Re: Polyfit with scaling

[Rolf Fabian]

Ben Abbott wrote:
> 
> On Jan 31, 2008, at 3:08 PM, Thomas Weber wrote:
> 
>> On 31/01/08 07:47 -0500, Ben Abbott wrote:
>>> It doesn't appear that the example has much comparative value.
>>
>> Honestly, I don't care what we change. However, what shouldn't  
>> happen is
>> that things just stay the same. If we have a better implementation in
>> wpolyfit, we should take that for polyfit. Having a better algorithm
>> lying around and not using is just a waste (actually, it's even worse:
>> we not even have a better algorithm, but it's also already  
>> implemented).
>>
>>      Thomas
> 
> ? hmmm ... I agree we  should use the better algorithm.
> 
> In any event, I've modified polyfit.m to optionally use the method  
> employed by wpolyfit. To do this; I wrote two new scripts. One to  
> offset a polynomial's dependent variable (polyshift), and one to scale  
> a polynomial's dependent variable (polyscale).
> 
> Below I compared the present version of polyfit to the one attached to  
> this email.
> 
> octave:1> x = 1:4;
> octave:2> y = polyval ([1, 1, 1], x);
> octave:3> polyfit (x, y, 2)
> ans =   1.00000   1.00000   1.00000
> octave:4> polyfit (x, y, 2, 1)
> ans =   1.00000   1.00000   1.00000
> octave:5> polyfit (x, y, 3)
> ans =   0   1   1   1
> octave:6> polyfit (x, y, 3, 1)
> ans =   5.2633e-16   1.0000e+00   1.0000e+00   1.0000e+00
> octave:7> polyfit (x, y, 4)
> ans =  -0.0020445   0.0204453   0.9284416   1.1022263   0.9509314
> octave:8> p = polyfit (x, y, 4, 1)
> p =    -5.0886    50.8856  -177.0996   255.4281  -121.1255
> octave:9> polyval (p, x)
> ans =    3.0000    7.0000   13.0000   21.0000
> octave:10> polyval ([1, 1, 1], x)
> ans =    3    7   13   21
> 
> Comparing to Matlab ...
> 
>  >> x = 1:4;
>  >> y = polyval([1, 1, 1], x);
>  >> polyfit(x, y, 2)
> ans =    1.0000    1.0000    1.0000
>  >> polyfit(x, y, 3)
> ans =    0.0000    1.0000    1.0000    1.0000
>  >> polyfit(x, y, 4)
> Warning: Polynomial is not unique; degree >= number of data points.
>  > In polyfit at 72
> ans =    0.0200   -0.2000    1.7000    0    1.4800
> 
> The offset and scaling appears to significantly modify the result when  
> the order of the fit exceeds to order of the original polynomial.
> 
> I am not an expert in such matters. How might we quantitatively  
> determine which algorithm is best?
> 
> I've attached the modified polyfit.m as well as the two new scripts it  
> relies upon.
> 
> Perhaps it is worth the effort for others to compare their results  
> with mine, and to compare Matlab's results on a computer other than an  
> Intel/ppc based Mac (each gave the same result).
> 
> Ben
> 

Dear Ben

Without testing it, I looked shortly at the code of 
your polyfit.m and found  close to the end:

  ## Return a row vector
  p = p(:)';

This is a bug because it converts complex column vectors
to complex conjugated polynomials ( row-vectors)

Rolf


-----
Rolf Fabian
<r dot fabian at jacobs-university dot de>

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