for f(x) = x3h4h4h+ah?[0-3h2] =12dxx244for f(x) = x4hshshsx*dx=+ah[0-4h312￥562I(x')= I2(x")j= 0,1,2,3soI(x)+ I2(x)上页So that integration formula has 3 algebraic precision下页返圆
上页 下页 返回 [0 3 ] 2 2 2 4 2 ah h h I = + − = h I x dx 0 3 4 4 h = 4 4 h = [0 4 ] 2 2 3 5 2 ah h h = I = + − h I x dx 0 4 5 5 h = 6 5 h = I(x ) = I2 (x ) j = 0,1,2,3 j j ( ) ( ) 4 2 4 I x I x so So that integration formula has 3 algebraic precision. =
3Interpolation QuadratureFormulaSetting a group of nodes a≤xo<x,<x2<... <xn≤b,according to the value of f(x) on these nodes, we canwork out interpolation function L,(x). Because we canwork out the original function of algebraic polynomialLn(x) easily, we can let I, = I'L,(x)dx as theapproximation of integration I = f' f(x)dx , so this kindof integration formulaAk =F'lk(x)dxcan be called as interpolation integration.According to integration of interpolation function lk (x)we can work out integration coefficient AkI, = ZAkf(xk)上页k=0下页返圆
上页 下页 返回 3 Interpolation Quadrature Formula . = b In a Ln (x)dx = = n k n k xk I A f 0 ( ) Setting a group of nodes a ≤ x0 < x1 < x2 < . < xn≤ b, according to the value of f(x) on these nodes, we can work out interpolation function Ln (x). Because we can work out the original function of algebraic polynomial Ln (x) easily, we can let as the approximation of integration , so this kind of integration formula can be called as interpolation integration. According to integration of interpolation function lk (x), we can work out integration coefficient Ak = b a I f (x)dx = b Ak a lk (x)dx
According to Interpolation Remainder Theorem.the remainder of interpolation quadrature formulais(s)RLf1=I-I, =o(x)dx; o(x)=(x-x,)(x-x)...(x-xnJa(n +1)!If the integration formula is interpolation, its remainder,for the Polynomial f (x) times ≤ n , its remainder R[f]equals to O, so the integration formula at that time has atleast n algebraic precisionTheorem 1 the necessary and sufficient condition forsuch integration formula like I, = ZAx(x)has at上页下页least n algebraic precision is it is interpolation返圆
上页 下页 返回 According to Interpolation Remainder Theorem, the remainder of interpolation quadrature formula is ( )d ; ( ) ( )( ) ( ) ( 1)! ( ) [ ] 0 1 ( 1) n b a n n x x x x x x x x x n f R f I I = − − − + = − = + = = n k n k k I A f x 0 ( ) If the integration formula is interpolation, its remainder, for the Polynomial f (x) times ≤ n , its remainder R[ f ] equals to 0, so the integration formula at that time has at least n algebraic precision Theorem 1 the necessary and sufficient condition for such integration formula like has at least n algebraic precision is it is interpolation
4 the stability and convergence of integrationformulaDefinition 2 in integration formula f' f(x)dx ~ ZA,(xx)k=0nZIfArf(x) = f" f(x)dxlimh→0k=0Among them h = max(x; -x;-1), so we call the integration1<i≤nformula is convergent上页下页返圆
上页 下页 返回 4 the stability and convergence of integration formula = = → b a n k k k h lim A f (x ) f (x)dx 0 0 max( ) 1 1 − = i − i i n h x x = b a n k k xk f x x A f 0 Definition 2 in integration formula ( )d ( ) If Among them , so we call the integration formula is convergent
Consider of that calculation of f (x) may have error,we actually getfk,namlyf(xk) = fk + Ok.Definition 3, for any > ,if s > O, provided thatZAf(x)-ZAJ<eIf(xk) - fkl ≤ 8(k = 0,1...,n),k=0k=0, we can call the integration formula is stable.Theorem 2if integration formula coefficient Ak > 0 (k = O,1...,n),we call this integration formula is stable上页下页返圆
上页 下页 返回 Consider of that calculation of f (xk ) may have error, we actually get − = = n k n k k k k k A f x A f 0 0 ~ ( ) , we can call the integration formula is stable. Theorem 2 if integration formula coefficient we call this integration formula is stable