fixes to fib - changed from (1, 1, 2, ..) to (0, 1, 1, 2, ...)

d874e90c · Paul McCarthy · 6866995e · d874e90c · d874e90c
Commit d874e90c authored 7 years ago by Paul McCarthy
--- a/advanced_topics/decorators.ipynb
+++ b/advanced_topics/decorators.ipynb
@@ -383,11 +383,11 @@
    "\n",
    "    if n in (0, 1):\n",
    "        print('fib({}) = {}'.format(n, n))\n",
-    "        return 1\n",
+    "        return n\n",
    "\n",
    "    twoback = 1\n",
    "    oneback = 1\n",
-    "    val     = 0\n",
+    "    val     = 1\n",
    "\n",
    "    for _ in range(2, n):\n",
    "\n",

 %% Cell type:markdown id: tags:

 # Decorators


 Remember that in Python, everything is an object, including functions. This
 means that we can do things like:


 - Pass a function as an argument to another function.
 - Create/define a function inside another function.
 - Write a function which returns another function.


 These abilities mean that we can do some neat things with functions in Python.


 * [Overview](#overview)
 * [Decorators on methods](#decorators-on-methods)
 * [Example - memoization](#example-memoization)
 * [Decorators with arguments](#decorators-with-arguments)
 * [Chaining decorators](#chaining-decorators)
 * [Decorator classes](#decorator-classes)
 * [Appendix: Functions are not special](#appendix-functions-are-not-special)
 * [Appendix: Closures](#appendix-closures)
 * [Appendix: Decorators without arguments versus decorators with arguments](#appendix-decorators-without-arguments-versus-decorators-with-arguments)
 * [Appendix: Per-instance decorators](#appendix-per-instance-decorators)
 * [Appendix: Preserving function metadata](#appendix-preserving-function-metadata)
 * [Appendix: Class decorators](#appendix-class-decorators)
 * [Useful references](#useful-references)


 <a class="anchor" id="overview"></a>
 ## Overview


 Let's say that we want a way to calculate the execution time of any function
 (this example might feel familiar to you if you have gone through the
 practical on operator overloading).


 Our first attempt at writing such a function might look like this:

 %% Cell type:code id: tags:

 ``` 
 import time
 def timeFunc(func, *args, **kwargs):

    start  = time.time()
    retval = func(*args, **kwargs)
    end    = time.time()

    print('Ran {} in {:0.2f} seconds'.format(func.__name__, end - start))

    return retval
 ```

 %% Cell type:markdown id: tags:

 The `timeFunc` function accepts another function, `func`, as its first
 argument. It calls `func`, passing it all of the other arguments, and then
 prints the time taken for `func` to complete:

 %% Cell type:code id: tags:

 ``` 
 import numpy        as np
 import numpy.linalg as npla

 def inverse(a):
    return npla.inv(a)

 data    = np.random.random((2000, 2000))
 invdata = timeFunc(inverse, data)
 ```

 %% Cell type:markdown id: tags:

 But this means that whenever we want to time something, we have to call the
 `timeFunc` function directly. Let's take advantage of the fact that we can
 define a function inside another funciton. Look at the next block of code
 carefully, and make sure you understand what our new `timeFunc` implementation
 is doing.

 %% Cell type:code id: tags:

 ``` 
 import time
 def timeFunc(func):

    def wrapperFunc(*args, **kwargs):

        start  = time.time()
        retval = func(*args, **kwargs)
        end    = time.time()

        print('Ran {} in {:0.2f} seconds'.format(func.__name__, end - start))

        return retval

    return wrapperFunc
 ```

 %% Cell type:markdown id: tags:

 This new `timeFunc` function is again passed a function `func`, but this time
 as its sole argument. It then creates and returns a new function,
 `wrapperFunc`. This `wrapperFunc` function calls and times the function that
 was passed to `timeFunc`.  But note that when `timeFunc` is called,
 `wrapperFunc` is _not_ called - it is only created and returned.


