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
Error Propagation f ImAr Estimated error Error in x leads to error in f(x) x土△→ fx)±△f(E)=fx)±f'(x)x-)=f(x)±f'(x)△c
Error Propagation f x f x f x f x x x f x f x x x x ~ ) ( ) ( ) ~ ) ( ) ( )( ~ ( ) ( ~ = − = Error in x leads to error in f(x)
Total Numerical Error Point of diminishing returns Total error Truncation error Round-off error log step size Trade-off between truncation and round-off errors
Trade-off between truncation and round-off errors Total Numerical Error