P-Adic numbers, basic: Difference between revisions
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__TOC__
=={{header|C++}}==
This example displays p-adic numbers in standard mathematical format, consisting of a possibly infinite list of digits extending leftwards from the p-adic point. p-adic numbers are given corrrect to O(prime^40) and rational reconstructions are accurate to O(prime^20).
<syntaxhighlight lang="c++">
#include <cmath>
#include <cstdint>
#include <iostream>
#include <numeric>
#include <stdexcept>
#include <string>
#include <vector>
class Rational {
public:
Rational(const int32_t& aNumerator, const int32_t& aDenominator) {
if ( aDenominator < 0 ) {
numerator = -aNumerator;
denominator = -aDenominator;
} else {
numerator = aNumerator;
denominator = aDenominator;
}
if ( aNumerator == 0 ) {
denominator = 1;
}
const uint32_t divisor = std::gcd(numerator, denominator);
numerator /= divisor;
denominator /= divisor;
}
std::string to_string() const {
return std::to_string(numerator) + " / " + std::to_string(denominator);
}
private:
int32_t numerator;
int32_t denominator;
};
class P_adic {
public:
// Create a P_adic number, with p = 'prime', from the given rational 'numerator' / 'denominator'.
P_adic(const uint32_t& prime, int32_t numerator, int32_t denominator) : prime(prime) {
if ( denominator == 0 ) {
throw std::invalid_argument("Denominator cannot be zero");
}
order = 0;
// Process rational zero
if ( numerator == 0 ) {
digits.assign(DIGITS_SIZE, 0);
order = ORDER_MAX;
return;
}
// Remove multiples of 'prime' and adjust the order of the P_adic number accordingly
while ( modulo_prime(numerator) == 0 ) {
numerator /= static_cast<int32_t>(prime);
order += 1;
}
while ( modulo_prime(denominator) == 0 ) {
denominator /= static_cast<int32_t>(prime);
order -= 1;
}
// Standard calculation of P_adic digits
const uint64_t inverse = modulo_inverse(denominator);
while ( digits.size() < DIGITS_SIZE ) {
const uint32_t digit = modulo_prime(numerator * inverse);
digits.emplace_back(digit);
numerator -= digit * denominator;
if ( numerator != 0 ) {
// The denominator is not a power of a prime
uint32_t count = 0;
while ( modulo_prime(numerator) == 0 ) {
numerator /= static_cast<int32_t>(prime);
count += 1;
}
for ( uint32_t i = count; i > 1; --i ) {
digits.emplace_back(0);
}
}
}
}
// Return the sum of this P_adic number with the given P_adic number.
P_adic add(P_adic other) {
if ( prime != other.prime ) {
throw std::invalid_argument("Cannot add p-adic's with different primes");
}
std::vector<uint32_t> this_digits = digits;
std::vector<uint32_t> other_digits = other.digits;
std::vector<uint32_t> result;
// Adjust the digits so that the P_adic points are aligned
for ( int32_t i = 0; i < -order + other.order; ++i ) {
other_digits.insert(other_digits.begin(), 0);
}
for ( int32_t i = 0; i < -other.order + order; ++i ) {
this_digits.insert(this_digits.begin(), 0);
}
// Standard digit by digit addition
uint32_t carry = 0;
for ( uint32_t i = 0; i < std::min(this_digits.size(), other_digits.size()); ++i ) {
const uint32_t sum = this_digits[i] + other_digits[i] + carry;
const uint32_t remainder = sum % prime;
carry = ( sum >= prime ) ? 1 : 0;
result.emplace_back(remainder);
}
return P_adic(prime, result, all_zero_digits(result) ? ORDER_MAX : std::min(order, other.order));
}
// Return the Rational representation of this P_adic number.
