Denote Y=[y1 y2...yl',C=[cI C2...cul'.Then the problem of solving the normal equation can be expressed as a matrix formula as follows: F'FC=F'Y. where F'F is an MX Msystem of linear equations,C is a vector with M unknowns coefficients,and F'Y is a known vector producted by the matrix F and the vector Y
Denote Y=[y1 y2 … yN]’，C=[c1 c2 … cM]’. Then the problem of solving the normal equation can be expressed as a matrix formula as follows: F’FC=F’Y, where F’F is an M×M system of linear equations, C is a vector with M unknowns coefficients, and F’Y is a known vector producted by the matrix F and the vector Y
Polynomial Fitting When the foregoing method is adapted to using functions f(x)x),for j=1,...,M+1,the function fx)will be a polynomial of degree M: fx)-Cj+czx+cx2+...+CvIxM. It is tempting to use a least-squares polynomial to fit data that are nonlinear.But if the data do not exhibit a polynomial nature,the resulting curve may exhibit large oscillations.This phenomenon,called polynomial wiggle,becomes more pronounced with higher-degree polynomials.For this reason we seldom use a polynomial of degree 6 or above unless it is known that the true function we are working with is a polynomial
Polynomial Fitting ◼ When the foregoing method is adapted to using functions {f j (x)=x j-1}, for j=1,…, M+1, the function f(x) will be a polynomial of degree M: f(x)=c1+c2x+c3x 2+…+cM+1x M. ◼ It is tempting to use a least-squares polynomial to fit data that are nonlinear. But if the data do not exhibit a polynomial nature, the resulting curve may exhibit large oscillations. This phenomenon, called polynomial wiggle, becomes more pronounced with higher-degree polynomials. For this reason we seldom use a polynomial of degree 6 or above unless it is known that the true function we are working with is a polynomial
Using Polynomials to Fit Data 30 40 20 30 f(x) P2(x) 20 P:(x) 10 10 0 0 2 -10 40 60 40 20 20 P(x) P:(x) 0 2 2 -20 -20 -40
Using Polynomials to Fit Data f (x) P2 (x) P5 P (x) 4 (x) P3 (x)
polynomial wiggle Cubic Splines Interpolation Lagrange Interpolation 9-10】 华南师范大学数学科学学院谢细玲
polynomial wiggle Cubic Splines Interpolation Lagrange Interpolation 华南师范大学数学科学学院 谢骊玲
4.3 Interpolation by Spline Functions Higher-degree polynomial interpolation for a set of N+1 points {(xkyis frequently unsatisfactory.A polynomial of degree Ncan have N-1 relative maxima and minima,and the graph can wiggle in order to pass through the points. Another method is to piece together the graphs of lower-degree polynomials S(x)and interpolate between the successive nodes (x) and (,)And the set of functions {S(x))forms a piecewise polynomial curve,which is denoted by S(x)
4.3 Interpolation by Spline Functions ◼ Higher-degree polynomial interpolation for a set of N+1 points {(𝑥𝑘, 𝑦𝑘)}𝑘=1 𝑁 is frequently unsatisfactory. A polynomial of degree N can have N-1 relative maxima and minima, and the graph can wiggle in order to pass through the points. ◼ Another method is to piece together the graphs of lower-degree polynomials Sk (x) and interpolate between the successive nodes (xk , yk ) and (xk+1, yk+1). And the set of functions {Sk (x)} forms a piecewise polynomial curve, which is denoted by S(x)