310 lines
9.3 KiB
C++
310 lines
9.3 KiB
C++
#include <algorithm>
|
|
#include <cmath>
|
|
#include <cstdint>
|
|
#include <iostream>
|
|
#include <stdexcept>
|
|
#include <vector>
|
|
|
|
class P_adic_square_root {
|
|
public:
|
|
// Create a P_adic_square_root number, with p = 'prime', from the given rational 'numerator' / 'denominator'.
|
|
P_adic_square_root(const uint32_t& prime, const uint32_t& precision, int32_t numerator, int32_t denominator)
|
|
: prime(prime), precision(precision) {
|
|
|
|
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_square_root number accordingly
|
|
while ( modulo(numerator, prime) == 0 ) {
|
|
numerator /= static_cast<int32_t>(prime);
|
|
order += 1;
|
|
}
|
|
|
|
while ( modulo(denominator, prime) == 0 ) {
|
|
denominator /= static_cast<int32_t>(prime);
|
|
order -= 1;
|
|
}
|
|
|
|
if ( ( order & 1 ) != 0 ) {
|
|
throw std::invalid_argument("Number does not have a square root in " + std::to_string(prime) + "-adic");
|
|
}
|
|
order >>= 1;
|
|
|
|
if ( prime == 2 ) {
|
|
square_root_even_prime(numerator, denominator);
|
|
} else {
|
|
square_root_odd_prime(numerator, denominator);
|
|
}
|
|
}
|
|
|
|
// Return the additive inverse of this P_adic_square_root number.
|
|
P_adic_square_root negate() {
|
|
if ( digits.empty() ) {
|
|
return *this;
|
|
}
|
|
|
|
std::vector<uint32_t> negated = digits;
|
|
negate_digits(negated);
|
|
|
|
return P_adic_square_root(prime, precision, negated, order);
|
|
}
|
|
|
|
// Return the product of this P_adic_square_root number and the given P_adic_square_root number.
|
|
P_adic_square_root multiply(P_adic_square_root other) {
|
|
if ( prime != other.prime ) {
|
|
throw std::invalid_argument("Cannot multiply p-adic's with different primes");
|
|
}
|
|
|
|
if ( digits.empty() || other.digits.empty() ) {
|
|
return P_adic_square_root(prime, precision, 0 , 1);
|
|
}
|
|
|
|
return P_adic_square_root(prime, precision, multiply(digits, other.digits), order + other.order);
|
|
}
|
|
|
|
// Return a string representation of this P_adic_square_root as a rational number.
|
|
std::string convertToRational() {
|
|
std::vector<uint32_t> numbers = digits;
|
|
|
|
if ( numbers.empty() ) {
|
|
return "0 / 1";
|
|
}
|
|
|
|
// Lagrange lattice basis reduction in two dimensions
|
|
int64_t series_sum = numbers.front();
|
|
int64_t maximum_prime = 1;
|
|
|
|
for ( uint32_t i = 1; i < precision; ++i ) {
|
|
maximum_prime *= prime;
|
|
series_sum += numbers[i] * maximum_prime;
|
|
}
|
|
|
|
std::vector<int64_t> one = { maximum_prime, series_sum };
|
|
std::vector<int64_t> two = { 0, 1 };
|
|
|
|
int64_t previous_norm = series_sum * series_sum + 1;
|
|
int64_t current_norm = previous_norm + 1;
|
|
uint32_t i = 0;
|
|
uint32_t j = 1;
|
|
|
|
while ( previous_norm < current_norm ) {
|
|
int64_t numerator = one[i] * one[j] + two[i] * two[j];
|
|
int64_t denominator = previous_norm;
|
|
current_norm = std::floor(static_cast<double>(numerator) / denominator + 0.5);
|
|
one[i] -= current_norm * one[j];
|
|
two[i] -= current_norm * two[j];
|
|
|
|
current_norm = previous_norm;
|
|
previous_norm = one[i] * one[i] + two[i] * two[i];
|
|
|
|
if ( previous_norm < current_norm ) {
|
|
std::swap(i, j);
|
|
}
|
|
}
|
|
|
|
int64_t x = one[j];
|
|
int64_t y = two[j];
|
|
if ( y < 0 ) {
|
|
y = -y;
|
|
x = -x;
|
|
}
|
|
|
|
if ( std::abs(one[i] * y - x * two[i]) != maximum_prime ) {
|
|
throw std::invalid_argument("Rational reconstruction failed.");
|
|
}
|
|
|
|
for ( int32_t k = order; k < 0; ++k ) {
|
|
y *= prime;
|
|
}
|
|
|
|
for ( int32_t k = order; k > 0; --k ) {
|
|
x *= prime;
|
|
}
|
|
|
|
return std::to_string(x) + " / " + std::to_string(y);
|
|
}
|
|
|
|
// Return a string representation of this P_adic_square_root.
|
|
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_square_root, 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_square_root ...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_square_root(
|
|
const uint32_t& prime, const uint32_t& precision, const std::vector<uint32_t>& digits, const int32_t& order)
|
|
: prime(prime), precision(precision), digits(digits), order(order) {
|
|
}
|
|
|
|
// Create a 2-adic number which is the square root of the rational 'numerator' / 'denominator'.
