Package org.hipparchus.dfp
Class DfpMath
- java.lang.Object
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- org.hipparchus.dfp.DfpMath
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public class DfpMath extends Object
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Method Summary
All Methods Static Methods Concrete Methods 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/8.0protected 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)
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.
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Method Detail
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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 belonga
- string representation to split- Returns:
- an array of two
Dfp
which sum is a
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split
protected static Dfp[] split(Dfp a)
- Parameters:
a
- number to split- Returns:
- two elements array containing the split number
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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 formb
- second factor of the multiplication, in split form- Returns:
- a × b, in split form
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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 formb
- divisor, in split form- Returns:
- a / b, in split form
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splitPow
protected static Dfp splitPow(Dfp[] base, int a)
Raise a split base to the a power.- Parameters:
base
- number to raisea
- power- Returns:
- basea
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pow
public static Dfp pow(Dfp base, int a)
Raises base to the power a by successive squaring.- Parameters:
base
- number to raisea
- power- Returns:
- basea
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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
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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
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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)
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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)
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pow
public static Dfp pow(Dfp x, Dfp y)
Computes x to the y power.Uses the following method:
- Set u = rint(y), v = y-u
- Compute a = v * ln(x)
- Compute b = rint( a/ln(2) )
- Compute c = a - b*ln(2)
- xy = xu * 2b * ec
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 raisedy
- power to which base should be raised- Returns:
- xy
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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)
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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)
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sin
public static Dfp sin(Dfp a)
computes the sine of the argument.- Parameters:
a
- number from which sine is desired- Returns:
- sin(a)
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cos
public static Dfp cos(Dfp a)
computes the cosine of the argument.- Parameters:
a
- number from which cosine is desired- Returns:
- cos(a)
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tan
public static Dfp tan(Dfp a)
computes the tangent of the argument.- Parameters:
a
- number from which tangent is desired- Returns:
- tan(a)
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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)
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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)
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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)
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