Obtaining approximate derivatives with variable x spacing in MATLAB? [duplicate] - arrays

Question

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Numerical derivative of a vector
(3 answers)
Closed 7 years ago.
I have two arrays: x and y. In practice, y is dependent on x, but both arrays are measured values. I would like to obtain the derivative of y with respect to x. If x were uniformly spaced (i.e. x=[1 2 3 4 5]), I could do something with diff like this:
h = 0.001;
x = -pi:h:pi;
f = sin(X);
y = diff(f)/h;
However, x is not uniformly spaced (i.e. x=[1 1.9 2.8 4.1]). How can I obtain the partial derivative of this data set?

Answer 1

A good way to do it is gradient,
dydx = gradient(y, x);
I like it because it returns a vector which is the same length as x and y. The downside though, is it's first order accurate. This can sometimes be a problem, a fix could be to write your own,
x = unique([linspace(0, 2*pi, 50), logspace(0, log10(2*pi), 50)]);
y = cos(x) ;
subplot(2,1,1) ;
plot(x, Gradient(y, x), x, gradient(y,x), x, -sin(x));
legend('2^{nd} order', '1^{st} order', 'exact') ;
subplot(2,1,2) ;
plot(x, Gradient(y, x) + sin(x), x, gradient(y,x) + sin(x));
legend('2^{nd} order - exact', '1^{st} order - exact')
With Gradient being
function dydx = Gradient(y,x)
y = y(:);
p = x(3:end) - x(2:end-1);
p = p(:);
m = x(2:end-1) - x(1:end-2);
m = m(:);
p1 = x(2) - x(1);
p2 = x(3) - x(1);
m1 = x(end) - x(end-1);
m2 = x(end) - x(end-2);
dydx = reshape([ ((-p1^2 + p2^2)*y(1) - p2^2*y(2) + p1^2*y(3))/(p1*(p1 - p2)*p2);
((-m.^2 + p.^2).*y(2:end-1) - p.^2.*y(1:end-2) + m.^2.*y(3:end))./(m.*p.*(m + p));
((m1^2 - m2^2)*y(end) + m2^2*y(end-1) - m1^2*y(end-2))/(m1^2*m2 - m1*m2^2) ...
], size(x));
end
Edit:
Improved it for multidimensional array and constant spacing support
function dydx = Gradient(y,x)
if length(y) < 3
dydx = gradient(y,x);
return
end
[~, n] = max(size(y));
N = ndims(y);
i = repmat({':'},1,N-1);
y = permute(y, [n, 1:n-1, n+1:N]);
if isscalar(x)
%"x" is actually a spacing value
p = x;
m = x;
p1 = x;
p2 = x;
m1 = x;
m2 = x;
else
if isvector(x)
x = repmat(x(:), size(y(1, i{:})));
else
x = permute(x, [n, 1:n-1, n+1:N]);
end
if all(size(x) ~= size(y))
error('Sizes of arrays must be the same.')
end
p = x(3:end, i{:}) - x(2:end-1, i{:});
m = x(2:end-1, i{:}) - x(1:end-2, i{:});
p1 = x(2, i{:}) - x(1, i{:});
p2 = x(3, i{:}) - x(1, i{:});
m1 = x(end, i{:}) - x(end-1, i{:});
m2 = x(end, i{:}) - x(end-2, i{:});
end
dydx = ipermute([ ((-p1.^2 + p2.^2).*y(1,i{:}) - p2.^2.*y(2,i{:}) + p1.^2.*y(3,i{:}))./(p1.*(p1 - p2).*p2);
((-m.^2 + p.^2).*y(2:end-1,i{:}) - p.^2.*y(1:end-2,i{:}) + m.^2.*y(3:end,i{:}))./(m.*p.*(m + p));
((m1.^2 - m2.^2).*y(end,i{:}) + m2.^2.*y(end-1,i{:}) - m1.^2.*y(end-2,i{:}))./(m1.^2.*m2 - m1.*m2.^2) ...
], [n, 1:n-1, n+1:N]);
end