Rational convert_to_rational() {
std::vector<uint32_t> numbers = digits;
// Zero
if ( numbers.empty() || all_zero_digits(numbers) ) {
return Rational(1, 0);
}
// Positive integer
if ( order >= 0 && ends_with(numbers, 0) ) {
for ( int32_t i = 0; i < order; ++i ) {
numbers.emplace(numbers.begin(), 0);
}
return Rational(convert_to_decimal(numbers), 1);
}
// Negative integer
if ( order >= 0 && ends_with(numbers, prime - 1) ) {
negate_digits(numbers);
for ( int32_t i = 0; i < order; ++i ) {
numbers.emplace(numbers.begin(), 0);
}
return Rational(-convert_to_decimal(numbers), 1);
}
// Rational
const P_adic copy(prime, digits, order);
P_adic sum(prime, digits, order);
int32_t denominator = 1;
do {
sum = sum.add(copy);
denominator += 1;
} while ( ! ( ends_with(sum.digits, 0) || ends_with(sum.digits, prime - 1) ) );
const bool negative = ends_with(sum.digits, 6);
if ( negative ) {
negate_digits(sum.digits);
}
int32_t numerator = negative ? -convert_to_decimal(sum.digits) : convert_to_decimal(sum.digits);
if ( order > 0 ) {
numerator *= std::pow(prime, order);
}
if ( order < 0 ) {
denominator *= std::pow(prime, -order);
}
return Rational(numerator, denominator);
}
// Return a string representation of this P_adic number.
std::string to_string() {
std::vector<uint32_t> numbers = digits;
pad_with_zeros(numbers);
std::string result = "";
for ( int64_t i = numbers.size() - 1; i >= 0; --i ) {
result += std::to_string(digits[i]);
}
if ( order >= 0 ) {
for ( int32_t i = 0; i < order; ++i ) {
result += "0";
}
result += ".0";
} else {
result.insert(result.length() + order, ".");
while ( result[result.length() - 1] == '0' ) {
result = result.substr(0, result.length() - 1);
}
}
return " ..." + result.substr(result.length() - PRECISION - 1);
}
private:
/**
* Create a P_adic, with p = 'prime', directly from a vector of digits.
*
* For example: with 'order' = 0, the vector [1, 2, 3, 4, 5] creates the p-adic ...54321.0,
* 'order' > 0 shifts the vector 'order' places to the left and
* 'order' < 0 shifts the vector 'order' places to the right.
*/
P_adic(const uint32_t& prime, const std::vector<uint32_t>& digits, const int32_t& order)
: prime(prime), digits(digits), order(order) {
}
// Transform the given vector of digits representing a P_adic number
// into a vector which represents the negation of the P_adic number.
void negate_digits(std::vector<uint32_t>& numbers) {
numbers[0] = modulo_prime(prime - numbers[0]);
for ( uint64_t i = 1; i < numbers.size(); ++i ) {
numbers[i] = prime - 1 - numbers[i];
}
}
// Return the multiplicative inverse of the given number modulo 'prime'.
uint32_t modulo_inverse(const uint32_t& number) const {
uint32_t inverse = 1;
while ( modulo_prime(inverse * number) != 1 ) {
inverse += 1;
}
return inverse;
}
// Return the given number modulo 'prime' in the range 0..'prime' - 1.
int32_t modulo_prime(const int64_t& number) const {
const int32_t div = static_cast<int32_t>(number % prime);
return ( div >= 0 ) ? div : div + prime;
}
// The given vector is padded on the right by zeros up to a maximum length of 'DIGITS_SIZE'.
void pad_with_zeros(std::vector<uint32_t>& vector) {
while ( vector.size() < DIGITS_SIZE ) {
vector.emplace_back(0);
}
}
// Return the given vector of base 'prime' integers converted to a decimal integer.
uint32_t convert_to_decimal(const std::vector<uint32_t>& numbers) const {
uint32_t decimal = 0;
uint32_t multiple = 1;
for ( const uint32_t& number : numbers ) {
decimal += number * multiple;
multiple *= prime;
}
return decimal;
}
// Return whether the given vector consists of all zeros.
bool all_zero_digits(const std::vector<uint32_t>& numbers) const {
for ( uint32_t number : numbers ) {
if ( number != 0 ) {
return false;
}
}
return true;
}
// Return whether the given vector ends with multiple instances of the given number.
