Numerical Methods: Finding Roots Department of Mathematics University of Leicester

Content MotivationChange of sign methodIterative methodNewton-Raphson method

Reasons for Finding Roots by Numerical Methods If the data is obtained from observations, it often won’t have an equation which accurately models Some equations are not easy to solve Can program a computer to solve equations for us Next Iterative method Newton- Raphson Change of sign method Motivation

Solving equations by change of sign This is also known as ‘Iteration by Bisection’ It is done by bisecting an interval we know the solution lies in repeatedly Next Iterative method Newton- Raphson Change of sign method Motivation

METHOD Find an interval in which the solution lies Split the interval into 2 equal parts Find the change of sign Repeat Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Step 3: Now we just keep repeating the process Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

So to 3 s.f. the solution is Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method ‘Iteration’ is the process of repeatedly using a previous result to obtain a new result Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Click on a seed value to see the cobweb: start here start here start here start here start here start here Clear Cobwebs Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

This gives us the solution to 3 d.p. Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation

Newton-Raphson Method Sometimes known as the Newton Method Named after Issac Newton and Joseph Raphson Iteratively finds successively better approximations to the roots Next Iterative method Newton- Raphson Change of sign method Motivation

Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation

Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation

Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation

Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation

Newton-Raphson Method So this means that the cube root of 37 is approximately Next Iterative method Newton- Raphson Change of sign method Motivation