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--- fp:lab02 [2021/03/18 16:58]
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+++ fp:lab02 [2023/03/10 10:16] (current)
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-====== Introduction to Haskell ======
+====== 2. Scala syntax and function definition ======
+** Objectives: **
+  * get yourself familiar with Scala syntax basics
+  * practice writing **tail-recursive** functions as an alternative to imperative **loops**
+  * keep your code clean and well-structured.
-===== 1. Functions in Haskell =====
+** Create a new Scala worksheet to write your solutions **
-In mathematics, functions have a domain an codomain. In Haskell, functions have **types** or **signatures**. They often can be omitted in Haskell, but can
-also be explicitly written as in:
-<code haskell>
+**2.1.** Write a tail-recursive function that computes the factorial of a natural number. Start from the code stub below:
-f :: Integer -> Integer -> Integer
-f x y = x + y
-</code>
-or:
+<code scala>
-<code haskell>
+def fact (n: Int): Int = {
-f :: Bool -> Bool
+   def aux_fact(i: Int, acc: Int): Int =
-f True = False
+       if (???) acc
-f False = True
+       else ???
+   ???
+}
 </code>
-The previous example illustrates that we can define functions by specifying a behaviour for given values.
+**2.2.** Implement a tail-recursive function that computes the greatest common divisor of a natural number:
-.1. Write a function together with its signature, which implements boolean AND:
+<code scala>
-<code haskell>
+def gcd(a: Int, b: Int): Int = ???
-myand :: Bool -> Bool -> Bool
-...
 </code>
-===== 2. If and conditionals (guards) =====
+**2.3.** Write a tail-recursive function takes an integer $math[n] and computes the value $math[1 + 2^2 + 3^2 + ... + (n-1)^2 + n^2]. (Hint: use inner functions).
-.1. Write an implementation for a function ''ifp'' which takes a boolean, expressions $math[e_1] and $math[e_2] and returns $math[e_1] if the boolean is true and $math[e_2] otherwise.
-<code haskell>
-ifp = ...
-</code>
-.2. Write a function which takes three integers and returns the largest. Hint - sometimes parentheses are **important in function calls**.
+<code scala>
-<code haskell>
+def sumSquares(n: Int): Int = ???
-f :: Integer -> Integer -> Integer -> Integer
 </code>
-In Haskell, we can use the if construct, in a manner almost identical to the above implementation, e.g.
+===== Newton's Square Root method =====
-<code haskell>
-f x = if x == 0 then 1 else 0
-</code>
-The previous function returns 1 if x is equal to 0 and 0 otherwise. A more elegant way is to use **guards**:
+A very fast way to numerically compute $math[\sqrt{a}], often used as a standard //sqrt(.)// implementation, relies on Newton's Square Root approximation. The main idea relies on starting with an estimate (often 1), and incrementally improving the estimate. More precisely:
-<code haskell>
+  * Start with $math[x_0 = 1].
-f x
+  * Compute $math[x_{n+1} = \displaystyle\frac{1}{2}(x_n+\frac{a}{x_n})]
-   | x == 0 = 1
-   | otherwise = 0
-</code>
-More generally, guards can be used as follows:
+**2.4.** Implement the function ''improve'' which takes an estimate $math[x_n] of $math[\sqrt{a}] and improves it (computes $math[x_{n+1}]).
-<code haskell>
+<code scala>
-<function_name> <parameters>
+def improve(xn: Double, a: Double): Double = ???
-  | <boolean_condition_1> = <expression_1>
-  | <boolean_condition_2> = <expression_2>
-  ...
-  | otherwise = <expression_n>
 </code>
-.3. Solve the previous exercise using guards.
+**2.5.** Implement the function ''nthGuess'' which starts with $math[x_0 = 1] and computes the nth estimate $math[x_n] of $math[\sqrt{a}]:
+<code scala>
-===== 3. Lists =====
+def nth_guess(n: Int, a: Double): Double = ???
