Package org.nevec.rjm

Class BigIntegerMath

java.lang.Object

org.nevec.rjm.BigIntegerMath

public class BigIntegerMath
extends java.lang.Object

BigInteger special functions and Number theory.

Since:

2009-08-06

Author:

Richard J. Mathar

Constructor Summary

Constructors

Constructor	Description
BigIntegerMath()

Method Summary

Modifier and Type	Method	Description
static java.math.BigInteger	binomial(int n, int k)	Evaluate binomial(n,k).
static java.math.BigInteger	binomial(java.math.BigInteger n, java.math.BigInteger k)	Evaluate binomial(n,k).
static Rational	centrlFactNumt(int n, int k)	The central factorial number t(n,k) number at the indices provided.
static Rational	centrlFactNumT(int n, int k)	The central factorial number T(n,k) number at the indices provided.
static java.math.BigInteger	core(java.math.BigInteger n)	Evaluate core(n).
static java.math.BigInteger	det(java.math.BigInteger[][] A)	Determinant of an integer square matrix.
static java.util.Vector<java.math.BigInteger>	divisors(java.math.BigInteger n)	Compute the list of positive divisors.
static int	isqrt(int n)	Evaluate floor(sqrt(n)).
static long	isqrt(long n)	Evaluate floor(sqrt(n)).
static java.math.BigInteger	isqrt(java.math.BigInteger n)	Evaluate floor(sqrt(n)).
static java.math.BigInteger	lcm(java.math.BigInteger a, java.math.BigInteger b)	The lowest common multiple
static java.math.BigInteger[][]	minor(java.math.BigInteger[][] A, int r, int c)	Minor of an integer matrix.
static java.math.BigInteger	sigma(int n)	Evaluate sigma(n).
static java.math.BigInteger	sigma(java.math.BigInteger n)	Evaluate sigma(n).
static java.math.BigInteger	sigmak(java.math.BigInteger n, int k)	Evaluate sigma_k(n).
static Rational[]	solve(java.math.BigInteger[][] A, java.math.BigInteger[] rhs)	Solve a linear system of equations.
static java.math.BigInteger	valueOf(java.util.Vector<java.math.BigInteger> c, java.math.BigInteger x)	Evaluate the value of an integer polynomial at some integer argument.

Methods inherited from class java.lang.Object

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

Constructor Detail

BigIntegerMath

public BigIntegerMath()

Method Detail

binomial

public static java.math.BigInteger binomial(int n,
                                            int k)

Evaluate binomial(n,k).

Parameters:

n - The upper index

k - The lower index

Returns:

The binomial coefficient

binomial

public static java.math.BigInteger binomial(java.math.BigInteger n,
                                            java.math.BigInteger k)

Evaluate binomial(n,k).

Parameters:

n - The upper index

k - The lower index

Returns:

The binomial coefficient

Since:

2008-10-15

sigmak

public static java.math.BigInteger sigmak(java.math.BigInteger n,
                                          int k)

Evaluate sigma_k(n).

Parameters:

n - the main argument which defines the divisors

k - the lower index, which defines the power

Returns:

The sum of the k-th powers of the positive divisors

sigma

public static java.math.BigInteger sigma(int n)

Evaluate sigma(n).

Parameters:

n - the argument for which divisors will be searched.

Returns:

the sigma function. Sum of the positive divisors of the argument.

Since:

2006-08-14

divisors

public static java.util.Vector<java.math.BigInteger> divisors(java.math.BigInteger n)

Compute the list of positive divisors.

Parameters:

n - The integer of which the divisors are to be found.

Returns:

The sorted list of positive divisors.

Since:

2010-08-27

sigma

public static java.math.BigInteger sigma(java.math.BigInteger n)

Evaluate sigma(n).

Parameters:

n - the argument for which divisors will be searched.

Returns:

the sigma function. Sum of the divisors of the argument.

