How to find data values that falls on line connecting loglog plot in matlab

I tried 'interp1' with 'linear' and 'pchip' to tried to pull more data values out of the line that matlab connected my data points. Neither of them look correct when i go to plot the new data values on a loglog plot.

The problem is obvious when using the example from the Matlab Help for 'loglog'

x = logspace(-1,2);

y = exp(x);

loglog(x,y,'-s');

Then find the interpolation to get more data points in the middle, the line isn't smooth:

xq = logspace(-1,2,500);

yq = interp1(x,y,xq,'linear'); %OR% yq = interp1(x,y,xq,'pchip');

figure; loglog(xq,yq);

I must be missing something easy. Thanks.

Answer 1

[Star Strider]

To get a smooth interpolation, it is necessary to do two transformations on it. First, interpolate log(y), and then take the exponential of the interpolated result:

yq = exp(interp1(x,log(y),xq,'linear'));

[Jeremy]

This works great!

[Star Strider]

Thank you!

Answer 2

[Sindar]

Your interpolation is linear. Compare:

loglog(x,y,'-s',xq,yq,'r*');

and

plot(x,y,'-s',xq,yq,'r*');

Log-log interpolation can be done "by hand":

yq = 10.^interp1(log10(x),log10(y),log10(xq),'linear');

[Sindar]

Since your data is smoother on a semilogy plot, use Star Strider's version. If you come upon data that is smoothest on loglog (e.g., y=x^5), you'll need to log the x-data as well. Regardless, exp() and log() are probably better than 10.^ and log10

[Jeremy]

Sorry I didn't give enough look at the data to see that your answer looks to give better results. I could see slight bends in the plot line when zoomed in. It seems to be that either of these 2 solve my problem:

yq = 10.^interp1(log10(x),log10(y),log10(xq),'linear'); %OR

yq = exp(interp1(log(x),log(y),log(xq),'linear'));

[Sindar]

Yes, they are mathematically equivalent. Numerically, I think exp(log(x)) is likely to be slightly closer to x than 10.^(log10(x)) -- I think more work has been put into calculating exp() accurately (and quickly) than 10.^