algebra.h

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#ifndef THEORETICA_ALGEBRA_H
#define THEORETICA_ALGEBRA_H
 
#include "../complex/complex_types.h"
#include "../core/core_traits.h"
#include "../core/error.h"
 
 
namespace theoretica {
 
 
    namespace algebra {
 
 
        // Operations involving one matrix or vector
 
 
        template<typename Matrix>
        inline Matrix& mat_error(Matrix& m) {
 
            using Type = matrix_element_t<Matrix>;
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    m(i, j) = (Type) (i == j ? nan() : 0);
 
            return m;
        }
 
 
        template<typename Vector>
        inline Vector& vec_error(Vector& v) {
 
            using Type = vector_element_t<Vector>;
 
            for (unsigned int i = 0; i < v.size(); ++i)
                v[i] = Type(nan());
 
            return v;
        }
 
 
        template<typename Vector>
        inline Vector make_error(unsigned int n) {
            Vector result;
            result.resize(n);
            return vec_error(result);
        }
 
 
        template<typename Matrix>
        inline Matrix make_error(unsigned int rows, unsigned int cols) {
            Matrix result;
            result.resize(rows, cols);
            return mat_error(result);
        }
 
 
        template<typename Matrix>
        inline Matrix& make_identity(Matrix& m) {
 
            using Type = matrix_element_t<Matrix>;
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    m(i, j) = (Type) (i == j ? 1 : 0);
 
            return m;
        }
 
 
        template<typename Matrix>
        inline Matrix& mat_zeroes(Matrix& m) {
 
            using Type = matrix_element_t<Matrix>;
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    m(i, j) = (Type) 0;
 
            return m;
        }
 
 
        template<typename Vector>
        inline Vector& vec_zeroes(Vector& v) {
 
            using Type = vector_element_t<Vector>;
 
            for (unsigned int i = 0; i < v.size(); ++i)
                v[i] = Type(0.0);
 
            return v;
        }
 
 
        template<typename Matrix1, typename Matrix2>
        inline Matrix1& mat_copy(Matrix1& dest, const Matrix2& src) {
 
            dest.resize(src.rows(), src.cols());
 
            for (unsigned int i = 0; i < src.rows(); ++i)
                for (unsigned int j = 0; j < src.cols(); ++j)
                    dest(i, j) = src(i, j);
 
            return dest;
        }
 
 
        template<typename Vector1, typename Vector2>
        inline Vector1& vec_copy(Vector1& dest, const Vector2& src) {
 
            dest.resize(src.size());
 
            for (unsigned int i = 0; i < src.size(); ++i)
                dest[i] = src[i];
 
            return dest;
        }
 
 
        template<typename Matrix>
        inline Matrix& mat_swap_rows(Matrix& A, unsigned int row1, unsigned int row2) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (row1 >= A.rows()) {
                TH_MATH_ERROR("algebra::mat_swap_rows", row1, MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if (row2 >= A.rows()) {
                TH_MATH_ERROR("algebra::mat_swap_rows", row2, MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if (row1 == row2)
                return A;
 
            for (unsigned int j = 0; j < A.cols(); ++j) {
 
                const Type tmp = A(row1, j);
 
                A(row1, j) = A(row2, j);
                A(row2, j) = tmp;
            }
 
            return A;
        }
 
 
        template<typename Matrix>
        inline Matrix& mat_swap_cols(Matrix& A, unsigned int col1, unsigned int col2) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (col1 >= A.cols()) {
                TH_MATH_ERROR("algebra::mat_swap_cols", col1, MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if (col2 >= A.cols()) {
                TH_MATH_ERROR("algebra::mat_swap_cols", col2, MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if (col1 == col2)
                return A;
 
            for (unsigned int i = 0; i < A.cols(); ++i) {
 
                const Type tmp = A(i, col1);
 
