lecture10 - Numerical Methods in Engineering ENGR 391 Curve...

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Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Curve fitting - Interpolation
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Highlights from last lecture Regression - general least-squares regression procedure (minimizing the residual^2) -des ire a curve (equat ion) to follow the general trend of the data -“bes t f it” -corre lat ion coeff ic ient, standard of deviation, error of the estimate Curve fitting - Linear least-squares fit -L i n e a r i z a t i o n o f n o n - l i n e a r r e l a tionship (exponential, power and saturation growth rate models) -P o l y n o m i a l l e a s t s q u a r e s r e g r e s s i o n -M u l t i p l e l i n e a r r e g r e s s i o n
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Interpolation •I n t e r p o l a t i o n : T h e E s t i m a t i o n o f i n t e r m e d i a t e v a l u e s between precise data points. •T h e m o s t c o m m o n m e t h o d i s : n n x a x a x a a x f ± ± ± ± ± 2 2 1 0 ) ( Monomial basis •Interpo la t ing curve goes through all of the data points
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Although there is one and only one n th-order polynomial that fits n+1 points, there are a variety of mathematical formats in which this polynomial can be expressed and the numerical methods are derived based on these different formats: •Monomial Polynomial •The Newton polynomial •The Lagrange polynomial Interpolation n n x a x a x a a x f ± ± ± ± ± 2 2 1 0 ) (
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Interpolating Polynomials Linear Interpolation given/using two data points •I s t h e s i m p l e s t f o r m o f i n t e r p o l a t i o n , c o n n e c t i n g t w o data points with a straight line. f 1 (x) designates that this is a first-order interpolating polynomial. ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 1 0 1 0 1 0 1 0 1 0 0 1 x x x x x f x f x f x f x x x f x f x x x f x f ± ± ± ² ± ± ± ± Linear-interpolation formula Slope and a finite divided difference approximation to 1 st derivative
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Example
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7 Quadratic Interpolation •I f t h r e e d a t a p o i n t s a r e a v a i l a b l e , t h e e s t i m a t e i s improved by introducing some curvature into the line connecting the points. •A s i m p l e p r o c e d u r e c a n b e u s e d t o d e t e r m i n e the values of the coefficients. ) )( ( ) ( ) ( 1 0 2 0 1 0 2 x x x x b x x b b x f ± ± ² ± ² 0 2 0 1 0 1 1 2 1 2 2 2 0 1 0 1 1 1 0 0 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( x x x x x f x f x x x f x f b x x x x x f x f b x x x f b x x ± ± ± ± ± ± ± ±
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Example
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•R e w r i t e t h e p o l y n o m i a l i n t h i s f o r m 0 0 2 1 1 1 0 1 1 0 1 1 0 1 2 2 0 1 1 0 0 0 1 1 1 0 0 1 2 1 0 0 1 0 0 ] , , , [ ] , , , [ ] , , , , [ ] , [ ] , [ ] , , [ ) ( ) ( ] , [ ] , , , , [ ] , , [ ] , [ ) ( ] , , , [ ) ( ) )( ( ] , , [ ) )( ( ] , [ ) ( ) ( ) ( x x x x x f x x x f x x x x f x x x x f x x f x x x f x x x f x f x x f x x x x f b x x x f b x x f b x f b x x x f x x x x x x x x x f x x x x x x f x x x f x f n n n n n n n k i k j j i k j i j i j i j i n n n n n n n ± ± ± ± ± ± ± ± ± ² ² ± ± ² ± ² ± ± ± ± ± ± ± ± ± ± ² ³ ² ³ ³ ³ Bracketed function evaluations are finite divided differences General Form of Newton’s Interpolating Polynomials Construct the Newton’s divided difference table
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0.658 1.0 3 0.224 0.8 2 0.0138 0.7 1 -0.177 0.6 0 f(x i ) x i i ) )( )( ( ) )( ( ) ( ) ( 2 1 0 3 1 0 2 0 1 0 3 x x x x x x b x x x x b x x b b x P ± ± ± ² ± ± ² ± ² N = 4 ³ (N-1) = Third-order polynomial Construct the Newton’s divided difference table Example:
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0.658
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lecture10 - Numerical Methods in Engineering ENGR 391 Curve...

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