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332
Task/P-Adic-numbers-basic/Java/p-adic-numbers-basic.java
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332
Task/P-Adic-numbers-basic/Java/p-adic-numbers-basic.java
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public final class PAdicNumbersBasic {
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public static void main(String[] args) {
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System.out.println("3-adic numbers:");
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Padic padicOne = new Padic(3, -5, 9);
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System.out.println("-5 / 9 => " + padicOne);
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Padic padicTwo = new Padic(3, 47, 12);
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System.out.println("47 / 12 => " + padicTwo);
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Padic sum = padicOne.add(padicTwo);
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System.out.println("sum => " + sum);
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System.out.println("Rational = " + sum.convertToRational());
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System.out.println();
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System.out.println("7-adic numbers:");
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padicOne = new Padic(7, 5, 8);
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System.out.println("5 / 8 => " + padicOne);
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padicTwo = new Padic(7, 353, 30809);
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System.out.println("353 / 30809 => " + padicTwo);
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sum = padicOne.add(padicTwo);
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System.out.println("sum => " + sum);
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System.out.println("Rational = " + sum.convertToRational());
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}
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}
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final class Padic {
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/**
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* Create a p-adic, with p = aPrime, from the given rational 'aNumerator' / 'aDenominator'.
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*/
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public Padic(int aPrime, int aNumerator, int aDenominator) {
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if ( aDenominator == 0 ) {
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throw new IllegalArgumentException("Denominator cannot be zero");
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}
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prime = aPrime;
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digits = new ArrayList<Integer>(DIGITS_SIZE);
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order = 0;
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// Process rational zero
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if ( aNumerator == 0 ) {
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order = MAX_ORDER;
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return;
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}
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// Remove multiples of 'prime' and adjust the order of the p-adic number accordingly
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while ( Math.floorMod(aNumerator, prime) == 0 ) {
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aNumerator /= prime;
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order += 1;
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}
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while ( Math.floorMod(aDenominator, prime) == 0 ) {
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aDenominator /= prime;
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order -= 1;
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}
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// Standard calculation of p-adic digits
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final long inverse = moduloInverse(aDenominator);
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while ( digits.size() < DIGITS_SIZE ) {
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final int digit = Math.floorMod(aNumerator * inverse, prime);
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digits.addLast(digit);
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aNumerator -= digit * aDenominator;
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if ( aNumerator != 0 ) {
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// The denominator is not a power of a prime
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int count = 0;
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while ( Math.floorMod(aNumerator, prime) == 0 ) {
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aNumerator /= prime;
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count += 1;
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}
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for ( int i = count; i > 1; i-- ) {
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digits.addLast(0);
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}
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}
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}
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}
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/**
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* Return the sum of this p-adic number and the given p-adic number.
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*/
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public Padic add(Padic aOther) {
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if ( prime != aOther.prime ) {
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throw new IllegalArgumentException("Cannot add p-adic's with different primes");
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}
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List<Integer> result = new ArrayList<Integer>();
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// Adjust the digits so that the p-adic points are aligned
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for ( int i = 0; i < -order + aOther.order; i++ ) {
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aOther.digits.addFirst(0);
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}
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for ( int i = 0; i < -aOther.order + order; i++ ) {
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digits.addFirst(0);
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}
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// Standard digit by digit addition
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int carry = 0;
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for ( int i = 0; i < Math.min(digits.size(), aOther.digits.size()); i++ ) {
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final int sum = digits.get(i) + aOther.digits.get(i) + carry;
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final int remainder = Math.floorMod(sum, prime);
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carry = ( sum >= prime ) ? 1 : 0;
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result.addLast(remainder);
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}
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// Reverse the changes made to the digits
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for ( int i = 0; i < -order + aOther.order; i++ ) {
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aOther.digits.removeFirst();
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}
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for ( int i = 0; i < -aOther.order + order; i++ ) {
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digits.removeFirst();
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}
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return new Padic(prime, result, allZeroDigits(result) ? MAX_ORDER : Math.min(order, aOther.order));
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}
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/**
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* Return the Rational representation of this p-adic number.
