Re: Multi Column Data file to plot, gradient plot, or gradient of gradient plot

[Przemek Klosowski]

On Mar 26, 2004, at 1:39 PM, Timothy Milstead wrote:

   > xy plot the gradient of the data as lines and then finally the 
   > gradient of the gradient as xy plot.

And Paul Kienzle suggested:

   This is the simple formula for a gradient,

           dy/dx ~= (y_i+1 - y_i)  /  (x_i+1 - x_i)
           d2y/dx2 ~= (y_i+1 - 2y_i - y_i-1) / (x_i+1 - x_i)^2

   You can come up with more sophisticated approximations.

and came up with this implementation

   plot(x(1:end-1),diff(y)./repmat(diff(x),1,columns(y)));
   plot(x(2:end-1),diff(y,2)./repmat(diff(x(2:end)).^2,1,columns(y)));

OK, so we have relatively simple formulas, and an implementation that
is not only nonobvious, but I think might also be incorrect.

At first, I thought that there should be built-in differential
operators (gradients, Laplacians, etc), but then, I thought that there
are multiple details---possibly higher-order corrections, etc, that
would be hard to provide in a simple and general way. Then, it
occurred to me that the 'filter()' function is great for constructing
differential operators. For instance,

             dydx = filter([-1,1],1,y) ./ filter([-1,1],1,x)

             d2ydx2 = filter ([1,-2,1],1,y) ./ filter ([1,-2,1],1,x)

(I gloss over what are the values of X that I calculate the derivatives
in, but you get the general drift).



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