                A(i, col1) = A(i, col2);
                A(i, col2) = tmp;
            }
 
            return A;
        }
 
 
        template<typename Matrix, typename Type = matrix_element_t<Matrix>>
        inline Matrix& mat_shift_diagonal(Matrix& A, const Type& sigma) {
 
            const unsigned int count = min(A.rows(), A.cols());
 
            for (unsigned int i = 0; i < count; ++i)
                A(i, i) += sigma;
 
            return A;
        }
 
 
        template<typename Type>
        inline auto pair_inner_product(const Type& v_i, const Type& w_i) {
            return v_i * w_i;
        }
 
        template<typename Type>
        inline auto pair_inner_product(const complex<Type>& v_i, const complex<Type>& w_i) {
            return v_i * conjugate(w_i);
        }
 
 
        template<typename Vector>
        inline auto sqr_norm(const Vector& v) {
 
            auto sum = pair_inner_product(v[0], v[0]);
 
            // Use conjugation for complex numbers
            for (unsigned int i = 1; i < v.size(); ++i)
                sum += pair_inner_product(v[i], v[i]);
 
            return sum;
        }
 
 
        template<typename Vector>
        inline auto norm(const Vector& v) {
 
            return vector_element_t<Vector>(sqrt(sqr_norm(v)));
        }
 
 
        template<typename Vector>
        inline Vector normalize(const Vector& v) {
 
            Vector r;
            r.resize(v.size());
            vec_copy(r, v);
 
            const auto m = norm(v);
 
            if(abs(m) < MACH_EPSILON) {
                TH_MATH_ERROR("algebra::normalize", m, MathError::DivByZero);
                vec_error(r);
                return r;
            }
 
            for (unsigned int i = 0; i < r.size(); ++i)
                r[i] /= m;
 
            return r;
        }
 
 
        template<typename Vector>
        inline Vector& make_normalized(Vector& v) {
 
            const auto m = norm(v);
 
            if(abs(m) < MACH_EPSILON) {
                TH_MATH_ERROR("algebra::make_normalized", m, MathError::DivByZero);
                vec_error(v);
                return v;
            }
 
            for (unsigned int i = 0; i < v.size(); ++i)
                v[i] /= m;
 
            return v;
        }
 
 
        template<typename Vector1, typename Vector2>
        inline auto dot(const Vector1& v, const Vector2& w) {
 
            if(v.size() != w.size()) {
                TH_MATH_ERROR("algebra::dot", v.size(), MathError::InvalidArgument);
                return vector_element_t<Vector1>(nan());
            }
 
            auto sum = pair_inner_product(v[0], w[0]);
 
            // Use conjugation for complex numbers
            for (unsigned int i = 1; i < v.size(); ++i)
                sum += pair_inner_product(v[i], w[i]);
 
            return sum;
        }
 
 
        template<typename Vector1, typename Vector2>
        inline Vector1 cross(const Vector1& v1, const Vector2& v2) {
 
            Vector1 v3;
            v3.resize(3);
 
            if(v1.size() != 3) {
                TH_MATH_ERROR("algebra::cross", v1.size(), MathError::InvalidArgument);
                return vec_error(v3);
            }
 
            if(v2.size() != 3) {
                TH_MATH_ERROR("algebra::cross", v2.size(), MathError::InvalidArgument);
                return vec_error(v3);
            }
 
            v3[0] = v1[1] * v2[2] - v1[2] * v2[1];
            v3[1] = v1[2] * v2[0] - v1[0] * v2[2];
            v3[2] = v1[0] * v2[1] - v1[1] * v2[0];
 
            return v3;
        }
 
 
        template<typename Matrix, typename MatrixT = Matrix>
        inline MatrixT transpose(const Matrix& m) {
 
            MatrixT res;
            res.resize(m.cols(), m.rows());
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    res(j, i) = m(i, j);
 
            return res;
        }
 
 
        template<typename Matrix>
        inline Matrix& make_transposed(Matrix& m) {
 
            if(m.rows() != m.cols()) {
                TH_MATH_ERROR("algebra::make_transposed", m.rows(), MathError::InvalidArgument);
                return mat_error(m);
            }
 
            for (unsigned int i = 0; i < m.rows(); ++i) {
                for (unsigned int j = 0; j < i; ++j) {
                    
                    const auto buff = m(i, j);
                    m(i, j) = m(j, i);
                    m(j, i) = buff;
                }
            }
 
            return m;
        }
 
        
        template<typename Matrix1, typename Matrix2>
        inline Matrix1& transpose(Matrix1& dest, const Matrix2& src) {
 
