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===== Matrices ======

==== Haskell implementation of foldl ====

In Haskell, ''foldl'' is implemented as follows:
<code>
foldl op acc []     = acc                  
foldl op acc (h:t) = foldl op (op acc h) t
</code>

Note the call of the ''op'' function.

==== An exercise in modular programming with higher-order functions ====

Let us consider the task of matrix multiplication in Haskell. Example:
<code>
  1   2   3            1   0   0         1+3  2  2+3      4   2   5
  4   5   6      x     0   1   1     =   4+6  5  5+6  =   10  5   11
  7   8   9            1   0   1         7+9  8  8+9      16  8   17
</code>


=== Matrix representation in Haskell ===

The basic representation building-block in Haskell is the list. We can represent matrices in Haskell as a list where each element is a row (hence a list of elements). Example:

<code haskell>
m = [[1,2,3],[4,5,6],[7,8,9]]
</code>

A good warmup exercise is to write a nice display function for matrices. We transform each element of a line into a string:

<code haskell>
displayline l = map (\e->(show e)++" ") l
</code>

The code can be improved for legibility:
<code haskell>
displayline = map ((++" ").show) 
</code>


Next, we fold the list into a string with a newline character: 

<code haskell>
displayline l = foldr (++) "\n" (map ((++" ").show) l)
</code>

and simplify again, by expressing the pipeline function calls as functional composition:
<code haskell>
displayline :: Show a => [a] -> [Char]
displayline = (foldr (++) "\n") .  (map ( (++ " ") . show ) )
</code>

Note that we need to explicitly state the type of display. Sometimes, in Haskell, an explicit type declaration is required. For now, we omit details.

Next, we apply the above process on all matrix lines:

<code haskell>
display :: Show a => [[a]] -> [[Char]]
display = 
    let bind = foldr (++) "\n"
    in bind.(map (bind . (map ((++" ") . show ) ) ) )
</code>

Notice that we have separated the binding process, because we reuse it.

Finally, we make all matrices displayable:
<code haskell>
instance (Show a) => Show [[a]] where
    show = 
        let bind = foldr (++) "\n"
        in bind.(map (bind.(map ((++" ").show ) ) ) )
</code>
More details about this implementation (e.g. instances, classes) will be given in future lectures.
        
=== Matrix multiplication ===

== Step 1: Transposition ==

Matrix multiplication operates on the **lines** of the first matrix and **columns** of the second. We transpose the second matrix, so that we now operate on lines on both matrices. The following code extracts **the first line** from a matrix ''m'':
<code haskell>
map head m
</code>
A matrix ''m'' **without** its first column is:
<code haskell>
map tail m
</code>
Finally transposition is given by:
<code haskell>
transpose ([]:_) = []
transpose m = (map head m) : transpose (map tail m)
</code>
Notice that the //basis case// corresponds to a **list containing empty lists**.

== Step 2: Computing multiplication ==

To compute the ''i,j''th **element** of the multiplication matrix, we need to multiply per element the ''i''th line by the ''j''th column:
<code haskell>
zipWith (*) li cj
</code>
and then add-up the values:
<code haskell>
foldr (+) 0 (zipWith (*) li cj)
</code>

To obtain the ''i''th **line** of the multiplication matrix, we need to repeat the above process **for each column of the second matrix**, in other words, for each line of its transposition:

<code haskell>
map (\col -> foldr (+) 0 (zipWith (*) li col) ) (transpose m2)
</code>

Finally, to obtain the multiplication matrix, we need to compute all its lines, hence:
<code haskell>
mult m1 m2 = 
  map (\line -> map (\col -> foldr (+) 0 (zipWith (*) line col) ) (transpose m2) ) m1
</code>

=== Matrices as images ===

A matrix can be used to represent a **rasterized image** (a collection of pixels). In this example, we consider that pixels can have values: ' ' (white), '.' (grey) and '*' (black).

Higher-order functions can be naturally used to represent image-transformations, for instance, flipping:

<code haskell>
flipH = map reverse
flipV = reverse
</code>

Rotations:
<code haskell>
rotate90left = flipV.transpose
rotate90right = flipH.transpose
</code>

The //negative// of an image:
<code haskell>
invert = map (map (\x->if x=='*' then ' ' else '*') )
</code>

Scaling an image horizontally:
<code haskell>
scalex = foldr (\h t->h:h:t) []    
</code>

Scaling vertically:
<code haskell>
scaley = map scalex
</code>

Balanced scale:
<code haskell>
scale = scalex . scaley
</code>

Or just a random sequence of operations:
<code haskell>
rand = foldr (.) id [rotate90left, invert, scale]
</code>

===== Function composition and function application =====

29. Sometimes it really helps to define function via function composition. For instance:
<code haskell>
f x = (g.h) x
	where g x = 2*x
	      h x = x + 1
</code>

What type does f have?
What type does g have?

What does the following function do?

<code haskell>
f x = ff x
	where g x = 2*x
	      h x = x + 1
	      ff = f.h
</code>
What does the following function do?
<code haskell>
f x = ff x
	where g x = 2*x
	      h x = x + 1
	      ff = h.f
</code>

30. Rewrite exercise 28 via function composition:
<code haskell>
f l = 
</code>

34. Sometimes it helps to use functions in infix form. For instance, instead of ''mod x 2'', we can write ''x `mod` 2''. We can also define infix functions:

   x 'f' y = x + y

Implement a function for transforming lists into pairs, in exercise 32. and use it in the infix form.

35. Just in the same way, we can treat infix functions as prefix. We do this using round parentheses. Test:
<code haskell>
		:t (+)
		:t (&&)
		:t (++)
</code>

36. What about function composition? Is it a special operator, or is it just a function as well? What does '':t (.)'' do?

37. What does the function below do? Take parts of the function below, and test them.
<code haskell>
f l = (((++) "?").((:) '>')) l
</code>

How about?
<code haskell>
f = (((:) '>').(++"?"))
</code>

38. Write a function of a single line, such that:
<code haskell>
f "Matei" = "{Matei}"
</code>

39. Use only ''(:)'', ''reverse'' and functional composition ''.'' to solve exercise 38 

40. What does the function ''($)'' to? 
    (Use :t, and make up some tests)

41. Write the following implementation using only ($):
<code haskell>
f "||" "Matei" = "||Matei||"
</code>

42.  Solve exercise 13. from [[pp:l02|Lab 2]] using ''$'' (''++'' and ''reverse'' are also permitted) and other improvements (treat the case when the list has fewer elements):