Talk:Lagrange polynomial

I did a complete overhaul of the page. At the moment it is quite technical, maybe someone could add a practical example. MathMartin 16:58, 25 Apr 2004 (UTC)

Good work, sorry for all the errors, they were mine... Nixdorf 18:18, 25 Apr 2004 (UTC)

No sweat, I have not found any big errors. The article was just a bit small and blurry. I hope it is better now. Is there any reason you used t instead of x as a variable ? MathMartin 22:55, 25 Apr 2004 (UTC)

Not really, use x if you prefer. I come from physics so I like to think of it as "time", but generally, perhaps x is better. Nixdorf 16:33, 26 Apr 2004 (UTC)

Some one shoud add that $\ell _{j}$ can also be writen as

$\ell _{j}={\frac {f(x)}{(x-x_{j})\,f'(x_{j})}}$

with

$f(x):=\prod _{i=0}^{k}(x-x_{i})$

Multiplying by four

Do we really have to multiply everything by four, so that we get integers? It gave me something of a double-take. I think it would be more intuitive the other way, so it is easier to figure out what numbers came from where. Sim 16:29, 6 March 2007 (UTC)

I assume you're talking about using factors like $(4x-3)$ in the construction of the Lagrange polynomial, instead of $(x-{\tfrac {3}{4}})$ , in which case, I agree that it's better to do it the other way. Please go ahead and change it. -- Jitse Niesen (talk) 09:36, 7 March 2007 (UTC)

Errors

Any chance of a brief discussion of errors for Lagrange interpolation, preferably by someone who understands it properly (i.e., not me!)? me_and 21:21, 9 June 2007 (UTC)

Efficient code

instead of

     for (int j = 0; j < degree; ++j) {
        // The i-th term has to be skipped
        if (j != i) {
           weight *= (desiredPos - pos[j]) / (pos[i] - pos[j]);
        }
     }

it is more efficient to do two for loops




     for (int j=0; j<i; j++) {
        weight *= (desiredPos - pos[j]) / (pos[i] - pos[j]); 
     }

     for(int j=i+1; j<=degree; j++) {
        weight *= (desiredPos - pos[j]) / (pos[i] - pos[j]); 
     }

This avoid the cost of a test at each iteration. —Preceding unsigned comment added by 199.212.17.130 (talk) 14:44, 6 August 2008 (UTC)

     prod = 1;
     for (int j=0; j<i; j++) {
        prod *= (pos[i] - pos[j]);
        weight *= (desiredPos - pos[j]);
     }

     for(int j=i+1; j<=degree; j++) {
        prod *= (pos[i] - pos[j]);
        weight *= (desiredPos - pos[j]);
     }
     weight /= prod;

This avoids the cost of a division each iteration. dila (talk) 08:09, 3 June 2009 (UTC)