public static BigRational operator *(BigRational num1, BigRational num2)
{
// a/b * c/d = ac/bd
BigRational retVal = new BigRational(num1.Numerator * num2.Numerator, num1.Denominator * num2.Denominator);
retVal.Reduce();
return retVal;
}
/// <summary>
/// Computes pi to the specified number of digits using the Machin Formula.
/// </summary>
/// <param name="numDigits">The number of digits to which pi should be calculated.</param>
/// <returns></returns>
public static BigRational GetPi(BigInteger numDigits)
{
// pi = 16 arctan(1/5) − 4 arctan(1/239)
BigRational oneFifth = new BigRational(1,5);
BigRational oneTwoThirtyNine = new BigRational(1, 239);
BigRational arcTanOneFifth = PiCalc.ArcTangent(oneFifth, PiCalc.GetConversionIterations(numDigits + 1, oneFifth)); // Start computing
BigRational arcTanOneTwoThirtyNine = PiCalc.ArcTangent(oneTwoThirtyNine, PiCalc.GetConversionIterations(numDigits + 1, oneTwoThirtyNine));
return (arcTanOneFifth * 16) - (arcTanOneTwoThirtyNine * 4);
}
/// <summary>
/// Calculates an inverse tangent (arctan).
/// </summary>
/// <param name="input">The number of which the arctan should be computed.</param>
/// <param name="iterations">The number of iterations of arctangent to compute.</param>
/// <returns></returns>
public static BigRational ArcTangent(BigRational input, BigInteger iterations)
{
BigRational retVal = input;
for (BigInteger i = 1; i < iterations; i++)
{
// arctan(x) = x - x^3/3 + x^5/5 ...
// = summation(n->infinity) (-1)^(n) * x^(2n+1)/(2n+1)
BigRational powRat = input.Pow((2 * i) + 1);
retVal += new BigRational(powRat.Numerator * (BigInteger)Math.Pow(-1d, (double)(i)), ((2 * i) + 1) * powRat.Denominator); // Code representation of the above formula.
if (i % 100 == 0)
{
Console.WriteLine("ArcTangent {0}: {1}/{2} iterations complete.", input, i, iterations); // Status update.
}
}
return retVal;
}
/// <summary>
/// Determines the number of iterations that must be
/// </summary>
/// <param name="digits">The number of correct digits </param>
/// <returns></returns>
public static BigInteger GetConversionIterations(BigInteger digits, BigRational q)
{
// http://turner.faculty.swau.edu/mathematics/materialslibrary/pi/calterms.html
// iterations ~= digits / (2 log10 (1/number to arctan))
return (BigInteger)((double)digits / (2 * Math.Log10((double)q.GetReciprocal())));
}
// Operator overloading
public static BigRational operator +(BigRational num1, BigRational num2)
{
if (num1.Denominator == num2.Denominator)
{
// The denominators are the same; just add the numerators and return.
return new BigRational(num1.Numerator + num2.Numerator, num1.Denominator);
}
else
{
// Make the denominator the same, reduce it, and return it.
BigRational retval = new BigRational ((num1.Numerator * num2.Denominator) + (num2.Numerator * num1.Denominator), num1.Denominator * num2.Denominator);
retval.Reduce();
return retval;
}
}
// I don't want to call it scalar multiplication... but it's scalar multiplication if you consider the BigRational a 1x2 matrix.
public static BigRational operator *(BigRational num1, BigInteger num2)
{
BigRational retVal = new BigRational(num1.Numerator * num2, num1.Denominator);
retVal.Reduce();
return retVal;
}
/// <summary>
/// Returns the BigRational raised to the specified power.
/// </summary>
/// <param name="power">An integer greater than or equal to one.</param>
/// <returns></returns>
public BigRational Pow(BigInteger power)
{
// Error checking
if (power < 1) throw new ArgumentException("BigRational.Pow only supports integer powers greater than or equal to one.");
BigRational retVal = new BigRational(Numerator, Denominator); // Create a copy of the current BigRational
for (BigInteger i = 1; i < power; i++) // Multiply the copy by itself power - 1 number of times
{
retVal.Numerator *= Numerator;
retVal.Denominator *= Denominator;
}
retVal.Reduce();
return retVal;
}
public static BigRational operator -(BigRational num1, BigRational num2)
{
// Same as + but we subtract here...
if (num1.Denominator == num2.Denominator)
{
// The denominators are the same; just add the numerators and return.
return new BigRational(num1.Numerator - num2.Numerator, num1.Denominator);
}
else
{
// Make the denominator the same, reduce it, and return it.
BigRational retval = new BigRational((num1.Numerator * num2.Denominator) - (num2.Numerator * num1.Denominator), num1.Denominator * num2.Denominator);
retval.Reduce();
return retval;
}
}
