《数值分析》课程PPT教学课件（英文版）Chapter 02 Numerical Integration 2.1 Introduction to Quadrature

Chapter 2 Numerical Integration

Chapter 2 Numerical Integration

(52= 3 dt≈4.8998922 y=f(t) 05 Figure 2.1 Figure 2.1 Area under the curve y= f(t) for0<t<5

Figure 2.1 Values of (a) 重(x) 1.00.2248052 2.01.1763426 3025522185 403.8770542 5.04.8998922 605.5858554 7.06.0031690 8.06.2396238 9.06.365739 10.06.4319219

2.1 Introduction to Quadrature

2.1 Introduction to Quadrature

Definition 2. 1 Suppose that a =I0<1<.<.M=b. A formula of the orm Q月=2,()=0(0+(n)+…+(y) with the property that f(ad r=Qf+ef 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 uaaratore nodes, an k=0 are ca lled 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 n, but for which E[Pn+#0 for some polynomial Pn+1(a)of degree n +1

Theorem 2. 1(Closed Newton-Cotes Quadrature Formula). Assume that Tk=a0+ kh are equally spaced nodes and fk=f(k. The first four closed Newton-Cotes quadrature formulas are 广"f (x)dx≈r(f0+f1) (the trapezoidal rule), f(a).c a o(o+4f1+f2)( Simpson's rule (25) f(a)d r ao(fo+3f1+3f2+f 3)(Si 3 lmpson's rule l),(2.6 广 f(x)dm≈(70+321+122+328+7f4)( Boole rule).(27) Figure 2.2(a) The trapezoidal rule integrates( b) Sinpson's rule integrates(c) Simpson's rule integrates(d) Boole's rule integrates

Corollary 2.1(Newton-Cotes Precision). Assume that f(a) is sufficiently differentiable; then EI for Newton-Cotes quadrature involves an appropriate higher derivative. The trapezoidal rule has degree of precision n= 1.If f∈C2a,b],th hen f(x)dx=n(6+f1)-1nf2( 2.8 Simpson's rule has degree of precision n=3. If f E C4 a, b, then 2 f(x)=(0+4f1+1)-of(c) 2.9 Simpson's& rule has degree of precision =3. If f eC [a, 6],then 3h 37 cda (f+3f1+3f2+f) f4(c).(2.10) Boole's rule has degree of precision n=5, If f E Ca, b), then 8h f(x)dx=-(70+32f1+12f2+32/+7f4)--f6(c).(2.1) 945

Example 2. 1. Consider the function f(a)=1 +e- sin(4r), the equally spaced spaced quadrature nodes o =0.0, 1=0.5,I2=1.0, 3=1.5 and 34= 2.0, and the corresponding function values fo 1.00000, f 2 0. 72159, f3=0.93765, and f= 1. 13390. Apply the various quadrature for mulas(2. 4)through(2.7)

Example 2.2 Consider the integration of the function f(ar)=1+e- sin(4.c) over the fixed interval [ a, b=[0, 1. Apply the various formulas(2. 4) throug (27)