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| * alternatively, you can use [[https://www.sublimetext.com/download | Sublime]], which is a simpler (faster) editor. You can add syntax highlighting plugins for Scala (see: [[https://scalameta.org/metals/docs/editors/sublime/ | scalameta ]]), as well as worksheet/REPL support (see: [[https://packagecontrol.io/packages/SublimeREPL | SublimeREPL]]). | * alternatively, you can use [[https://www.sublimetext.com/download | Sublime]], which is a simpler (faster) editor. You can add syntax highlighting plugins for Scala (see: [[https://scalameta.org/metals/docs/editors/sublime/ | scalameta ]]), as well as worksheet/REPL support (see: [[https://packagecontrol.io/packages/SublimeREPL | SublimeREPL]]). | ||
| - | ===== Online exercises ===== | ||
| - | |||
| - | Consider the following code: | ||
| - | <code scala> | ||
| - | def fact(n: Integer): Integer = | ||
| - | if (n == 1) 1 | ||
| - | else n*fact(n-1) | ||
| - | </code> | ||
| - | |||
| - | The function ''fact'' is not tail recursive and will quickly fill the call stack. Starting from the code stub below, implement a tail-recursive factorial: | ||
| - | <code scala> | ||
| - | def fact (n: Integer): Integer = { | ||
| - | def aux_fact(n: Integer, acc: Integer): Integer = | ||
| - | if (???) acc | ||
| - | else ??? | ||
| - | aux_fact(n,1) | ||
| - | } | ||
| - | </code> | ||
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| - | ===== Homework - Newton's Square Root method ===== | ||
| - | |||
| - | * Video (Newton's Square Root solution) | ||
| - | |||
| - | A very fast way to numerically compute $math[\sqrt{a}], is using Newton's Square Root approximation: | ||
| - | * Start with $math[x_0 = 1]. | ||
| - | * Compute $math[x_{n+1} = \displaystyle\frac{1}{2}(x_n+\frac{a}{x_n})] | ||
| - | |||
| - | Implement the following function: | ||
| - | <code scala> | ||
| - | //given a guess x_n, it computes x_{n+1} | ||
| - | def improve(xn: Integer, a: Integer): Integer = ??? | ||
| - | |||
| - | // computes the nth estimation of sqrt(a) | ||
| - | def nth_guess(n: Integer, a: Integer): Integer = | ||
| - | if (n == 0) ??? | ||
| - | else ??? | ||
| - | </code> | ||
| - | |||
| - | Note that ''nth_guess'' will return a more accurate estimation for smaller a, than for larger a's. We may waste a lot of iterations on small numbers, and spend insufficient ones on large numbers. Thus, instead of ''nth_guess'', implement ''acceptable(xn,a)'' 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: Integer, a: Integer): Boolean = ??? | ||
| - | </code> | ||