 Let's use our new `timeFunc` implementation:

 %% Cell type:code id: tags:

 ``` 
 import numpy        as np
 import numpy.linalg as npla

 def inverse(a):
    return npla.inv(a)

 data    = np.random.random((2000, 2000))
 inverse = timeFunc(inverse)
 invdata = inverse(data)
 ```

 %% Cell type:markdown id: tags:

 Here, we did the following:


 1. We defined a function called `inverse`:

  > ```
  > def inverse(a):
  >     return npla.inv(a)
  > ```

 2. We passed the `inverse` function to the `timeFunc` function, and
   re-assigned the return value of `timeFunc` back to `inverse`:

  > ```
  > inverse = timeFunc(inverse)
  > ```

 3. We called the new `inverse` function:

  > ```
  > invdata = inverse(data)
  > ```


 So now the `inverse` variable refers to an instantiation of `wrapperFunc`,
 which holds a reference to the original definition of `inverse`.


 > If this is not clear, take a break now and read through the appendix on how
 > [functions are not special](#appendix-functions-are-not-special).


 Guess what? We have just created a __decorator__. A decorator is simply a
 function which accepts a function as its input, and returns another function
 as its output. In the example above, we have _decorated_ the `inverse`
 function with the `timeFunc` decorator.


 Python provides an alternative syntax for decorating one function with
 another, using the `@` character. The approach that we used to decorate
 `inverse` above:

 %% Cell type:code id: tags:

 ``` 
 def inverse(a):
    return npla.inv(a)

 inverse = timeFunc(inverse)
 invdata = inverse(data)
 ```

 %% Cell type:markdown id: tags:

 is semantically equivalent to this:

 %% Cell type:code id: tags:

 ``` 
 @timeFunc
 def inverse(a):
    return npla.inv(a)

 invdata = inverse(data)
 ```

 %% Cell type:markdown id: tags:

 <a class="anchor" id="decorators-on-methods"></a>
 ## Decorators on methods


 Applying a decorator to the methods of a class works in the same way:

 %% Cell type:code id: tags:

 ``` 
 import numpy.linalg as npla

 class MiscMaths(object):

    @timeFunc
    def inverse(self, a):
        return npla.inv(a)
 ```

 %% Cell type:markdown id: tags:

 Now, the `inverse` method of all `MiscMaths` instances will be timed:

 %% Cell type:code id: tags:

 ``` 
 mm1 = MiscMaths()
 mm2 = MiscMaths()

 i1 = mm1.inverse(np.random.random((1000, 1000)))
 i2 = mm2.inverse(np.random.random((1500, 1500)))
 ```

 %% Cell type:markdown id: tags:

 Note that only one `timeFunc` decorator was created here - the `timeFunc`
 function was only called once - when the `MiscMaths` class was defined.  This
 might be clearer if we re-write the above code in the following (equivalent)
 manner:

 %% Cell type:code id: tags:

 ``` 
 class MiscMaths(object):
    def inverse(self, a):
        return npla.inv(a)

 MiscMaths.inverse = timeFunc(MiscMaths.inverse)
 ```

 %% Cell type:markdown id: tags:

 So only one `wrapperFunc` function exists, and this function is _shared_ by
 all instances of the `MiscMaths` class - (such as the `mm1` and `mm2`
 instances in the example above). In many cases this is not a problem, but
 there can be situations where you need each instance of your class to have its
 own unique decorator.


 > If you are interested in solutions to this problem, take a look at the
 > appendix on [per-instance decorators](#appendix-per-instance-decorators).


 <a class="anchor" id="example-memoization"></a>
 ## Example - memoization


 Let's move onto another example.
 [Meowmoization](https://en.wikipedia.org/wiki/Memoization) is a common
 performance optimisation technique used in cats. I mean software. Essentially,
 memoization refers to the process of maintaining a cache for a function which
 performs some expensive calculation. When the function is executed with a set
 of inputs, the calculation is performed, and then a copy of the inputs and the
 result are cached. If the function is called again with the same inputs, the
 cached result can be returned.


