Numerical Analysis/Exercises and quiz
In this section we are going to look at a method for approximating solutions to equations. We all know that equations need to be solved on occasion and in fact we’ve solved quite a few equations ourselves to this point. we were able to actually find the solutions, but it’s not always possible to do that exactly and/or do the work by hand. That is where this application comes into play. So, let’s see what this application is all about.
Let’s suppose that we want to approximate the solution to f(x0) and let’s also suppose that we have somehow found an initial approximation to this solution say, x0. This initial approximation is probably not all that good and so we’d like to find a better approximation. This is easy enough to do. First we will get the tangent line to f(x) at x0.
y=f(x0)+f'(x0)(x-x0).
Then we get the formula xn+1=xn-f(xn)/f'(xn).
Example:
1)Find a nonzero root of equation x2-sin x using newton's method.
Solution:
Let us take our initial guess as .5
f'(x)=2x-cos x.
now we will use newton's formula to find x2
x2=.5-sin(.5)2(.5)-cos(.5)=2.374
In this way we get x3=1.4702,x4=1.0594,x5=0.9057,x6=0.8727,x7=0.8766,x8=0.8767
2)Find an approximation to 51/2 to ten decimal places with starting point as 2.
51/2 is the root of f(x)=x2-5
so now using the formula,
we get,x1=2
x2=2.25
x3=2.236111111111
x4=2.236067977915
x5=2.236067977499
x6=2.236067977499
Exercise
Use Newton's method to approxmate the zero's of function f(x) = x3+2x+2
Solution:
0.358
Method Fails: In some cases the conditions on function necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. In such cases a different method, such as bisection, should be used to obtain a better estimate for the zero to use as an initial point.
Example:Use x0=1 to find the approximation to the solution to f(x)=x1/3 .
using Newton's Formula xn+1=-2xn.
x1=-2
x2=4
x3=-8
x4=16 ..
Because each step produces new approximation that is far from the root , method fails.
Exercise
Find the roots of the polynomial f(x)=ex-2x with starting point as 1
Solution:
Method fails
Quiz[edit | edit source]