-The following code examples illustrate the usage of lists:
-<code haskell>
--- defining a new list
-l = [1,2,3]
--- adds x to the beginning of a list l. The operator : is called 'cons' and it is infix (just like +)
-cons x l = x:l
--- the previous list can be defined also as:
-lp = 1:2:3:[]
--- function head returns the first element of a list
-first_of_l = head l
--- function tail removes the first element of a list and returns the result
-remove_first_of_l = tail l
--- the infix operator ++ concatenates lists.
-lpp = l ++ lp
 </code>
-.1. Implement reversal
+Note that:
+  * for smaller $math[a], there is no need to compute $math[n] estimations as $math[(x_n)_n] converges quite fast to $math[\sqrt{a}].
-.2. Write a function which extracts the third to last integer from a list and returns ''True'', if that number is odd (hint: the function ''mod'' may be useful), and false otherwise. If the list has fewer than three elements, the function should also return false.
-<code>
+**2.6.** Thus, implement the function ''acceptable'' which returns ''true'' iff $math[\mid x_n^2 - a \mid \leq 0.001]. (Hint, google the ''abs'' function in Scala. Don't forget to import ''scala.math._'').
-            V
+<code scala>
-   f [3,4,5,2,3,9] = False
+  def acceptable(xn: Double, a: Double): Boolean = ???
-   f [3,4,2,1,4,4] = True
 </code>
-Using ''head'' and ''tail'' can become tedious if we need to extract specific inner elements of a list. Alternatively, we can use **list patterns** in function definition, in order to explore the structure of a list. A few examples are given below.
+**2.7.** Implement the function ''mySqrt'' which computes the square root of an integer ''a''. Modify the previous implementations to fit the following code structure:
+<code scala>
-<code haskell>
+def mySqrt(a: Double): Double = {
--- this function uses patterns to test if a list is empty. The pattern 'x:xs' refers to a list where the first element is x and the rest of the list is xs
+   def improve(xn: Double): Double = ???
-is_empty [] = True
+   def acceptable(xn: Double): Boolean = ???
-is_empty (x:xs) = False
+   def tailSqrt(estimate: Double): Double = ???
+   ???
+}
 </code>
-.3. Implement the previous exercise using patterns
+**2.8. (!) **  Try out your code for: ''2.0e50'' (which is $math[2.0\cdot 10^{50}]) or ''2.0e-50''. The code will likely take a very long time to finish. The reason is that $math[xn^2 - a] will suffer from rounding error which may be larger than 0.001. Can you find a different implementation for the function ''acceptable'' which takes that into account? (Hint: the code is just as simple as the original one).
-.4. Implement a function which returns the sum of integers from a list.
-.5. Implement a function which takes a list of booleans and returns false if **at least** one boolean from the list is false.
-.6. Implement a function which filters out all odd numbers from a list.
-.7. Implement a function which takes a list of booleans and returns a list of integers. In the latter, (''True'' becomes ''1'' and ''False'' becomes ''0''). Example: ''f [False, True, False] = [0,1,0]''.
-We can construct more complicated patterns from simpler ones:
-<code haskell>
--- this function returns true if the list given as parameter has at least tree elements.
-f (x:y:z:xs) = True
-f _ = False
-</code>
-===== 4. Strings =====
-The following example illustrates the usage of ''Char'' and ''String''. In Haskell, ''String'' is a type-alias for ''[Char]'' (strings are lists of chars and can be introspected using pattern matching as well).
-<code haskell>
--- this function returns true if the parameter is the character 'a'
-f 'a' = True
-f _ = False
--- this function returns true if the given string has the SECOND character equal to 'b' and false otherwise (or if the string has less than two characters)
-g (_:'b':xs) = True
-g _ = False
-</code>
-.1. Write a function which removes all empty strings from a list.
-.2. Write a function which removes all strings of size smaller than 3. Do **not** use the builtin function ''length''.
-.3. Write a function which removes all strings having the third letter equal to 'a'.
-.4. Write a function which:
-    * removes all strings which are not **names**. A **name** always starts with an uppercase.
-    * removes the last name from the resulting list of names. A name always comprises of the first name followed by the last name.