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--- pp:other-exercises [2021/03/13 14:39]
pdmatei created
+++ pp:other-exercises [2021/03/21 08:14] (current)
pdmatei
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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>
+   2   3            1   0   0         1+3  2  2+3      4   2   5
+   5   6      x     0   1   1     =   4+6  5  5+6  =   10  5   11
+   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 =====
@@ Line 30: / Line 190: @@
 <code haskell>
 f l =
-</code>
-. Write a function which takes a list of pairs - student-name and grade, removes those with grades less than 5, splits the name in substrings, and adds one bonus point to people with three names. Example:
-<code haskell>
-    f [("Dan Matei Popovici",9),("Mihai",4),("Andrei Alex",6)] =
-    	[(["Dan", "Matei", "Popovici"],10),(["Andrei,Alex"],6)]
-</code>
-. Write the signature and implement the following function:
-<code haskell>
-   f ["Matei", "Mihai"] ["Popovici","Dumitru"] = [("Matei","Popovici"), ("Mihai","Dumitru")]
-</code>
-. Implement the following function:
-<code haskell>
-   f ["Matei", "Mihai"] ["Popovici","Dumitru"] = ["MPopovici", "MDumitru"]
 </code>