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- See the set up page to install Jupyter, Python and all necessary libraries
- Please direct feedback to contact@quantecon.org or the discourse forum

# Numba¶

In addition to what’s in Anaconda, this lecture will need the following libraries

```
!pip install quantecon
```

## Overview¶

In our lecture on NumPy we learned one method to improve speed and efficiency in numerical work

That method, called *vectorization*, involved sending array processing operations in batch to efficient low level code

This clever idea dates back to Matlab, which uses it extensively

Unfortunately, vectorization is limited and has several weaknesses

One weakness is that it is highly memory intensive

Another problem is that only some algorithms can be vectorized

In the last few years, a new Python library called Numba has appeared that solves many of these problems

It does so through something called **just in time (JIT) compilation**

JIT compilation is effective in many numerical settings and can generate extremely fast, efficient code

It can also do other tricks such as facilitate multithreading (a form of parallelization well suited to numerical work)

### The Need for Speed¶

To understand what Numba does and why, we need some background knowledge

Let’s start by thinking about higher level languages, such as Python

These languages are optimized for humans

This means that the programmer can leave many details to the runtime environment

- specifying variable types
- memory allocation/deallocation, etc.

The upside is that, compared to low-level languages, Python is typically faster to write, less error prone and easier to debug

The downside is that Python is harder to optimize — that is, turn into fast machine code — than languages like C or Fortran

Indeed, the standard implementation of Python (called CPython) cannot match the speed of compiled languages such as C or Fortran

Does that mean that we should just switch to C or Fortran for everything?

The answer is no, no and one hundred times no

High productivity languages should be chosen over high speed languages for the great majority of scientific computing tasks

This is because

- Of any given program, relatively few lines are ever going to be time-critical
- For those lines of code that
*are*time-critical, we can achieve C-like speed using a combination of NumPy and Numba

This lecture provides a guide

## Where are the Bottlenecks?¶

Let’s start by trying to understand why high level languages like Python are slower than compiled code

### Dynamic Typing¶

```
a, b = 10, 10
a + b
```

Even for this simple operation, the Python interpreter has a fair bit of work to do

For example, in the statement `a + b`

, the interpreter has to know which
operation to invoke

If `a`

and `b`

are strings, then `a + b`

requires string concatenation

```
a, b = 'foo', 'bar'
a + b
```

If `a`

and `b`

are lists, then `a + b`

requires list concatenation

```
a, b = ['foo'], ['bar']
a + b
```

(We say that the operator `+`

is *overloaded* — its action depends on the
type of the objects on which it acts)

As a result, Python must check the type of the objects and then call the correct operation

This involves substantial overheads

#### Static Types¶

Compiled languages avoid these overheads with explicit, static types

For example, consider the following C code, which sums the integers from 1 to 10

```
#include <stdio.h>
int main(void) {
int i;
int sum = 0;
for (i = 1; i <= 10; i++) {
sum = sum + i;
}
printf("sum = %d\n", sum);
return 0;
}
```

The variables `i`

and `sum`

are explicitly declared to be integers

Hence, the meaning of addition here is completely unambiguous

### Data Access¶

Another drag on speed for high level languages is data access

To illustrate, let’s consider the problem of summing some data — say, a collection of integers

#### Summing with Compiled Code¶

In C or Fortran, these integers would typically be stored in an array, which is a simple data structure for storing homogeneous data

Such an array is stored in a single contiguous block of memory

- In modern computers, memory addresses are allocated to each byte (one byte = 8 bits)
- For example, a 64 bit integer is stored in 8 bytes of memory
- An array of $ n $ such integers occupies $ 8n $
**consecutive**memory slots

Moreover, the compiler is made aware of the data type by the programmer

- In this case 64 bit integers

Hence, each successive data point can be accessed by shifting forward in memory space by a known and fixed amount

- In this case 8 bytes

#### Summing in Pure Python¶

Python tries to replicate these ideas to some degree

For example, in the standard Python implementation (CPython), list elements are placed in memory locations that are in a sense contiguous

However, these list elements are more like pointers to data rather than actual data

Hence, there is still overhead involved in accessing the data values themselves

This is a considerable drag on speed

In fact, it’s generally true that memory traffic is a major culprit when it comes to slow execution

Let’s look at some ways around these problems

## Vectorization¶

Vectorization is about sending batches of related operations to native machine code

