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

You might think this would work:

import numpy as npn = 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])

2019 Update. The Numba code broke with the new version of numba. Changing dtype="float32" to dtype=np.float32 solved it.

I performed some benchmarks and in 2019 using Numba is the first option people should try to accelerate recursive functions in Numpy (adjusted proposal of Aronstef). Numba is already preinstalled in the Anaconda package and has one of the fastest times (about 20 times faster than any Python). In 2019 Python supports @numba annotations without additional steps (at least versions 3.6, 3.7, and 3.8). Here are three benchmarks: performed on 2019-12-05, 2018-10-20 and 2016-05-18.

And, as mentioned by Jaffe, in 2018 it is still not possible to vectorize recursive functions. I checked the vectorization by Aronstef and it does NOT work.

Benchmarks sorted by execution time:

-------------------------------------------|Variant        |2019-12 |2018-10 |2016-05 |-------------------------------------------|Pure C         |   na   |   na   | 2.75 ms||C extension    |   na   |   na   | 6.22 ms||Cython float32 | 0.55 ms| 1.01 ms|   na   ||Cython float64 | 0.54 ms| 1.05 ms| 6.26 ms||Fortran f2py   | 4.65 ms|   na   | 6.78 ms||Numba float32  |73.0  ms| 2.81 ms|   na   ||(Aronstef)     |        |        |        ||Numba float32v2| 1.82 ms| 2.81 ms|   na   ||Numba float64  |78.9  ms| 5.28 ms|   na   ||Numba float64v2| 4.49 ms| 5.28 ms|   na   ||Append to list |73.3  ms|48.2  ms|91.0  ms||Using a.item() |36.9  ms|58.3  ms|74.4  ms||np.fromiter()  |60.8  ms|60.0  ms|78.1  ms||Loop over Numpy|71.3  ms|71.9  ms|87.9  ms||(Jaffe)        |        |        |        ||Loop over Numpy|74.6  ms|74.4  ms|   na   ||(Aronstef)     |        |        |        |-------------------------------------------

Corresponding code is provided at the end of the answer.

It seems that with time Numba and Cython times get better. Now both of them are faster than Fortran f2py. Cython is faster 8.6 times now and Numba 32bit is faster 2.5 times. Fortran was very hard to debug and compile in 2016. So now there is no reason to use Fortran at all.

I did not check Pure C and C extension in 2019 and 2018, because it is not easy to compile them in Jupyter notebooks.

I had the following setup in 2019:

Processor: Intel i5-9600K 3.70GHzVersions:Python:  3.8.0Numba:  0.46.0Cython: 0.29.14Numpy:  1.17.4

I had the following setup in 2018:

Processor: Intel i7-7500U 2.7GHzVersions:Python:  3.7.0Numba:  0.39.0Cython: 0.28.5Numpy:  1.15.1

The recommended Numba code using float32 (adjusted Aronstef):

@numba.jit("float32[:](float32[:], float32[:])", nopython=True, nogil=True)def calc_py_jit32v2(Tm_, tau_):    tt = np.empty(len(Tm_),dtype=np.float32)    tt[0] = Tm_[0]    for i in range(1, len(Tm_)):        tt[i] = Tm_[i] - (tt[i-1] + Tm_[i])**(-tau_[i])    return tt[1:]

All the other code:

Data creation (like Aronstef + Mike T comment):

np.random.seed(0)n = 100000Tm = np.cumsum(np.random.uniform(0.1, 1, size=n).astype('float64'))tau = np.random.uniform(-1, 0, size=n).astype('float64')ar = np.column_stack([Tm,tau])Tm32 = Tm.astype('float32')tau32 = tau.astype('float32')Tm_l = list(Tm)tau_l = list(tau)

The code in 2016 was slightly different as I used abs() function to prevent nans and not the variant of Mike T. In 2018 the function is exactly the same as OP (Original Poster) wrote.