 This is a perfect problem to tackle with decorators:

 %% Cell type:code id: tags:

 ``` 
 def memoize(func):

    cache = {}

    def wrapper(*args):

        # is there a value in the cache
        # for this set of inputs?
        cached = cache.get(args, None)

        # If not, call the function,
        # and cache the result.
        if cached is None:
            cached      = func(*args)
            cache[args] = cached
        else:
            print('Cached {}({}): {}'.format(func.__name__, args, cached))

        return cached

    return wrapper
 ```

 %% Cell type:markdown id: tags:

 We can now use our `memoize` decorator to add a memoization cache to any
 function.  Let's memoize a function which generates the $n^{th}$ number in the
 [Fibonacci series](https://en.wikipedia.org/wiki/Fibonacci_number):

 %% Cell type:code id: tags:

 ``` 
 @memoize
 def fib(n):

    if n in (0, 1):
        print('fib({}) = {}'.format(n, n))
-        return 1
+        return n

    twoback = 1
    oneback = 1
-    val     = 0
+    val     = 1

    for _ in range(2, n):

        val     = oneback + twoback
        twoback = oneback
        oneback = val

    print('fib({}) = {}'.format(n, val))

    return val
 ```

 %% Cell type:markdown id: tags:

 For a given input, when `fib` is called the first time, it will calculate the
 $n^{th}$ Fibonacci number:

 %% Cell type:code id: tags:

 ``` 
 for i in range(10):
    fib(i)
 ```

 %% Cell type:markdown id: tags:

 However, on repeated calls with the same input, the calculation is skipped,
 and instead the result is retrieved from the memoization cache:

 %% Cell type:code id: tags:

 ``` 
 for i in range(10):
    fib(i)
 ```

 %% Cell type:markdown id: tags:

 > If you are wondering how the `wrapper` function is able to access the
 > `cache` variable, refer to the [appendix on closures](#appendix-closures).


 <a class="anchor" id="decorators-with-arguments"></a>
 ## Decorators with arguments


 Continuing with our memoization example, let's say that we want to place a
 limit on the maximum size that our cache can grow to. For example, the output
 of our function might have large memory requirements, so we can only afford to
 store a handful of pre-calculated results. It would be nice to be able to
 specify the maximum cache size when we define our function to be memoized,
 like so:


 > ```
 > # cache at most 10 results
 > @limitedMemoize(10):
 > def fib(n):
 >     ...
 > ```


 In order to support this, our `memoize` decorator function needs to be
 modified - it is currently written to accept a function as its sole argument,
 but we need it to accept a cache size limit.

 %% Cell type:code id: tags:

 ``` 
 from collections import OrderedDict

 def limitedMemoize(maxSize):

    cache = OrderedDict()

    def decorator(func):
        def wrapper(*args):

            # is there a value in the cache
            # for this set of inputs?
            cached = cache.get(args, None)

            # If not, call the function,
            # and cache the result.
            if cached is None:

                cached = func(*args)

                # If the cache has grown too big,
                # remove the oldest item. In practice
                # it would make more sense to remove
                # the item with the oldest access
                # time, but this is good enough for
                # an introduction!
                if len(cache) >= maxSize:
                    cache.popitem(last=False)

                cache[args] = cached
            else:
                print('Cached {}({}): {}'.format(func.__name__, args, cached))

            return cached
        return wrapper
    return decorator
 ```

 %% Cell type:markdown id: tags:

 > We used the handy
 > [`collections.OrderedDict`](https://docs.python.org/3.5/library/collections.html#collections.OrderedDict)
 > class here which preserves the insertion order of key-value pairs.


 This is starting to look a little complicated - we now have _three_ layers of
 functions. This is necessary when you wish to write a decorator which accepts
 arguments (refer to the
 [appendix](#appendix-decorators-without-arguments-versus-decorators-with-arguments)
 for more details).


 But this `limitedMemoize` decorator is used in essentially the same way as our
 earlier `memoize` decorator:

 %% Cell type:code id: tags:

 ``` 
 @limitedMemoize(5)
 def fib(n):

    if n in (1, 2):
        print('fib({}) = 1'.format(n))
        return 1

    twoback = 1
    oneback = 1
    val     = 0

    for _ in range(2, n):

        val     = oneback + twoback
        twoback = oneback
        oneback = val

    print('fib({}) = {}'.format(n, val))

    return val
 ```

 %% Cell type:markdown id: tags:

 Except that now, the `fib` function will only cache up to 5 values.

