Is it possible to vectorize recursive calculation of a NumPy array where each element depends on the previous one?

python numpy vectorization
tnt picture tnt · Dec 10, 2010 · Viewed 8.4k times · Source 
T(i) = Tm(i) + (T(i-1)-Tm(i))**(-tau(i))

Tm and tau are NumPy vectors of the same length that have been previously calculated, and the desire is to create a new vector T. The i is included only to indicate the element index for what is desired.

Is a for loop necessary for this case?

Answer

Andrew Jaffe picture Andrew Jaffe · Dec 10, 2010

You might think this would work: 

import numpy as np
n = len(Tm)
t = np.empty(n)

t[0] = 0  # or whatever the initial condition is 
t[1:] = Tm[1:] + (t[0:n-1] - Tm[1:])**(-tau[1:])

but it doesn't: you can't actually do recursion in numpy this way (since numpy calculates the whole RHS and then assigns it to the LHS).

So unless you can come up with a non-recursive version of this formula, you're stuck with an explicit loop:

tt = np.empty(n)
tt[0] = 0.
for i in range(1,n):
    tt[i] = Tm[i] + (tt[i-1] - Tm[i])**(-tau[i])

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