Cython float32 using Jupyter %% magic. The function can be used directly in Python. Cython needs a C++ compiler in which Python was compiled. Installation of the right version of Visual C++ compiler (for Windows) could be problematic:

%%cythonimport cythonimport numpy as npcimport numpy as npfrom numpy cimport ndarraycdef extern from "math.h":    np.float32_t exp(np.float32_t m)@cython.boundscheck(False)@cython.wraparound(False)@cython.infer_types(True)@cython.initializedcheck(False)def cy_loop32(np.float32_t[:] Tm,np.float32_t[:] tau,int alen):    cdef np.float32_t[:] T=np.empty(alen, dtype=np.float32)    cdef int i    T[0]=0.0    for i in range(1,alen):        T[i] = Tm[i] + (T[i-1] - Tm[i])**(-tau[i])    return T

Cython float64 using Jupyter %% magic. The function can be used directly in Python:

%%cythoncdef extern from "math.h":    double exp(double m)import cythonimport numpy as npcimport numpy as npfrom numpy cimport ndarray@cython.boundscheck(False)@cython.wraparound(False)@cython.infer_types(True)@cython.initializedcheck(False)def cy_loop(double[:] Tm,double[:] tau,int alen):    cdef double[:] T=np.empty(alen)    cdef int i    T[0]=0.0    for i in range(1,alen):        T[i] = Tm[i] + (T[i-1] - Tm[i])**(-tau[i])    return T

Numba float64:

@numba.jit("float64[:](float64[:], float64[:])", nopython=False, nogil=True)def calc_py_jitv2(Tm_, tau_):    tt = np.empty(len(Tm_),dtype=np.float64)    tt[0] = Tm_[0]    for i in range(1, len(Tm_)):        tt[i] = Tm_[i] - (tt[i-1] + Tm_[i])**(-tau_[i])    return tt[1:]

Append to list. Fastest non-compiled solution:

def rec_py_loop(Tm,tau,alen):     T = [Tm[0]]     for i in range(1,alen):        T.append(Tm[i] - (T[i-1] + Tm[i])**(-tau[i]))     return np.array(T)

Using a.item():

def rec_numpy_loop_item(Tm_,tau_):    n_ = len(Tm_)    tt=np.empty(n_)    Ti=tt.item    Tis=tt.itemset    Tmi=Tm_.item    taui=tau_.item    Tis(0,Tm_[0])    for i in range(1,n_):        Tis(i,Tmi(i) - (Ti(i-1) + Tmi(i))**(-taui(i)))    return tt[1:]

np.fromiter():

def it(Tm,tau):    T=Tm[0]    i=0    while True:        yield T        i+=1        T=Tm[i] - (T + Tm[i])**(-tau[i])def rec_numpy_iter(Tm,tau,alen):    return np.fromiter(it(Tm,tau), np.float64, alen)[1:]

Loop over Numpy (based on the Jaffe's idea):

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

Loop over Numpy (Aronstef's code). On my computer float64 is the default type for np.empty.

def calc_py(Tm_, tau_):    tt = np.empty(len(Tm_),dtype="float64")    tt[0] = Tm_[0]    for i in range(1, len(Tm_)):        tt[i] = (Tm_[i] - (tt[i-1] + Tm_[i])**(-tau_[i]))    return tt[1:]

Pure C without using Python at all. Version from year 2016 (with fabs() function):

