Find the roots of the equation using the bisection method f(x)= (e^-x) -x, such

Question

Find the roots of the equation using the bisection method f(x)= (e^-x) -x, such that xl=-1 and xu=4 and the error is less than 0.5
(By hand without a program) -please explain what you are doing so I can apply this method to other problems. Find the roots of the equation using the bisection method f(x)= (e^-x) -x, such that xl=-1 and xu=4 and the error is less than 0.5
(By hand without a program) -please explain what you are doing so I can apply this method to other problems.
(By hand without a program) -please explain what you are doing so I can apply this method to other problems.

Explanation / Answer

Bisection Method:

In a closed interval [a,b], for a transcendental equation f(x); if f(a) & f(b) are of opposite signs,

then there must be a value 'c' for which f(x)=0; i.e. there lies a root of that equation at x=c.

In bisection method, the above said fact is used to find the root of a given equation ( within a given error tolerance)

Steps:

1. In this method we bisect the given closed interval [ xl, xu ] at point c where xm = ( xu - xl ) / 2 and find the value of f(x) at the point xm.

2. now find the value of f(xl), f(xu) and f(xm)

if f(xm)=0, then stop the iteration and desired result is x=xm

3. now check the sign of f(xl) * f(xm) ,

if sign is -ve, then upper boundry xu is replaced by xm, i.e xu = xm

if sign is +ve, then lower boundry xl is replaced by xm, i.e xl = xm

4. check the error e = | xu-xl |

if e> given error tolerance then continue iteration steps(1-4)

if e<=given error tolerance, then stop the iteration and desired result is x=xm

replacement

for next iteration

In the above table; we stopped after 4th iteration as the given error tolerance is achieved. i.e 0.875 - 0.5625 < 0.5

So, the solution for given problem is x = 0.5625 .

Iteration Table xl xu xm f(x) |x=xl f(x) |x=xu f(x) |x=xm f(xl)*f(xm)

replacement

for next iteration

-1 4 1.5 3.7182 -3.9816 -1.2768 -ve xu=xm -1 1.5 0.25 3.7182 -1.2768 0.5288 +ve xl=xm 0.25 1.5 .0875 0.5288 -1.2768 -0.4581 -ve xu=xm 0.25 0.875 0.5625 0.5288 -0.4581 0.0072 +ve xl=xm 0.5625 0.875

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