How to find the polynomial coefficients of two Array

I have such data from a thesis as below:

x=[1/1800,1/60,1/3,1,2,4,8,24];
y1=[0.7291,3.1663,4.3217,6.5508,9.1189,16.2679,30.3524,44.7320];
y2=[0.4572,1.7790,5.2580,13.7910,30.1775,64.9784,109.7704,136.0388];

The author said he can get the chart as below after smooth processing the data.

Also,he can get the Polynomial coefficients:

r1 = 3. 550, 1. 788, 0. 6734, - 0. 09003, 0. 004448, 9. 396e-5, 6. 675e-7
r2=  1. 709, 8. 714, 4. 178, - 0. 8004, 0. 05731, - 0. 001856, 2. 277e- 5

I can figure out the chart by using

xi = 0:0.1:25;
yi1 = interp1(x,y1,xi,'pchip');
yi2 = interp1(x,y2,xi,'pchip');
plot(xi,yi1,'b:',xi,yi2,'r-');

This is the picture:

However, i get the Polynomial coefficients as below:

py1= polyfit(xi,yi1,6);
py2= polyfit(xi,yi2,6);

py1 =

    0.0000   -0.0002    0.0056   -0.0851    0.4458    3.1673    1.4525
py2 =

    0.0000   -0.0016    0.0498   -0.6844    3.3783   10.8772    0.1662

the Polynomial coefficients were quite different from the author's.

I don't konw if i missed any steps or which step is wrong.

I think maybe the error is happened when smooth processing the data.The author didn't tell which method he used to processing the data, while i just used the interp1 method.

I want to get the same coeffcients as the author.Any sugguestion about the processing method?

Any help would be kindly, many thanks!

[Matt J]

I think maybe the error is happened when smooth processing the data.The author didn't tell which method he used to processing the data, while i just used the interp1 method.

Interpolation doesn't smooth the data in any way.

Answer 1

[Matt J]

the Polynomial coefficients were quite different from the author's.

You're using a very small amount of data to perform the fit (only 1 more data point than the number of unknown coefficients).

In addition, it's a 6th-order polynomial. Fitting at an order that high is starting to get fairly ill-conditioned,

>> cond(vander(0:6))

ans =

   1.5973e+06

[Alex]

Thanks very much! Yor are right, i can't fit it directly with polynomial. My finnal method is using interp1(x,Lind,xi,'pchip') to smooth the line, then using 6th-order polynomial to get the coefficients.