# scipy.optimize¶

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.

## bisect¶

`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
```

### Performance¶

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
```

## fmin¶

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
```

## newton¶

`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
```