Apache Commons Math 3.6.1 · org.apache.commons.math3.dfp.DfpMath

Modifier and Type	Field and Description
private static final String	POW_TRAP Name for traps triggered by pow.

Access	Constructor and Description
private	DfpMath() Private Constructor.

Method Summary

Modifier and Type	Method and Description
public static Dfp	Returns: acos(a) acos(Dfp number from which arc-cosine is desired a) computes the arc-cosine of the argument.
public static Dfp	Returns: asin(a) asin(final Dfp number from which arc-sine is desired a) computes the arc-sine of the argument.
public static Dfp	Returns: atan(a) atan(final Dfp number from which arc-tangent is desired 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+xsqrt(2) - x) + PI/8.0
protected static Dfp	Returns: atan(a) atanInternal(final Dfp number from which arc-tangent is desired a) computes the arc-tangent of the argument.
public static Dfp	Returns: cos(a) cos(Dfp number from which cosine is desired a) computes the cosine of the argument.
protected static Dfp	Returns: cos(a) cosInternal(Dfp[] number from which cosine is desired, in split form a) Computes cos(a) Used when 0 < a < pi/4.
public static Dfp	Returns: e^a exp(final Dfp power at which e should be raised a) Computes e to the given power.
protected static Dfp	Returns: e^a expInternal(final Dfp power at which e should be raised a) Computes e to the given power.
public static Dfp	Returns: log(a) log(Dfp number from which logarithm is requested a) Returns the natural logarithm of a.
protected static Dfp[]	Returns: log(a) logInternal(final Dfp[] number from which logarithm is requested, in split form a) Computes the natural log of a number between 0 and 2.
public static Dfp	Returns: base^a pow(Dfp number to raise base, int power a) Raises base to the power a by successive squaring.
public static Dfp	Returns: x^y pow(Dfp base to be raised x, final Dfp power to which base should be raised y) Computes x to the y power.
public static Dfp	Returns: sin(a) sin(final Dfp number from which sine is desired a) computes the sine of the argument.
protected static Dfp	Returns: sin(a) sinInternal(Dfp[] number from which sine is desired, in split form a) Computes sin(a) Used when 0 < a < pi/4.
protected static Dfp[]	Returns: an array of two `Dfp` which sum is a split(final DfpField field to which the Dfp must belong field, final String string representation to split a) Breaks a string representation up into two dfp's.
protected static Dfp[]	Returns: two elements array containing the split number split(final Dfp number to split a) Splits a `Dfp` into 2 `Dfp`'s such that their sum is equal to the input `Dfp`.
protected static Dfp[]	Returns: a / b, in split form splitDiv(final Dfp[] dividend, in split form a, final Dfp[] divisor, in split form b) Divide two numbers that are split in to two pieces that are meant to be added together.
protected static Dfp[]	Returns: a × b, in split form splitMult(final Dfp[] first factor of the multiplication, in split form a, final Dfp[] second factor of the multiplication, in split form b) Multiply two numbers that are split in to two pieces that are meant to be added together.
protected static Dfp	Returns: base^a splitPow(final Dfp[] number to raise base, int power a) Raise a split base to the a power.
public static Dfp	Returns: tan(a) tan(final Dfp number from which tangent is desired a) computes the tangent of the argument.

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

POW_TRAP	back to summary
private static final String POW_TRAP Name for traps triggered by pow.

DfpMath	back to summary
private DfpMath() Private Constructor.

Method Detail

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

acos

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public static Dfp acos(Dfp a)

computes the arc-cosine of the argument.

Parameters

a:Dfp: number from which arc-cosine is desired

Returns:Dfp

acos(a)

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

asin

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public static Dfp asin(final Dfp a)

computes the arc-sine of the argument.

Parameters

a:Dfp: number from which arc-sine is desired

Returns:Dfp

asin(a)

atan	back to summary
public static Dfp atan(final 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+xsqrt(2) - x) + PI/8.0 Parameters a:Dfp number from which arc-tangent is desired Returns:Dfp atan(a)

atan

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public static Dfp atan(final 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:Dfp: number from which arc-tangent is desired

Returns:Dfp

atan(a)

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

atanInternal

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protected static Dfp atanInternal(final Dfp a)

computes the arc-tangent of the argument.

Parameters

a:Dfp: number from which arc-tangent is desired

Returns:Dfp

atan(a)

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

cos

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public static Dfp cos(Dfp a)

computes the cosine of the argument.

