Package org.apfloat

Class AprationalMath

java.lang.Object

org.apfloat.AprationalMath

public class AprationalMath
extends Object

Various mathematical functions for arbitrary precision rational numbers.

Version:

1.14.0

Author:

Mikko Tommila

Method Summary

Modifier and Type	Method	Description
static Aprational	abs(Aprational x)	Absolute value.
static Aprational	bernoulli(long n)	Returns the specified Bernoulli number.
static Aprational	bernoulli(long n, int radix)	Returns the specified Bernoulli number in the given radix.
static Aprational	binomial(Aprational n, Aprational k)	Binomial coefficient.
static Apint[]	continuedFraction(Aprational x, int n)	Generates the first n terms in the continued fraction representation of x.
static Aprational[]	convergents(Aprational x, int n)	Generates the first n convergents corresponding to the continued fraction of x.
static Aprational	copySign(Aprational x, Aprational y)	Copy sign from one argument to another.
static Aprational	harmonicNumber(Apint n)	Harmonic number.
static Aprational	harmonicNumber(Apint n, Apint r)	Generalized harmonic number.
static Aprational	max(Aprational x, Aprational y)	Returns the greater of the two values.
static Aprational	min(Aprational x, Aprational y)	Returns the smaller of the two values.
static Aprational	negate(Aprational x)	Deprecated. Use Aprational.negate().
static Aprational	pochhammer(Aprational x, Apint n)	Pochhammer symbol.
static Aprational	pow(Aprational x, long n)	Integer power.
static Aprational	product(Aprational... x)	Product of numbers.
static Apfloat	round(Aprational x, long precision, RoundingMode roundingMode)	Deprecated. Use roundToPrecision(Aprational,long,RoundingMode).
static Apint	roundToInteger(Aprational x, RoundingMode roundingMode)	Rounds x to integer using the specified rounding mode.
static Aprational	roundToMultiple(Aprational x, Aprational y, RoundingMode roundingMode)	Rounds x to the nearest multiple of y using the specified rounding mode.
static Apfloat	roundToPlaces(Aprational x, long places, RoundingMode roundingMode)	Rounds x to the specified number of places using the specified rounding mode.
static Apfloat	roundToPrecision(Aprational x, long precision, RoundingMode roundingMode)	Rounds the given number to the specified precision with the specified rounding mode.
static Aprational	scale(Aprational x, long scale)	Multiply by a power of the radix.
static Aprational	sum(Aprational... x)	Sum of numbers.

Methods inherited from class java.lang.Object

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

Method Details

pow

public static Aprational pow(Aprational x,
 long n)
                      throws ArithmeticException,
ApfloatRuntimeException

Integer power.

Parameters:

x - Base of the power operator.

n - Exponent of the power operator.

Returns:

x to the n:th power, that is xn.

Throws:

ArithmeticException - If both x and n are zero.

ApfloatRuntimeException

negate

@Deprecated
public static Aprational negate(Aprational x)
                         throws ApfloatRuntimeException

Deprecated.

Use Aprational.negate().

Returns an aprational whose value is -x.

Parameters:

x - The argument.

Returns:

-x.

Throws:

ApfloatRuntimeException

abs

public static Aprational abs(Aprational x)
                      throws ApfloatRuntimeException

Absolute value.

Parameters:

x - The argument.

Returns:

Absolute value of x.

Throws:

ApfloatRuntimeException

copySign

public static Aprational copySign(Aprational x,
 Aprational y)
                           throws ApfloatRuntimeException

Copy sign from one argument to another.

Parameters:

x - The value whose sign is to be adjusted.

y - The value whose sign is to be used.

Returns:

x with its sign changed to match the sign of y.

Throws:

ApfloatRuntimeException

Since:

1.1

scale

public static Aprational scale(Aprational x,
 long scale)
                        throws ApfloatRuntimeException

Multiply by a power of the radix. Note that this method is prone to intermediate overflow errors. Also, scaling by a very large negative number won't result in an underflow and a zero result, but an overflow of the denominator and an exception thrown.

Parameters:

x - The argument.

scale - The scaling factor.

Returns:

x * x.radix()scale.

Throws:

ApfloatRuntimeException

round

@Deprecated
public static Apfloat round(Aprational x,
 long precision,
 RoundingMode roundingMode)
                     throws IllegalArgumentException,
ArithmeticException,
ApfloatRuntimeException

Deprecated.

Use roundToPrecision(Aprational,long,RoundingMode).

Rounds the given number to the specified precision with the specified rounding mode.

Parameters:

x - The number to round.

precision - The precision to round to.

roundingMode - The rounding mode to use.

Returns:

The rounded number.

Throws:

IllegalArgumentException - If precision is less than zero or zero.

ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.

ApfloatRuntimeException

Since:

1.7.0

roundToPrecision

public static Apfloat roundToPrecision(Aprational x,
 long precision,
 RoundingMode roundingMode)
                                throws IllegalArgumentException,
ArithmeticException,
ApfloatRuntimeException

Rounds the given number to the specified precision with the specified rounding mode.

Parameters:

x - The number to round.

precision - The precision to round to.

roundingMode - The rounding mode to use.

Returns:

The rounded number.

Throws:

IllegalArgumentException - If precision is less than zero or zero.

ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.

ApfloatRuntimeException

Since:

1.11.0

roundToInteger

public static Apint roundToInteger(Aprational x,
 RoundingMode roundingMode)
                            throws IllegalArgumentException,
ArithmeticException,
ApfloatRuntimeException

Rounds x to integer using the specified rounding mode.

