modular_linalg.H

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:
 
  The above copyright notice and this permission notice shall be included in all
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  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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*/
 
# ifndef MODULAR_LINALG_H
# define MODULAR_LINALG_H
 
# include <cstdint>
# include <algorithm>
# include <optional>
# include <type_traits>
# include <ah-errors.H>
# include <tpl_array.H>
# include <tpl_dynMat.H>
# include <modular_arithmetic.H>
# include <primality.H>
# include <concepts>
# include <utility>
 
namespace Aleph
{
  template <typename T>
  concept SparseMatrix = requires(const T & m, T & mut_m, size_t r, size_t c, uint64_t v)
    {
      { m.rows() } -> std::convertible_to<size_t>;
      { m.cols() } -> std::convertible_to<size_t>;
      { m.read_ne(r, c) } -> std::convertible_to<uint64_t>;
      { mut_m.write(r, c, v) };
      { mut_m.traverse_allocated([](uint64_t &) { return true; }) } -> std::convertible_to<bool>;
      { mut_m.set_dimension(r, c) };
    };
 
  template <typename T>
  concept DenseMatrix = std::is_same_v<T, Array<Array<uint64_t>>>;
 
  template <typename T>
  concept ModularMatrixBackend = DenseMatrix<T> or SparseMatrix<T>;
 
  template <DenseMatrix MatrixT>
  [[nodiscard]] constexpr std::pair<size_t, size_t>
  matrix_dims(const MatrixT & m) noexcept
  {
    const size_t rows = m.size();
    return {rows, rows > 0 ? m[0].size() : 0};
  }
 
  template <SparseMatrix MatrixT>
  [[nodiscard]] constexpr std::pair<size_t, size_t>
  matrix_dims(const MatrixT & m) noexcept
  {
    return {m.rows(), m.cols()};
  }
 
  template <ModularMatrixBackend MatrixT>
  class Modular_Matrix
  {
    MatrixT mat; 
    uint64_t mod; 
 
    // Private helpers for matrix element access using constrained overloads
    static uint64_t get_val(const MatrixT & m, size_t r, size_t c)
    {
      if constexpr (DenseMatrix<MatrixT>)
        return m[r][c];
      else
        return m.read_ne(r, c);
    }
 
    static void set_val(MatrixT & m, size_t r, size_t c, uint64_t v)
    {
      if constexpr (DenseMatrix<MatrixT>)
        m(r)(c) = v;
      else
        {
          // Avoid materializing zero entries in sparse matrices.
          // Only write if value is non-zero or if entry already exists.
          if (v != 0 or m.read_ne(r, c) != 0)
            m.write(r, c, v);
        }
    }
 
    static void swap_rows(MatrixT & m, size_t r1, size_t r2)
    {
      if constexpr (DenseMatrix<MatrixT>)
        std::swap(m(r1), m(r2));
      else
        {
          const size_t cols = m.cols();
          for (size_t j = 0; j < cols; ++j)
            {
              const uint64_t v1 = m.read_ne(r1, j);
              const uint64_t v2 = m.read_ne(r2, j);
 
              // Avoid redundant writes that can densify sparse backends
              if (v1 == v2)
                continue;
 
              set_val(m, r1, j, v2);
              set_val(m, r2, j, v1);
            }
        }
    }
 
  public:
    Modular_Matrix(const MatrixT & m, const uint64_t p)
      : mat(m), mod(p)
    {
      ah_invalid_argument_if(not miller_rabin(mod))
        << "Modular_Matrix: modulus " << mod << " must be prime";
 
      // Normalize initial values
      if constexpr (DenseMatrix<MatrixT>)
        {
          const auto [n_rows, n_cols] = matrix_dims(mat);
          if (n_rows > 0)
            for (size_t i = 0; i < n_rows; ++i)
              {
                ah_invalid_argument_if(mat[i].size() != n_cols)
                    << "Modular_Matrix: ragged matrix input is not supported";
                for (size_t j = 0; j < n_cols; ++j)
                  mat(i)(j) %= mod;
              }
        }
      else // SparseMatrix
        {
          // Use traverse_allocated to efficiently normalize only stored entries
          mat.traverse_allocated([this](uint64_t & val)
                                   {
                                     val %= mod;
                                     return true;
                                   });
        }
    }
 
  private:
    // Internal constructor to skip validation when data is already normalized
    struct skip_validation_t
    {};
 
    Modular_Matrix(MatrixT && m, uint64_t p, skip_validation_t)
      : mat(std::move(m)), mod(p) {}
 
  public:
    const MatrixT &get() const noexcept { return mat; }
 
    [[nodiscard]] uint64_t determinant() const
    {
      const auto [n_rows, n_cols] = matrix_dims(mat);
 
      ah_domain_error_if(n_rows == 0 or n_rows != n_cols)
          << "Determinant requires a square matrix";
 
