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<t<5
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<.<M=b. A fo ormula of the form Q=∑(x)=0(0)+-n,()+…+f(mM)(21) with the property that f(x)dm=Q月+E[升 2.2) is called a numerical integration or quadrature formula. The term E()is called the truncation error for integration. The values ak ko are called the quadrature nodes, and fwk ko are called the weights
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< n, but for which E[Pn+#0 for some polynomial Pn+1(a) of degree m+1
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