bool ends_with(const std::vector<uint32_t>& numbers, const uint32_t& number) const {
for ( uint64_t i = numbers.size() - 1; i >= numbers.size() - PRECISION / 2; --i ) {
if ( numbers[i] != number ) {
return false;
}
}
return true;
}
uint32_t prime;
std::vector<uint32_t> digits;
int32_t order;
static const uint32_t PRECISION = 40;
static const uint32_t ORDER_MAX = 1'000;
static const uint32_t DIGITS_SIZE = PRECISION + 5;
};
int main() {
std::cout << "3-adic numbers:" << std::endl;
P_adic padic_one(3, -2, 87);
std::cout << "-2 / 87 => " << padic_one.to_string() << std::endl;
P_adic padic_two(3, 4, 97);
std::cout << "4 / 97 => " << padic_two.to_string() << std::endl;
P_adic sum = padic_one.add(padic_two);
std::cout << "sum => " << sum.to_string() << std::endl;
std::cout << "Rational = " << sum.convert_to_rational().to_string() << std::endl;
std::cout << std::endl;
std::cout << "7-adic numbers:" << std::endl;
padic_one = P_adic(7, 5, 8);
std::cout << "5 / 8 => " << padic_one.to_string() << std::endl;
padic_two = P_adic(7, 353, 30809);
std::cout << "353 / 30809 => " << padic_two.to_string() << std::endl;
sum = padic_one.add(padic_two);
std::cout << "sum => " << sum.to_string() << std::endl;
std::cout << "Rational = " << sum.convert_to_rational().to_string() << std::endl;
std::cout << std::endl;
}
</syntaxhighlight>
{{ out }}
<pre>
3-adic numbers:
-2 / 87 => ...101020111222001212021110002210102011122.2
4 / 97 => ...022220111100202001010001200002111122021.0
sum => ...201011000022210220101111202212220210220.2
Rational = 154 / 8439
7-adic numbers:
5 / 8 => ...424242424242424242424242424242424242425.0
353 / 30809 => ...560462505550343461155520004023663643455.0
sum => ...315035233123101033613062431266421216213.0
Rational = 156869 / 246472
</pre>
=={{header|FreeBASIC}}==
Line 1,360 ⟶ 1,686:
λ> let x = 2/7 in (2*x - x^2) / 3 :: Rational
8 % 49</pre>
=={{header|Java}}==
This example displays p-adic numbers in standard mathematical format, consisting of a possibly infinite list of digits extending leftwards from the p-adic point. p-adic numbers are given correct to O(prime^40) and the rational reconstruction is correct to O(prime^20).
<syntaxhighlight lang="java">
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.stream.Collectors;
public final class PAdicNumbersBasic {
public static void main(String[] args) {
System.out.println("3-adic numbers:");
Padic padicOne = new Padic(3, -5, 9);
System.out.println("-5 / 9 => " + padicOne);
Padic padicTwo = new Padic(3, 47, 12);
System.out.println("47 / 12 => " + padicTwo);
Padic sum = padicOne.add(padicTwo);
System.out.println("sum => " + sum);
System.out.println("Rational = " + sum.convertToRational());
System.out.println();
System.out.println("7-adic numbers:");
padicOne = new Padic(7, 5, 8);
System.out.println("5 / 8 => " + padicOne);
padicTwo = new Padic(7, 353, 30809);
System.out.println("353 / 30809 => " + padicTwo);
sum = padicOne.add(padicTwo);
System.out.println("sum => " + sum);
System.out.println("Rational = " + sum.convertToRational());
}
}
final class Padic {
/**
* Create a p-adic, with p = aPrime, from the given rational 'aNumerator' / 'aDenominator'.
