Trying to vectorize iterative calculation with numpy

Question

I am trying to make some piece of code more efficient by using the vectorized form in numpy. Let me show you an example so you know what I mean.

Given the following

Answer 1

A linear recurrence such as this can be computed using scipy.signal.lfilter:

In [19]: from scipy.signal import lfilter

In [20]: num = np.array([1.0])

In [21]: alpha = 2.0

In [22]: den = np.array([1.0, -alpha])

In [23]: a = np.zeros((4,4))

In [24]: a[0,:] = [1,2,3,4]

In [25]: lfilter(num, den, a, axis=0)
Out[25]: 
array([[  1.,   2.,   3.,   4.],
       [  2.,   4.,   6.,   8.],
       [  4.,   8.,  12.,  16.],
       [  8.,  16.,  24.,  32.]])

See the following for more details: python recursive vectorization with timeseries, Recursive definitions in Pandas

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Note that using lfilter really only makes sense if you are solving a nonhomogeneous problem such as x[i+1] = alpha*x[i] + u[i], where u is a given input array. For the simple recurrence a[i+1] = alpha*a[i], you can use the exact solution a[i] = a[0]*alpha**i. The solution for multiple initial values can be vectorized using broadcasting. For example,

In [271]: alpha = 2.0

In [272]: a0 = np.array([1, 2, 3, 4])

In [273]: n = 5

In [274]: a0 * (alpha**np.arange(n).reshape(-1, 1))
Out[274]: 
array([[  1.,   2.,   3.,   4.],
       [  2.,   4.,   6.,   8.],
       [  4.,   8.,  12.,  16.],
       [  8.,  16.,  24.,  32.],
       [ 16.,  32.,  48.,  64.]])