[Scilab-users] Polynomial fitting

[Federico Miyara]

Stéphane,

> Yeah, but really badly conditionned compared to the above method which 
> is based on orthogonal tranformations (X=Q*R factorization). With your 
> below method you solve a linear system with X'*X matrix which has a 
> condition number which is the square of the condition number of the R 
> matrix issued from the Q*R factorization of X.

Thanks for the clarification!

Is it posible to predict in what kind of cases will the bad conditioning 
impact on the result? I've compared the results for several cases and 
they are exactly the same.

However, I've noticed that when taking the inverse of a matrix whose 
components span several orders of magnitude there are important errors, 
but if I manage to normalize the poblem the errors are substantially reduced

Regards,

Federico


> S.
>
>>
>> The basic algorithm I use is (n = desired degree):
>>
>> // Initialize matrix X
>> X = ones(length(x), n+1);
>> // Compute Vandermonde's matrix
>> for k =2:n+1
>>    X(:,k) = X(:,k-1).*x;
>> end
>> // Apply the Moore-Penrose pseudoinverse matrix and
>> // multiply by the dependent data vector to get the
>> // least squares approximation of the polynomial
>> // coefficients
>> A = inv(X'*X)*X'*y;
>>
>> I've seen some discussion regarding the need for a polyfit function 
>> in Scilab. The main argument against such a function is that it is 
>> unnecessary since it is a particular case of the backslash division. 
>> This is true, but the above example shows that users' implementations 
>> are not always optimized, and as it is such a frequent problem, it 
>> would be nice to have a native polyfit (or whatever it may be called) 
>> function.
>>
>> Regards,
>>
>> Federico Miyara
>>
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> -- 
> Stéphane Mottelet
> Ingénieur de recherche
> EA 4297 Transformations Intégrées de la Matière Renouvelable
> Département Génie des Procédés Industriels
> Sorbonne Universités - Université de Technologie de Compiègne
> CS 60319, 60203 Compiègne cedex
> Tel : +33(0)344234688
> http://www.utc.fr/~mottelet
>
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