Ch.5NumericalDifferentiationandIntegration.ppt

1、Ch. 5 Numerical Differentiation and Integration,Using Lagrange Interpolating Polynomial,Numerical Differentiation,We know that the derivative of the function f at x0 is,This formula gives an obvious way to generate an approximation to f (x); simply compute,for small values of h. Although this may be

2、 obvious, it is not very successful, due to roundoff error. But it is certainly the place to start.,We construct the first Lagrange interpolating polynomial P1(x) for f determined by x0 and x1, with its error term:,Differentiating gives,Forward-difference,Backward-difference,This formula is known as

3、 the forward-difference formula if h0 and the backward-difference formula if h0.,To obtain general derivative approximation formulas, we use the theorem of the Lagrange interpolating polynomial remainder term,Differentiating this expression gives the first derivative formula,which is called an (n+1)

4、-point formula to approximate f (xj),and the second derivative formula,In general, using more evaluation points produces greater accuracy, although the number of functional evaluations and growth of roundoff error discourages this somewhat. The most common formulas are those involving three and five

5、 evaluation points.,We first derive some useful three-point formulas and consider aspects of their errors.,Three-Point Formulas,Recalling the second-order version of Lagrange interpolation,Since,Similarly,Hence, from the equation,We get,and from the expression,We get,Three-Point Formulas,Elements of

6、 Numerical Integration,to approximate .,The methods of quadrature are based on the Lagrange interpolation polynomials. We first select a set of distinct nodes x0, x1, , xn from the interval a,b. Then we integrate the Lagrange interpolating polynomial,The basic method involved in approximating is cal

7、led numerical quadrature. It uses a sum,and its truncation error over a, b to obtain,Let us consider formulas produced by using first, second, and third Lagrange polynomials with equally spaced nodes, This gives the Trapezoidal rule, Simpsons 1/3 rule, and Simpsons 3/8 rule.,Trapezoidal Rule (using

8、first Lagrange polynomials, 2 points, 1 interval):,Derivation of Simpsons 1/3 Rule,Substituting the agreement x with x = x0 + th, we have,Finally,Newton-Cotes Formulas (n= 4, 5 ),The trapezoidal rule and both of Simpsons rules are members of a family of integrating equations known as the Newton-Cote

9、s closed integration formulas. However, it must be stressed that, the high-degree (that is, greater than four-point) Newton-Cotes formulas are generally unsuitable for use over large integration intervals because of the oscillatory nature of high-degree polynomials. Simpsons rules are sufficient for

10、 most applications. Accuracy can be improved by using the multiple-application version. Furthermore, when the function is known and high accuracy is required, methods such as Romberg integration or Gauss quadrature offer viable and attractive alternatives.,Composite Numerical Integration,Now we disc

11、uss a piecewise approach to numerical integration that uses the low-order Newton-Cotes formulas. These are the techniques most often applied.,Composite Simpsons 1/3 Rule:,Composite Trapezoidal Rule,Simpsons 1/3 rule is usually the method of preference because it attains third-order accuracy. But we

12、note that the method can be employed only if the number of intervals is even. The Simpsons Three-Eighths rule, however, has utility when the number of segments, the intervals, is odd. An alternative would be to apply Simpsons 3/8 rule to the last three segments.,Adaptive Quadrature Methods,Matlab ha

13、s two main functions for quadrature, quad, and quadl to approximate the definite integrals . Both routines use adaptive quadrature.,Romberg Integration,Romberg integration uses the composite Trapezoidal rule to give preliminary approximations and then applies the Richardson extrapolation process, th

14、at is, decreases the step size successively, to improve the approximation.,Richardson extrapolation is used to generate high-accuracy results while using low-order formulas. Extrapolation can be applied whenever it is known that an approximation technique has an error term with a predictable form, o

15、ne that depends on a parameter, usually the step size h. Suppose that for each number h 0 we have a formulathat approximates an unknown value M and the truncation error involved with the approximation has the form,for some collection of unknown constants K1, K2, K3, .,The truncation error is O(h), that is,The object of extrapolation is to find an easy way to combine the rather inaccurate O(h), approximations in an appropriate way to produce formulas with a higher-order truncation error.,Now we consider the result when we replace the parameter h by half its value.,Subtraction