Parameters

a:Dfp: number from which cosine is desired

Returns:Dfp

cos(a)

cosInternal	back to summary
protected static Dfp cosInternal(Dfp[] a) Computes cos(a) Used when 0 < a < pi/4. Uses the classic Taylor series for cosine. 1 - x2/2! + x4/4! ... Parameters a:Dfp[] number from which cosine is desired, in split form Returns:Dfp cos(a)

cosInternal

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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:Dfp[]: number from which cosine is desired, in split form

Returns:Dfp

cos(a)

exp	back to summary
public static Dfp exp(final 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:Dfp power at which e should be raised Returns:Dfp e^a

exp

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public static Dfp exp(final 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:Dfp: power at which e should be raised

Returns:Dfp

e^a

expInternal	back to summary
protected static Dfp expInternal(final Dfp a) Computes e to the given power. Where -1 < a < 1. Use the classic Taylor series. 1 + x2/2! + x3/3! + x**4/4! ... Parameters a:Dfp power at which e should be raised Returns:Dfp e^a

expInternal

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protected static Dfp expInternal(final 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:Dfp: power at which e should be raised

Returns:Dfp

e^a

log	back to summary
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:Dfp number from which logarithm is requested Returns:Dfp log(a)

log

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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:Dfp: number from which logarithm is requested

Returns:Dfp

log(a)

logInternal	back to summary
protected static Dfp[] logInternal(final 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:Dfp[] number from which logarithm is requested, in split form Returns:Dfp[] log(a)

logInternal

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protected static Dfp[] logInternal(final 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:Dfp[]: number from which logarithm is requested, in split form

Returns:Dfp[]

log(a)

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

pow

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public static Dfp pow(Dfp base, int a)

Raises base to the power a by successive squaring.

Parameters

base:Dfp: number to raise
a:int: power

Returns:Dfp

base^a

pow	back to summary
public static Dfp pow(Dfp x, final 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 - bln(2) x^y = x^u 2^b * e^c if \|y\| > 1e8, then we compute by exp(yln(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:Dfp base to be raised y:Dfp power to which base should be raised Returns:Dfp x^y

pow

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public static Dfp pow(Dfp x, final 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)
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:Dfp: base to be raised
y:Dfp: power to which base should be raised

Returns:Dfp

x^y

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

sin

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public static Dfp sin(final Dfp a)

computes the sine of the argument.

Parameters

a:Dfp: number from which sine is desired

Returns:Dfp

sin(a)

sinInternal	back to summary
protected static Dfp sinInternal(Dfp[] a) Computes sin(a) Used when 0 < a < pi/4. Uses the classic Taylor series. x - x3/3! + x5/5! ... Parameters a:Dfp[] number from which sine is desired, in split form Returns:Dfp sin(a)

sinInternal

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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:Dfp[]: number from which sine is desired, in split form

Returns:Dfp

sin(a)

split back to summary

split	back to summary
protected static Dfp[] split(final DfpField field, final 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:DfpField field to which the Dfp must belong a:String string representation to split Returns:Dfp[] an array of two `Dfp` which sum is a

protected static Dfp[] split(final DfpField field, final 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:DfpField: field to which the Dfp must belong
a:String: string representation to split

Returns:Dfp[]

an array of two Dfp which sum is a

split back to summary

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

protected static Dfp[] split(final Dfp a)

Splits a Dfp into 2 Dfp's such that their sum is equal to the input Dfp.

Parameters

a:Dfp: number to split

Returns:Dfp[]

two elements array containing the split number

splitDiv	back to summary
protected static Dfp[] splitDiv(final Dfp[] a, final 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:Dfp[] dividend, in split form b:Dfp[] divisor, in split form Returns:Dfp[] a / b, in split form

splitDiv

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protected static Dfp[] splitDiv(final Dfp[] a, final 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:Dfp[]: dividend, in split form
b:Dfp[]: divisor, in split form

Returns:Dfp[]

a / b, in split form

splitMult	back to summary
protected static Dfp[] splitMult(final Dfp[] a, final 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:Dfp[] first factor of the multiplication, in split form b:Dfp[] second factor of the multiplication, in split form Returns:Dfp[] a × b, in split form

splitMult

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protected static Dfp[] splitMult(final Dfp[] a, final 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:Dfp[]: first factor of the multiplication, in split form
b:Dfp[]: second factor of the multiplication, in split form

Returns:Dfp[]

a × b, in split form

splitPow	back to summary
protected static Dfp splitPow(final Dfp[] base, int a) Raise a split base to the a power. Parameters base:Dfp[] number to raise a:int power Returns:Dfp base^a

splitPow

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protected static Dfp splitPow(final Dfp[] base, int a)

Raise a split base to the a power.

Parameters

base:Dfp[]: number to raise
a:int: power

Returns:Dfp

base^a

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

tan

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public static Dfp tan(final Dfp a)

computes the tangent of the argument.

Parameters

a:Dfp: number from which tangent is desired

Returns:Dfp

tan(a)

public Class DfpMath

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