//############################################################################
// Programmation IV - 2002 - Week 02
//############################################################################
// $Id$
object M0 {
def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b);
def factorial(n: Int): Int = if (n == 0) 1 else n * factorial(n - 1);
Console.println(gcd(14,21));
Console.println(factorial(5));
Console.println;
}
//############################################################################
object M1 {
def cube(x: Int): Double = x * x * x;
def sumInts(a: Int, b: Int): Double = if (a > b) 0
else a + sumInts(a + 1, b);
def sumCubes(a: Int, b: Int): Double = if (a > b) 0
else cube(a) + sumCubes(a + 1, b);
def sumReciprocals(a: Int, b: Int): Double = if (a > b) 0
else 1.0/a + sumReciprocals(a + 1, b);
def sumPi(n: Int): Double = {
def element(x: Int): Double = 4.0/(4*x+1) - 4.0/(4*x-1);
def sumElements(a: Int, b: Int): Double =
if (a > b) 0
else element(a) + sumElements(a + 1, b);
4 + sumElements(1,n)
}
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M2 {
def id(x: Int): Double = x;
def cube(x: Int): Double = x * x * x;
def reciprocal(x: Int): Double = 1.0/x;
def sum(f: Int => Double, a: Int, b: Int): Double =
if (a > b) 0
else f(a) + sum(f, a + 1, b);
def sumInts(a: Int, b: Int): Double = sum(id, a, b);
def sumCubes(a: Int, b: Int): Double = sum(cube, a, b);
def sumReciprocals(a: Int, b: Int): Double = sum(reciprocal, a, b);
def sumPi(n: Int): Double = {
def element(x: Int): Double = 4.0/(4*x+1) - 4.0/(4*x-1);
4 + sum(element, 1, n)
}
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M3 {
def sum(f: Int => Double, a: Int, b: Int): Double =
if (a > b) 0
else f(a) + sum(f, a + 1, b);
def sumInts(a: Int, b: Int): Double = sum((xXXXXX => xXXXXX), a, b);
def sumCubes(a: Int, b: Int): Double = sum((x => x * x * x), a, b);
def sumReciprocals(a: Int, b: Int): Double = sum((x => 1.0/x), a, b);
def sumPi(n: Int): Double = 4 + sum((x => 4.0/(4*x+1) - 4.0/(4*x-1)), 1, n);
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M4 {
def sum(f: Int => Double): (Int, Int) => Double = {
def sumF(a: Int, b: Int): Double =
if (a > b) 0
else f(a) + sumF(a + 1, b);
sumF
}
def sumInts = sum(x => x);
def sumCubes = sum(x => x * x * x);
def sumReciprocals = sum(x => 1.0/x);
def sumPi = (n: Int => 4 + sum(x => 4.0/(4*x+1) - 4.0/(4*x-1))(1, n));
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M5 {
def sum(f: Int => Double): (Int, Int) => Double = { (a, b) =>
if (a > b) 0
else f(a) + sum(f)(a + 1, b)
}
def sumInts = sum(x => x);
def sumCubes = sum(x => x * x * x);
def sumReciprocals = sum(x => 1.0/x);
def sumPi = (n: Int => 4 + sum(x => 4.0/(4*x+1) - 4.0/(4*x-1))(1, n));
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M6 {
def sum(f: Int => Double)(a: Int, b: Int): Double =
if (a > b) 0
else f(a) + sum(f)(a + 1, b);
def sumInts = &sum(x => x);
def sumCubes = &sum(x => x * x * x);
def sumReciprocals = &sum(x => 1.0/x);
def sumPi = (n: Int => 4 + sum(x => 4.0/(4*x+1) - 4.0/(4*x-1))(1, n));
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M7 {
def sum(f: Int => Double)(a: Int, b: Int): Double = {
def iter(a: Int, result: Double): Double =
if (a > b) result
else iter(a + 1, f(a) + result);
iter(a, 0)
}
def sumInts = &sum(x => x);
def sumCubes = &sum(x => x * x * x);
def sumReciprocals = &sum(x => 1.0/x);
def sumPi = (n: Int => 4 + sum(x => 4.0/(4*x+1) - 4.0/(4*x-1))(1, n));
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println;
}