 %% Cell type:code id: tags:

 ``` 
 fib(10)
 fib(11)
 fib(12)
 fib(13)
 fib(14)
 print('The result for 10 should come from the cache')
 fib(10)
 fib(15)
 print('The result for 10 should no longer be cached')
 fib(10)
 ```

 %% Cell type:markdown id: tags:

 <a class="anchor" id="chaining-decorators"></a>
 ## Chaining decorators


 Decorators can easily be chained, or nested:

 %% Cell type:code id: tags:

 ``` 
 import time

 @timeFunc
 @memoize
 def expensiveFunc(n):
    time.sleep(n)
    return n
 ```

 %% Cell type:markdown id: tags:

 > Remember that this is semantically equivalent to the following:
 >
 > ```
 > def expensiveFunc(n):
 >     time.sleep(n)
 >     return n
 >
 > expensiveFunc = timeFunc(memoize(expensiveFunc))
 > ```


 Now we can see the effect of our memoization layer on performance:

 %% Cell type:code id: tags:

 ``` 
 expensiveFunc(0.5)
 expensiveFunc(1)
 expensiveFunc(1)
 ```

 %% Cell type:markdown id: tags:

 <a class="anchor" id="decorator-classes"></a>
 ## Decorator classes


 By now, you will have gained the impression that a decorator is a function
 which _decorates_ another function. But if you went through the practical on
 operator overloading, you might remember the special `__call__` method, that
 allows an object to be called as if it were a function.


 This feature allows us to write our decorators as classes, instead of
 functions. This can be handy if you are writing a decorator that has
 complicated behaviour, and/or needs to maintain some sort of state which
 cannot be easily or elegantly written using nested functions.


 As an example, let's say we are writing a framework for unit testing. We want
 to be able to "mark" our test functions like so, so they can be easily
 identified and executed:


 > ```
 > @unitTest
 > def testblerk():
 >     """tests the blerk algorithm."""
 >     ...
 > ```


 With a decorator like this, we wouldn't need to worry about where our tests
 are located - they will all be detected because we have marked them as test
 functions. What does this `unitTest` decorator look like?

 %% Cell type:code id: tags:

 ``` 
 class TestRegistry(object):

    def __init__(self):
        self.testFuncs = []

    def __call__(self, func):
        self.testFuncs.append(func)

    def listTests(self):
        print('All registered tests:')
        for test in self.testFuncs:
            print(' ', test.__name__)

    def runTests(self):
        for test in self.testFuncs:
            print('Running test {:10s} ... '.format(test.__name__), end='')
            try:
                test()
                print('passed!')
            except Exception as e:
                print('failed!')

 # Create our test registry
 registry = TestRegistry()

 # Alias our registry to "unitTest"
 # so that we can register tests
 # with a "@unitTest" decorator.
 unitTest = registry
 ```

 %% Cell type:markdown id: tags:

 So we've defined a class, `TestRegistry`, and created an instance of it,
 `registry`, which will manage all of our unit tests. Now, in order to "mark"
 any function as being a unit test, we just need to use the `unitTest`
 decorator (which is simply a reference to our `TestRegistry` instance):

 %% Cell type:code id: tags:

 ``` 
 @unitTest
 def testFoo():
    assert 'a' in 'bcde'

 @unitTest
 def testBar():
    assert 1 > 0

 @unitTest
 def testBlerk():
    assert 9 % 2 == 0
 ```

 %% Cell type:markdown id: tags:

 Now that these functions have been registered with our `TestRegistry`
 instance, we can run them all:

 %% Cell type:code id: tags:

 ``` 
 registry.listTests()
 registry.runTests()
 ```

 %% Cell type:markdown id: tags:

 > Unit testing is something which you must do! This is __especially__
 > important in an interpreted language such as Python, where there is no
 > compiler to catch all of your mistakes.
 >
 > Python has a built-in
 > [`unittest`](https://docs.python.org/3.5/library/unittest.html) module,
 > however the third-party [`pytest`](https://docs.pytest.org/en/latest/) and
 > [`nose`](http://nose2.readthedocs.io/en/latest/) are popular.  It is also
 > wise to combine your unit tests with
 > [`coverage`](https://coverage.readthedocs.io/en/coverage-4.5.1/), which
 > tells you how much of your code was executed, or _covered_ when your
 > tests were run.