Parameters:

x - The number to round.

roundingMode - The rounding mode to use.

Returns:

The rounded number.

Throws:

ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.

IllegalArgumentException

ApfloatRuntimeException

Since:

1.11.0

roundToPlaces

public static Apfloat roundToPlaces(Aprational x,
 long places,
 RoundingMode roundingMode)
                             throws IllegalArgumentException,
ArithmeticException,
ApfloatRuntimeException

Rounds x to the specified number of places using the specified rounding mode.

Parameters:

x - The number to round.

places - The number of places to round to (in base 10, the number of decimal places).

roundingMode - The rounding mode to use.

Returns:

The rounded number.

Throws:

ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.

IllegalArgumentException

ApfloatRuntimeException

Since:

1.11.0

roundToMultiple

public static Aprational roundToMultiple(Aprational x,
 Aprational y,
 RoundingMode roundingMode)
                                  throws IllegalArgumentException,
ArithmeticException,
ApfloatRuntimeException

Rounds x to the nearest multiple of y using the specified rounding mode.

Parameters:

x - The number to round.

y - The integer multiple to round to.

roundingMode - The rounding mode to use.

Returns:

The rounded number.

Throws:

ArithmeticException - If rounding is necessary (result is not exact) and rounding mode is RoundingMode.UNNECESSARY.

IllegalArgumentException

ApfloatRuntimeException

Since:

1.11.0

product

public static Aprational product(Aprational... x)
                          throws ApfloatRuntimeException

Product of numbers. This method may perform significantly better than simply multiplying the numbers sequentially.

If there are no arguments, the return value is 1.

Parameters:

x - The argument(s).

Returns:

The product of the given numbers.

Throws:

ApfloatRuntimeException

Since:

1.3

sum

public static Aprational sum(Aprational... x)
                      throws ApfloatRuntimeException

Sum of numbers. This method may perform significantly better than simply adding the numbers sequentially.

If there are no arguments, the return value is 0.

Parameters:

x - The argument(s).

Returns:

The sum of the given numbers.

Throws:

ApfloatRuntimeException

Since:

1.3

continuedFraction

public static Apint[] continuedFraction(Aprational x,
 int n)

Generates the first n terms in the continued fraction representation of x.

Note that the result length might be less than n, depending on the input value.

Parameters:

x - The number whose continued fraction terms should be generated.

n - The maximum number of terms to generate.

Returns:

The continued fraction.

Throws:

IllegalArgumentException - If n is less than one.

Since:

1.12.0

convergents

public static Aprational[] convergents(Aprational x,
 int n)

Generates the first n convergents corresponding to the continued fraction of x.

Note that the result length might be less than n, depending on the input value.

Parameters:

x - The number whose continued fraction convergents should be generated.

n - The maximum number of convergents to generate.

Returns:

The convergents.

Throws:

IllegalArgumentException - If n is less than one.

Since:

1.12.0

max

public static Aprational max(Aprational x,
 Aprational y)
                      throws ApfloatRuntimeException

Returns the greater of the two values.

Parameters:

x - An argument.

y - Another argument.

Returns:

The greater of the two values.

Throws:

ApfloatRuntimeException

Since:

1.9.0

min

public static Aprational min(Aprational x,
 Aprational y)
                      throws ApfloatRuntimeException

Returns the smaller of the two values.

Parameters:

x - An argument.

y - Another argument.

Returns:

The smaller of the two values.

Throws:

ApfloatRuntimeException

Since:

1.9.0

binomial

public static Aprational binomial(Aprational n,
 Aprational k)
                           throws ArithmeticException,
ApfloatRuntimeException

Binomial coefficient.

Parameters:

n - The first argument.

k - The second argument.

Returns:

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$

Throws:

ArithmeticException - If the result is not finite or not a rational number.

ApfloatRuntimeException

Since:

1.11.0

pochhammer

public static Aprational pochhammer(Aprational x,
 Apint n)

Pochhammer symbol.

Parameters:

x - The first argument.

n - The second argument.

Returns:

(x)n

Since:

1.13.0

Implementation notes:

This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the pochhammer symbol.

bernoulli

public static Aprational bernoulli(long n)
                            throws IllegalArgumentException,
ApfloatRuntimeException

Returns the specified Bernoulli number. The default radix is used.

Parameters:

n - The argument.

Returns:

The Bernoulli number Bn.

Throws:

IllegalArgumentException - If n < 0.

ApfloatRuntimeException

Since:

1.11.0

bernoulli

public static Aprational bernoulli(long n,
 int radix)
                            throws IllegalArgumentException,
NumberFormatException,
ApfloatRuntimeException

Returns the specified Bernoulli number in the given radix.

Parameters:

n - The argument.

radix - The radix.

Returns:

The Bernoulli number Bn.

Throws:

IllegalArgumentException - If n < 0.

NumberFormatException - If the radix is not valid.

ApfloatRuntimeException

Since:

1.11.0

harmonicNumber

public static Aprational harmonicNumber(Apint n)
                                 throws ArithmeticException,
ApfloatRuntimeException

Harmonic number.

Parameters:

n - The argument.

Returns:

H_n

Throws:

ArithmeticException - If n is negative.

ApfloatRuntimeException

Since:

1.14.0

harmonicNumber

public static Aprational harmonicNumber(Apint n,
 Apint r)
                                 throws ArithmeticException,
ApfloatRuntimeException

Generalized harmonic number.

Parameters:

n - The first argument.

r - The second argument.

Returns:

H_n^(r)

Throws:

ArithmeticException - If n is negative and r is positive.

ApfloatRuntimeException

Since:

1.14.0