Chapter 4 Numerical Integration andNumerical DifferentiationSectionl the Introduction of Practical ProblemSection2 Mechanical Quadrature Method andAlgebraic PrecisionSection3 Newton-Cotes Quadrature FormulaSection4 Compound MultiplicativeSection5 Romberg Quadrature FormulaSection6 Gaussian Quadrature FormulaSection7 Numerical Differentiation上页下页返圆
上页 下页 返回 Chapter 4 Numerical Integration and Numerical Differentiation Section1 the Introduction of Practical Problem Section3 Newton—Cotes Quadrature Formula Section4 Compound Multiplicative Section2 Mechanical Quadrature Method and Algebraic Precision Section5 Romberg Quadrature Formula Section6 Gaussian Quadrature Formula Section7 Numerical Differentiation
s1 the Introduction of Practical ProblemShenzhou-VI spacecraft ran 5 laps on the elliptical orbit, whichorbit inclination angle is 42.4 degrees , height of perigee 200 kmheight of apogee 347km. Try to calculate the travel kilometers ofShenzhou-VI.the perimeter of elliptical orbit is the main factor上页下页返圆
上页 下页 返回 §1 the Introduction of Practical Problem Shenzhou-VI spacecraft ran 5 laps on the elliptical orbit, which orbit inclination angle is 42.4 degrees , height of perigee 200 km, height of apogee 347km. Try to calculate the travel kilometers of Shenzhou-VI. the perimeter of elliptical orbit is the main factor
According to the elliptical parametric equation and arclength formula of curve. we can calculate the perimeter ofelliptical orbit2Tcos? tdtL = 4a2aThis is a definite integral, we can just work out itsvalue.上页下页返圆
上页 下页 返回 This is a definite integral, we can just work out its value. tdt a c L a = − 2 0 2 2 2 4 1 cos According to the elliptical parametric equation and arc length formula of curve, we can calculate the perimeter of elliptical orbit
According to I = ['f(x)dx, if we can find out Integrand function f(x) and original function F(x), then we can work out thefollowing Newton-Leibniz formula[~ f(x)dx = F(b) - F(a)But considering of large number of integrand functions f(x)actually it's difficult in practical problems上页下页返圆
上页 下页 返回 f (x)dx F(b) F(a) b a = − According to , if we can find out Integrand function f (x) and original function F(x)， then we can work out the following Newton—Leibniz formula = b a I f(x )dx But considering of large number of integrand functions f (x), actually it’s difficult in practical problems
sinx(1 ) suppose that f(x) is, sin x?,We can not find out theYelementary function of original function( 2 ) When f (x) is based on an numerical measurements to calculatea sheet of data, we can not use Newton-Leibniz formula directly(3 ) When the structure of f(x) is complicated, we would findthat it's difficult to work out original function. At that time, it'snecessary to study integral numerical problem.上页下页返圆
上页 下页 返回 x sin x （1）suppose that f (x) is , sin x 2 ， We can not find out the elementary function of original function （2） When f (x) is based on an numerical measurements to calculate a sheet of data, we can not use Newton—Leibniz formula directly. （3） When the structure of f (x) is complicated, we would find that it’s difficult to work out original function. At that time, it’s necessary to study integral numerical problem
$2 Mechanical Quadrature Method and Algebraic Precisionthe basic concept of numerical quadratureAccording to integral mean value theorem, there is a point inintegral region section [a, b], so we can get['f(x)dx = (b - a)f()In another word, the rectangular area at the bottom of b-a andheightf(E) is equals to curved trapezoid area.yf（）Eb上页下页返圆
上页 下页 返回 §2 Mechanical Quadrature Method and Algebraic Precision the basic concept of numerical quadrature f(x )dx (b a)f( ) b a = − According to integral mean value theorem, there is a point in integral region section [a, b], so we can get In another word, the rectangular area at the bottom of b-a and height is equals to curved trapezoid area. f()
But the problem is we generally don't know the exactlocation of point S, so it's hard to work out preciselyWe can work outf(e), it can be called as averageheight of section region[a,b]So if we can find out an algorithm of averageheightf(E), we can work out numerical quadraturemethod correspondingly上页下页返圆
上页 下页 返回 But the problem is we generally don’t know the exact location of point , so it’s hard to work out precisely. We can work out , it can be called as average height of section region[a,b] So if we can find out an algorithm of average height , we can work out numerical quadrature method correspondingly. f() f()
If we use the arithmetic average of f(a) andf(x)f(b) as the approximate of average height f()f(b)the quadrature formula we get is our familiar f(a)batrapezoid formula0Lf(a)+ f(b))2上页下页返圆
上页 下页 返回 If we use the arithmetic average of f(a) and f(b) as the approximate of average height f(ξ), the quadrature formula we get is our familiar trapezoid formula [ ( ) ( )] 2 f a f b b a T + − = f(x) a b f(a) f(b)
a+bIf we use f(c) the height of interval midpointc2as the approximate of average height , we can workout so called mid-rectangle formula (rectangularformula for short)a+bR=(b-a)fAnother common Simpson formulab-(a+b)f(a)+4fl+ f(b)S上页6下页返圆
上页 下页 返回 + = − 2 ( ) a b R b a f Another common Simpson formula + + + − = ( ) 2 ( ) 4 6 f b a b f a f b a S If we use f(c) the height of interval midpoint as the approximate of average height , we can work out so called mid-rectangle formula (rectangular formula for short) 2 a b c + =
More generally, we can select some nodes xk on thesection[a,bl properly, and work out the weighted average f (xk ) asthe approximate of average height f(), so the quadrature formulacanbeseenbelow.nZF"f(x)dx~Akf(xk)k=0上页下页返圆
上页 下页 返回 More generally, we can select some nodes xk on the section[a,b] properly, and work out the weighted average f (xk ) as the approximate of average height f ( ξ ) , so the quadrature formula can be seen below. = b a n k k xk f x x A f 0 ( )d ( )