Question

Use Bisection method to calculate the first root for :
f(d)= 257d^2-640d=0
a=0.0000 b=5.0000
Tolerance=0.0500

Answer 1

Conditions

Use Bisection method to calculate the first root for :

f(d)= 257d^2-640d=0

a=0.0000 b=5.0000

Tolerance=0.0500

Solution

The bisection method in mathematics is a root-finding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the binary search method or the dichotomy method

The method is applicable when we wish to solve the equation $f(x) = 0$ for the real variable $x$, where $f$ is a continuous function defined on an interval $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs. In this case $a$ and $b$ are said to bracket a root since, by the intermediate value theorem, the $f$ must have at least one root in the interval $(a, b)$.

At each step the method divides the interval in two by computing the midpoint $c = (a + b) / 2$ of the interval and the value of the function $f(c)$ at that point. Unless $c$ is itself a root (which is very unlikely, but possible) there are now two possibilities: either $f(a)$ and $f(c)$ have opposite signs and bracket a root, or $f(c)$ and $f(b)$ have opposite signs and bracket a root. The method selects the subinterval that is a bracket as a new interval to be used in the next step. In this way the interval that contains a zero of $f$ is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.

Explicitly, if $f(a)$ and $f(c)$ are opposite signs, then the method sets $c$ as the new value for $b$, and if $f(b)$ and $f(c)$ are opposite signs then the method sets $c$ as the new $a$. (If $f(c) = 0$ then $c$ may be taken as the solution and the process stops.) In both cases, the new $f(a)$ and $f(b)$ have opposite signs, so the method is applicable to this smaller interval.

$$f(a) = f(0) = 0$$ $$f(b) = f(5) = 3225$$

Because the function is continuous, there must be a root within the interval. (Actually, it's obvious to notice, that the root is $x = 640 / 257 \approx 2,4902723735408560311284046692607$)

In the first iteration, the end points of the interval which brackets the root are $\alpha = 0$ and $b = 5$ .

So, $c = \frac{a + b}{2} = 2.5$ .

$f(c) = 6.25$

As $f(c) > 0$ and $f(c) < f(b)$ so the next b-point will be c.

New $c = \frac{0 + 2.5}{2} = 1.25$

$f(c) = -398.438$

As $f(c) < 0$ so the new a is 1.25, the new b is 2.5

New $c = \frac{1.25 + 2.5}{2} = 1.875$

$f(c) = -296.484$

$\alpha = 1.875$

$\mathrm{b} = 2.5$

New $c = \frac{1.875 + 2.5}{2} = 2.1875$

And so on and so for:

As the tolerance was 0.05, we have a root $x = 2,490234375$

We can compare this value with a value 2,4902723735408560311284046692607, which was noticed before. As we see, the calculator's value is very close to the value, given by Bisection method.