Functions in the optimize
module can be called by prepending them by scipy.optimize.
. The module defines the following three functions:
Note that routines that work with user-defined functions still have to call the underlying python
code, and therefore, gains in speed are not as significant as with other vectorised operations. As a rule of thumb, a factor of two can be expected, when compared to an optimised python
implementation.
scipy
: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.bisect.html
bisect
finds the root of a function of one variable using a simple bisection routine. It takes three positional arguments, the function itself, and two starting points. The function must have opposite signs at the starting points. Returned is the position of the root.
Two keyword arguments, xtol
, and maxiter
can be supplied to control the accuracy, and the number of bisections, respectively.
# code to be run in micropython
from ulab import scipy as spy
def f(x):
return x*x - 1
print(spy.optimize.bisect(f, 0, 4))
print('only 8 bisections: ', spy.optimize.bisect(f, 0, 4, maxiter=8))
print('with 0.1 accuracy: ', spy.optimize.bisect(f, 0, 4, xtol=0.1))
0.9999997615814209 only 8 bisections: 0.984375 with 0.1 accuracy: 0.9375
Since the bisect
routine calls user-defined python
functions, the speed gain is only about a factor of two, if compared to a purely python
implementation.
# code to be run in micropython
from ulab import scipy as spy
def f(x):
return (x-1)*(x-1) - 2.0
def bisect(f, a, b, xtol=2.4e-7, maxiter=100):
if f(a) * f(b) > 0:
raise ValueError
rtb = a if f(a) < 0.0 else b
dx = b - a if f(a) < 0.0 else a - b
for i in range(maxiter):
dx *= 0.5
x_mid = rtb + dx
mid_value = f(x_mid)
if mid_value < 0:
rtb = x_mid
if abs(dx) < xtol:
break
return rtb
@timeit
def bisect_scipy(f, a, b):
return spy.optimize.bisect(f, a, b)
@timeit
def bisect_timed(f, a, b):
return bisect(f, a, b)
print('bisect running in python')
bisect_timed(f, 3, 2)
print('bisect running in C')
bisect_scipy(f, 3, 2)
bisect running in python execution time: 1270 us bisect running in C execution time: 642 us
scipy
: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin.html
The fmin
function finds the position of the minimum of a user-defined function by using the downhill simplex method. Requires two positional arguments, the function, and the initial value. Three keyword arguments, xatol
, fatol
, and maxiter
stipulate conditions for stopping.
# code to be run in micropython
from ulab import scipy as spy
def f(x):
return (x-1)**2 - 1
print(spy.optimize.fmin(f, 3.0))
print(spy.optimize.fmin(f, 3.0, xatol=0.1))
0.9996093749999952 1.199999999999996
scipy
:https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.newton.html
newton
finds a zero of a real, user-defined function using the Newton-Raphson (or secant or Halley’s) method. The routine requires two positional arguments, the function, and the initial value. Three keyword arguments can be supplied to control the iteration. These are the absolute and relative tolerances tol
, and rtol
, respectively, and the number of iterations before stopping, maxiter
. The function retuns a single scalar, the position of the root.
# code to be run in micropython
from ulab import scipy as spy
def f(x):
return x*x*x - 2.0
print(spy.optimize.newton(f, 3., tol=0.001, rtol=0.01))
1.260135727246117