Added sequential LU factorization from tnl-mhfem

// Copyright (c) 2004-2022 Tomáš Oberhuber et al.

//

// This file is part of TNL - Template Numerical Library (https://tnl-project.org/)

//

// SPDX-License-Identifier: MIT

// Implemented by: Jakub Klinkovský

#pragma once

#include <TNL/Assert.h>

#include <TNL/Cuda/CudaCallable.h>

namespace TNL {

namespace Matrices {

namespace Factorization {

template< typename Matrix >

__cuda_callable__

void

LU_sequential_factorize( Matrix& A )

{

   using IndexType = typename Matrix::IndexType;

   static_assert( Matrix::getRows() == Matrix::getColumns(), "LU factorization is possible only for square matrices" );

   constexpr IndexType n = Matrix::getRows();

   for( IndexType k = 0; k < n; k++ ) {

      const auto pivot = A( k, k );

      // Update the (k+1 .. n-1)-th rows

      for( IndexType j = k + 1; j < n; j++ ) {

         const auto factor = A( j, k ) / pivot;

         // Update elements of k-th column below pivot (i.e. elements of L)

         A( j, k ) = factor;

         // Subtract k-th row from j-th

         for( IndexType i = k + 1; i < n; i++ ) {

            A( j, i ) = A( j, i ) - factor * A( k, i );

         }

      }

   }

}

template< typename Matrix, typename Vector >

__cuda_callable__

void

LU_sequential_solve_inplace( const Matrix& A, Vector& x )

{

   using IndexType = typename Matrix::IndexType;

   static_assert( std::is_signed< IndexType >::value, "LU got a matrix with an unsigned index type (2nd for loop won't work)" );

   static_assert( Matrix::getRows() == Matrix::getColumns(), "LU factorization is possible only for square matrices" );

   constexpr IndexType n = Matrix::getRows();

   // Forward substitution

   for( IndexType k = 1; k < n; k++ )

      for( IndexType j = 0; j < k; j++ )

         x[ k ] -= A( k, j ) * x[ j ];

   // Back substitution

   for( IndexType k = n - 1; k >= 0; k-- ) {

      for( IndexType j = n - 1; j > k; j-- )

         x[ k ] -= A( k, j ) * x[ j ];

      x[ k ] /= A( k, k );

   }

}

template< typename Matrix, typename Vector1, typename Vector2 >

__cuda_callable__

void

LU_sequential_solve( const Matrix& A, const Vector1& b, Vector2& x )

{

   // Copy right hand side

   x = b;

   // Solve in-place

   LU_sequential_solve_inplace( A, x );

}

}  // namespace Factorization

}  // namespace Matrices

}  // namespace TNL