- The machine code itself is typically compiled from carefully optimized C or Fortran

This can greatly accelerate many (but not all) numerical computations

### Operations on Arrays¶

```
import random
import numpy as np
import quantecon as qe
```

Now let’s try this non-vectorized code

```
qe.util.tic() # Start timing
n = 100_000
sum = 0
for i in range(n):
x = random.uniform(0, 1)
sum += x**2
qe.util.toc() # End timing
```

Now compare this vectorized code

```
qe.util.tic()
n = 100_000
x = np.random.uniform(0, 1, n)
np.sum(x**2)
qe.util.toc()
```

The second code block — which achieves the same thing as the first — runs much faster

The reason is that in the second implementation we have broken the loop down into three basic operations

- draw
`n`

uniforms - square them
- sum them

These are sent as batch operators to optimized machine code

Apart from minor overheads associated with sending data back and forth, the result is C or Fortran-like speed

When we run batch operations on arrays like this, we say that the code is *vectorized*

Vectorized code is typically fast and efficient

It is also surprisingly flexible, in the sense that many operations can be vectorized

The next section illustrates this point

```
np.cos(1.0)
```

```
np.cos(np.linspace(0, 1, 3))
```

By exploiting ufuncs, many operations can be vectorized

For example, consider the problem of maximizing a function $ f $ of two variables $ (x,y) $ over the square $ [-a, a] \times [-a, a] $

For $ f $ and $ a $ let’s choose

$$ f(x,y) = \frac{\cos(x^2 + y^2)}{1 + x^2 + y^2} \quad \text{and} \quad a = 3 $$Here’s a plot of $ f $

```
import matplotlib.pyplot as plt
%matplotlib inline
from mpl_toolkits.mplot3d.axes3d import Axes3D
from matplotlib import cm
def f(x, y):
return np.cos(x**2 + y**2) / (1 + x**2 + y**2)
xgrid = np.linspace(-3, 3, 50)
ygrid = xgrid
x, y = np.meshgrid(xgrid, ygrid)
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(x,
y,
f(x, y),
rstride=2, cstride=2,
cmap=cm.jet,
alpha=0.7,
linewidth=0.25)
ax.set_zlim(-0.5, 1.0)
plt.show()
```

To maximize it, we’re going to use a naive grid search:

- Evaluate $ f $ for all $ (x,y) $ in a grid on the square
- Return the maximum of observed values

Here’s a non-vectorized version that uses Python loops

```
def f(x, y):
return np.cos(x**2 + y**2) / (1 + x**2 + y**2)
grid = np.linspace(-3, 3, 1000)
m = -np.inf
qe.tic()
for x in grid:
for y in grid:
z = f(x, y)
if z > m:
m = z
qe.toc()
```

And here’s a vectorized version

```
def f(x, y):
return np.cos(x**2 + y**2) / (1 + x**2 + y**2)
grid = np.linspace(-3, 3, 1000)
x, y = np.meshgrid(grid, grid)
qe.tic()
np.max(f(x, y))
qe.toc()
```

### Pros and Cons of Vectorization¶

At its best, vectorization yields fast, simple code

However, it’s not without disadvantages

One issue is that it can be highly memory intensive

For example, the vectorized maximization routine above is far more memory intensive than the non-vectorized version that preceded it

Another issue is that not all algorithms can be vectorized

In these kinds of settings, we need to go back to loops

Fortunately, there are nice ways to speed up Python loops

## Numba¶

One exciting development in this direction is Numba

Numba aims to automatically compile functions to native machine code instructions on the fly

The process isn’t flawless, since Numba needs to infer type information on all variables to generate pure machine instructions

Such inference isn’t possible in every setting

But for simple routines Numba infers types very well

Moreover, the “hot loops” at the heart of our code that we need to speed up are often such simple routines

### Prerequisites¶

If you followed our set up instructions, then Numba should be installed

Make sure you have the latest version of Anaconda by running `conda update anaconda`

from a terminal (Mac, Linux) / Anaconda command prompt (Windows)

### An Example¶

Let’s consider some problems that are difficult to vectorize

One is generating the trajectory of a difference equation given an initial condition