//############################################################################
object M8 {
def product(f: Int => Double)(a: Int, step: Int, b: Int): Double =
if (a > b) 1
else f(a) * product(f)(a + step, step, b);
def productPi = (n: Int => product(x=>4.0*x*x/(2*x-1)/(2*x-1))(1,1,n)/n);
val pi = 2 * product(x => x * x)(2, 2, 40) / product(x => x * x)(1, 2,40)/40;
Console.println("pi = " + productPi(20));
Console.println("pi = " + pi);
Console.println;
}
//############################################################################
object M9 {
def accumulate[t](combiner: (t, t) => t, nullValue: t, f: Int => t,
next: Int => Int)(a: Int, b: Int): t = if (a > b) nullValue
else combiner(f(a), accumulate(combiner, nullValue, f, next)(next(a), b));
def inc(x: Int) = x + 1;
def sum(f: Int => Double): (Int, Int) => Double = accumulate((x: Double, y: Double) => x + y,
0d, f, inc);
def product(f: Int => Double): (Int, Int) => Double = accumulate((x: Double, y: Double) => x * y,
1d, f, inc);
def sumInts = sum(x => x);
def sumCubes = sum(x => x * x * x);
def sumReciprocals = sum(x => 1.0/x);
def sumPi = (n: Int => 4 + sum(x => 4.0/(4*x+1) - 4.0/(4*x-1))(1, n));
def productPi = (n: Int => product(x=>4.0*x*x/(2*x-1)/(2*x-1))(1,n)/n);
val pi = 2*product(x => 2*x*2*x)(1,20)/product(x =>(2*x-1)*(2*x-1))(1,20)/40;
Console.println(sumInts(1,4));
Console.println(sumCubes(1,4));
Console.println(sumReciprocals(1,4));
Console.println(sumCubes(1, 10) + sumReciprocals(10, 20));
Console.println("pi = " + sumPi(20));
Console.println("pi = " + productPi(20));
Console.println("pi = " + pi);
Console.println;
}
//############################################################################
object MA {
val tolerance = 0.0001;
def abs(x: Double) = if (x < 0) -x else x;
def isCloseEnough(x: Double, y: Double) = abs((x - y) / x) < tolerance;
def fixedPoint(f: Double => Double)(firstGuess: Double) = {
def iterate(guess: Double): Double = {
val next = f(guess);
Console.println(next);
if (isCloseEnough(guess, next)) next
else iterate(next)
}
iterate(firstGuess)
}
def sqrt(x: Double) = fixedPoint(y => (y + x / y) / 2)(1.0);
Console.println("sqrt(2) = " + sqrt(2));
Console.println
}
//############################################################################
object MB {
val tolerance = 0.0001;
def abs(x: Double) = if (x < 0) -x else x;
def isCloseEnough(x: Double, y: Double) = abs((x - y) / x) < tolerance;
def fixedPoint(f: Double => Double)(firstGuess: Double) = {
def iterate(guess: Double): Double = {
val next = f(guess);
Console.println(next);
if (isCloseEnough(guess, next)) next
else iterate(next)
}
iterate(firstGuess)
}
def averageDamp(f: Double => Double)(x: Double) = (x + f(x)) / 2;
def sqrt(x: Double) = fixedPoint(averageDamp(y => x/y))(1.0);
Console.println("sqrt(2) = " + sqrt(2));
Console.println
}
//############################################################################
object MC {
def sum(f: Int => Double)(a: Int, b: Int): Double = {
def iter(a: Int, result: Double): Double = {
if (a > b) result
else iter(a + 1, result + f(a))
}
iter(a, 0)
}
def product(f: Int => Double)(a: Int, b: Int): Double = {
def iter(a: Int, result: Double): Double = {
if (a > b) result
else iter(a + 1, result * f(a))
}
iter(a, 1)
}
def factorial(n: Int) = product(x => x)(1 , n);
Console.println(
"1 + 2 + .. + 5 = " + sum(x => x)(1, 5));
Console.println(
"1 * 2 * .. * 5 = " + product(x => x)(1, 5));
Console.println;
Console.println(
"1^2 + 2^2 + .. + 5^2 = " + sum(x => x*x)(1, 5));
Console.println(
"1^2 * 2^2 * .. * 5^2 = " + product(x => x*x)(1, 5));
