lagrange interpolation, .m

can anyone explain me how to use this program

http://www.mathworks.com/matlabcentral/fileexchange/899-lagrange-polynomial-interpolation

function y=lagrange(x,pointx,pointy)
%
%LAGRANGE   approx a point-defined function using the Lagrange polynomial interpolation
%
%      LAGRANGE(X,POINTX,POINTY) approx the function definited by the points:
%      P1=(POINTX(1),POINTY(1)), P2=(POINTX(2),POINTY(2)), ..., PN(POINTX(N),POINTY(N))
%      and calculate it in each elements of X
%
%      If POINTX and POINTY have different number of elements the function will return the NaN value
%
%      function wrote by: Calzino
%      7-oct-2001
%
n=size(pointx,2);
L=ones(n,size(x,2));
if (size(pointx,2)~=size(pointy,2))
   fprintf(1,'\nERROR!\nPOINTX and POINTY must have the same number of elements\n');
   y=NaN;
else
   for i=1:n
      for j=1:n
         if (i~=j)
            L(i,:)=L(i,:).*(x-pointx(j))/(pointx(i)-pointx(j));
         end
      end
   end
   y=0;
   for i=1:n
      y=y+pointy(i)*L(i,:);
   end
end

[Jatin Arora]

How to run this code

[Hardipsinh Jadeja]

If size of pointx and pointy is same size then why not print the statement

Answer 1

[Matt Tearle]

pointx and pointy are two vectors of data values, x is a vector of points where you want to interpolate. For example:

x = 0:10;
y = x.^2;
xx = linspace(0,10);
yy = lagrange(xx,x,y);
plot(x,y,'o',xx,yy,'.')

As an aside, with no offense intended to Calzino, there are other options available for interpolation. Firstly, of course, interp1 is a standard MATLAB function, with options for linear, cubic spline, and PCHIP interpolation. Cleve Moler (aka The Guy Who Wrote MATLAB) also has a Lagrange interpolation function available for download.

[buxZED]

still dont get it

given the points x0 , x1 , x2 , x3 , x4

and the values f(x0 ), f(x1 ), f(x2 ), f(x3 ), f(x4 )

how can i get an 4th order accurat function f in position x

just point me in the right derection

[Jan]

I think, Matt Tearle's example does point in the right direction already. But I can repeat it:

y = lagrange(x, [x0,x1,x2,x3,x4], [f(x0),f(x1),f(x2),f(x3),f(x4)]);

This is exactly found in the help section of the function.

[Matt Tearle]

Right, what Jan said. In my example,x and y are vectors of the points x0, x1, ..., x4 and f(x0), ..., f(x4). The new point you're calling x is what I called xx. I used a vector of points, but it could be a single value.

[Mudra Dave]

"Jan Simon on 17 Mar 2011 I think, Matt Tearle's example does point in the right direction already. But I can repeat it: y = lagrange(x, [x0,x1,x2,x3,x4], [f(x0),f(x1),f(x2),f(x3),f(x4)]); This is exactly found in the help section of the function." I gave some values like , y = lagrange(x, [1,2,3,4], [2,4,6,8]) which returned, 2 4 6 8 10

what does this means?

[Jan]

@Mudra: What does what mean? Please open a new thread for a new question. Then provide ay many details as required to repdoduce the problem.

[Russell]

@Matt Tearle

That link is dead, I don't suppose you have an updated one?

[Walter Roberson]

The code is included in https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/interp.pdf

Note: the File Exchange has some more advanced polyinterp functions.

Answer 2

[Matt Fig]

This is really a question for the author of the program. I believe it is also bad etiquette to post somebody's code like that without permission.

Did you try to contact the author?

[Matt Tearle]

It's from File Exchange, so I don't seem any great harm in posting it.

[Matt Fig]

Ah, but I wasn't talking about harm, just polite behavior. The author should have been contacted first, that's all.

[Matt Tearle]

Fair call. I guess it does open the door for people to bash the author's code in a separate location, which would be uncool.

Answer 3

[SAM Arani]

%% Lagrangian interpolation
clear;clc;close all;
X=[-3 -2.5 -1 0 2 3.75 4.25 7];
Y=(sqrt(1+abs(X)));
xq=min(X):0.1:max(X);
f=(sqrt(1+abs(xq)));
syms x

S=0;
for i=1:length(X)
temp=X;
A=temp(i);
temp(i)=[];
L=prod((x-temp)./(A-temp),'all');
S=(L*Y(i))+S;
L=[];
end

figure()
fplot(S,'black--',[min(X) max(X)]);
hold on
F=interp1(X,Y,xq);
plot(xq,F,"bo");
hold on
plot(xq,f,"r*");
legend("Lagrangian","interp1","f(x)",'Location','north');
xlabel(" X axis ");
ylabel(" Y axis");
title("Lagrangian interpolation VS interp1-MatlabFunction")

Above we can see an easy way to implement lagrangian interpolation which has been checked with matlab interp1() function;

From MohammadReza Arani

[e-mail]

[image-pro]

how to imaplement this method for image?

[Walter Roberson]

@image-pro

X = 1:numel(YourImage);
Y = double(YourImage(:));

This is not likely to give good results.

Perhaps you want something different than this? Perhaps the image has a curve already drawn in it, and you want to extract the points from the image and then do interpolation on the curve?

[image-pro]

yes, but how to code all this?

[Walter Roberson]

See https://www.mathworks.com/matlabcentral/fileexchange/?term=tag:%22digitize%22 for a number of File Exchange contributions that try to extract data from images of graphs.

Answer 4

[norah]

how can i find error bound ?

Answer 5

[John]

function Y = Lagrange_371(x,y,X)
n = length(x) - 1;
Y = 0;
for i = 0:n
    prod = 1; 
    for j = 0:n
        if i ~= j
            prod = prod.*(X - x(j+1))./(x(i+1) - x(j+1));
        end
    end 
    Y = Y + prod*y(i+1);
end
end

[Oussama]

bonsoir ,comment appliquer cette fonction sur un exemple

Answer 6

[MUHAMMAD IQRAM HAFIZ]

function P = lagrange_interpolation_3point(x1, y1, x2, y2, x3, y3, x)

% Compute the Lagrange basis polynomials

L1 = ((x - x2) .* (x - x3)) / ((x1 - x2) * (x1 - x3));

L2 = ((x - x1) .* (x - x3)) / ((x2 - x1) * (x2 - x3));

L3 = ((x - x1) .* (x - x2)) / ((x3 - x1) * (x3 - x2));

% Compute the Lagrange polynomial

P = y1 * L1 + y2 * L2 + y3 * L3;

end

% Given points

x0 = 0; y0 = 0;

x1 = 2; y1 = 10;

x2 = 4; y2 = 20;

x3 = 8; y3 = 100;

% Point at which to evaluate the polynomial

x = 4.5;

% Calculate the interpolation polynomial at x = 4.5

P = lagrange_interpolation_3point(x1, y1, x2, y2, x3, y3, x);

% Display the result

fprintf('The interpolated value at x = %.1f is P = %.2f\n', x, P);