Polynomial interpolation

　　Polynomial interpolation function

%% polynomial interpolation 
%% Description: precision of accuracy, the larger the finer the image, attribute is an attribute value, when it is known that the unknown function expression function value is 1, otherwise 0 
function Polynomial_interpolation the PI = (F, X-, Precision, attribute) 
X-Sort = (X-); 
IF attribute == 0 
    [m, n-] = size (X-); max = mAX ([m, n-]); 
    X-the RESHAPE = (X-,. 1, mAX); error = []; 
    for I =. 1: mAX 
        the Y (I) = SUBS (F, X-(I)); 
    End 
    Y_value = Double (the Y); 
    A = min (X-); B = max (X-); 
    T = A : (BA) / Precision: B; 
    T = zeros (. 1, Precision +. 1); 
    Yreal = SUBS (F, T); 
    Coe = VPA (Polynomial_interpolation_cofficient (F, X-, attribute),. 4); 
    for I =. 1: . 1: Precision. 1 + 
        T (I) = Polynomial_value (Coe, T (I)); 
    End 
    for I =. 1: MAX
        error (I) = ABS (the Y (I) -Polynomial_value (Coe, X-(I))); 
    End 
        
    %% plotted 
    H = Figure; 
    SET (H, 'Color', 'W'); 
    [HAX, hLine1, hLine2] = plotyy (T, T, X-, the Y, 'Plot', 'STEM'); 
    title ( 'polynomial interpolation'); 
    the xlabel ( 'Variable X'); 
    ylabel (HAX (. 1), 'Variable'); 
    ylabel (HAX (2), 'Variable'); 
    Grid ON 
    HOLD ON 
    Plot (T, Yreal); 
    Legend ( 'Yreal: real image', 'Y: fitting polynomial image', 'T: actual data'); 
    
    % % display coordinates 
    for I =. 1: MAX 
        text (X-(I), Y_value (I), [ '(', num2str (X-(I)), ',', num2str (Y_value (I)), ')'] , 'Color', [0.02 0.79 0.99]); 
    End 
    
    DISP ( 'Error value '); error 
ELSEIF attribute. 1 ==
    [m,n] = size(X);MAX = max([m,n]);X = reshape(X,1,MAX);f = reshape(f,1,MAX);
    a = min(X);b = max(X);
    t = a:(b-a)/precision:b;
    T = zeros(1,precision+1);
    Coe = vpa(Polynomial_interpolation_cofficient(f,X,attribute),4);
    for i = 1:1:precision+1
        T(i) = Polynomial_value(Coe,t(i));
    end
    
    h=figure;
    set(h,'color','w');
    plot(t,T,'b',X,f,'g*');
    grid on
    title('多项式插值');
    xlabel('Variable x');
    ylabel('Variable y');
    legend('Y:拟合多项式图像','T:已知数据');
    
    for i = 1:MAX
        text(X(i),f(i),['(',num2str(X(i)),',',num2str(f(i)),')'],'color',[0.02 0.79 0.99]);
    than 
than 

syms x;
PI=vpa(Polynomial_value(Coe,x),4);
than

　　2. The polynomial function value

%%多项式函数值
function PV = Polynomial_value(P,t)
[m,n] = size(P);
MAX = max([m,n]);
sum = 0;
for i = MAX:-1:1
    sum = sum+P(i)*Power_function(i-1,t);
end
PV = sum;
%%幂函数
    function pf = Power_function(index,t)
        pf = t.^index;
    end
end

　　3. polynomial coefficients

%% This function of the given number of nodes minus one polynomial coefficients 
%% Description: attribute is an attribute value, when it is known that the unknown function expression function value is 1, otherwise 0 
function Polynomial_interpolation_cofficient the PIC = (F, X- , attribute) 
Global mAX; Global m; n-Global; Global I; 
X-Sort = (X-); 
IF attribute == 0 
    [m, n-] = size (X-); max = mAX ([m, n-]); 
    X- the RESHAPE = (X-,. 1, MAX); 
    A = zeros (MAX, MAX); the Y = zeros (. 1, MAX); 
    for I =. 1: MAX 
        . A (:, I) = (X-') ^ (I- . 1); 
        the Y (I) = SUBS (F, X-(I)); 
    End 
    Coefficient = VPA (the RESHAPE (A \ (the Y '),. 1, MAX),. 4); 
ELSEIF attribute. 1 == 
    [m, n-] = size (X-); mAX = max ([m, n-]); the PIC = Cell (. 1, mAX +. 1); 
    X-= the RESHAPE (X-,. 1, mAX); 
    A = zeros (mAX, mAX); the Y = the RESHAPE (F,. 1, MAX); 
    for I =. 1: MAX
        A (:, I) = (X-') ^ (I-. 1);. 
    End 
    Coefficient = VPA (the RESHAPE (A \ (the Y'),. 1, MAX),. 4); 
End 
DISP ( 'the maximum number of n =' ); 
mAX. 1- 
the PIC = Coefficient; 
%% value polynomial 
    function Polynomial_value the PV = (P, T) 
        [m, n-] = size (P); 
        mAX = max ([m, n-]); 
        SUM = 0; 
        for MAX = I: -1:. 1 
            SUM = SUM + P (I) * Power_function (. 1-I, T); 
        End 
        the PV = SUM; 
        %% power function 
        function Power_function PF = (index, T) 
            PF = T ^ index. ; 
        End 
    End 
End

　　4. An example

All Clear 
CLC 
Precision = 500; 
X-=. 1:. 1:. 6; 
Rl = the RESHAPE (RAND (. 6), 1,36); 
R2 = the RESHAPE (RAND (12 is), 1,144); 
R & lt = zeros (1,6); 
= I. 1 for:. 6 
    R & lt (I) Rl = (I *. 6) R2 * (24 * I) * 100; 
End 

%% known function 
disp ( 'known polynomial fit function'); 
syms X; 
F * SiN X = (X ^ 2) * exp (the -X-^ 2) + log (ABS (SiN (X))); 
Polynomial_interpolation (F, X-, Precision, 0) 

%% known function value 
disp ( 'known fitting a polynomial function value '); 
Polynomial_interpolation (R & lt, X-, Precision,. 1)

　　result

Known function of polynomial fitting 
maximum number = n- 
ANS = 
     . 5 
error value 
error = 
   1.0E-08 * 
    0.0066 0.0092 0.0027 0.0473 0.1507 .3463 
ANS = 
0.1248 ^. 5 * X - 2.291 * 15.64 * + X ^ X ^. 4. 3 - 48.61 * x ^ 2 + 66.56 * x - 31.29 
polynomial fitting a known function of the value of 
the maximum number of times = n- 
ANS = 
     . 5 
ANS = 
- 1.993 * X ^ X ^ *. 5. 4 31.02 + - 175.7 + 444.1 * X * X ^. 3 ^ 2 - 491.6 * x + 201.5

Polynomial interpolation

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