inflection point on sum of sinusoids

Hi, I have a problem that may have a neat solution but I'm not certain of the approach. I'd like to find the inflection points of a function which is the sum (superposition) of sinusoids (tidal data).Its formulated like this:

y = A1*sin(w1*t + p1) + A2*sin(w2*t + p2) + ... + An*sin(wn*t + pn)

where A1..An is the component Amplitude; w1..wn is the component frequency; p1..pn is the component phase. And each of these is known.

I think you can find the inflection by finding where the second derivative of y is zero. ie

y'' = 0 = -A1*w1^2*sin(w1*t +p1)+ A1*cos(w1*t +p1) + ..

Is there a fast way to determine the values of t (within a range) for which y'' =0? The timeseries is long (around 1 million points) so recreating the entire timeseries is time consuming.

Answer 1

[Andrew Newell]

If you define a vector A = [A1 A2 ... An], and analogous vectors w and p, then in MATLAB

y = sum(A.*sin(w*t+p));

and the second derivative is

ypp = -sum(A.*w.^2.*sin(w*t+p));

(yours is incorrect). Given A, w and p, you can define an anonymous function

f = @(t) -sum(A.*w.^2.*sin(w*t+p));

and then use fzero to find a root, e.g.,

inflection_point = fzero(f,[x0 x1])

for a solution between x0 and x1.

[Ben]

Thanks Andrew, for the code and for correcting my maths! I was hoping that it would give an array of inflection points but I can easily add a loop as the times between inflections are almost regular. Cheers