fr.cnes.sirius.patrius.math.dfp

Class DfpMath

java.lang.Object

fr.cnes.sirius.patrius.math.dfp.DfpMath

public final class DfpMath
extends Object

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

Since:

2.2

Version:

$Id: DfpMath.java 18108 2017-10-04 06:45:27Z bignon $

Method Summary

Methods

Modifier and Type	Method and Description
static Dfp	acos(Dfp aIn) 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 aIn) 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 aIn) 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 aIn)

Raise a split base to the a power.

Parameters:

base - number to raise

aIn - power

Returns:

base^a

pow

public static Dfp pow(Dfp base,
      int aIn)

Raises base to the power a by successive squaring.

Parameters:

base - number to raise

aIn - power

Returns:

base^a

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 e^m. We return e*ⁿ × e^m

Parameters:

a - power at which e should be raised

Returns:

e^a

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:

e^a

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. x^y = x^u * 2^b * e^c

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:

x^y

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 aIn)

computes the arc-cosine of the argument.

Parameters:

aIn - number from which arc-cosine is desired

Returns:

acos(a)