// File: solvers.cxx 
// Written by: Michael Main (main@colorado.edu) on Sep 16, 2011
// The purpose of this program is to show
// how to create functions that solve equations

//-----------------------------------------------------------------------------
// Directives:
#include <cassert>
#include <iostream>
#include <cmath>
#include <cstdlib>
#include <graphics.h>
using namespace std;
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
// Function Prototypes.

// The dist function finds the distance between two points (x1, y1) and
// (x2, y2) with integer coordinates, for instance two pixels on the screen.   
double distance(double x1, double y1, double x2, double y2);

// This function computes an approximation to e by adding the reciprocals
// of the first n factorials.
double e_approximation(int s);

// The fahr_to_cell function converts Fahrenheit temperature f to Celsius.
// Precondition: f >= -459.67
double fahr_to_cel (double f);

// This function computes an approximation to pi using a n-random points
// that are dropped on a square of size s by s.
// Note: This is a different algorithm than the one you use in Homework 3!
// Precondition: n > 0
double pi_approximation(int s, int n);

// Use an iterative algorithm to solve x = a*x + b. Again, do not use this
// in real life (it is too slow and it works only when 0 <= a < 1).
// But it does illustrate iteration nicely.
// Precondition: 0 <= a < 1.
double solution(double a, double b);

// Use an iterative algorithm along with the sqrt function to find the
// cube root of x.
// Precondition: 0 <= x
double third_root(double z);
//-----------------------------------------------------------------------------

 
//-----------------------------------------------------------------------------
int main()
{
    //double x;

    // Illustrate getting input from the console input:
    //cout << "Please enter a value: ";
    //cin >> x;
    //cout << "You chose " << x << endl;

    // Use that input as the argument to several functions:
    //cout << "Converted to fahrenheit: " << fahr_to_cel(x) << endl;
    //cout << "Sine of that value is: " << sin(x) << endl;
    //cout << "Absolute value is: " << abs(x) << endl;

    // Now just test the other functions with hard-coded arguments:
    cout << "distance: "
	 << distance(50, 70, 250, 20) << endl;
    cout << "solution: "
	 << solution(0.5, 4.0) << endl;
    cout << "cube root: "
	 << third_root(1000) << endl;
    cout << "Pi estimate: "
	 << pi_approximation(400, 20000) << endl;
    //cout << "Approximation of e: "
    //     << e_approximation(400) << endl;

    return EXIT_SUCCESS;
}
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
double distance(double x1, double y1, double x2, double y2)
{
    return pow(pow(x1-x2, 2) + pow(y1-y2, 2), 0.5);
}
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
double e_approximation(int n)
{
    double denominator = 1.0;
    double sum = 0.0;
    int many_terms = 0;

    while (many_terms < n)
    {
	sum += 1.0/denominator;
	++many_terms;
	denominator = denominator*many_terms;
    }

    return sum;
}
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
double fahr_to_cel (double f)
{
    return 0;
}
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
double pi_approximation(int s, int n)
{
    int count = 0; //How many points we have looked at so far
    int count_red = 0; // How many red points have we seen
    int px, py; // Coordinates of a pixel
    
    initwindow(s, s, "PI!!!");
    setfillstyle(SOLID_FILL, RED);
    setcolor(RED);
    fillellipse(s/2, s/2, s/2, s/2);

    while (count < n)
    {
	px = rand( ) % s; // Pick a random x pixel from 0 to s-1
	py = rand( ) % s; // Pick a random y pixel

	if (getpixel(px, py) == RED)
	{
	    ++count_red;
	}
	++count;
    }

    closegraph();
    return 4.0*count_red/count;
}
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
double solution(double a, double b)
{
    assert(a != 1);
    //return b / (1-a);

    double x = 42;
    double right = a*x + b;
    double diff = abs(right - x);
    while (diff > 0.0001)
    {
	x = right;
	right = a*x + b;
	diff = abs(right - x);
    }
    return x;
    
}
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
double third_root(double z)
{
    double x = 42;
    double right = sqrt(z/x);
    double diff = abs(right - x);
    while (diff > 0.0001)
    {
	x = right;
	right = sqrt(z/x);
	diff = abs(right - x);
    }
    return x;
}
//-----------------------------------------------------------------------------