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fp:lab02 [2022/02/24 11:31] pdmatei |
fp:lab02 [2023/03/10 10:16] (current) pdmatei |
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| - | 2.1. Write a tail-recursive function that computes the factorial of a natural number. Start from the code stub below: | + | **2.1.** Write a tail-recursive function that computes the factorial of a natural number. Start from the code stub below: |
| <code scala> | <code scala> | ||
| - | def fact (n: Integer): Integer = { | + | def fact (n: Int): Int = { |
| - | def aux_fact(n: Integer, acc: Integer): Integer = | + | def aux_fact(i: Int, acc: Int): Int = |
| if (???) acc | if (???) acc | ||
| else ??? | else ??? | ||
| Line 20: | Line 20: | ||
| </code> | </code> | ||
| - | 2.2. Implement a tail-recursive function that computes the greatest common divisor of a natural number: | + | **2.2.** Implement a tail-recursive function that computes the greatest common divisor of a natural number: |
| <code scala> | <code scala> | ||
| Line 26: | Line 26: | ||
| </code> | </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). | + | **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> | <code scala> | ||
| Line 38: | Line 38: | ||
| * Compute $math[x_{n+1} = \displaystyle\frac{1}{2}(x_n+\frac{a}{x_n})] | * 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}]). | + | **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> | <code scala> | ||
| def improve(xn: Double, a: Double): Double = ??? | def improve(xn: Double, a: Double): Double = ??? | ||
| </code> | </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}]: | + | **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> | <code scala> | ||
| - | def nth_guess(n: Double, a: Double): Double = ??? | + | def nth_guess(n: Int, a: Double): Double = ??? |
| </code> | </code> | ||
| Note that: | 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}]. | + | * 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._''). | + | **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> | <code scala> | ||
| def acceptable(xn: Double, a: Double): Boolean = ??? | def acceptable(xn: Double, a: Double): Boolean = ??? | ||
| </code> | </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: | + | **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 scala> | ||
| def mySqrt(a: Double): Double = { | def mySqrt(a: Double): Double = { | ||
| Line 67: | Line 67: | ||
| } | } | ||
| </code> | </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). | ||
| + | |||