over the closed interval [<a>, <b>]. The expression <expr> may be an equation, in which case find_root seeks a root of lhs(<expr>) - rhs(<expr>).

Given that Maxima can evaluate <expr> or <f> over [<a>, <b>] and that <expr> or <f> is continuous, find_root is guaranteed to find the root, or one of the roots if there is more than one.

find_root initially applies binary search. If the function in question appears to be smooth enough, find_root applies linear interpolation instead.

The accuracy of find_root is governed by find_root_abs and find_root_rel. find_root stops when the function in question evaluates to something less than or equal to find_root_abs, or if successive approximants <x_0>, <x_1> differ by no more than find_root_rel * max(abs(x_0), abs(x_1)). The default values of find_root_abs and find_root_rel are both zero.

find_root 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 find_root is governed by find_root_error. When find_root_error is true, find_root prints an error message. Otherwise find_root returns the value of find_root_error. The default value of find_root_error is true.

If <f> evaluates to something other than a number at any step in the search algorithm, find_root returns a partially-evaluated find_root expression.

The order of <a> and <b> is ignored; the region in which a root is sought is [min(<a>, <b>), max(<a>, <b>)].

Examples:

          (%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

There are also some inexact matches for find_root. Try ?? find_root to see them.

(%o1)                                true
(%i2) 

Find_root Example

Related Examples

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