Lecture12

# Lecture12 - Forward difference x i 1 x i x i 1 x h T r u e...

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Unformatted text preview: Forward difference x i- 1 x i x i+1 x h T r u e d e r i v a t i v e A p p r o x i m a t i o n Backward difference x i- 1 x i x i+1 x h T r u e d e r i v a t i v e A p p r o x i m a t i o n Centered difference x i- 1 x i x i+1 x 2h T r u e d e r i v a t i v e A p p r o x i m a t i o n First Derivatives ❧ Forward difference ❧ Backward difference ❧ Central difference 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 i i 1 i i 1 i i 1 i i i 1 i i 1 i i 1 i i 1 i x x y y x x ) x ( f ) x ( f ) x ( f x x y y x x ) x ( f ) x ( f ) x ( f x x y y x x ) x ( f ) x ( f ) x ( f- +- +- +- +---- + + + +-- =-- 2245 ′-- =-- 2245 ′-- =-- 2245 ′ ) x ( f ′ i-2 i-1 i i+1 i+2 x y Truncation Errors ❧ Uniform grid spacing (Taylor Series) + ′ ′ ′- ′ ′ + ′- =- = + ′ ′ ′ + ′ ′ + ′ + = + =- + ) x ( f ! 3 h ) x ( f ! 2 h ) x ( f h ) x ( f ) h x ( f ) x ( f ) x ( f ! 3 h ) x ( f ! 2 h ) x ( f h ) x ( f ) h x ( f ) x ( f i 3 i 2 i i i 1 i i 3 i 2 i i i 1 i ′ ′ ′-- = ′ ′ ′ +- = ′ ′ ′-- = ′- +- + ) ) ( ) ( ) ( ) ( : ) ( ) ( ) ( ) ( : ) ( ) ( ) ( ) ( : 2 3 2 1 i 1 i i 2 1 i i i 1 i 1 i i O(h f 6 h h 2 x f x f x f central O(h) f 2 h h x f x f x f backward O(h) f 2 h h x f x f x f forward ξ ξ ξ Example: First Derivatives ❧ Use forward and backward difference approximations to estimate the first derivative of...
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Lecture12 - Forward difference x i 1 x i x i 1 x h T r u e...

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