 <a class="anchor" id="appendix-functions-are-not-special"></a>
 ## Appendix: Functions are not special


 When we write a statement like this:

 %% Cell type:code id: tags:

 ``` 
 a = [1, 2, 3]
 ```

 %% Cell type:markdown id: tags:

 the variable `a` is a reference to a `list`. We can create a new reference to
 the same list, and delete `a`:

 %% Cell type:code id: tags:

 ``` 
 b = a
 del a
 ```

 %% Cell type:markdown id: tags:

 Deleting `a` doesn't affect the list at all - the list still exists, and is
 now referred to by a variable called `b`.

 %% Cell type:code id: tags:

 ``` 
 print('b: ', b)
 ```

 %% Cell type:markdown id: tags:

 `a` has, however, been deleted:

 %% Cell type:code id: tags:

 ``` 
 print('a: ', a)
 ```

 %% Cell type:markdown id: tags:

 The variables `a` and `b` are just references to a list that is sitting in
 memory somewhere - renaming or removing a reference does not have any effect
 upon the list<sup>2</sup>.


 If you are familiar with C or C++, you can think of a variable in Python as
 like a `void *` pointer - it is just a pointer of an unspecified type, which
 is pointing to some item in memory (which does have a specific type). Deleting
 the pointer does not have any effect upon the item to which it was pointing.


 > <sup>2</sup> Until no more references to the list exist, at which point it
 > will be
 > [garbage-collected](https://www.quora.com/How-does-garbage-collection-in-Python-work-What-are-the-pros-and-cons).


 Now, functions in Python work in _exactly_ the same way as variables.  When we
 define a function like this:

 %% Cell type:code id: tags:

 ``` 
 def inverse(a):
    return npla.inv(a)

 print(inverse)
 ```

 %% Cell type:markdown id: tags:

 there is nothing special about the name `inverse` - `inverse` is just a
 reference to a function that resides somewhere in memory. We can create a new
 reference to this function:

 %% Cell type:code id: tags:

 ``` 
 inv2 = inverse
 ```

 %% Cell type:markdown id: tags:

 And delete the old reference:

 %% Cell type:code id: tags:

 ``` 
 del inverse
 ```

 %% Cell type:markdown id: tags:

 But the function still exists, and is still callable, via our second
 reference:

 %% Cell type:code id: tags:

 ``` 
 print(inv2)
 data    = np.random.random((10, 10))
 invdata = inv2(data)
 ```

 %% Cell type:markdown id: tags:

 So there is nothing special about functions in Python - they are just items
 that reside somewhere in memory, and to which we can create as many references
 as we like.


 > If it bothers you that `print(inv2)` resulted in
 > `<function inverse at ...>`, and not `<function inv2 at ...>`, then refer to
 > the appendix on
 > [preserving function metdata](#appendix-preserving-function-metadata).


 <a class="anchor" id="appendix-closures"></a>
 ## Appendix: Closures


 Whenever we define or use a decorator, we are taking advantage of a concept
 called a [_closure_][wiki-closure]. Take a second to re-familiarise yourself
 with our `memoize` decorator function from earlier - when `memoize` is called,
 it creates and returns a function called `wrapper`:


 [wiki-closure]: https://en.wikipedia.org/wiki/Closure_(computer_programming)

 %% Cell type:code id: tags:

 ``` 
 def memoize(func):

    cache = {}

    def wrapper(*args):

        # is there a value in the cache
        # for this set of inputs?
        cached = cache.get(args, None)

        # If not, call the function,
        # and cache the result.
        if cached is None:
            cached      = func(*args)
            cache[args] = cached
        else:
            print('Cached {}({}): {}'.format(func.__name__, args, cached))

        return cached

    return wrapper
 ```

 %% Cell type:markdown id: tags:

 Then `wrapper` is executed at some arbitrary point in the future. But how does
 it have access to `cache`, defined within the scope of the `memoize` function,
 after the execution of `memoize` has ended?

