The slope of the function at x0 is then

(x0) = (x0) + (x0) = [ f(x1) f(x0)] + (x0)

Forward difference approximation is obtained when the slope of the interpolating polynomial estimates the derivative of the function at x0 as shown in Figure 3.3-2. In term of the forward difference operator

(x0) = + (x0)

where f(xi) = f(xi+1) f(xi).

Figure 3.3-2 Derivatives of the function at x0 and at x1.

The error for the derivative can be estimated by taking derivative of the error

E1(x) = ()

(x0) = [ (x x0)(x x1) ()

(x0) = () [(x x0)(x x1)

(x0) = () [(x x1) + (x x0) = h () = O(h)

Backward difference approximation is obtained when the slope of the interpolating polynomial estimates the derivative of the function at x1 as shown in Figure 3.3-2.

(x1) = (x1) + (x1) = [ f(x1) f(x0)] + (x1)

The error term has the form

(x1) = () [(x x1) + (x x0) = h () = O(h)

The error in the backward difference approximation, while having the same form as that in the forward difference approximation, has a different sign. In term of the backward difference operator

Figure 3.3-3 Derivative of the function at x1 is estimated by a second-degree polynomial.

Let h = x1 x0 = x2 x1, the three points interpolating polynomial over this interval is

P2(x) = L2,0(x) f(x0) + L2,1(x)f(x1) + L2,2(x)f(x2)

P2(x) = f(x0) + f(x1) + f(x2)

P2(x) = [(x x1)(x x2)f(x0) 2(x x0)(x x2)f(x1) + (x x0)(x x1) f(x2)]

The function f(x) can be expressed in terms of its approximating polynomial with an error as

f(x) = P2(x) + E2(x)

The slope of the function at x1 is then

(xi) = + O(h)

where f(xi) = f(xi) f(xi-1)

Central difference approximation is obtained when the slope of the interpolating polynomial estimates the derivative of the function at the midpoint x1 as shown in Figure 3.3-3.

P1(x) = f(x0) + f(x1)

Let h = (x1 x0), then

P1(x) = [(x1 x) f(x0) + (x x0) f(x1)]

Figure 3.3-1 Approximating by first-degree polynomial with error E1(x).

The function f(x) can be expressed in terms of its approximating polynomial with an error as

f(x) = P1(x) + E1(x) (x1) = (x1) + (x1) = [ f(x2) f(x0)] + (x1)

The error term in this difference approximation is

(x1) = [ (x x0)(x x1)(x x2) ()

(x1) = h2 () = O(h2)

The three points approximation is accurate to O(h2). In term of the central difference operator

(x1) = [f(x1 + ) + f(x1 )] + O(h2)

where the central difference operator is defined as

f(xi) = f(xi + ) (xi )

f(x1 + ) + f(x1 ) = f(x1 + h) f(x1) + f(x1) f(x1 h) = f(x2) f(x0)

Finite difference approximation of higher order derivatives can also be obtained(x) = P2(x) + E2(x)

The second derivative of the function at x1 is then

(x1) = (x1) + (x1)

After some algebra

(x1) = [f(x0) 2f(x1) + f(x2)] + (x1)

The error term can be evaluated to yield

(x1) = h2 () = O(h2)

The central difference can also be written in terms of the central difference operator

(x1) = + O(h2)

where

2f(x1) = [f(x1)] = [ f(x1 + ) f(x1 )] = f(x2) f(x1) [f(x1) f(x0)]

So we have INTEGRATION , Prof. Orasanu

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