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Numerical2003solutions - ΗΛΕΚΤΡΟΛΟΓΟΙ...

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ΗΛΕΚΤΡΟΛΟΓΟΙ ΜΗΧΑΝΙΚΟΙ ΚΑΙ ΜΗΧΑΝΙΚΟΙ ΥΠΟΛΟΓΙΣΤΩΝ ΕΞΕΤΑΣΗ ΑΡΙΘΜΗΤΙΚΗΣ ΑΝΑΛΥΣΗΣ 17/2/2003 1. Για να υπολογίσου μ ε μ ια προσέγγιση του αριθ μ ού ln3 x = θεωρού μ ε την εξίσωση ( ) 3 0 x f x e = = ( α ) Υπολογίστε τις τρεις πρώτες προσεγγίσεις του x που παράγει η μ έθοδος της διχοτό μ ησης για την αριθ μ ητική επίλυση της ( ) 0 f x = μ ε αρχικό διάστη μ α [0,2]. Πόσο το πολύ απέχει η τρίτη προσέγγιση από το x ; [1] ( β ) ∆είξτε ότι η ακολουθία 1 0.1( 3), 0,1,... k x k k x x e k + = = συγκλίνει στο x για κάθε 0 [0,2] x και ότι αν k k x x ε = , τότε 1 0.7, k k k ε ε + → ∞ . [1.5] ( γ ) Με 0 0 x = να γίνει μ ια επανάληψη της μ εθόδου Newton-Raphson για την επίλυση της ( ) 3 0 x f x
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