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*/
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public Rational convertToRational() {
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List<Integer> numbers = new ArrayList<Integer>(digits);
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// Zero
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if ( allZeroDigits(numbers) ) {
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return new Rational(0, 1);
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}
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// Positive integer
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if ( order >= 0 && endsWith(numbers, 0) ) {
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for ( int i = 0; i < order; i++ ) {
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numbers.addFirst(0);
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}
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return new Rational(convertToDecimal(numbers), 1);
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}
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// Negative integer
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if ( order >= 0 && endsWith(numbers, prime - 1) ) {
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negateList(numbers);
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for ( int i = 0; i < order; i++ ) {
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numbers.addFirst(0);
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}
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return new Rational(-convertToDecimal(numbers), 1);
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}
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// Rational
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Padic sum = new Padic(prime, digits, order);
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Padic self = new Padic(prime, digits, order);
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int denominator = 1;
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do {
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sum = sum.add(self);
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denominator += 1;
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} while ( ! ( endsWith(sum.digits, 0) || endsWith(sum.digits, prime - 1) ) );
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final boolean negative = endsWith(sum.digits, prime - 1);
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if ( negative ) {
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negateList(sum.digits);
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}
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int numerator = negative ? -convertToDecimal(sum.digits) : convertToDecimal(sum.digits);
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if ( order > 0 ) {
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numerator *= Math.pow(prime, order);
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}
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if ( order < 0 ) {
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denominator *= Math.pow(prime, -order);
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}
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return new Rational(numerator, denominator);
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}
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/**
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* Return a string representation of this p-adic.
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*/
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public String toString() {
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while ( digits.size() > PRECISION ) {
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digits.removeLast();
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}
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padWithZeros(digits);
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StringBuilder builder = new StringBuilder();
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for ( int i = digits.size() - 1; i >= 0; i-- ) {
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builder.append(digits.get(i));
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}
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if ( order >= 0 ) {
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for ( int i = 0; i < order; i++ ) {
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builder.append("0");
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builder.deleteCharAt(0);
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}
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builder.append(".0");
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} else {
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builder.insert(builder.length() + order, ".");
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}
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return " ..." + builder.toString();
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}
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// PRIVATE //
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/**
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* Create a p-adic, with p = 'aPrime', directly from a list of digits.
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*
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* With 'aOrder' = 0, the list [1, 2, 3, 4, 5] creates the p-adic ...54321.0
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* 'aOrder' > 0 shifts the list 'aOrder' places to the left and
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* 'aOrder' < 0 shifts the list 'aOrder' places to the right.
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*/
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private Padic(int aPrime, List<Integer> aDigits, int aOrder) {
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prime = aPrime;
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digits = new ArrayList<Integer>(aDigits);
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padWithZeros(digits);
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order = aOrder;
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}
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/**
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* Return the multiplicative inverse of the given decimal number modulo 'prime'.
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*/
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private int moduloInverse(int aNumber) {
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int inverse = 1;
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while ( Math.floorMod(inverse * aNumber, prime) != 1 ) {
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inverse += 1;
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}
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return inverse;
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}
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/**
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* Transform the given list of digits representing a p-adic number
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* into a list which represents the negation of the p-adic number.
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*/
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private void negateList(List<Integer> aDigits) {
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aDigits.set(0, ( prime - aDigits.get(0) ) % prime);
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for ( int i = 1; i < aDigits.size(); i++ ) {
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aDigits.set(i, prime - 1 - aDigits.get(i));
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}
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}
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/**
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* Return the given list of base 'prime' integers converted to a decimal integer.
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*/
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private int convertToDecimal(List<Integer> aNumbers) {
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int decimal = 0;
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int multiple = 1;
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for ( int number : aNumbers ) {
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decimal += number * multiple;
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multiple *= prime;
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}
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return decimal;
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}
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/**
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* Return whether the given list consists of all zeros.
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*/
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private static boolean allZeroDigits(List<Integer> aList) {
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return aList.stream().allMatch( i -> i == 0 );
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}
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/**
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* The given list is padded on the right by zeros up to a maximum length of 'PRECISION'.
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*/
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private static void padWithZeros(List<Integer> aList) {
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while ( aList.size() < PRECISION ) {
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aList.addLast(0);
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}
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}
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/**
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* Return whether the given list ends with multiple instances of the given number.
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*/
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private static boolean endsWith(List<Integer> aDigits, int aDigit) {
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for ( int i = aDigits.size() - 1; i >= aDigits.size() - PRECISION / 2; i-- ) {
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if ( aDigits.get(i) != aDigit ) {
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return false;
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}
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}
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return true;
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}
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private static class Rational {
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public Rational(int aNumerator, int aDenominator) {
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if ( aDenominator < 0 ) {
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numerator = -aNumerator;
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denominator = -aDenominator;
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} else {
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numerator = aNumerator;
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denominator = aDenominator;
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}
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if ( aNumerator == 0 ) {
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denominator = 1;
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}
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final int gcd = gcd(numerator, denominator);
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numerator /= gcd;
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denominator /= gcd;
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}
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public String toString() {
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return numerator + " / " + denominator;
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}
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private int gcd(int aOne, int aTwo) {
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if ( aTwo == 0 ) {
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return Math.abs(aOne);
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}
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return gcd(aTwo, Math.floorMod(aOne, aTwo));
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}
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private int numerator;
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private int denominator;
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}
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private List<Integer> digits;
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private int order;
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private final int prime;
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private static final int MAX_ORDER = 1_000;
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private static final int PRECISION = 40;
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private static final int DIGITS_SIZE = PRECISION + 5;
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}
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