Let’s take the difference equation to be the quadratic map

$$ x_{t+1} = 4 x_t (1 - x_t) $$Here’s the plot of a typical trajectory, starting from $ x_0 = 0.1 $, with $ t $ on the x-axis

```
def qm(x0, n):
x = np.empty(n+1)
x[0] = x0
for t in range(n):
x[t+1] = 4 * x[t] * (1 - x[t])
return x
x = qm(0.1, 250)
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(x, 'b-', lw=2, alpha=0.8)
ax.set_xlabel('time', fontsize=16)
plt.show()
```

To speed this up using Numba is trivial using Numba’s `jit`

function

```
from numba import jit
qm_numba = jit(qm) # qm_numba is now a 'compiled' version of qm
```

Let’s time and compare identical function calls across these two versions:

```
qe.util.tic()
qm(0.1, int(10**5))
time1 = qe.util.toc()
```

```
qe.util.tic()
qm_numba(0.1, int(10**5))
time2 = qe.util.toc()
```

```
qe.util.tic()
qm_numba(0.1, int(10**5))
time2 = qe.util.toc()
```

```
time1 / time2 # Calculate speed gain
```

That’s a speed increase of two orders of magnitude!

Your mileage will of course vary depending on hardware and so on

Nonetheless, two orders of magnitude is huge relative to how simple and clear the implementation is

#### Decorator Notation¶

If you don’t need a separate name for the “numbafied” version of `qm`

,
you can just put `@jit`

before the function

```
@jit
def qm(x0, n):
x = np.empty(n+1)
x[0] = x0
for t in range(n):
x[t+1] = 4 * x[t] * (1 - x[t])
return x
```

This is equivalent to `qm = jit(qm)`

### How and When it Works¶

Numba attempts to generate fast machine code using the infrastructure provided by the LLVM Project

It does this by inferring type information on the fly

As you can imagine, this is easier for simple Python objects (simple scalar data types, such as floats, integers, etc.)

Numba also plays well with NumPy arrays, which it treats as typed memory regions

In an ideal setting, Numba can infer all necessary type information

This allows it to generate native machine code, without having to call the Python runtime environment

In such a setting, Numba will be on par with machine code from low level languages

When Numba cannot infer all type information, some Python objects are given generic `object`

status, and some code is generated using the Python runtime

In this second setting, Numba typically provides only minor speed gains — or none at all

Hence, it’s prudent when using Numba to focus on speeding up small, time-critical snippets of code

This will give you much better performance than blanketing your Python programs with `@jit`

statements

#### A Gotcha: Global Variables¶

Consider the following example

```
a = 1
@jit
def add_x(x):
return a + x
print(add_x(10))
```

```
a = 2
print(add_x(10))
```

Notice that changing the global had no effect on the value returned by the function

When Numba compiles machine code for functions, it treats global variables as constants to ensure type stability

### Numba for vectorization¶

Numba can also be used to create custom ufuncs with the @vectorize decorator

To illustrate the advantage of using Numba to vectorize a function, we return to a maximization problem discussed above

```
from numba import vectorize
@vectorize
def f_vec(x, y):
return np.cos(x**2 + y**2) / (1 + x**2 + y**2)
grid = np.linspace(-3, 3, 1000)
x, y = np.meshgrid(grid, grid)
np.max(f_vec(x, y)) # Run once to compile
qe.tic()
np.max(f_vec(x, y))
qe.toc()
```

This is faster than our vectorized version using NumPy’s ufuncs

Why should that be? After all, anything vectorized with NumPy will be running in fast C or Fortran code

The reason is that it’s much less memory intensive

For example, when NumPy computes `np.cos(x**2 + y**2)`

it first creates the
intermediate arrays `x**2`

and `y**2`

, then it creates the array `np.cos(x**2 + y**2)`

In our `@vectorize`

version using Numba, the entire operator is reduced to a
single vectorized process and none of these intermediate arrays are created

We can gain further speed improvements using Numba’s automatic parallelization
feature by specifying `target='parallel'`

In this case, we need to specify the types of our inputs and outputs

```
@vectorize('float64(float64, float64)', target='parallel')
def f_vec(x, y):
return np.cos(x**2 + y**2) / (1 + x**2 + y**2)
np.max(f_vec(x, y)) # Run once to compile
qe.tic()
np.max(f_vec(x, y))
qe.toc()
```

This is a striking speed up with very little effort