Package org.hipparchus.dfp

Class DfpMath

java.lang.Object
org.hipparchus.dfp.DfpMath
public class DfpMath extends Object
Mathematical routines for use with Dfp. The constants are defined in DfpField

  • Method Summary Link icon

    Modifier and Type
    Method
    Description
    static Dfp
    acos(Dfp a)
    computes the arc-cosine of the argument.
    static Dfp
    asin(Dfp a)
    computes the arc-sine of the argument.
    static Dfp
    atan(Dfp a)
    computes the arc tangent of the argument Uses the typical taylor series but may reduce arguments using the following identity tan(x+y) = (tan(x) + tan(y)) / (1 - tan(x)*tan(y)) since tan(PI/8) = sqrt(2)-1, atan(x) = atan( (x - sqrt(2) + 1) / (1+x*sqrt(2) - x) + PI/8.0
    protected static Dfp
    atanInternal(Dfp a)
    computes the arc-tangent of the argument.
    static Dfp
    cos(Dfp a)
    computes the cosine of the argument.
    protected static Dfp
    cosInternal(Dfp[] a)
    Computes cos(a) Used when 0 < a < pi/4.
    static Dfp
    exp(Dfp a)
    Computes e to the given power.
    protected static Dfp
    expInternal(Dfp a)
    Computes e to the given power.
    static Dfp
    log(Dfp a)
    Returns the natural logarithm of a.
    protected static Dfp[]
    logInternal(Dfp[] a)
    Computes the natural log of a number between 0 and 2.
    static Dfp
    pow(Dfp base, int a)
    Raises base to the power a by successive squaring.
    static Dfp
    pow(Dfp x, Dfp y)
    Computes x to the y power.
    static Dfp
    sin(Dfp a)
    computes the sine of the argument.
    protected static Dfp
    sinInternal(Dfp[] a)
    Computes sin(a) Used when 0 < a < pi/4.
    protected static Dfp[]
    split(Dfp a)
    Splits a Dfp into 2 Dfp's such that their sum is equal to the input Dfp.
    protected static Dfp[]
    split(DfpField field, String a)
    Breaks a string representation up into two dfp's.
    protected static Dfp[]
    splitDiv(Dfp[] a, Dfp[] b)
    Divide two numbers that are split in to two pieces that are meant to be added together.
    protected static Dfp[]
    splitMult(Dfp[] a, Dfp[] b)
    Multiply two numbers that are split in to two pieces that are meant to be added together.
    protected static Dfp
    splitPow(Dfp[] base, int a)
    Raise a split base to the a power.
    static Dfp
    tan(Dfp a)
    computes the tangent of the argument.

    Methods inherited from class java.lang.Object Link icon

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Method Details Link icon

    • split Link icon

      protected static Dfp[] split(DfpField field, String a)
      Breaks a string representation up into two dfp's.

      The two dfp are such that the sum of them is equivalent to the input string, but has higher precision than using a single dfp. This is useful for improving accuracy of exponentiation and critical multiplies.

      Parameters:
      field - field to which the Dfp must belong
      a - string representation to split
      Returns:
      an array of two Dfp which sum is a

    • split Link icon

      protected static Dfp[] split(Dfp a)
      Splits a Dfp into 2 Dfp's such that their sum is equal to the input Dfp.
      Parameters:
      a - number to split
      Returns:
      two elements array containing the split number

    • splitMult Link icon

      protected static Dfp[] splitMult(Dfp[] a, Dfp[] b)
      Multiply two numbers that are split in to two pieces that are meant to be added together. Use binomial multiplication so ab = a0 b0 + a0 b1 + a1 b0 + a1 b1 Store the first term in result0, the rest in result1
      Parameters:
      a - first factor of the multiplication, in split form
      b - second factor of the multiplication, in split form
      Returns:
      a × b, in split form

    • splitDiv Link icon

      protected static Dfp[] splitDiv(Dfp[] a, Dfp[] b)
      Divide two numbers that are split in to two pieces that are meant to be added together. Inverse of split multiply above: (a+b) / (c+d) = (a/c) + ( (bc-ad)/(c**2+cd) )
      Parameters:
      a - dividend, in split form
      b - divisor, in split form
      Returns:
      a / b, in split form

    • splitPow Link icon

      protected static Dfp splitPow(Dfp[] base, int a)
      Raise a split base to the a power.
      Parameters:
      base - number to raise
      a - power
      Returns:
      basea

    • pow Link icon

      public static Dfp pow(Dfp base, int a)
      Raises base to the power a by successive squaring.
      Parameters:
      base - number to raise
      a - power
      Returns:
      basea

    • exp Link icon

      public static Dfp exp(Dfp a)
      Computes e to the given power. a is broken into two parts, such that a = n+m where n is an integer. We use pow() to compute en and a Taylor series to compute em. We return e*n × em
      Parameters:
      a - power at which e should be raised
      Returns:
      ea

    • expInternal Link icon

      protected static Dfp expInternal(Dfp a)
      Computes e to the given power. Where -1 < a < 1. Use the classic Taylor series. 1 + x**2/2! + x**3/3! + x**4/4! ...
      Parameters:
      a - power at which e should be raised
      Returns:
      ea

    • log Link icon

      public static Dfp log(Dfp a)
      Returns the natural logarithm of a. a is first split into three parts such that a = (10000^h)(2^j)k. ln(a) is computed by ln(a) = ln(5)*h + ln(2)*(h+j) + ln(k) k is in the range 2/3 < k < 4/3 and is passed on to a series expansion.
      Parameters:
      a - number from which logarithm is requested
      Returns:
      log(a)