Since:

2006-08-14

isqrt

public static int isqrt(int n)

Evaluate floor(sqrt(n)).

Parameters:

n - The non-negative argument.

Returns:

The integer square root. The square root rounded down.

Since:

2010-08-27

isqrt

public static long isqrt(long n)

Evaluate floor(sqrt(n)).

Parameters:

n - The non-negative argument. Arguments less than zero throw an ArithmeticException.

Returns:

The integer square root, the square root rounded down.

Since:

2010-08-27

isqrt

public static java.math.BigInteger isqrt(java.math.BigInteger n)

Evaluate floor(sqrt(n)).

Parameters:

n - The non-negative argument. Arguments less than zero throw an ArithmeticException.

Returns:

The integer square root, the square root rounded down.

Since:

2011-02-12

core

public static java.math.BigInteger core(java.math.BigInteger n)

Evaluate core(n). Returns the smallest positive integer m such that n/m is a perfect square.

Parameters:

n - The non-negative argument.

Returns:

The square-free part of n.

Since:

2011-02-12

minor

public static java.math.BigInteger[][] minor(java.math.BigInteger[][] A,
                                             int r,
                                             int c)
                                      throws java.lang.ArithmeticException

Minor of an integer matrix.

Parameters:

A - The matrix.

r - The row index of the row to be removed (0-based). An exception is thrown if this is outside the range 0 to the upper row index of A.

c - The column index of the column to be removed (0-based). An exception is thrown if this is outside the range 0 to the upper column index of A.

Returns:

The depleted matrix. This is not a deep copy but contains references to the original.

Throws:

java.lang.ArithmeticException

Since:

2010-08-27

det

public static java.math.BigInteger det(java.math.BigInteger[][] A)
                                throws java.lang.ArithmeticException

Determinant of an integer square matrix.

Parameters:

A - The square matrix. If column and row dimensions are unequal, an ArithmeticException is thrown.

Returns:

The determinant.

Throws:

java.lang.ArithmeticException

Since:

2010-08-27

solve

public static Rational[] solve(java.math.BigInteger[][] A,
                               java.math.BigInteger[] rhs)
                        throws java.lang.ArithmeticException

Solve a linear system of equations.

Parameters:

A - The square matrix. If it is not of full rank, an ArithmeticException is thrown.

rhs - The right hand side. The length of this vector must match the matrix size; else an ArithmeticException is thrown.

Returns:

The vector of x in A*x=rhs.

Throws:

java.lang.ArithmeticException

Since:

2010-08-28

lcm

public static java.math.BigInteger lcm(java.math.BigInteger a,
                                       java.math.BigInteger b)

The lowest common multiple

Parameters:

a - The first argument

b - The second argument

Returns:

lcm(|a|,|b|)

Since:

2010-08-27

valueOf

public static java.math.BigInteger valueOf(java.util.Vector<java.math.BigInteger> c,
                                           java.math.BigInteger x)

Evaluate the value of an integer polynomial at some integer argument.

Parameters:

c - Represents the coefficients c[0]+c[1]*x+c[2]*x^2+.. of the polynomial

x - The abscissa point of the evaluation

Returns:

The polynomial value.

Since:

2010-08-27

centrlFactNumt

public static Rational centrlFactNumt(int n,
                                      int k)

The central factorial number t(n,k) number at the indices provided.

Parameters:

n - the first parameter, non-negative.

k - the second index, non-negative.

Returns:

t(n,k)

Since:

2009-08-06

See Also:

P. L. Butzer et al, Num. Funct. Anal. Opt. 10 (5)( 1989) 419-488

centrlFactNumT

public static Rational centrlFactNumT(int n,
                                      int k)

The central factorial number T(n,k) number at the indices provided.

Parameters:

n - the first parameter, non-negative.

k - the second index, non-negative.

Returns:

T(n,k)

Since:

2009-08-06

See Also:

P. L. Butzer et al, Num. Funct. Anal. Opt. 10 (5)( 1989) 419-488