            // Check that the two matrices have the correct
            // number of rows and columns
            if(src.rows() != dest.cols()) {
                TH_MATH_ERROR("algebra::transpose", src.rows(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            if(src.cols() != dest.rows()) {
                TH_MATH_ERROR("algebra::transpose", src.cols(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            // Overwrite dest with the transpose of src
            for (unsigned int i = 0; i < src.rows(); ++i)
                for (unsigned int j = 0; j < src.cols(); ++j)
                    dest(j, i) = src(i, j);
 
            return dest;
        }
 
 
        template<typename Matrix, typename MatrixT = Matrix>
        inline MatrixT hermitian(const Matrix& m) {
 
            MatrixT res;
            res.resize(m.cols(), m.rows());
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    res(j, i) = conjugate(m(i, j));
 
            return res;
        }
 
 
        template<typename Matrix>
        inline Matrix& make_hermitian(Matrix& m) {
 
            if(m.rows() != m.cols()) {
                TH_MATH_ERROR("algebra::hermitian", m.rows(), MathError::InvalidArgument);
                return mat_error(m);
            }
 
            for (unsigned int i = 0; i < m.rows(); ++i) {
                for (unsigned int j = 0; j < i; ++j) {
                    
                    const auto buff = m(i, j);
                    m(i, j) = conjugate(m(j, i));
                    m(j, i) = conjugate(buff);
                }
            }
 
            return m;
        }
 
 
        template<typename Matrix1, typename Matrix2>
        inline Matrix1& hermitian(Matrix1& dest, const Matrix2& src) {
 
            // Check that the two matrices have the correct
            // number of rows and columns
            if(src.rows() != dest.cols()) {
                TH_MATH_ERROR("algebra::hermitian", src.rows(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            if(src.cols() != dest.rows()) {
                TH_MATH_ERROR("algebra::hermitian", src.cols(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            // Overwrite dest with the transpose of src
            for (unsigned int i = 0; i < src.rows(); ++i)
                for (unsigned int j = 0; j < src.cols(); ++j)
                    dest(j, i) = conjugate(src(i, j));
 
            return dest;
        }
 
 
        template<typename Matrix>
        inline auto trace(const Matrix& m) {
 
            auto sum = m(0, 0);
            const size_t n = min(m.rows(), m.cols());
 
            for (unsigned int i = 1; i < n; ++i)
                sum += m(i, i);
 
            return sum;
        }
 
 
        template<typename Matrix>
        inline auto diagonal_product(const Matrix& m) {
 
            auto mul = m(0, 0);
            const size_t n = min(m.rows(), m.cols());
 
            for (unsigned int i = 1; i < n; ++i)
                mul *= m(i, i);
 
            return mul;
        }
 
 
        template<typename Field, typename Matrix>
        inline Matrix& mat_scalmul(Field a, Matrix& m) {
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    m(i, j) *= a;
 
            return m;
        }
 
 
        template<typename Field, typename Matrix1, typename Matrix2>
        inline Matrix1& mat_scalmul(Matrix1& dest, Field a, const Matrix2& src) {
 
            if(src.rows() != dest.rows()) {
                TH_MATH_ERROR("algebra::mat_scalmul", src.rows(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            if(src.cols() != dest.cols()) {
                TH_MATH_ERROR("algebra::mat_scalmul", src.cols(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            for (unsigned int i = 0; i < src.rows(); ++i)
                for (unsigned int j = 0; j < src.cols(); ++j)
                    dest(i, j) = a * src(i, j);
 
            return dest;
        }
 
 
        template<typename Field, typename Vector>
        inline Vector& vec_scalmul(Field a, Vector& v) {
 
            for (unsigned int i = 0; i < v.size(); ++i)
                v[i] *= a;
 
            return v;
        }
 
 
        template<typename Field, typename Vector1, typename Vector2>
        inline Vector1& vec_scalmul(Vector1& dest, Field a, const Vector2& src) {
 
            if(src.size() != dest.size()) {
                TH_MATH_ERROR("algebra::vec_scalmul", src.size(), MathError::InvalidArgument);
                return vec_error(dest);
            }
 
            for (unsigned int i = 0; i < src.size(); ++i)
                dest[i] = a * src[i];
 
            return dest;
        }
 
 
        // Operations involving a matrix and a vector
 
 
        template<typename Matrix, typename Vector>
        inline Vector& apply_transform(const Matrix& A, Vector& v) {
 
            if(v.size() != A.cols()) {
                TH_MATH_ERROR("algebra::apply_transform", v.size(), MathError::InvalidArgument);
                return vec_error(v);
            }
 