Console.println;
Console.println(
"factorial(0) = " + factorial(0));
Console.println(
"factorial(1) = " + factorial(1));
Console.println(
"factorial(2) = " + factorial(2));
Console.println(
"factorial(3) = " + factorial(3));
Console.println(
"factorial(4) = " + factorial(4));
Console.println(
"factorial(5) = " + factorial(5));
Console.println;
}
//############################################################################
object MD {
def reduce(op: (Double,Double) => Double, zero:Double)(f: Int => Double)(a: Int,b: Int): Double = {
def iter(a: Int, result: Double): Double = {
if (a > b) result
else iter(a + 1, op(result, f(a)))
}
iter(a, zero)
}
def plus (x:Double,y:Double) = x+y;
val sum: (Int => Double) => (Int, Int) => Double = reduce(plus , 0);
def times(x:Double,y:Double) = x*y;
val product: (Int => Double) => (Int, Int) => Double = reduce(times, 1);
def factorial(n: Int) = product(x => x)(1 , n);
Console.println(
"1 + 2 + .. + 5 = " + sum(x => x)(1, 5));
Console.println(
"1 * 2 * .. * 5 = " + product(x => x)(1, 5));
Console.println;
Console.println(
"1^2 + 2^2 + .. + 5^2 = " + sum(x => x*x)(1, 5));
Console.println(
"1^2 * 2^2 * .. * 5^2 = " + product(x => x*x)(1, 5));
Console.println;
Console.println(
"factorial(0) = " + factorial(0));
Console.println(
"factorial(1) = " + factorial(1));
Console.println(
"factorial(2) = " + factorial(2));
Console.println(
"factorial(3) = " + factorial(3));
Console.println(
"factorial(4) = " + factorial(4));
Console.println(
"factorial(5) = " + factorial(5));
Console.println;
}
//############################################################################
object ME {
def reduce(op: (Double,Double) => Double, zero:Double)(f: Int => Double)(a: Int,b: Int): Double = {
def iter(a: Int, result: Double): Double = {
if (a > b) result
else iter(a + 1, op(result, f(a)))
}
iter(a, zero)
}
def sum: (Int => Double) => (Int, Int) => Double = reduce((x,y) => x + y, 0);
def product: (Int => Double) => (Int, Int) => Double = reduce((x,y) => x * y, 1);
def factorial(n: Int) = product(x => x)(1 , n);
Console.println(
"1 + 2 + .. + 5 = " + sum(x => x)(1, 5));
Console.println(
"1 * 2 * .. * 5 = " + product(x => x)(1, 5));
Console.println;
Console.println(
"1^2 + 2^2 + .. + 5^2 = " + sum(x => x*x)(1, 5));
Console.println(
"1^2 * 2^2 * .. * 5^2 = " + product(x => x*x)(1, 5));
Console.println;
Console.println(
"factorial(0) = " + factorial(0));
Console.println(
"factorial(1) = " + factorial(1));
Console.println(
"factorial(2) = " + factorial(2));
Console.println(
"factorial(3) = " + factorial(3));
Console.println(
"factorial(4) = " + factorial(4));
Console.println(
"factorial(5) = " + factorial(5));
Console.println;
}
//############################################################################
object MF {
def fib(x: Int): Int =
if (x <= 1) x
else fib(x - 2) + fib(x - 1);
Console.println("fib(0) = " + fib(0));
Console.println("fib(1) = " + fib(1));
Console.println("fib(2) = " + fib(2));
Console.println("fib(3) = " + fib(3));
Console.println("fib(4) = " + fib(4));
Console.println("fib(5) = " + fib(5));
Console.println("fib(6) = " + fib(6));
Console.println("fib(7) = " + fib(7));
Console.println("fib(8) = " + fib(8));
Console.println("fib(9) = " + fib(9));
}
//############################################################################
object MG {
def fib(x: Int) = {
def loop(n: Int, prev: Int, fibn: Int): Int =
if (n == x) fibn
else loop(n + 1, fibn, fibn + prev);
if (x == 0) 0 else loop(1, 0, 1)
}
