《数值分析》课程PPT教学课件（英文版）Chapter 02 Numerical Integration 2.2 Composite Trapezoidal and Simpson’s Rule

2.2 Composite Trapezoidal and Simpson's Rule

2.2 Composite Trapezoidal and Simpson’s Rule

heorem 2.2 (Composite Trapezoidal Rule). Suppose that the inter- val la b is subdivided into M subintervals [ok, k+1 of width h=(b-a)/M by using the equally spaced nodes ck =a+hh, for k=0, 1,..., M. The composite trapezoidal rule for M subintervals can be expressed in any of three equivalent ways r(,b)=∑(f(xk-1)+f(x) k-1 (219) or T(f,h)=x(6+2f1+22+23+…+2fM-2+2fM-1+f or T(, h)=o(f(a)+f(b))+h2 f(ak) (2.21) k=1 This is an approximation to the integral of f(a)over [a, bJ, and we write f()d≈r(f,b (222)

Example2.5. Consider f(a)=2+sin(2v/). Use the composite trapezoidal rule with 11 sample points to compute an approximation to the integral of f(a) taken over[1,6

Theorem 2.3( Composite Simpson Rule). Suppose that [a, b is subdi vided into 2M subintervals [ak, Ck+1] of equal width h=(b-a)/(2M) by using k=a+hh for k=0, 1, .. 2M. The composite Simpson rule for 2M subintervals can be expressed in any of three equivalent ways S(,b)=3>(a22)+4f(x2-)+(a)(2 or S(,2310+4f1+22+4+…+212M-2+4f2M-1+f2)(2.25) 4h M S(,b)=(f(a)+f(b)+3∑f(x2)+3∑f(21).(20 This is an approximation to the integral of f(a)over a, bl, and we write f()d≈S(f,b) (227)

Example 2.6. Consider f(a)=2+sin(2v/r). Use the composite Simp son rule with 11 sample points to compute an approximation to the integral of f(a) taken over 1,6