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 nodal points
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 nodal points
True derivative Approximation Numerical Differentiation Forward difference (a) Backward difference True derivative Centered difference Approximation 2
Forward difference Backward difference Centered difference Numerical Differentiation
Centered difference True derivative Approximation 2h Xi-1 Xi Xi+1
Centered difference xi−1 xi xi+1 x 2h
First Derivatives f'(x) i-2 -1 i计l i+2 ■Forward difference 了x)f4小-fs-4-y 七+1-X, xi41- Backward difference x))f( Xi-Xi-1 Xi-xi-1 ■Central difference f(x)f(xu)-f(x)=yu-y i+1-i-1 Xi+l-Xi-1
First Derivatives ◼ Forward difference ◼ Backward difference ◼ Central difference f ( x ) i-2 i-1 i i+1 i+2 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 1 i 1 i i 1 i i 1 i i 1 i x x y y x x f ( x ) f ( x ) f ( x ) x x y y x x f ( x ) f ( x ) f ( x ) x x y y x x f ( x ) f ( x ) f ( x ) + − + − + − + − − − − − + + + + − − = − − − − = − − − − = − − x y
Truncation Errors Uniform grid spacing h2 fx)=x,+)=x+(+2∫x,+x+ h2 d=0-)=0代-x+2x)了R fonvard:P()f(a)() O h 2 backward:frx))=f)-f小+gfr5,) 0(h) h central f(x)=f)-f( f"(5) 02) 2h 6
Truncation Errors ◼ Uniform grid spacing = − = − + − + = + = + + + + − + f ( x ) 3! h f ( x ) 2! h f ( x ) f ( x h ) f ( x ) hf ( x ) f ( x ) 3! h f ( x ) 2! h f ( x ) f ( x h ) f ( x ) hf ( x ) i 3 i 2 i 1 i i i i 3 i 2 i 1 i i i − − = + − = − − = + − − + ( ) ) ( ) ( ) : ( ) ( ) ( ) ( ) : ( ) ( ) ( ) ( ) : ( ) 2 3 2 i 1 i 1 i 2 i i 1 i 1 i 1 i i f O(h 6 h 2h f x f x central f x f O(h) 2 h h f x f x backward f x f O(h) 2 h h f x f x forward f x