$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 ( )