1. interpolation function
%% n order interpolation polynomial %% X is an interpolation nodes, n being the number of interpolation polynomial, if the function expression is known attribute is zero, but unknown function expression is a known function of the value function IPn = Interpolation_polynomials_of_degree_n(X,Y,precision,attribute) global MAX;global m;global n;global i; X = sort(X); [m,n] = size(X);MAX = max([m,n]);error = []; if attribute == 0 F = ones(1,MAX); for i = 1:MAX F(i) = subs(Y,X(i)); end sum = 0; for i = 1:MAX sum = sum+F(i)*Interpolation_basis_fun(X,i-1); end IPn = vpa(collect(sum),4); for i = 1:MAX error(i) = abs(F(i)-subs(sum,X(i))); end %% mapping h=figure; set(h,'color','w'); t = min(X):(max(X)-min(X))/precision:max(X); Yreal = subs(Y,t); T = subs(sum,t); plot(t,Yreal,'b',t,T,'g',X,F,'r*'); grid on title ( 'Lagrange'); xlabel('Variable x'); ylabel ( 'Variable y'); legend ( 'Yreal: real image', 'T: fitting polynomial image', 'F: actual data'); %% display coordinates for i = 1:MAX text(X(i),F(i),['(',num2str(X(i)),',',num2str(F(i)),')'],'color',[0.02 0.79 0.99]); end disp ( 'error value'); error elseif attribute == 1 sum = 0; for i = 1:MAX sum = sum+Y(i)*Interpolation_basis_fun(X,i-1); end IPn = vpa(collect(sum),4); h=figure; set(h,'color','w'); t = min(X):(max(X)-min(X))/precision:max(X); T = subs(sum,t); plot(X,Y,'g*',t,T,'b'); grid on title ( 'Lagrange'); xlabel('Variable x'); ylabel ( 'Variable y'); legend ( 'Y: known data', 'T: fitting polynomial image "); for i = 1:MAX text(X(i),Y(i),['(',num2str(X(i)),',',num2str(Y(i)),')'],'color',[0.02 0.79 0.99]); end end end
2. Basis Functions
Interpolation basis functions %% function IBF = Interpolation_basis_fun(X,k) [m,n] = size(X);MAX = max([m,n]); X = sort(X); mult_x = 1; mult_v = 1; for i = 1:MAX syms x; if i ~= k+1 mult_v = mult_v * (X (k + 1) X (i)); mult_x = mult_x*(x-X(i)); end end IBF = mult_x / mult_v; end
3. The interpolation remainder of error bounds
%% interpolation remainder of the margin of error (the remainder calculated only known function expression) function MI = More_than_the_interpolation(X,f,xi,precision) X = sort(X); a = min(X);b = max(X); disp ( 'xi interval should be the following:'); [a,b] [m,n] = size(X);MAX = max([m,n]); Df = diff(f,MAX);Df_value = subs(Df,xi); MI = vpa(collect(Df_value*omiga(X)/factorial(MAX)),4); %% error limit Df_max = max(subs(Df,X)); R_x = vpa(collect(Df_max*abs(omiga(X))/factorial(MAX)),4); DISP ( 'error limits:'); R_x %% plot area t = a:(b-a)/precision:b; T1 = subs(R_x,t); T2 = subs(MI,t); h=figure; set(h,'color','w'); plot(t,T1,'r--',t,T2,'g'); grid on title ( 'error image'); xlabel('Variable x'); ylabel ( 'Variable y'); legend ( 'T1: the upper limit of errors', 'T2: specified error limit'); I = Factorial function (n) if n == 0 I = 1; else Factorial I = (n-1) * n; end end end
4. continually multiply polynomials
function ox = omiga(X) [m,n] = size(X);MAX = max([m,n]); syms x; More = 1; for i = 1:MAX * = more than (xx (i)); end ox = more; end
5. Examples
clear all clc precision=500; X=1:1:9; R1=reshape(rand(9),1,9^2); R2=reshape(rand(18),1,18^2); R=zeros(1,9); for i=1:9 R(i)=R1(9*i)*R2(18*i)*100; end %% known function DISP ( 'known function expression'); syms x; f=x*exp(-x^2)+log(abs(exp(x)+precision*sin(x))); Interpolation_polynomials_of_degree_n(X,f,precision,0) %% known function values DISP ( 'known function value'); Interpolation_polynomials_of_degree_n(X,R,precision,1)
result
Expression of known function Error value error = 0 0 0 0 0 0 0 0 0 years = 1.621e-5*x^8 + 0.002542*x^7 - 0.1033*x^6 + 1.566*x^5 - 12.15*x^4 + 52.22*x^3 - 122.5*x^2 + 141.6*x - 54.27 Known function values years = - 0.06151*x^8 + 2.428*x^7 - 40.08*x^6 + 359.3*x^5 - 1899.0*x^4 + 6000.0*x^3 - 10950.0*x^2 + 10420.0*x - 3849.0
Image below