Vectorization time-varying recursive linear function
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Bruno Luong
on 27 Aug 2020
Commented: David Goodmanson
on 29 Aug 2020
I try to vectorize this simple recursive relation (all quantities are scalars)
x_{0} = 0;
x_{n} = x_{n-1}*a_{n} + b_{n} for n=1,2,...,N
In MATLAB code it can be carried out by for loop
% test inputs
b=rand(1,10);
a=0.9+zeros(size(b));
xk=0;
x=zeros(size(b));
for k=1:length(x)
xk = a(k)*xk+b(k);
x(k) = xk;
end
For a(:) constant this can be vectorized by IIR filter
ac = unique(a);
if length(ac)==1
x = filter(1, [1 -ac], b);
end
I would though it could have some time-varying IIR filter that I can use to vectorize the case where a is time-dependent.
But I couldn't find anywhere such stock function. anyone have an idea?
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Accepted Answer
David Goodmanson
on 28 Aug 2020
Edited: David Goodmanson
on 28 Aug 2020
Hi Bruno,
a = rand(1,50);
b = rand(1,50);
% method 1
xk = 0;
x = zeros(1,50);
for k = 1:50
xk = a(k)*xk + b(k);
x(k) = xk;
end
% method 2
cpa = cumprod([1 a(2:end)])
x1 = filter(1,[1 -1],b./cpa).*cpa;
max(abs(x1-x))
ans = 4.4409e-16
2 Comments
David Goodmanson
on 29 Aug 2020
Hi Bruno,
Also, if one of the a's is nonzero but very small, there are probably going to be numerical accuracy issues. It's unfortunate that Matlab apparently does not have a built-in function for this type of iteration.
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