            Vector res;
            res.resize(v.size());
            vec_zeroes(res);
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    res[i] += A(i, j) * v[j];
 
            vec_copy(res, v);
            return v;
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector transform(const Matrix& A, const Vector& v) {
 
            Vector res;
            res.resize(v.size());
 
            if(v.size() != A.cols()) {
                TH_MATH_ERROR("algebra::transform", v.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            vec_zeroes(res);
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    res[i] += A(i, j) * v[j];
 
            return res;
        }
 
 
        template<typename Matrix, typename Vector1, typename Vector2>
        inline Vector1& transform(Vector1& res, const Matrix& A, const Vector2& v) {
 
            if(v.size() != A.cols()) {
                TH_MATH_ERROR("algebra::transform", v.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            if(res.size() != v.size()) {
                TH_MATH_ERROR("algebra::transform", res.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            vec_zeroes(res);
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    res[i] += A(i, j) * v[j];
 
            return res;
        }
 
 
        // Operations involving multiple matrices or vectors
 
 
        template<typename Matrix1, typename Matrix2>
        inline Matrix2& mat_sum(Matrix1& A, const Matrix2& B) {
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_sum", A.rows(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_sum", A.cols(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    A(i, j) = A(i, j) + B(i, j);
 
            return A;
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3>
        inline Matrix1& mat_sum(Matrix1& res, const Matrix2& A, const Matrix3& B) {
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_sum", A.rows(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_sum", A.cols(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(res.rows() != A.rows()) {
                TH_MATH_ERROR("algebra::mat_sum", res.rows(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(res.cols() != A.cols()) {
                TH_MATH_ERROR("algebra::mat_sum", res.cols(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    res(i, j) = A(i, j) + B(i, j);
 
            return res;
        }
 
 
        template<typename Matrix1, typename Matrix2>
        inline Matrix2& mat_diff(Matrix1& A, const Matrix2& B) {
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_diff", A.rows(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_diff", A.cols(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    A(i, j) = A(i, j) - B(i, j);
 
            return A;
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3>
        inline Matrix1& mat_diff(Matrix1& res, const Matrix2& A, const Matrix3& B) {
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_diff", A.rows(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_diff", A.cols(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(res.rows() != A.rows()) {
                TH_MATH_ERROR("algebra::mat_diff", res.rows(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(res.cols() != A.cols()) {
                TH_MATH_ERROR("algebra::mat_diff", res.cols(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    res(i, j) = A(i, j) - B(i, j);
 
            return res;
        }
 
 
        template<typename Field1, typename Matrix1, typename Field2, typename Matrix2>
        inline Matrix2& mat_lincomb(
            Field1 alpha, Matrix1& A, Field2 beta, const Matrix2& B) {
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_lincomb", A.rows(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_lincomb", A.cols(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    A(i, j) = A(i, j) * alpha + B(i, j) * beta;
 
            return A;
        }
 
 
        template<typename Matrix1, typename Field1, typename Matrix2,
            typename Field2, typename Matrix3>
        inline Matrix1& mat_lincomb(
            Matrix1& res, Field1 alpha, const Matrix2& A, Field2 beta, const Matrix3& B) {
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_lincomb", A.rows(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_lincomb", A.cols(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            if(res.rows() != A.rows()) {
                TH_MATH_ERROR("algebra::mat_lincomb", res.rows(), MathError::InvalidArgument);
                return mat_error(res);
                return res;
            }
 
            if(res.cols() != A.cols()) {
                TH_MATH_ERROR("algebra::mat_lincomb", res.cols(), MathError::InvalidArgument);
                return mat_error(res);
            }
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    res(i, j) = A(i, j) * alpha + B(i, j) * beta;
 
            return A;
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3 = Matrix1>
        inline Matrix3 mat_mul(const Matrix1& A, const Matrix2& B) {
 