*/
public Padic(int aPrime, int aNumerator, int aDenominator) {
if ( aDenominator == 0 ) {
throw new IllegalArgumentException("Denominator cannot be zero");
}
prime = aPrime;
digits = new ArrayList<Integer>(DIGITS_SIZE);
order = 0;
// Process rational zero
if ( aNumerator == 0 ) {
order = MAX_ORDER;
return;
}
// Remove multiples of 'prime' and adjust the order of the p-adic number accordingly
while ( Math.floorMod(aNumerator, prime) == 0 ) {
aNumerator /= prime;
order += 1;
}
while ( Math.floorMod(aDenominator, prime) == 0 ) {
aDenominator /= prime;
order -= 1;
}
// Standard calculation of p-adic digits
final long inverse = moduloInverse(aDenominator);
while ( digits.size() < DIGITS_SIZE ) {
final int digit = Math.floorMod(aNumerator * inverse, prime);
digits.addLast(digit);
aNumerator -= digit * aDenominator;
if ( aNumerator != 0 ) {
// The denominator is not a power of a prime
int count = 0;
while ( Math.floorMod(aNumerator, prime) == 0 ) {
aNumerator /= prime;
count += 1;
}
for ( int i = count; i > 1; i-- ) {
digits.addLast(0);
}
}
}
}
/**
* Return the sum of this p-adic number and the given p-adic number.
*/
public Padic add(Padic aOther) {
if ( prime != aOther.prime ) {
throw new IllegalArgumentException("Cannot add p-adic's with different primes");
}
List<Integer> result = new ArrayList<Integer>();
// Adjust the digits so that the p-adic points are aligned
for ( int i = 0; i < -order + aOther.order; i++ ) {
aOther.digits.addFirst(0);
}
for ( int i = 0; i < -aOther.order + order; i++ ) {
digits.addFirst(0);
}
// Standard digit by digit addition
int carry = 0;
for ( int i = 0; i < Math.min(digits.size(), aOther.digits.size()); i++ ) {
final int sum = digits.get(i) + aOther.digits.get(i) + carry;
final int remainder = Math.floorMod(sum, prime);
carry = ( sum >= prime ) ? 1 : 0;
result.addLast(remainder);
}
// Reverse the changes made to the digits
for ( int i = 0; i < -order + aOther.order; i++ ) {
aOther.digits.removeFirst();
}
for ( int i = 0; i < -aOther.order + order; i++ ) {
digits.removeFirst();
}
return new Padic(prime, result, allZeroDigits(result) ? MAX_ORDER : Math.min(order, aOther.order));
}
/**
* Return the Rational representation of this p-adic number.
*/
public Rational convertToRational() {
List<Integer> numbers = new ArrayList<Integer>(digits);
// Zero
if ( numbers.isEmpty() || allZeroDigits(numbers) ) {
return new Rational(0, 1);
}
// Positive integer
if ( order >= 0 && endsWith(numbers, 0) ) {
for ( int i = 0; i < order; i++ ) {
numbers.addFirst(0);
}
return new Rational(convertToDecimal(numbers), 1);
}
// Negative integer
if ( order >= 0 && endsWith(numbers, prime - 1) ) {
negateList(numbers);
for ( int i = 0; i < order; i++ ) {
numbers.addFirst(0);
}
return new Rational(-convertToDecimal(numbers), 1);
}
// Rational
Padic sum = new Padic(prime, digits, order);
Padic self = new Padic(prime, digits, order);
int denominator = 1;
do {
sum = sum.add(self);
denominator += 1;
} while ( ! ( endsWith(sum.digits, 0) || endsWith(sum.digits, prime - 1) ) );
final boolean negative = endsWith(sum.digits, prime - 1);
if ( negative ) {
negateList(sum.digits);
}
int numerator = negative ? -convertToDecimal(sum.digits) : convertToDecimal(sum.digits);
if ( order > 0 ) {
numerator *= Math.pow(prime, order);
}
if ( order < 0 ) {
denominator *= Math.pow(prime, -order);
}
return new Rational(numerator, denominator);
}
/**
* Return a string representation of this p-adic.