#include <stdio.h>#include <math.h>#include <stdlib.h>#include <windows.h>#include <sys\timeb.h> double randn() {    double u = rand();    if (u > 0.5) {        return sqrt(-1.57079632679*log(1.0 - pow(2.0 * u - 1, 2)));    }    else {        return -sqrt(-1.57079632679*log(1.0 - pow(1 - 2.0 * u,2)));    }}void rec_pure_c(double *Tm, double *tau, int alen, double *T){    for (int i = 1; i < alen; i++)    {        T[i] = Tm[i] + pow(fabs(T[i - 1] - Tm[i]), (-tau[i]));    }}int main() {    int N = 100000;    double *Tm= calloc(N, sizeof *Tm);    double *tau = calloc(N, sizeof *tau);    double *T = calloc(N, sizeof *T);    double time = 0;    double sumtime = 0;    for (int i = 0; i < N; i++)    {        Tm[i] = randn();        tau[i] = randn();    }    LARGE_INTEGER StartingTime, EndingTime, ElapsedMicroseconds;    LARGE_INTEGER Frequency;    for (int j = 0; j < 1000; j++)    {        for (int i = 0; i < 3; i++)        {            QueryPerformanceFrequency(&Frequency);            QueryPerformanceCounter(&StartingTime);            rec_pure_c(Tm, tau, N, T);            QueryPerformanceCounter(&EndingTime);            ElapsedMicroseconds.QuadPart = EndingTime.QuadPart - StartingTime.QuadPart;            ElapsedMicroseconds.QuadPart *= 1000000;            ElapsedMicroseconds.QuadPart /= Frequency.QuadPart;            if (i == 0)                time = (double)ElapsedMicroseconds.QuadPart / 1000;            else {                if (time > (double)ElapsedMicroseconds.QuadPart / 1000)                    time = (double)ElapsedMicroseconds.QuadPart / 1000;            }        }        sumtime += time;    }    printf("1000 loops,best of 3: %.3f ms per loop\n",sumtime/1000);    free(Tm);    free(tau);    free(T);}

Fortran f2py. Function can be used from Python. Version from year 2016 (with abs() function):

subroutine rec_fortran(tm,tau,alen,result)    integer*8, intent(in) :: alen    real*8, dimension(alen), intent(in) :: tm    real*8, dimension(alen), intent(in) :: tau    real*8, dimension(alen) :: res    real*8, dimension(alen), intent(out) :: result    res(1)=0    do i=2,alen        res(i) = tm(i) + (abs(res(i-1) - tm(i)))**(-tau(i))    end do    result=res    end subroutine rec_fortran

Update: 21-10-2018I have corrected my answer based on comments.

It is possible to vectorize operations on vectors as long as the calculation is not recursive. Because a recursive operation depends on the previous calculated value it is not possible to parallel process the operation.This does therefore not work:

def calc_vect(Tm_, tau_):    return Tm_[1:] - (Tm_[:-1] + Tm_[1:]) ** (-tau_[1:])

Since (serial processing / a loop) is necessary, the best performance is gained by moving as close as possible to optimized machine code, therefore Numba and Cython are the best answers here.

A Numba approach can be achieves as follows:

init_string = """from math import powimport numpy as npfrom numba import jit, float32np.random.seed(0)n = 100000Tm = np.cumsum(np.random.uniform(0.1, 1, size=n).astype('float32'))tau = np.random.uniform(-1, 0, size=n).astype('float32')def calc_python(Tm_, tau_): tt = np.empty(len(Tm_)) tt[0] = Tm_[0] for i in range(1, len(Tm_)):     tt[i] = Tm_[i] - pow(tt[i-1] + Tm_[i], -tau_[i]) return tt@jit(float32[:](float32[:], float32[:]), nopython=False, nogil=True)def calc_numba(Tm_, tau_):  tt = np.empty(len(Tm_))  tt[0] = Tm_[0]  for i in range(1, len(Tm_)):      tt[i] = Tm_[i] - pow(tt[i-1] + Tm_[i], -tau_[i])  return tt"""import timeitpy_time = timeit.timeit('calc_python(Tm, tau)', init_string, number=100)numba_time = timeit.timeit('calc_numba(Tm, tau)', init_string, number=100)print("Python Solution: {}".format(py_time))print("Numba Soltution: {}".format(numba_time))

Timeit comparison of the Python and Numba functions:

Python Solution: 54.58057559299999Numba Soltution: 1.1389029540000024