3-d order derivative

Dear all,

there is the following problem with the calculation of a 3-d order derivative.

I have two vectors of lambda and refractive index, respectively. I take the 3-d order derivative using a gradient().

dndl=gradient(n)./gradient(lambda);
d2ndl2=gradient(dndl)./gradient(lambda);
d3ndl3=gradient(d2ndl2)./gradient(lambda);

When I use a relatively small number of points (for example 3000) , I get a smooth plot.

In the case of more points (30 000) there is some oscillation in the plot.

What is the reason of such behavior?

Thank you a lot.

[Matt J]

What do you mean by "use more points"? If it's a different input array why expect the same results?

[Max Demesh]

What do you mean by "use more points"?

I mean, that I make the differences smaller and increase the accuracy.

Moreover, if I find the analytical function and then take the 3-d order derivative, I obtain smooth plots in any case.

Answer 1

[Catalytic]

If the points are too close together, the difference between neighbours will be so small as to be dominated by floating point errors

[Max Demesh]

It seems to be true. Using non SI base units do not help.

I guess in this case there is no way to solve this issue.

P.S. For the 2-nd derivative there is no problem for any number of points.

Answer 2

[Matt J]

You could try diff(x,3)

[Max Demesh]

It does not work too, because diff() uses a forward difference, but the gradient() uses a central difference.

[Matt J]

Why care whether its forward or central? For a smooth curve, it should work out the same.

Answer 3

[Max Demesh]

The answer is simple: there is nenessery to use 128-bit floating point arithmetic.

https://blogs.mathworks.com/cleve/2017/05/22/quadruple-precision-128-bit-floating-point-arithmetic/#7b575c04-ef9c-466e-a586-ca699e81b772

or https://www.advanpix.com/.