Chapter 3 Interpolation methodSection 1IntroductionSection 2 Lagrange interpolationSection 3 Deviation and Newton interpolationSection 4 Difference and Equidistant interpolationSection 5Hermite interpolationSection 6Piecewise low-order interpolationSection 7Cubic spline interpolation上页下页返圆
上页 下页 返回 Chapter 3 Interpolation method Section 1 Introduction Section 2 Lagrange interpolation Section 3 Deviation and Newton interpolation Section 4 Difference and Equidistant interpolation Section 5 Hermite interpolation Section 6 Piecewise low-order interpolation Section 7 Cubic spline interpolation
$1 IntroductionThe interpolation method is an important numerical method iswidely used in theoretical research and engineering practice.The definitionofinterpolation method:If function y = f (x) in the interval [a, b] are defined, andknown the value of yo, yi, ... , yn is a ≤ Xo<xi<... <xn ≤b, ifhaving a simplefunction P(x), letP(x;) =yi(i = 0,1,...,n)is right, P(x) is called the interpolation function of f(x), spot XoXi, ... , Xn is called interpolation node , the interval [a, b] is calledInterpolation interval, The method of solving interpolation上页functionP(x)calledinterpolationmethod.下页返圆
上页 下页 返回 The interpolation method is an important numerical method is widely used in theoretical research and engineering practice. The definition of interpolation method ： If function y = f (x) in the interval [a, b] are defined, and known the value of y0 , y1 , . , yn is a ≤ x0 < x1< . < xn ≤ b, if having a simple function P(x), let P(xi ) = yi (i = 0,1,.,n) is right，P(x) is called the interpolation function of f (x), spot x0 , x1 , . , xn is called interpolation node , the interval [a, b] is called Interpolation interval, The method of solving interpolation function P(x) called interpolation method. §1 Introduction
The geometric significance ofy=f(x)interpolationmethod:y=P(x)Getting the solution of curvey=P (x) , let it through given n+1points (xi,yi), i= O,l,...,n, andusing it approximate the known curve yXnXI=f (x) .上页下页返圆
上页 下页 返回 The geometric significance of interpolation method： Getting the solution of curve y = P (x) ，let it through given n+1 points (xi , yi )， i = 0,1,.,n，and using it approximate the known curve y = f (x)
$ 2 Lagrange interpolationDLinearinterpolation and Parabolic interpolation()Linearinterpolationyy=L(x)y=f(x)-Used straight line y = L, (x) approximat e curveiVk+1y = f(x), L,(x) is called Linear interpolat ionpolynomial , it can beindicated byTwo points linear equations4XkXk+1xXk+1 -XX-XkL;(x) = Yk+lXk+1 -XkXk+1 -Xk上页下页返圆
上页 下页 返回 1) Linear interpolation and Parabolic interpolation (1) Linear interpolation 1 1 1 1 1 1 1 ( ) Two points linear equations polynomial ( ) ( ) Linear interpolat ion straight line ( ) approximat e curve + + + + − − + − − = = = k k k k k k k k y x x x x y x x x x L x it can beindicated by y f x L x is called Used y L x ， ， §2 Lagrange interpolation
L, (x) is composed of two linear functionsX-XkX-X+L, 1k+I(x)lk(x) =Xk-Xk+1Xk+1-XkI (x) and Ik+i (x) are also Linear interpolat ion polynomial sInthe node need: l(x)= 1, l(xk+)= O;Ik+1(x) = O, lk+1(xk+1) = 1. calling l,(x) and lk+1(x) areLinear interpolat ion basic functions .上页下页返圆
上页 下页 返回 Linear interpolat ion basic functions . ( ) 0 ( ) 1. ( ) ( ) In the node need ( ) 1 ( ) 0 ( ) ( ) Linear interpolat ion polynomial s ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 l x l x calling l x and l x are l x l x l x and l x are also x x x x l x x x x x l x L x is composed of two linear functions k k k k k k k k k k k k k k k k k k k k + + + + + + + + + + = = = = − − = − − = ， ： ， ； ， ，