            Matrix3 R;
            R.resize(A.rows(), B.cols());
 
            if(A.cols() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_mul", A.cols(), MathError::InvalidArgument);
                return mat_error(R);
            }
 
            mat_zeroes(R);
            
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < B.cols(); ++j)
                    for (unsigned int k = 0; k < A.cols(); ++k)
                        R(i, j) += A(i, k) * B(k, j);
 
            return R;
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3>
        inline Matrix1& mat_mul(Matrix1& R, const Matrix2& A, const Matrix3& B) {
            
            if(R.rows() != A.rows()) {
                TH_MATH_ERROR("algebra::mat_mul", R.rows(), MathError::InvalidArgument);
                return mat_error(R);
            }
 
            if(R.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_mul", R.cols(), MathError::InvalidArgument);
                return mat_error(R);
            }
 
            if(A.cols() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_mul", A.cols(), MathError::InvalidArgument);
                return mat_error(R);
            }
 
            mat_zeroes(R);
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < B.cols(); ++j)
                    for (unsigned int k = 0; k < A.cols(); ++k)
                        R(i, j) += A(i, k) * B(k, j);
 
            return R;
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3 = Matrix1>
        inline Matrix3 mat_transpose_mul(const Matrix1& A, const Matrix2& B) {
 
            Matrix3 R;
            R.resize(A.cols(), B.cols());
 
            if(A.rows() != B.rows()) {
                TH_MATH_ERROR("algebra::mat_transpose_mul", A.rows(), MathError::InvalidArgument);
                return mat_error(R);
            }
 
            mat_zeroes(R);
 
            for (unsigned int i = 0; i < R.rows(); ++i)
                for (unsigned int j = 0; j < R.cols(); ++j)
                    for (unsigned int k = 0; k < A.rows(); ++k)
                        R(i, j) += A(k, i) * B(k, j);
 
            return R;
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3 = Matrix1>
        inline Matrix3 mat_mul_transpose(const Matrix1& A, const Matrix2& B) {
 
            Matrix3 R;
            R.resize(A.rows(), B.rows());
 
            if(A.cols() != B.cols()) {
                TH_MATH_ERROR("algebra::mat_mul_transpose", A.rows(), MathError::InvalidArgument);
                return mat_error(R);
            }
 
            mat_zeroes(R);
 
            for (unsigned int i = 0; i < R.rows(); ++i)
                for (unsigned int j = 0; j < R.cols(); ++j)
                    for (unsigned int k = 0; k < A.cols(); ++k)
                        R(i, j) += A(i, k) * B(j, k);
 
            return R;
        }
 
 
        template<typename Matrix1, typename Matrix2>
        inline bool mat_equals(
            const Matrix1& A, const Matrix2& B, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            if(A.rows() != B.rows() || A.cols() != B.cols())
                return false;
 
            for (unsigned int i = 0; i < A.rows(); ++i) {
                for (unsigned int j = 0; j < A.cols(); ++j) {
            
                    const auto diff = abs(A(i, j) - B(i, j));
 
                    if(diff > tolerance || is_nan(diff))
                        return false;
                }
            }
 
            return true;
        }
 
 
        template<typename Vector1, typename Vector2>
        inline Vector2& vec_sum(Vector1& v1, const Vector2& v2) {
 
            if(v1.size() != v2.size()) {
                TH_MATH_ERROR("algebra::vec_sum", v1.size(), MathError::InvalidArgument);
                return vec_error(v1);
            }
 
            for (unsigned int i = 0; i < v1.size(); ++i)
                v1[i] = v1[i] + v2[i];
 
            return v1;
        }
 
 
        template<typename Vector1, typename Vector2, typename Vector3>
        inline Vector1& vec_sum(Vector1& res, const Vector2& v1, const Vector3& v2) {
 