      MatrixT a = mat;
      uint64_t det = 1;
 
      for (size_t i = 0; i < n_rows; ++i)
        {
          size_t pivot = i;
          if (get_val(a, i, i) == 0)
            {
              for (size_t j = i + 1; j < n_rows; ++j)
                if (get_val(a, j, i) != 0)
                  {
                    pivot = j;
                    break;
                  }
            }
 
          if (get_val(a, pivot, i) == 0)
            return 0; // Singular matrix
 
          if (i != pivot)
            {
              swap_rows(a, i, pivot);
              det = (mod - det) % mod;
            }
 
          const uint64_t diag = get_val(a, i, i);
          det = mod_mul(det, diag, mod);
          const uint64_t inv = mod_inv(diag, mod);
 
          for (size_t j = i + 1; j < n_rows; ++j)
            if (const uint64_t val_ji = get_val(a, j, i); val_ji != 0)
              {
                const uint64_t factor = mod_mul(val_ji, inv, mod);
                for (size_t k = i; k < n_cols; ++k)
                  if (const uint64_t val_ik = get_val(a, i, k); val_ik != 0)
                    {
                      const uint64_t term = mod_mul(factor, val_ik, mod);
                      const uint64_t current = get_val(a, j, k);
                      const uint64_t res = (current >= term) ? (current - term) : (mod - (term - current));
                      set_val(a, j, k, res);
                    }
              }
        }
      return det;
    }
 
    [[nodiscard]] std::optional<Modular_Matrix<MatrixT>> inverse() const
    {
      const auto [n_rows, n_cols] = matrix_dims(mat);
 
      ah_domain_error_if(n_rows == 0 or n_rows != n_cols)
          << "Inverse requires a square matrix";
 
      MatrixT a = mat;
      MatrixT inv;
 
      if constexpr (DenseMatrix<MatrixT>)
        {
          inv.reserve(n_rows);
          for (size_t i = 0; i < n_rows; ++i)
            {
              Array<uint64_t> row;
              row.reserve(n_rows);
              for (size_t j = 0; j < n_rows; ++j)
                row.append(i == j ? 1 : 0);
              inv.append(std::move(row));
            }
        }
      else
        {
          inv.set_dimension(n_rows, n_rows);
          for (size_t i = 0; i < n_rows; ++i)
            inv.write(i, i, 1);
        }
 
      for (size_t i = 0; i < n_rows; ++i)
        {
          size_t pivot = i;
          if (get_val(a, i, i) == 0)
            for (size_t j = i + 1; j < n_rows; ++j)
              if (get_val(a, j, i) != 0)
                {
                  pivot = j;
                  break;
                }
 
          if (get_val(a, pivot, i) == 0)
            return std::nullopt; // Singular matrix
 
          if (i != pivot)
            {
              swap_rows(a, i, pivot);
              swap_rows(inv, i, pivot);
            }
 
          const uint64_t pivot_val = get_val(a, i, i);
          const uint64_t pivot_inv = mod_inv(pivot_val, mod);
          for (size_t j = 0; j < n_rows; ++j)
            {
              if (const uint64_t val = get_val(a, i, j); val != 0)
                set_val(a, i, j, mod_mul(val, pivot_inv, mod));
              if (const uint64_t val = get_val(inv, i, j); val != 0)
                set_val(inv, i, j, mod_mul(val, pivot_inv, mod));
            }
 
          for (size_t j = 0; j < n_rows; ++j)
            if (i != j)
              if (const uint64_t factor = get_val(a, j, i); factor != 0)
                for (size_t k = 0; k < n_rows; ++k)
                  {
                    if (const uint64_t val_ik = get_val(a, i, k); val_ik != 0)
                      {
                        const uint64_t term_a = mod_mul(factor, val_ik, mod);
                        const uint64_t curr_a = get_val(a, j, k);
                        set_val(a, j, k, (curr_a >= term_a) ? (curr_a - term_a) : (mod - (term_a - curr_a)));
                      }
 
                    if (const uint64_t val_inv_ik = get_val(inv, i, k); val_inv_ik != 0)
                      {
                        const uint64_t term_inv = mod_mul(factor, val_inv_ik, mod);
                        const uint64_t curr_inv = get_val(inv, j, k);
                        set_val(inv, j, k, (curr_inv >= term_inv) ?
                                             (curr_inv - term_inv) :
                                             (mod - (term_inv - curr_inv)));
                      }
                  }
        }
 
      return Modular_Matrix<MatrixT>(std::move(inv), mod, skip_validation_t{});
    }
  };
 
  using ModularMatrix = Modular_Matrix<Array<Array<uint64_t>>>;
 
  using ModularSparseMatrix = Modular_Matrix<DynMatrix<uint64_t>>;
} // namespace Aleph
 
# endif // MODULAR_LINALG_H