HW3_Solutions

HW3_Solutions - MAE 107 Computational Methods Summer...

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Unformatted text preview: MAE 107 Computational Methods Summer Session II 2009 Homework # 3 Problem 1 10 points 1. FalsePos.m Solution: function [x,count] = FalsePos(F,a,b,er_final) % Performs bracketing test with the False Positive algorithm to find a root % of the functin F(x) over the interval [a,b] within the relative error % tolerence er. % Check if [a,b] is a true bracket if F(a)*F(b) >= 0 error([The interval [, num2str(a),,,num2str(b),] is not a bracket!]) end % Define an initial relative error er = 1; % Define an ititial guess of the root location x_old = F(b); % Setup iteration counter count = 0; while abs(er) > er_final % Define the false positive step x = b - (F(b)*(a-b))/(F(a)-F(b)); % if F(new point) has different sign than F(b) then make new bracket % [new_point,b], else make new bracket [a,new_point] if abs(F(x))<1e-14, % in case x is exactly at the root, to exit the loop break; elseif F(x)*F(b) < 0 a = x; else b = x; end 1 % Calculate the reative error er = (x - x_old)/x; % Redefine the old point as the newly calculated point for er % calculation on next iteration x_old = x; % Iterate the counter. count = count + 1; end Verifying the solution as requested: f ( x ) = x 2 5 over the interval [ , 4 ] (and with error 10 6 ) >> [x,count] = FalsePos(@(x) x.2-5,0,4,1e-6) x = 2.2361 count = 13 We can clearly see convergence toward 5 2. find roots.m Solution: function [x] = find_roots(F,A,B,Ns,er_final) % find_roots performs a successivly incremental search on the function F % over the interval [A,B] with Ns sub-sets. If a possible location is % found, the FalsePos function is called to find the root within a % relative tolerence of er_final. % Break the interval [A,B] into Ns subspaces; l = linspace(A,B,Ns+1); % Pre-define the output vector as a null x = ; % Check for a bracket for index = 1:length(l)-1 a = l(index) b = l(index+1); 2 % If F(a)*F(b) < 0 then we have a bracket if F(a)*F(b) < 0 % Now perform a FalsePos search [Xn,its] = FalsePos(F,a,b,er_final); % Store the root in a vector...
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HW3_Solutions - MAE 107 Computational Methods Summer...

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