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--- fp:lab02 [2021/03/18 10:28]
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+++ fp:lab02 [2023/03/10 10:16] (current)
pdmatei
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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.
+** Create a new Scala worksheet to write your solutions **
+**2.1.** Write a tail-recursive function that computes the factorial of a natural number. Start from the code stub below:
+<code scala>
+def fact (n: Int): Int = {
+   def aux_fact(i: Int, acc: Int): Int =
+       if (???) acc
+       else ???
+   ???
+}
+</code>
+**2.2.** Implement a tail-recursive function that computes the greatest common divisor of a natural number:
+<code scala>
+def gcd(a: Int, b: Int): Int = ???
+</code>
+**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 scala>
+def sumSquares(n: Int): Int = ???
+</code>
+===== Newton's Square Root method =====
+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})]
+**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>
+**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>
+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>
+**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>
+def mySqrt(a: Double): Double = {
+   def improve(xn: Double): Double = ???
+   def acceptable(xn: Double): Boolean = ???
+   def tailSqrt(estimate: Double): Double = ???
+   ???
+}
+</code>
+**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).