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# examples - Numerical Methods Natural Sciences Tripos 1B...

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Stuart Dalziel, DAMTP Numerical Methods Natural Sciences Tripos 1B Lent Term 1999 Problem Sheet 1 1. ROOT FINDING 1.1 Roots of a cubic Consider the solution to f ( x ) = 0.5 where f ( x ) = x 3 . Choosing initial guesses of x a = 0 and x b = 1 , (a) Write down an expression to show how the error ε n in the bisection method decreases with subsequent iterations. (b) Using the bisection method, determine the solution to four decimal places. Does the number of iterations this took agree with the predicted number? (c) Repeat this calculation using the Linear interpolation and secant methods. How do the results compare after each iteration? (d) Repeat the calculations using the Newton-Raphson method, starting from (i) x 0 = 1 , (ii) x 0 = 0 and (iii) x 0 = 3 . Comment on your results. 1.2 Direct iteration (a) Illustrate graphically the direct iteration method and show cases of monotonic convergence, oscillatory convergence and divergence. (b) Derive the convergence criteria for the direct iteration method for any arbitrary function. (c) Using only the four basic operations ( +–*/ ), write down a scheme which will solve f ( x ) = 0.5 , where f ( x )=sin x . Starting with x =90degrees= π /2 radians , use this scheme to compute the solution to six decimal places. Compare the convergence of your calculation with your analysis of the convergence. 1.3 Direct iteration Consider the roots of f ( x ) = 0 for some continuous function f ( x ) . (a) By rerranging and adding iteration labels ( i.e. change x into x n or x n +1 ), derive a direct iteration equation in the form x n +1 = g ( x n ) from ( x - x ) h ( x ) + f ( x ) = 0 . (b) Prove that the quadratic convergence can be obtained with a particular choice of h ( x ) and explain how this is related to the Newton-Raphson method. 2. LINEAR EQUATIONS 2.1 Gaussian elimination (a) Use Gaussian elimination to solve 1 2 3 2 3 4 3 4 5 6 9 12 1 2 3 = x x x

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NST1B Numerical Methods Lent 1999 Stuart Dalziel, DAMTP (b) Explain what is meant by pivoting. Why are these techniques used? Outline the differences between partial pivoting and full pivoting. (c) Evaluate the solution of 1 2 3 1 2 4 2 2 5 6 7 9 1 2 3 = x x x using both partial and full pivoting. Why is pivoting necessary? 2.2 Banded matrices Prove that if a matrix contains only zeroes outside some band centred around the leading diagonal, then performing Gaussian elimination (without pivoting) will maintain these zeros. 2.3 Tridiagonal matrices Develop an algorithm for solving a tridiagonal system of equations. 2.4 Over determined systems* (a) Explain what is meant by the term “over determined system”.
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examples - Numerical Methods Natural Sciences Tripos 1B...

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