 %% Cell type:code id: tags:

 ``` 
 def nby2(n):
    return n * 2

 # wrapper function is created here (and
 # assigned back to the nby2 reference)
 nby2 = memoize(nby2)

 # wrapper function is executed here
 print('nby2(2): ', nby2(2))
 print('nby2(2): ', nby2(2))
 ```

 %% Cell type:markdown id: tags:

 The trick is that whenever a nested function is defined in Python, the scope
 in which it is defined is preserved for that function's lifetime. So `wrapper`
 has access to all of the variables within the `memoize` function's scope, that
 were defined at the time that `wrapper` was created (which was when we called
 `memoize`).  This is why `wrapper` is able to access `cache`, even though at
 the time that `wrapper` is called, the execution of `memoize` has long since
 finished.


 This is what is known as a
 [_closure_](https://www.geeksforgeeks.org/python-closures/). Closures are a
 fundamental, and extremely powerful, aspect of Python and other high level
 languages. So there's your answer,
 [fishbulb](https://www.youtube.com/watch?v=CiAaEPcnlOg).


 <a class="anchor" id="appendix-decorators-without-arguments-versus-decorators-with-arguments"></a>
 ## Appendix: Decorators without arguments versus decorators with arguments


 There are three ways to invoke a decorator with the `@` notation:


 1. Naming it, e.g. `@mydecorator`
 2. Calling it, e.g. `@mydecorator()`
 3. Calling it, and passing it arguments, e.g. `@mydecorator(1, 2, 3)`


 Python expects a decorator function to behave differently in the second and
 third scenarios, when compared to the first:

 %% Cell type:code id: tags:

 ``` 
 def decorator(*args):
    print('  decorator({})'.format(args))
    def wrapper(*args):
        print('    wrapper({})'.format(args))
    return wrapper

 print('Scenario #1: @decorator')
 @decorator
 def noop():
    pass

 print('\nScenario #2: @decorator()')
 @decorator()
 def noop():
    pass

 print('\nScenario #3: @decorator(1, 2, 3)')
 @decorator(1, 2, 3)
 def noop():
    pass
 ```

 %% Cell type:markdown id: tags:

 So if a decorator is "named" (scenario 1), only the decorator function
 (`decorator` in the example above) is called, and is passed the decorated
 function.


 But if a decorator function is "called" (scenarios 2 or 3), both the decorator
 function (`decorator`), __and its return value__ (`wrapper`) are called - the
 decorator function is passed the arguments that were provided, and its return
 value is passed the decorated function.


 This is why, if you are writing a decorator function which expects arguments,
 you must use three layers of functions, like so:

 %% Cell type:code id: tags:

 ``` 
 def decorator(*args):
    def realDecorator(func):
        def wrapper(*args, **kwargs):
            return func(*args, **kwargs)
        return wrapper
    return realDecorator
 ```

 %% Cell type:markdown id: tags:

 > The author of this practical is angry about this, as he does not understand
 > why the Python language designers couldn't allow a decorator function to be
 > passed both the decorated function, and any arguments that were passed when
 > the decorator was invoked, like so:
 >
 > ```
 > def decorator(func, *args, **kwargs): # args/kwargs here contain
 >                                       # whatever is passed to the
 >                                       # decorator
 >
 >     def wrapper(*args, **kwargs):     # args/kwargs here contain
 >                                       # whatever is passed to the
 >                                       # decorated function
 >          return func(*args, **kwargs)
 >
 >     return wrapper
 > ```


 <a class="anchor" id="appendix-per-instance-decorators"></a>
 ## Appendix: Per-instance decorators


 In the section on [decorating methods](#decorators-on-methods), you saw
 that when a decorator is applied to a method of a class,  that decorator
 is invoked just once, and shared by all instances of the class. Consider this
 example:

 %% Cell type:code id: tags:

 ``` 
 def decorator(func):
    print('Decorating {} function'.format(func.__name__))
    def wrapper(*args, **kwargs):
        print('Calling decorated function {}'.format(func.__name__))
        return func(*args, **kwargs)
    return wrapper

 class MiscMaths(object):

    @decorator
    def add(self, a, b):
        return a + b
 ```

 %% Cell type:markdown id: tags:

 Note that `decorator` was called at the time that the `MiscMaths` class was
 defined. Now, all `MiscMaths` instances share the same `wrapper` function:

 %% Cell type:code id: tags:

 ``` 
 mm1 = MiscMaths()
 mm2 = MiscMaths()

 print('1 + 2 =', mm1.add(1, 2))
 print('3 + 4 =', mm2.add(3, 4))
 ```

