How to Calculate the Second Derivative of a Curve in Matlab?

[anahita]

I have a set of data as follows, How can I calculate the second derivative of the curve obtained from these data.

x=[0.1;0.07;0.05;0.03;0]; r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];

 

[Image Analyst]

How about fitting it to a curve, then calculating the second derivative analytically from the fitted curve formula? Like this:

x=[0.1;0.07;0.05;0.03;0];
r=[-98.9407;-105.7183;-111.2423;-116.0320;-120.0462];
plot(x, r, 'r*-', 'LineWidth', 2);
% Fit to a cubic
coefficients = polyfit(x, r, 4)
xFit = linspace(min(x), max(x), 100);
rFit = polyval(coefficients, xFit);
hold on;
plot(xFit, rFit, 'b-', 'LineWidth', 2);
grid on;
legend('Actual', 'fit');

% Set up figure properties:
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
% Get rid of tool bar and pulldown menus that are along top of figure.
set(gcf, 'Toolbar', 'none', 'Menu', 'none');
% Give a name to the title bar.
set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off')

% Equation is a1*x^4 + a2*x^3 + a3*x^2 + a4*x + a5
% First derivative is 4*a1*x^3 + 3*a2*x^2 + 2*a3*x + a4
% Second derivative is 12*a1*x^2 + 6*a2*x + 2*a3
a1 = coefficients(1)
a2 = coefficients(2)
a3 = coefficients(3)
deriv2 = 12*a1*xFit.^2 + 6*a2*xFit + 2*a3

 

[anahita]

Thanks.
But how to calculate second derivative graph x on r?

 

[Image Analyst]

What does that mean? I showed you how to calculate the second derivative of r with respect to x. Is that not what you want? Is r the independent variable instead of x? If so, just swap x and r in my code. Or do you just need to know how to call plot()?

plot(xFit, deriv2, 'b-');

 

[Irene Kaminkowa]

anahita,
you can siply use finite differencies: https://en.wikipedia.org/wiki/Finite_difference

 

[Image Analyst]

anahita,
you can siply use finite differencies: https://en.wikipedia.org/wiki/Finite_difference

You can implement this method with conv(), though with such a small array as the poster posted, the edge effects will reduce the valid region to about one element. However, for a longer vector (more elements), conv() is a good way to take the numerical/empirical derivative (vs. the analytical derivative as in my answer).