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--- fp:lab02 [2021/03/18 10:45]
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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.
-===== 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>
-.  Write a function together with its signature, which implements boolean AND:
+**2.2.** Implement a tail-recursive function that computes the greatest common divisor of a natural number:
-<code haskell>
-myand :: Bool -> Bool -> Bool
-...
-</code>
-. 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 scala>
-<code haskell>
+def gcd(a: Int, b: Int): Int = ???
-ifp = ...
 </code>
-. Write a function which takes three integers and returns the largest. Hint - sometimes parentheses are **important in function calls**.
+**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).
-<code haskell>
-f :: Integer -> Integer -> Integer -> Integer
+<code scala>
+def sumSquares(n: Int): Int = ???
 </code>
-<<We have to explain the syntax of if>>
+===== Newton's Square Root method =====
-<<Explain syntax of 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:
+  * Start with $math[x_0 = 1].
+  * Compute $math[x_{n+1} = \displaystyle\frac{1}{2}(x_n+\frac{a}{x_n})]
-x. Solve the previous exercise using guards
+**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 scala>
+def improve(xn: Double, a: Double): Double = ???
+</code>
-<<Explain lists, head, tail>>
+**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>
+def nth_guess(n: Int, a: Double): Double = ???
+</code>
-x. 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}].
+**2.6.** 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._'').
+<code scala>
+  def acceptable(xn: Double, a: Double): Boolean = ???
+</code>
-. Write a function which extracts the third to last number from a list and returns ''True'', if that number is odd (hint: the function ''mod'' may be useful)
+**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>
+<code scala>
-            V
+def mySqrt(a: Double): Double = {
-   f [3,4,5,2,3,9] = False
+   def improve(xn: Double): Double = ???
-   f [3,4,2,1,4,4] = True
+   def acceptable(xn: Double): Boolean = ???
+   def tailSqrt(estimate: Double): Double = ???
+   ???
+}
 </code>
-<<Explain pattern matching>>
+**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).
-x. Implement the previous exercise using patterns
-. Implement a function which returns the sum of integers from a list.
-. Implement a function which takes a list of booleans and returns false if **at least** one boolean from the list is false.
-. Implement a function which filters out all odd numbers from a list.
-. 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]''.
-<<Explain char and strings>>
-x. Write a function which removes all empty strings from a list.
-x. Write a function which removes all strings of size smaller than 3. Do **not** use the builtin function ''length''.
-x. Write a function which removes all strings having the third letter equal to 'a'.
-x. 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.