 %% Cell type:markdown id: tags:

 This is not an issue in many cases, but it can be problematic in some. Imagine
 if we have a decorator called `ensureNumeric`, which makes sure that arguments
 passed to a function are numbers:

 %% Cell type:code id: tags:

 ``` 
 def ensureNumeric(func):
    def wrapper(*args):
        args = tuple([float(a) for a in args])
        return func(*args)
    return wrapper
 ```

 %% Cell type:markdown id: tags:

 This all looks well and good - we can use it to decorate a numeric function,
 allowing strings to be passed in as well:

 %% Cell type:code id: tags:

 ``` 
 @ensureNumeric
 def mul(a, b):
    return a * b

 print(mul( 2,   3))
 print(mul('5', '10'))
 ```

 %% Cell type:markdown id: tags:

 But what will happen when we try to decorate a method of a class?

 %% Cell type:code id: tags:

 ``` 
 class MiscMaths(object):

    @ensureNumeric
    def add(self, a, b):
        return a + b

 mm = MiscMaths()
 print(mm.add('5', 10))
 ```

 %% Cell type:markdown id: tags:

 What happened here?? Remember that the first argument passed to any instance
 method is the instance itself (the `self` argument). Well, the `MiscMaths`
 instance was passed to the `wrapper` function, which then tried to convert it
 into a `float`.  So we can't actually apply the `ensureNumeric` function as a
 decorator on a method in this way.


 There are a few potential solutions here. We could modify the `ensureNumeric`
 function, so that the `wrapper` ignores the first argument. But this would
 mean that we couldn't use `ensureNumeric` with standalone functions.


 But we _can_ manually apply the `ensureNumeric` decorator to `MiscMaths`
 instances when they are initialised.  We can't use the nice `@ensureNumeric`
 syntax to apply our decorators, but this is a viable approach:

 %% Cell type:code id: tags:

 ``` 
 class MiscMaths(object):

    def __init__(self):
        self.add = ensureNumeric(self.add)

    def add(self, a, b):
        return a + b

 mm = MiscMaths()
 print(mm.add('5', 10))
 ```

 %% Cell type:markdown id: tags:

 Another approach is to use a second decorator, which dynamically creates the
 real decorator when it is accessed on an instance. This requires the use of an
 advanced Python technique called
 [_descriptors_](https://docs.python.org/3.5/howto/descriptor.html), which is
 beyond the scope of this practical. But if you are interested, you can see an
 implementation of this approach
 [here](https://git.fmrib.ox.ac.uk/fsl/fslpy/blob/1.6.8/fsl/utils/memoize.py#L249).


 <a class="anchor" id="appendix-preserving-function-metadata"></a>
 ## Appendix: Preserving function metadata


 You may have noticed that when we decorate a function, some of its properties
 are lost. Consider this function:

 %% Cell type:code id: tags:

 ``` 
 def add2(a, b):
    """Adds two numbers together."""
    return a + b
 ```

 %% Cell type:markdown id: tags:

 The `add2` function is an object which has some attributes, e.g.:

 %% Cell type:code id: tags:

 ``` 
 print('Name: ', add2.__name__)
 print('Help: ', add2.__doc__)
 ```

 %% Cell type:markdown id: tags:

 However, when we apply a decorator to `add2`:

 %% Cell type:code id: tags:

 ``` 
 def decorator(func):
    def wrapper(*args, **kwargs):
        """Internal wrapper function for decorator."""
        print('Calling decorated function {}'.format(func.__name__))
        return func(*args, **kwargs)
    return wrapper


 @decorator
 def add2(a, b):
    """Adds two numbers together."""
    return a + b
 ```

 %% Cell type:markdown id: tags:

 Those attributes are lost, and instead we get the attributes of the `wrapper`
 function:

 %% Cell type:code id: tags:

 ``` 
 print('Name: ', add2.__name__)
 print('Help: ', add2.__doc__)
 ```

 %% Cell type:markdown id: tags:

 While this may be inconsequential in most situations, it can be quite annoying
 in some, such as when we are automatically [generating
 documentation](http://www.sphinx-doc.org/) for our code.


