NumPy

this notebook is based on the SciPy NumPy tutorial

NumPy provides a multidimensional array class as well as a large number of functions that operate on arrays.

NumPy arrays allow you to write fast (optimized) code that works on arrays of data. To do this, there are some restrictions on arrays:

• all elements are of the same data type (e.g. float)
• the size of the array is fixed in memory, and specified when you create the array (e.g., you cannot grow the array like you do with lists)

The nice part is that arithmetic operations work on entire arrays—this means that you can avoid writing loops in python (which tend to be slow). Instead the "looping" is done in the underlying compiled code

Array Creation and Properties

There are a lot of ways to create arrays. Let's look at a few

Here we create an array using arange and then change its shape to be 3 rows and 5 columns. Note the row-major ordering—you'll see that the rows are together in the inner [] (more on this in a bit)

an array is an object of the ndarray class, and it has methods that know how to work on the array data. Here, .reshape() is one of the many methods.

A NumPy array has a lot of meta-data associated with it describing its shape, datatype, etc.

we can also create an array from a list

we can create a multi-dimensional array of a specified size initialized all to 0 easily, using the zeros() function.

There is also an analogous ones() and empty() array routine. Note that here we can explicitly set the datatype for the array in this function if we wish.

Unlike lists in python, all of the elements of a numpy array are of the same datatype

linspace creates an array with evenly space numbers. The endpoint optional argument specifies whether the upper range is in the array or not

Analogous to linspace(), there is a logspace() function that creates an array with elements equally spaced in log. Use help(np.logspace) to see the arguments, and create an array with 10 elements from $10^{-6}$ to $10^3$.

we can also initialize an array based on a function

Array Operations

most operations (+, -, *, /) will work on an entire array at once, element-by-element.

Note that that the multiplication operator is not a matrix multiply (there is a new operator in python 3.5+, @, to do matrix multiplicaiton.

Let's create a simply array to start with

Multiplication by a scalar multiplies every element

adding two arrays adds element-by-element

multiplying two arrays multiplies element-by-element

We can think of our 2-d array as a 3 x 5 matrix (3 rows, 5 columns). We can take the transpose to geta 5 x 3 matrix, and then we can do a matrix multiplication

What do you think 1./a will do? Try it and see

We can sum the elements of an array:

sum() takes an optional argument, axis=N, where N is the axis to sum over. Sum the elements of a across rows to create an array with just the sum along each column of a.

We can also easily get the extrema

universal functions

universal functions work element-by-element. Let's create a new array scaled by pi

We will often want to write our own function that operates on an array and returns a new array. We can do this just like we did with functions previously—the key is to use the methods from the np module to do any operations, since they work on, and return, arrays.

Write a simple function that returns $\sin(2\pi x)$ for an input array x. Then test it by passing in an array x that you create via linspace()

Slicing

slicing works very similarly to how we saw with strings. Remember, python uses 0-based indexing

Let's create this array from the image:

Now look at accessing a single element vs. a range (using slicing)

Giving a single (0-based) index just references a single value

Giving a range uses the range of the edges to return the values

The : can be used to specify all of the elements in that dimension

Multidimensional Arrays

Multidimensional arrays are stored in a contiguous space in memory -- this means that the columns / rows need to be unraveled (flattened) so that it can be thought of as a single one-dimensional array. Different programming languages do this via different conventions:

Storage order:

• Python/C use row-major storage: rows are stored one after the other
• Fortran/matlab use column-major storage: columns are stored one after another

The ordering matters when

• passing arrays between languages (we'll talk about this later this semester)

• looping over arrays -- you want to access elements that are next to one-another in memory

  • e.g, in Fortran:

    double precision :: A(M,N)
    do j = 1, N
     do i = 1, M
        A(i,j) = …
     enddo
    enddo

  • in C

    double A[M][N];
    for (i = 0; i < M; i++) {
     for (j = 0; j < N; j++) {
        A[i][j] = …
     }
    }

In python, using NumPy, we'll try to avoid explicit loops over elements as much as possible

Let's look at multidimensional arrays:

Notice that the output of a shows the row-major storage. The rows are grouped together in the inner [...]

Giving a single index (0-based) for each dimension just references a single value in the array

Doing slices will access a range of elements. Think of the start and stop in the slice as referencing the left-edge of the slots in the array.

Access a specific column

Sometimes we want a one-dimensional view into the array -- here we see the memory layout (row-major) more explicitly

we can also iterate -- this is done over the first axis (rows)

or element by element

Generally speaking, we want to avoid looping over the elements of an array in python—this is slow. Instead we want to write and use functions that operate on the entire array at once.

Consider the array defined as:

• using slice notation, create an array that consists of only the 4's in q (this will be called a view, as we'll see shortly)
• zero out all of the elements in your view
• how does q change?

  ---

Copying Arrays

simply using "=" does not make a copy, but much like with lists, you will just have multiple names pointing to the same ndarray object

Therefore, we need to understand if two arrays, A and B point to:

• the same array, including shape and data/memory space
• the same data/memory space, but perhaps different shapes (a view)
• a separate cpy of the data (i.e. stored completely separately in memory)

All of these are possible:

• B = A

  this is assignment. No copy is made. A and B point to the same data in memory and share the same shape, etc. They are just two different labels for the same object in memory

• B = A[:]

  this is a view or shallow copy. The shape info for A and B are stored independently, but both point to the same memory location for data

• B = A.copy()

  this is a deep copy. A completely separate object will be created in memory, with a completely separate location in memory.

Let's look at examples

Here is assignment—we can just use the is operator to test for equality

Since b and a are the same, changes to the shape of one are reflected in the other—no copy is made.

a shallow copy creates a new view into the array—the data is the same, but the array properties can be different

since the underlying data is the same memory, changing an element of one is reflected in the other

Even slices into an array are just views, still pointing to the same memory

There are lots of ways to inquire if two arrays are the same, views, own their own data, etc

to make a copy of the data of the array that you can deal with independently of the original, you need a deep copy

Boolean Indexing

There are lots of fun ways to index arrays to access only those elements that meet a certain condition

Here we set all the elements in the array that are > 4 to zero

and now, all the zeros to -1

if we have 2 tests, we need to use logical_and() or logical_or()

Our test that we index the array with returns a boolean array of the same shape:

Avoiding Loops

In general, you want to avoid loops over elements on an array.

Here, let's create 1-d x and y coordinates and then try to fill some larger array

we'll time out code

Now let's instead do this using all array syntax. First will extend our 1-d coordinate arrays to be 2-d. NumPy has a function for this (meshgrid())

Numerical differencing on NumPy arrays

Now we want to construct a derivative, $$ \frac{d f}{dx} $$

We want to do this without loops—we'll use views into arrays offset from one another. Recall from calculus that a derivative is approximately: $$ \frac{df}{dx} = \frac{f(x+h) - f(x)}{h} $$ Here, we'll take $h$ to be a single adjacent element