|
|
void square_root_even_prime(const int32_t& numerator, const int32_t& denominator) {
|
|
if ( modulo(numerator * denominator, 8) != 1 ) {
|
|
throw std::invalid_argument("Number does not have a square root in 2-adic");
|
|
}
|
|
|
|
// First digit
|
|
uint64_t sum = 1;
|
|
digits.emplace_back(sum);
|
|
|
|
// Further digits
|
|
while ( digits.size() < digits_size ) {
|
|
int64_t factor = denominator * sum * sum - numerator;
|
|
uint32_t valuation = 0;
|
|
while ( modulo(factor, 2) == 0 ) {
|
|
factor /= 2;
|
|
valuation += 1;
|
|
}
|
|
|
|
sum += std::pow(2, valuation - 1);
|
|
|
|
for ( uint32_t i = digits.size(); i < valuation - 1; ++i ) {
|
|
digits.emplace_back(0);
|
|
}
|
|
digits.emplace_back(1);
|
|
}
|
|
}
|
|
|
|
// Create a p-adic number, with an odd prime number, p = 'prime',
|
|
// which is the p-adic square root of the given rational 'numerator' / 'denominator'.
|
|
void square_root_odd_prime(const int32_t& numerator, const int32_t& denominator) {
|
|
// First digit
|
|
int32_t first_digit = 0;
|
|
for ( int32_t i = 1; i < prime && first_digit == 0; ++i ) {
|
|
if ( modulo(denominator * i * i - numerator, prime) == 0 ) {
|
|
first_digit = i;
|
|
}
|
|
}
|
|
|
|
if ( first_digit == 0 ) {
|
|
throw std::invalid_argument("Number does not have a square root in " + std::to_string(prime) + "-adic");
|
|
}
|
|
|
|
digits.emplace_back(first_digit);
|
|
|
|
// Further digits
|
|
const uint64_t coefficient = modulo_inverse(modulo(2 * denominator * first_digit, prime));
|
|
uint64_t sum = first_digit;
|
|
for ( uint32_t i = 2; i < digits_size; ++i ) {
|
|
int64_t next_sum = sum - ( coefficient * ( denominator * sum * sum - numerator ) );
|
|
next_sum = modulo(next_sum, static_cast<uint64_t>(std::pow(prime, i)));
|
|
next_sum -= sum;
|
|
sum += next_sum;
|
|
|
|
const uint32_t digit = next_sum / std::pow(prime, i - 1);
|
|
digits.emplace_back(digit);
|
|
}
|
|
}
|
|
|
|
// Return the list obtained by multiplying the digits of the given two lists,
|
|
// where the digits in each list are regarded as forming a single number in reverse.
|
|
// For example 12 * 13 = 156 is computed as [2, 1] * [3, 1] = [6, 5, 1].
|
|
std::vector<uint32_t> multiply(const std::vector<uint32_t>& one, const std::vector<uint32_t>& two) {
|
|
std::vector<uint32_t> product(one.size() + two.size(), 0);
|
|
for ( uint32_t b = 0; b < two.size(); ++b ) {
|
|
uint32_t carry = 0;
|
|
for ( uint32_t a = 0; a < one.size(); ++a ) {
|
|
product[a + b] += one[a] * two[b] + carry;
|
|
carry = product[a + b] / prime;
|
|
product[a + b] %= prime;
|
|
}
|
|
product[b + one.size()] = carry;
|
|
}
|
|
|
|
return std::vector(product.begin(), product.begin() + digits_size);
|
|
}
|
|
|
|
// 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(inverse * number, prime) != 1 ) {
|
|
inverse += 1;
|
|
}
|
|
return inverse;
|
|
}
|
|
|
|
// Return the given number modulo 'prime' in the range 0..'prime' - 1.
|
|
int32_t modulo(const int64_t& number, const int64_t& modulus) const {
|
|
const int32_t div = static_cast<int32_t>(number % modulus);
|
|
return ( div >= 0 ) ? div : div + modulus;
|
|
}
|
|
|
|
// 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 - numbers[0], prime);
|
|
for ( uint64_t i = 1; i < numbers.size(); ++i ) {
|
|
numbers[i] = prime - 1 - numbers[i];
|
|
}
|
|
}
|
|
|
|
// 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);
|
|
}
|
|
}
|
|
|
|
const int32_t prime;
|
|
const uint32_t precision;
|
|
const uint32_t digits_size = precision + 5;
|
|
|
|
std::vector<uint32_t> digits;
|
|
int32_t order;
|
|
|
|
static const uint32_t ORDER_MAX = 1'000;
|
|
};
|
|
|
|
int main() {
|
|
std::vector<std::vector<int32_t>> tests = { { 2, 20, 497, 10496 }, { 5, 14, 86, 25 }, { 7, 10, -19, 1 } };
|
|
|
|
for ( const std::vector<int32_t>& test : tests ) {
|
|
std::cout << "Number: " << test[2] << " / " << test[3] << " in " << test[0] << "-adic" << std::endl;
|
|
P_adic_square_root square_root(test[0], test[1], test[2], test[3]);
|
|
std::cout << "The two square roots are:" << std::endl;
|
|
std::cout << " " << square_root.to_string() << std::endl;
|
|
std::cout << " " << square_root.negate().to_string() << std::endl;
|
|
P_adic_square_root square = square_root.multiply(square_root);
|
|
std::cout << "The p-adic value is " << square.to_string() << std::endl;
|
|
std::cout << "The rational value is " << square.convertToRational() << std::endl;
|
|
std::cout << std::endl;
|
|
}
|
|
}
|