Chapter 2 Numerical Integrato
Chapter 2 Numerical Integration
3 Φ(5)=Jet dt≈4.8998922 y=f(t) Figure 2.1 Figure 2.1 Area under the curve y=f(t) for0
figure2 1 Values of重(x) 重(x) 1.00.2248052 2.01.1763426 3.02.5522185 4.03.8770542 5.048998922 6.055858554 7.06.0031690 8.06.2396238 9.06.3665739 10.06.4319219
2. 1 Introduction to Quadrature
2.1 Introduction to Quadrature
Definition 2. 1 Suppose that a=.0 1
Definition 2.2. The degree of precision of a quadrature formula is the positive integer n such that E[P=0 for all polynomials Pi(a)of degree i
Theorem 2.1(Closed Newton-Cotes Quadrature Formula). Assume that k =co+ kh are equally spaced nodes and fr= f(k). The first fo our closed Newton-Cotes quadrature formulas are 1 f(x)da≈x(fo+f1) (the trapezoidal rule) f(a)d. c s(fo +4f1+f2)(Si lmpson s rule (2.5) 厂f 3h (a)d ac a o(fo+3f 1+3f2+ f3)(Simpsons grule),(2.6) 2h f(x)dr≈:(7+32f1+12f2+32g+7f4)( Boole rule.(2.7) Figure 2.2(a) The trapezoidal rule integrates(b) Sinpson's rule integrates(c) Simpsons rule integrates(d) Boole's rule integrates
Corollary 2. 1(Newton-Cotes Precision). Assume that f(a)is sufficiently differentiable; then E[f] for Newton-Cotes quadrature involves an appropriate higher derivative. The trapezoidal rule has degree of precision n= 1. If f∈Ca,b],then f (a d r=o(o +f1)-of)(c) 2.8 Simpson's rule has degree of precision n=3. If f E C4a, b],then f(x)=(+4f1+f2)-of(c) 2.9 Simpson's rule has degree of precision =3. If f E C[a, b],then °f(x)=8(fo+3+3+3) 3h 3h (A)(C) 2.10 Boole's rule has degree of precision n= 5, If f E Cola, b, then 2h 8 f(a)d=45(7+32h+1212+32+7 f(6)(c).(2.11) 945
Example 2. 1. Consider the function f(a)=1+e-sin(4. e equally spaced spaced quadrature nodes o =0.0, 31=0.5, 2=1.0, 33=1.5 and a4= 2.0, and the corresponding function values fo= 1.00000, f2 0. 72159, f3=0.93765, and f4= 1. 13390. Apply the various quadrature for- mulas(2. 4)through(2.7)
Example 2. 2 Consider the integration of the function f(a)=1+e- sin(4. over the fixed interval a, b=[0, 1. Apply the various formulas(2.4)through 2.7