x, is quadrature node: Ak is quadrature coefficient,can also be called as weight of adjoint node xk :Weight Ak is only related with the selection of nodeXk, but has nothing to do with the specific form ofintegrand function f(x).This kind of numerical integration method can be calledas mechanical quadrature, whose characteristic is changeintegral evaluation problem into calculation of the valueof functions, preventing the difficulty of searching fororiginal function based on Newton-Leibniz formula上页下页返圆
上页 下页 返回 xk is quadrature node: Ak is quadrature coefficient, can also be called as weight of adjoint node xk . Weight Ak is only related with the selection of node xk , but has nothing to do with the specific form of integrand function f(x). This kind of numerical integration method can be called as mechanical quadrature, whose characteristic is change integral evaluation problem into calculation of the value of functions, preventing the difficulty of searching for original function based on Newton—Leibniz formula
2the Concept of Algebraic PrecisionNumerical quadrature method can only work outapproximate solution, in order to ensure accuracy, wenaturally hope quadrature formula can be established bylarge number of functions that is as many as possible. Thiscan also be called the conception of algebra precision.Assuming quadrature formula is" f(x)dx ~ZAf(x)k=0上页下页返圆
上页 下页 返回 b a f (x)dx = n k k xk A f 0 ( ) Assuming quadrature formula is: 2 the Concept of Algebraic Precision Numerical quadrature method can only work out approximate solution, in order to ensure accuracy, we naturally hope quadrature formula can be established by large number of functions that is as many as possible. This can also be called the conception of algebra precision
If any PolynomialP(x)(i≤ m) can beestablished correctlyh[ P(x)dx=ZA,P(x)Namelyi=0,1,...,mk=0aHowever based on m+1 order polynomial, itcan not be established correctly, namelym+1[xm+1dx ±>Akxnk=0So we call that quadrature formula has m-上页order algebraic precision.下页返圆
上页 下页 返回 If any Polynomial can be established correctly, ( )( ) P x i m i Namely 0 ( ) ( ) b n i k i k k a P x dx A P x = = i m = 0,1,., However based on m+1 order polynomial, it can not be established correctly, namely 1 1 0 n b m m k k a k x dx A x + + = So we call that quadrature formula has morder algebraic precision
Generally, in order to get quadrature formulaJ' f(x)dx ~ ZAkf(x) has the m order algebraic precision.we should fet it be correctly established on f(x) = 1, x , ...xm, that requiresZA,=b-a;ZAX=(b2-(bm+)am+1ZArx"m+1nZ.m+1tm+1￥AdxAkXk1k=0上页So that quadrature formula has the m algebraic precision下页返圆
上页 下页 返回 = b a n k k xk f x x A f 0 ( )d ( ) − + = = − = − + + ( ) . 1 1 ( ); 2 1 ; 1 1 2 2 m m m k k k k k b a m A x A x b a A b a + b a m x dx 1 = + n k m Ak xk 0 1 So that quadrature formula has the m algebraic precision. Generally, in order to get quadrature formula has the m order algebraic precision. we should let it be correctly established on f (x) = 1, x，.， x m , that requires
(x)Example1According toTrapezium formula(b)b-af(a)J' f(x)dx ~[f(a) + f(b))2baInspect of its algebraic precisionSolutionCheck the formula whether be established correctlygradually上页下页返圆
上页 下页 返回 f(x) a b f(a) f(b) Solution： Example 1： [ ( ) ( )] 2 ( ) f a f b b a f x dx b a + − Inspect of its algebraic precision According to Trapezium formula Check the formula whether be established correctly gradually