(a) Bisection Method: Let fe C[a, b] and suppose f(a) f(b) 0. Prove that the bisection...

 

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(a) Bisection Method: Let fe C[a, b] and suppose f(a) f(b) 0. Prove that the bisection method generates a sequence {pn} approximating the root p off with the property Pn-p| 2(b-a), n 1 I [7 marks] (b) Existence of a Fixed Point: Prove that if g C[a, b] and g(x) = [a, b] V rE [a, b] then the function g has a fixed point in [a, b]. [7 marks] (c) Newton-Raphson: Using Matlab syntax write down an appropriate algo- rithm for finding the root of an equation, f(x), using the Newton-Raphson method. Be as accurate as you can regarding the Matlab syntax. Include an fprintf() statement at every iteration showing the step number, the current estimation of the root and the associated absolute error relative to the previous step. Finally, include a final fprintf() statement at the end giving the number of steps required, the root and the absolute error. [6 marks] (a) Bisection Method: Let fe C[a, b] and suppose f(a) f(b) 0. Prove that the bisection method generates a sequence {pn} approximating the root p off with the property Pn-p| 2(b-a), n 1 I [7 marks] (b) Existence of a Fixed Point: Prove that if g C[a, b] and g(x) = [a, b] V rE [a, b] then the function g has a fixed point in [a, b]. [7 marks] (c) Newton-Raphson: Using Matlab syntax write down an appropriate algo- rithm for finding the root of an equation, f(x), using the Newton-Raphson method. Be as accurate as you can regarding the Matlab syntax. Include an fprintf() statement at every iteration showing the step number, the current estimation of the root and the associated absolute error relative to the previous step. Finally, include a final fprintf() statement at the end giving the number of steps required, the root and the absolute error. [6 marks]