 Fortunately, there is a workaround, available in the built-in
 [`functools`](https://docs.python.org/3.5/library/functools.html#functools.wraps)
 module:

 %% Cell type:code id: tags:

 ``` 
 import functools

 def decorator(func):
    @functools.wraps(func)
    def wrapper(*args, **kwargs):
        """Internal wrapper function for decorator."""
        print('Calling decorated function {}'.format(func.__name__))
        return func(*args, **kwargs)
    return wrapper

 @decorator
 def add2(a, b):
    """Adds two numbers together."""
    return a + b
 ```

 %% Cell type:markdown id: tags:

 We have applied the `@functools.wraps` decorator to our internal `wrapper`
 function - this will essentially replace the `wrapper` function metdata with
 the metadata from our `func` function. So our `add2` name and documentation is
 now preserved:

 %% Cell type:code id: tags:

 ``` 
 print('Name: ', add2.__name__)
 print('Help: ', add2.__doc__)
 ```

 %% Cell type:markdown id: tags:

 <a class="anchor" id="appendix-class-decorators"></a>
 ## Appendix: Class decorators


 > Not to be confused with [_decorator classes_](#decorator-classes)!


 In this practical, we have shown how decorators can be applied to functions
 and methods. But decorators can in fact also be applied to _classes_. This is
 a fairly niche feature that you are probably not likely to need, so we will
 only cover it briefly.


 Imagine that we want all objects in our application to have a globally unique
 (within the application) identifier. We could use a decorator which contains
 the logic for generating unique IDs, and defines the interface that we can
 use on an instance to obtain its ID:

 %% Cell type:code id: tags:

 ``` 
 import random

 allIds = set()

 def uniqueID(cls):
    class subclass(cls):
        def getUniqueID(self):

            uid = getattr(self, '_uid', None)

            if uid is not None:
                return uid

            while uid is None or uid in set():
                uid = random.randint(1, 100)

            self._uid = uid
            return uid

    return subclass
 ```

 %% Cell type:markdown id: tags:

 Now we can use the `@uniqueID` decorator on any class that we need to
 have a unique ID:

 %% Cell type:code id: tags:

 ``` 
 @uniqueID
 class Foo(object):
    pass

 @uniqueID
 class Bar(object):
    pass
 ```

 %% Cell type:markdown id: tags:

 All instances of these classes will have a `getUniqueID` method:

 %% Cell type:code id: tags:

 ``` 
 f1 = Foo()
 f2 = Foo()
 b1 = Bar()
 b2 = Bar()

 print('f1: ', f1.getUniqueID())
 print('f2: ', f2.getUniqueID())
 print('b1: ', b1.getUniqueID())
 print('b2: ', b2.getUniqueID())
 ```

 %% Cell type:markdown id: tags:

 <a class="anchor" id="useful-references"></a>
 ## Useful references


 * [Understanding decorators in 12 easy steps](http://simeonfranklin.com/blog/2012/jul/1/python-decorators-in-12-steps/)
 * [The decorators they won't tell you about](https://github.com/hchasestevens/hchasestevens.github.io/blob/master/notebooks/the-decorators-they-wont-tell-you-about.ipynb)
 * [Closures - Wikipedia][wiki-closure]
 * [Closures in Python](https://www.geeksforgeeks.org/python-closures/)
 * [Garbage collection in Python](https://www.quora.com/How-does-garbage-collection-in-Python-work-What-are-the-pros-and-cons)

 [wiki-closure]: https://en.wikipedia.org/wiki/Closure_(computer_programming)

--- a/advanced_topics/decorators.md
+++ b/advanced_topics/decorators.md
@@ -292,11 +292,11 @@ def fib(n):

    if n in (0, 1):
        print('fib({}) = {}'.format(n, n))
-        return 1
+        return n

    twoback = 1
    oneback = 1
-    val     = 0
+    val     = 1

    for _ in range(2, n):

@@ -417,13 +417,13 @@ earlier `memoize` decorator:
 @limitedMemoize(5)
 def fib(n):

-    if n in (1, 2):
+    if n in (0, 1):
        print('fib({}) = 1'.format(n))
-        return 1
+        return n

    twoback = 1
    oneback = 1
-    val     = 0
+    val     = 1

    for _ in range(2, n):