*/
public String toString() {
List<Integer> numbers = new ArrayList<Integer>(digits);
padWithZeros(numbers);
Collections.reverse(numbers);
String numberString = numbers.stream().map(String::valueOf).collect(Collectors.joining());
StringBuilder builder = new StringBuilder(numberString);
if ( order >= 0 ) {
for ( int i = 0; i < order; i++ ) {
builder.append("0");
}
builder.append(".0");
} else {
builder.insert(builder.length() + order, ".");
while ( builder.toString().endsWith("0") ) {
builder.deleteCharAt(builder.length() - 1);
}
}
return " ..." + builder.toString().substring(builder.length() - PRECISION - 1);
}
// PRIVATE //
/**
* Create a p-adic, with p = 'aPrime', directly from a list of digits.
*
* With 'aOrder' = 0, the list [1, 2, 3, 4, 5] creates the p-adic ...54321.0
* 'aOrder' > 0 shifts the list 'aOrder' places to the left and
* 'aOrder' < 0 shifts the list 'aOrder' places to the right.
*/
private Padic(int aPrime, List<Integer> aDigits, int aOrder) {
prime = aPrime;
digits = new ArrayList<Integer>(aDigits);
order = aOrder;
}
/**
* Return the multiplicative inverse of the given decimal number modulo 'prime'.
*/
private int moduloInverse(int aNumber) {
int inverse = 1;
while ( Math.floorMod(inverse * aNumber, prime) != 1 ) {
inverse += 1;
}
return inverse;
}
/**
* Transform the given list of digits representing a p-adic number
* into a list which represents the negation of the p-adic number.
*/
private void negateList(List<Integer> aDigits) {
aDigits.set(0, Math.floorMod(prime - aDigits.get(0), prime));
for ( int i = 1; i < aDigits.size(); i++ ) {
aDigits.set(i, prime - 1 - aDigits.get(i));
}
}
/**
* Return the given list of base 'prime' integers converted to a decimal integer.
*/
private int convertToDecimal(List<Integer> aNumbers) {
int decimal = 0;
int multiple = 1;
for ( int number : aNumbers ) {
decimal += number * multiple;
multiple *= prime;
}
return decimal;
}
/**
* Return whether the given list consists of all zeros.
*/
private static boolean allZeroDigits(List<Integer> aList) {
return aList.stream().allMatch( i -> i == 0 );
}
/**
* The given list is padded on the right by zeros up to a maximum length of 'PRECISION'.
*/
private static void padWithZeros(List<Integer> aList) {
while ( aList.size() < DIGITS_SIZE ) {
aList.addLast(0);
}
}
/**
* Return whether the given list ends with multiple instances of the given number.
*/
private static boolean endsWith(List<Integer> aDigits, int aDigit) {
for ( int i = aDigits.size() - 1; i >= aDigits.size() - PRECISION / 2; i-- ) {
if ( aDigits.get(i) != aDigit ) {
return false;
}
}
return true;
}
private static class Rational {
public Rational(int aNumerator, int aDenominator) {
if ( aDenominator < 0 ) {
numerator = -aNumerator;
denominator = -aDenominator;
} else {
numerator = aNumerator;
denominator = aDenominator;
}
if ( aNumerator == 0 ) {
denominator = 1;
}
final int gcd = gcd(numerator, denominator);
numerator /= gcd;
denominator /= gcd;
}
public String toString() {
return numerator + " / " + denominator;
}
private int gcd(int aOne, int aTwo) {
if ( aTwo == 0 ) {
return Math.abs(aOne);
}
return gcd(aTwo, Math.floorMod(aOne, aTwo));
}
private int numerator;
private int denominator;
}
private List<Integer> digits;
private int order;
private final int prime;
private static final int MAX_ORDER = 1_000;
private static final int PRECISION = 40;
private static final int DIGITS_SIZE = PRECISION + 5;
}
</syntaxhighlight>
{{ out }}
<pre>
3-adic numbers:
-5 / 9 => ...22222222222222222222222222222222222222.11
47 / 12 => ...020202020202020202020202020202020202101.2
sum => ...20202020202020202020202020202020202101.01
Rational = 121 / 36
7-adic numbers:
5 / 8 => ...424242424242424242424242424242424242425.0
353 / 30809 => ...560462505550343461155520004023663643455.0
sum => ...315035233123101033613062431266421216213.0
Rational = 156869 / 246472
</pre>
=={{header|Julia}}==
| |||