org.apache.commons.math3.dfp
Class DfpMath

java.lang.Object
  extended by org.apache.commons.math3.dfp.DfpMath

public class DfpMath
extends Object

Mathematical routines for use with Dfp. The constants are defined in DfpField

Since:
2.2
Version:
$Id: DfpMath.java 1462503 2013-03-29 15:48:27Z luc $

Method Summary
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 x, Dfp y)
          Computes x to the y power.
static Dfp pow(Dfp base, int a)
          Raises base to the power a by successive squaring.
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
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Method Detail

split

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

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

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

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

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

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

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

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

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

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

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

Parameters:
x - base to be raised
y - power to which base should be raised
Returns:
xy

sinInternal

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

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

public static Dfp sin(Dfp a)
computes the sine of the argument.

Parameters:
a - number from which sine is desired
Returns:
sin(a)

cos

public static Dfp cos(Dfp a)
computes the cosine of the argument.

Parameters:
a - number from which cosine is desired
Returns:
cos(a)

tan

public static Dfp tan(Dfp a)
computes the tangent of the argument.

Parameters:
a - number from which tangent is desired
Returns:
tan(a)

atanInternal

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

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

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

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)


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