Console.println("fib(0) = " + fib(0));
Console.println("fib(1) = " + fib(1));
Console.println("fib(2) = " + fib(2));
Console.println("fib(3) = " + fib(3));
Console.println("fib(4) = " + fib(4));
Console.println("fib(5) = " + fib(5));
Console.println("fib(6) = " + fib(6));
Console.println("fib(7) = " + fib(7));
Console.println("fib(8) = " + fib(8));
Console.println("fib(9) = " + fib(9));
}
//############################################################################
object MH {
def power(x: Double, y: Int): Double =
if (y <= 0) 1
else if (y % 2 == 0) power(x * x, y / 2)
else x * power(x, y - 1);
Console.println("power(0,0) = " + power(0,0));
Console.println("power(0,1) = " + power(0,1));
Console.println("power(0,2) = " + power(0,2));
Console.println("power(0,3) = " + power(0,3));
Console.println("power(0,4) = " + power(0,4));
Console.println("power(0,5) = " + power(0,5));
Console.println("power(0,6) = " + power(0,6));
Console.println("power(0,7) = " + power(0,7));
Console.println("power(0,8) = " + power(0,8));
Console.println;
Console.println("power(1,0) = " + power(1,0));
Console.println("power(1,1) = " + power(1,1));
Console.println("power(1,2) = " + power(1,2));
Console.println("power(1,3) = " + power(1,3));
Console.println("power(1,4) = " + power(1,4));
Console.println("power(1,5) = " + power(1,5));
Console.println("power(1,6) = " + power(1,6));
Console.println("power(1,7) = " + power(1,7));
Console.println("power(1,8) = " + power(1,8));
Console.println;
Console.println("power(2,0) = " + power(2,0));
Console.println("power(2,1) = " + power(2,1));
Console.println("power(2,2) = " + power(2,2));
Console.println("power(2,3) = " + power(2,3));
Console.println("power(2,4) = " + power(2,4));
Console.println("power(2,5) = " + power(2,5));
Console.println("power(2,6) = " + power(2,6));
Console.println("power(2,7) = " + power(2,7));
Console.println("power(2,8) = " + power(2,8));
Console.println;
Console.println("power(3,0) = " + power(3,0));
Console.println("power(3,1) = " + power(3,1));
Console.println("power(3,2) = " + power(3,2));
Console.println("power(3,3) = " + power(3,3));
Console.println("power(3,4) = " + power(3,4));
Console.println("power(3,5) = " + power(3,5));
Console.println("power(3,6) = " + power(3,6));
Console.println("power(3,7) = " + power(3,7));
Console.println("power(3,8) = " + power(3,8));
Console.println;
Console.println("power(4,0) = " + power(4,0));
Console.println("power(4,1) = " + power(4,1));
Console.println("power(4,2) = " + power(4,2));
Console.println("power(4,3) = " + power(4,3));
Console.println("power(4,4) = " + power(4,4));
Console.println("power(4,5) = " + power(4,5));
Console.println("power(4,6) = " + power(4,6));
Console.println("power(4,7) = " + power(4,7));
Console.println("power(4,8) = " + power(4,8));
Console.println;
Console.println("power(5,0) = " + power(5,0));
Console.println("power(5,1) = " + power(5,1));
Console.println("power(5,2) = " + power(5,2));
Console.println("power(5,3) = " + power(5,3));
Console.println("power(5,4) = " + power(5,4));
Console.println("power(5,5) = " + power(5,5));
Console.println("power(5,6) = " + power(5,6));
Console.println("power(5,7) = " + power(5,7));
Console.println("power(5,8) = " + power(5,8));
Console.println;
}
//############################################################################
object Test {
def main(args: Array[String]): Unit = {
M0;
M1;
M2;
M3;
M4;
M5;
M6;
M7;
M8;
M9;
MA;
MB;
MC;
MD;
ME;
MF;
MG;
MH;
()
}
}
//############################################################################