# Baby Steps - Algebra¶

## Adding two Scalars¶

To get us started with Theano and get a feel of what we’re working with, let’s make a simple function: add two numbers together. Here is how you do it:

```>>> import numpy
>>> import theano.tensor as T
>>> from theano import function
>>> x = T.dscalar('x')
>>> y = T.dscalar('y')
>>> z = x + y
>>> f = function([x, y], z)
```

And now that we’ve created our function we can use it:

```>>> f(2, 3)
array(5.0)
>>> numpy.allclose(f(16.3, 12.1), 28.4)
True
```

Let’s break this down into several steps. The first step is to define two symbols (Variables) representing the quantities that you want to add. Note that from now on, we will use the term Variable to mean “symbol” (in other words, x, y, z are all Variable objects). The output of the function f is a `numpy.ndarray` with zero dimensions.

If you are following along and typing into an interpreter, you may have noticed that there was a slight delay in executing the `function` instruction. Behind the scene, f was being compiled into C code.

Step 1

```>>> x = T.dscalar('x')
>>> y = T.dscalar('y')
```

In Theano, all symbols must be typed. In particular, `T.dscalar` is the type we assign to “0-dimensional arrays (scalar) of doubles (d)”. It is a Theano Type.

`dscalar` is not a class. Therefore, neither x nor y are actually instances of `dscalar`. They are instances of `TensorVariable`. x and y are, however, assigned the theano Type `dscalar` in their `type` field, as you can see here:

```>>> type(x)
<class 'theano.tensor.var.TensorVariable'>
>>> x.type
TensorType(float64, scalar)
>>> T.dscalar
TensorType(float64, scalar)
>>> x.type is T.dscalar
True
```

By calling `T.dscalar` with a string argument, you create a Variable representing a floating-point scalar quantity with the given name. If you provide no argument, the symbol will be unnamed. Names are not required, but they can help debugging.

More will be said in a moment regarding Theano’s inner structure. You could also learn more by looking into Graph Structures.

Step 2

The second step is to combine x and y into their sum z:

```>>> z = x + y
```

z is yet another Variable which represents the addition of x and y. You can use the pp function to pretty-print out the computation associated to z.

```>>> from theano import pp
>>> print(pp(z))
(x + y)
```

Step 3

The last step is to create a function taking x and y as inputs and giving z as output:

```>>> f = function([x, y], z)
```

The first argument to `function` is a list of Variables that will be provided as inputs to the function. The second argument is a single Variable or a list of Variables. For either case, the second argument is what we want to see as output when we apply the function. f may then be used like a normal Python function.

Note

As a shortcut, you can skip step 3, and just use a variable’s `eval` method. The `eval()` method is not as flexible as `function()` but it can do everything we’ve covered in the tutorial so far. It has the added benefit of not requiring you to import `function()` . Here is how `eval()` works:

```>>> import numpy
>>> import theano.tensor as T
>>> x = T.dscalar('x')
>>> y = T.dscalar('y')
>>> z = x + y
>>> numpy.allclose(z.eval({x : 16.3, y : 12.1}), 28.4)
True
```

We passed `eval()` a dictionary mapping symbolic theano variables to the values to substitute for them, and it returned the numerical value of the expression.

`eval()` will be slow the first time you call it on a variable – it needs to call `function()` to compile the expression behind the scenes. Subsequent calls to `eval()` on that same variable will be fast, because the variable caches the compiled function.

## Adding two Matrices¶

You might already have guessed how to do this. Indeed, the only change from the previous example is that you need to instantiate x and y using the matrix Types:

```>>> x = T.dmatrix('x')
>>> y = T.dmatrix('y')
>>> z = x + y
>>> f = function([x, y], z)
```

`dmatrix` is the Type for matrices of doubles. Then we can use our new function on 2D arrays:

```>>> f([[1, 2], [3, 4]], [[10, 20], [30, 40]])
array([[ 11.,  22.],
[ 33.,  44.]])
```

The variable is a NumPy array. We can also use NumPy arrays directly as inputs:

```>>> import numpy
>>> f(numpy.array([[1, 2], [3, 4]]), numpy.array([[10, 20], [30, 40]]))
array([[ 11.,  22.],
[ 33.,  44.]])
```

It is possible to add scalars to matrices, vectors to matrices, scalars to vectors, etc. The behavior of these operations is defined by broadcasting.

The following types are available:

• byte: `bscalar, bvector, bmatrix, brow, bcol, btensor3, btensor4, btensor5, btensor6, btensor7`

• 16-bit integers: `wscalar, wvector, wmatrix, wrow, wcol, wtensor3, wtensor4, wtensor5, wtensor6, wtensor7`

• 32-bit integers: `iscalar, ivector, imatrix, irow, icol, itensor3, itensor4, itensor5, itensor6, itensor7`

• 64-bit integers: `lscalar, lvector, lmatrix, lrow, lcol, ltensor3, ltensor4, ltensor5, ltensor6, ltensor7`

• float: `fscalar, fvector, fmatrix, frow, fcol, ftensor3, ftensor4, ftensor5, ftensor6, ftensor7`

• double: `dscalar, dvector, dmatrix, drow, dcol, dtensor3, dtensor4, dtensor5, dtensor6, dtensor7`

• complex: `cscalar, cvector, cmatrix, crow, ccol, ctensor3, ctensor4, ctensor5, ctensor6, ctensor7`

The previous list is not exhaustive and a guide to all types compatible with NumPy arrays may be found here: tensor creation.

Note

You, the user—not the system architecture—have to choose whether your program will use 32- or 64-bit integers (`i` prefix vs. the `l` prefix) and floats (`f` prefix vs. the `d` prefix).

## Exercise¶

```import theano
a = theano.tensor.vector() # declare variable
out = a + a ** 10               # build symbolic expression
f = theano.function([a], out)   # compile function
print(f([0, 1, 2]))
```
```[    0.     2.  1026.]
```

Modify and execute this code to compute this expression: a ** 2 + b ** 2 + 2 * a * b.

`Solution`