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Transcript of Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester

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www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester Slide 2 Content MotivationChange of sign methodIterative methodNewton-Raphson method Slide 3 Reasons for Finding Roots by Numerical Methods If the data is obtained from observations, it often wont 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 Slide 4 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 Slide 5 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 Slide 6 Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation Slide 7 Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation Slide 8 Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation Slide 9 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 Slide 10 Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation Slide 11 So to 3 s.f. the solution is Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation Slide 12 Solving equations by change of sign Next Iterative method Newton- Raphson Change of sign method Motivation Slide 13 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 Slide 14 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 15 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 16 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 17 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 18 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 19 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 Slide 20 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 21 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 22 This gives us the solution to 3 d.p. Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 23 Solving using iterative method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 24 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 Slide 25 Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 26 Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 27 Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 28 Newton-Raphson Method Next Iterative method Newton- Raphson Change of sign method Motivation Slide 29 Newton-Raphson Method So this means that the cube root of 37 is approximately 3.3322 Next Iterative method Newton- Raphson Change of sign method Motivation Slide 30