numerical_differentiation - Num erical Differentiation...

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Numerical Differentiation ° Estimate the derivatives (slope, curvature, etc.) of a function by using the function values at only a set of discrete points ° Ordinary differential equation (ODE) ° Partial differential equation (PDE) ° Represent the function by Taylor polynomials or Lagrange interpolation ° Evaluate the derivatives of the interpolation polynomial at selected (unevenly distributed) nodal points
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0 ( ) ( ) Differentiation lim ( ) ( ) ( ) Difference ( ) x y f(x) dy f x x f x dx x x x y f x x f x dy x x x x dx ∆ → = +∆ - = +∆ - +∆ - = +∆ - A smaller step (or h) results in a smaller error x
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x i-3 x i-2 x i-1 x i x i+1 x i+2 x i+3 h x = Evenly distributed points along the x-axis Distance between two neighboring points is the same, i.e. h. x 1 x 2 x 3 Unevenly distributed points along the x-axis
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Lagrange Interpolation ° 1st-order Lagrange polynomial (Two (Two-Point Point ) ) ( ) ( ) ( ) ( ) ( ) ( 1 0 1 0 0 1 0 1 1 1 0 0 1 x f x x x x x f x x x x x f x L x f L x f - - + + - - = + + = ° Second-order Lagrange polynomial (Three (Three- Point ) ) x ( f ) x x )( x x ( ) x x )( x x ( ) x ( f ) x x )( x x ( ) x x )( x x ( ) x ( f ) x x )( x x ( ) x x )( x x ( ) x ( f 2 1 2 0 2 1 0 1 2 1 0 1 2 0 0 2 0 1 0 2 1 2 - - - - + + - - - - + + + - - - - =
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Formula ± Lagrange interpolation polynomial for unequally spaced data ) ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 i 1 i i 1 i 1 i i i 1 i 1 i x f x x x x x f x L x f x L x f x L x f - + + + + + + + - - - - - - - = + + + + = ) ( ) )( ( ) )( ( ) ( ) )( ( ) )( ( ) )( ( 1 i i 1 i 1 i 1 i i 1 i i 1 i i 1 i i 1 i 1 i 1 i 1 i i 1 i x f x x x x x x x x x f x x x x x x x x x x x x + + + + - + + - + + - + + - + + - - - - - - + + - - - - + +
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Lagrange Interpolation L 2 (x)f(x 2 ) L 0 (x)f(x 0 ) L 1 (x)f(x 1 ) x 0 x 1 x 2
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Formula ± Lagrange interpolation polynomial for unequally spaced data ) )( ( ) )( ( ) )( ( ) ( ) )( ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 i i 1 i i 1 i 1 i i 1 i 1 i i 1 i 1 i i 1 i 1 i 1 i i i 1 i 1 i x x x x x x x x x x x x x f x x x x x x x x x f x f x L x f x L x f x L x f - - - - - - + + - - - - = + + = + + - + + - + + - - + + - + + + + - - ± First derivative ) )( ( ) ( i 1 i 1 i 1 i i 1 i 1 i x x x x x f - - + + + + - + + - + + ) )( ( ) ( ) )( ( ) ( ) )( ( ) ( ) ( i 1 i 1 i 1 i i 1 i 1 i 1 i i 1 i i 1 i 1 i i 1 i 1 i i 1 i 1 i i 1 i x x x x x x x 2 x f x x x x x x x 2 x f x x x x x x x 2 x f x f - - - - + + - - - - + + - - - - = + + + - + + + - + + + + - + + - + + - - + + -
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Lagrange polynomial (Two (Two-Point ) Point ) ) ( ) ( ) ( ) ( ) ( ) ( 1 0 0 1 1 1 0 0 1 x f x x x f x x x f x L x f L x f - - + - - = + = 0 1 1 0 x x x x
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