    • logInternal Link icon

      protected static Dfp[] logInternal(Dfp[] a)
      Computes the natural log of a number between 0 and 2. Let f(x) = ln(x), We know that f'(x) = 1/x, thus from Taylor's theorum we have: ----- n+1 n f(x) = \ (-1) (x - 1) / ---------------- for 1 <= n <= infinity ----- n or 2 3 4 (x-1) (x-1) (x-1) ln(x) = (x-1) - ----- + ------ - ------ + ... 2 3 4 alternatively, 2 3 4 x x x ln(x+1) = x - - + - - - + ... 2 3 4 This series can be used to compute ln(x), but it converges too slowly. If we substitute -x for x above, we get 2 3 4 x x x ln(1-x) = -x - - - - - - + ... 2 3 4 Note that all terms are now negative. Because the even powered ones absorbed the sign. Now, subtract the series above from the previous one to get ln(x+1) - ln(1-x). Note the even terms cancel out leaving only the odd ones 3 5 7 2x 2x 2x ln(x+1) - ln(x-1) = 2x + --- + --- + ---- + ... 3 5 7 By the property of logarithms that ln(a) - ln(b) = ln (a/b) we have: 3 5 7 x+1 / x x x \ ln ----- = 2 * | x + ---- + ---- + ---- + ... | x-1 \ 3 5 7 / But now we want to find ln(a), so we need to find the value of x such that a = (x+1)/(x-1). This is easily solved to find that x = (a-1)/(a+1).
      Parameters:
      a - number from which logarithm is requested, in split form
      Returns:
      log(a)

    • pow Link icon

      public static Dfp pow(Dfp x, Dfp y)
      Computes x to the y power.

      Uses the following method:

      1. Set u = rint(y), v = y-u
      2. Compute a = v * ln(x)
      3. Compute b = rint( a/ln(2) )
      4. Compute c = a - b*ln(2)
      5. xy = xu * 2b * ec
      if |y| > 1e8, then we compute by exp(y*ln(x))

      Special Cases

      • if y is 0.0 or -0.0 then result is 1.0
      • if y is 1.0 then result is x
      • if y is NaN then result is NaN
      • if x is NaN and y is not zero then result is NaN
      • if |x| > 1.0 and y is +Infinity then result is +Infinity
      • if |x| < 1.0 and y is -Infinity then result is +Infinity
      • if |x| > 1.0 and y is -Infinity then result is +0
      • if |x| < 1.0 and y is +Infinity then result is +0
      • if |x| = 1.0 and y is +/-Infinity then result is NaN
      • if x = +0 and y > 0 then result is +0
      • if x = +Inf and y < 0 then result is +0
      • if x = +0 and y < 0 then result is +Inf
      • if x = +Inf and y > 0 then result is +Inf
      • if x = -0 and y > 0, finite, not odd integer then result is +0
      • if x = -0 and y < 0, finite, and odd integer then result is -Inf
      • if x = -Inf and y > 0, finite, and odd integer then result is -Inf
      • if x = -0 and y < 0, not finite odd integer then result is +Inf
      • if x = -Inf and y > 0, not finite odd integer then result is +Inf
      • if x < 0 and y > 0, finite, and odd integer then result is -(|x|y)
      • if x < 0 and y > 0, finite, and not integer then result is NaN
      Parameters:
      x - base to be raised
      y - power to which base should be raised
      Returns:
      xy

    • sinInternal Link icon

      protected static Dfp sinInternal(Dfp[] a)
      Computes sin(a) Used when 0 < a < pi/4. Uses the classic Taylor series. x - x**3/3! + x**5/5! ...
      Parameters:
      a - number from which sine is desired, in split form
      Returns:
      sin(a)

    • cosInternal Link icon

      protected static Dfp cosInternal(Dfp[] a)
      Computes cos(a) Used when 0 < a < pi/4. Uses the classic Taylor series for cosine. 1 - x**2/2! + x**4/4! ...
      Parameters:
      a - number from which cosine is desired, in split form
      Returns:
      cos(a)

    • sin Link icon

      public static Dfp sin(Dfp a)
      computes the sine of the argument.
      Parameters:
      a - number from which sine is desired
      Returns:
      sin(a)

    • cos Link icon

      public static Dfp cos(Dfp a)
      computes the cosine of the argument.
      Parameters:
      a - number from which cosine is desired
      Returns:
      cos(a)

    • tan Link icon

      public static Dfp tan(Dfp a)
      computes the tangent of the argument.
      Parameters:
      a - number from which tangent is desired
      Returns:
      tan(a)

    • atanInternal Link icon

      protected static Dfp atanInternal(Dfp a)
      computes the arc-tangent of the argument.
      Parameters:
      a - number from which arc-tangent is desired
      Returns:
      atan(a)

    • atan Link icon

      public static Dfp atan(Dfp a)
      computes the arc tangent of the argument Uses the typical taylor series but may reduce arguments using the following identity tan(x+y) = (tan(x) + tan(y)) / (1 - tan(x)*tan(y)) since tan(PI/8) = sqrt(2)-1, atan(x) = atan( (x - sqrt(2) + 1) / (1+x*sqrt(2) - x) + PI/8.0
      Parameters:
      a - number from which arc-tangent is desired
      Returns:
      atan(a)

    • asin Link icon

      public static Dfp asin(Dfp a)
      computes the arc-sine of the argument.
      Parameters:
      a - number from which arc-sine is desired
      Returns:
      asin(a)

    • acos Link icon

      public static Dfp acos(Dfp a)
      computes the arc-cosine of the argument.
      Parameters:
      a - number from which arc-cosine is desired
      Returns:
      acos(a)