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# hwk1 - -3 How many iterations were needed(provide the...

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Homework # 1 Due Thursday, 01/17 NOTE: Each time you are ask to write a code which gives the solution with accuracy , it means that you have to use the stopping criterium | p n - p n - 1 | ≤ Note that this does not guaranty that | p n - p | ≤ !!! So you are actually not sure to have found the solution with accuracy !!! #1 p51 (do it by hand) Plot with MATLAB the function f ( x ) = x - 2 - x on [ - 5 , 5]. Use the command “GRID ON” so that there is a grid on the plot. (Provide this plot) Looking at this plot, is there a zero in [0,1]? Zoom in many times and give an estimate for this zero. #5a) p51 (provide the MATLAB code) #7 p51 (provide the MATLAB code) #10 a) c) p51 (this is a theoretical problem, do not use MATLAB, explain clearly your answer) Let f ( x ) = ln( x + 2) 1. Sketch the graph of f ( x ). 2. Find f ([0 , 2]) 3. Is it true that f ([0 , 2]) [0 , 2]? What can you conclude about the existence of a fixed point? 4. Find max x [0 , 2] | f ( x ) | . 5. Find a number k (0 , 1) such that | f ( x ) | ≤ k for all x [0 , 2] 6. What can you conclude about using a fixed point iteration. 7. Find the fixed point with accuracy 10
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Unformatted text preview: -3 . How many iterations were needed? (provide the MAT-LAB code) 8. Use the theoretical result given in class ( | p-p n | ≤ k n ( b-a )) to estimate the number of iterations required to achieve 10-3 accuracy. Why is this number diﬀerent than the one obtained in 7. ? • Let f ( x ) = e-x 1. Sketch the graph of f ( x ). 2. Find f ([ 1 3 , 1]) 3. Is it true that f ([ 1 3 , 1]) ⊂ [ 1 3 , 1]? What can you conclude about the existence of a ﬁxed point? 4. Find max x ∈ [ 1 3 , 1] | f ( x ) | . 5. Find a number k ∈ (0 , 1) such that | f ( x ) | ≤ k for all x ∈ ± 1 3 , 1 ² 6. What can you conclude about using a ﬁxed point iteration. 7. Find the ﬁxed point with accuracy 10-3 . How many iterations were needed? (provide the MAT-LAB code) 8. Use the theoretical result given in class ( | p-p n | ≤ k n ( b-a )) to estimate the number of iterations required to achieve 10-3 accuracy. 1...
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