Root Finder Interact

## Description

Here is a Sage interact that estimates the roots of a function using the bisection method. The user may input the function and the initial endpoints of the estimation range.

## Sage Cell

#### Code

```
def bisect_method(f, a, b, eps):
try:
f = f._fast_float_(f.variables()[0])
except AttributeError:
pass
intervals = [(a,b)]
two = float(2); eps = float(eps)
while True:
c = (a+b)/two
fa = f(a); fb = f(b); fc = f(c)
if abs(fc) < eps: return c, intervals
if fa*fc < 0:
a, b = a, c
elif fc*fb < 0:
a, b = c, b
else:
raise ValueError("f must have a sign change in the interval (%s,%s)"%(a,b))
intervals.append((a,b))
pretty_print(html("<h1>Double Precision Root Finding Using Bisection</h1>"))
@interact
def _(f = cos(x) - x, a = float(0), b = float(1), eps=(-3,(-16, -1))):
eps = 10^eps
print("eps = %s" % float(eps))
try:
c, intervals = bisect_method(f, a, b, eps)
except ValueError:
print("f must have opposite sign at the endpoints of the interval")
show(plot(f, a, b, color='red'), xmin=a, xmax=b)
else:
print("root =", c)
print("f(c) = %r" % f(x=c))
print("iterations =", len(intervals))
P = plot(f, a, b, color='red')
h = (P.ymax() - P.ymin())/ (1.5*len(intervals))
L = sum(line([(c,h*i), (d,h*i)]) for i, (c,d) in enumerate(intervals) )
L += sum(line([(c,h*i-h/4), (c,h*i+h/4)]) for i, (c,d) in enumerate(intervals) )
L += sum(line([(d,h*i-h/4), (d,h*i+h/4)]) for i, (c,d) in enumerate(intervals) )
show(P + L, xmin=a, xmax=b)
```

## Options

None

## Tags

Primary Tags: Single Variable Calculus: Limits and continuity

Secondary Tags: Limits and continuity: Applications - other

## Related Cells

None

## Attribute

Permalink: https://wiki.sagemath.org/interact/calculus#Root_Finding_Using_Bisection

Author: William Stein

Date: 19 Jul 2020 01:53

Submitted by: Zane Corbiere