![321.png](https://www.mathworks.com/matlabcentral/answers/uploaded_files/247033/321.png)
calling gradient multiple times
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Jiawei Gong
am 6 Nov. 2019
Kommentiert: darova
am 6 Nov. 2019
MATLAB suggests not to call gradient multiple times, but I did it for sin function. As shown in the figure, the first order derivative is fine, the second has one boundary point off, and the third two points off. I really appreciate if someone can quantatively explain it from the mathematical point of view.
h = .1; % time step
t = 0:h:10; % time array
y = sin(t); % sin function
dy = gradient(y,h);
ddy = gradient(dy,h);
dddy = gradient(ddy,h);
plot(t,[y;dy;ddy;dddy]);
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/246873/image.jpeg)
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darova
am 6 Nov. 2019
It's because of calculation first and last point. gradient does it this way:
first_grad = (y2-y1)/dx;
second_grad = 1/2*(y4-y3)/dx + 1/2*(y3-y2)/dx;
third_grad = 1/2*(y5-y4)/dx + 1/2*(y4-y3)/dx;
%% ...
last_grad = (ylast-ylast_1)/dx;
BLack curve and black points is your original data. Red points are points where gradient/derivative is calculated. Pay attention that first and last points are not at the beginning and end
![321.png](https://www.mathworks.com/matlabcentral/answers/uploaded_files/247033/321.png)
Second time you use gradient you use the same step (again). But for the first and last points step/2 should be used.
Interesting fact
If you have not regular step (h is not equal) gradient can't handle it. Better use diff
% generate some data
x = sort(10*rand(100,1));
y = sin(x);
dy1 = gradient(y,x);
dy2 = diff(y)./diff(x);
x2 = x(2:end)-diff(x)/2; % actual x for diff/derivative
plot(x,y) % original data
hold on
plot(x,dy1,'g') % derivative obtained with gradient
plot(x2,dy2,'.m') % derivative obtained with diff
plot(x,cos(x),'k') % exact derivative
hold off
legend('orignal data','gradient','diff','exact')
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