            if(v1.size() != v2.size()) {
                TH_MATH_ERROR("algebra::vec_sum", v1.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            if(res.size() != v1.size()) {
                TH_MATH_ERROR("algebra::vec_sum", res.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            for (unsigned int i = 0; i < v1.size(); ++i)
                res[i] = v1[i] + v2[i];
 
            return res;
        }
 
 
        template<typename Vector1, typename Vector2>
        inline Vector2& vec_diff(Vector1& v1, const Vector2& v2) {
 
            if(v1.size() != v2.size()) {
                TH_MATH_ERROR("algebra::vec_diff", v1.size(), MathError::InvalidArgument);
                return vec_error(v1);
            }
 
            for (unsigned int i = 0; i < v1.size(); ++i)
                v1[i] = v1[i] - v2[i];
 
            return v1;
        }
 
 
        template<typename Vector1, typename Vector2, typename Vector3>
        inline Vector1& vec_diff(Vector1& res, const Vector2& v1, const Vector3& v2) {
 
            if(v1.size() != v2.size()) {
                TH_MATH_ERROR("algebra::vec_diff", v1.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            if(res.size() != v1.size()) {
                TH_MATH_ERROR("algebra::vec_diff", res.size(), MathError::InvalidArgument);
                return vec_error(res);
            }
 
            for (unsigned int i = 0; i < v1.size(); ++i)
                res[i] = v1[i] - v2[i];
 
            return res;
        }
 
 
        // Matrix properties
 
 
        template<typename Matrix>
        inline bool is_square(const Matrix& m) {
            return (m.rows() == m.cols());
        }
 
 
        template<typename Matrix>
        inline bool is_diagonal(const Matrix& m, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    if(i != j && abs(m(i, j)) > tolerance)
                        return false;
 
            return true;
        }
 
        template<typename Matrix>
        inline bool is_symmetric(const Matrix& m, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            if(!is_square(m))
                return false;
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < m.cols(); ++j)
                    if(abs(m(i, j) - m(j, i)) > tolerance)
                        return false;
 
            return true;
        }
 
 
        template<typename Matrix>
        inline bool is_lower_triangular(const Matrix& m, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            if(!is_square(m))
                return false;
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = i + 1; j < m.cols(); ++j)
                    if (abs(m(i, j)) > tolerance)
                        return false;
 
            return true;
        }
 
 
        template<typename Matrix>
        inline bool is_upper_triangular(const Matrix& m, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            if(!is_square(m))
                return false;
 
            for (unsigned int i = 0; i < m.rows(); ++i)
                for (unsigned int j = 0; j < i; ++j)
                    if (abs(m(i, j)) > tolerance)
                        return false;
 
            return true;
        }
 
 
        template<typename Matrix, enable_matrix<Matrix> = true>
        inline real density(const Matrix& A, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            long unsigned int n = 0;
 
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = 0; j < A.cols(); ++j)
                    if (abs(A(i, j)) > tolerance)
                        n++;
 
            return (real(n) / A.rows()) / A.cols();
        }
 
 
        template<typename Matrix, enable_matrix<Matrix> = true>
        inline real sparsity(const Matrix& A, real tolerance = ALGEBRA_ELEMENT_TOL) {
 
            return 1.0 - density(A, tolerance);
        }
 
 
        // Matrix decompositions
 
 
        template<typename Matrix1, typename Matrix2, typename Matrix3>
        inline void decompose_lu(const Matrix1& A, Matrix2& L, Matrix3& U) {
 
            // Check the shapes of A, L and U
            unsigned int err = 0;
 
            // Make sure to allocate L and U if they are empty
            L.resize(A.rows(), A.cols());
            U.resize(A.rows(), A.cols());
 
            if (!is_square(A))
                err = A.rows();
 
            if (A.rows() != L.rows())
                err = L.rows();
 
            if (A.cols() != L.cols())
                err = L.cols();
 
            if (A.rows() != U.rows())
                err = U.rows();
 
            if (A.cols() != U.cols())
                err = U.cols();
 
            if (err) {
                TH_MATH_ERROR("algebra::decompose_lu", err, MathError::InvalidArgument);
                mat_error(L); mat_error(U);
                return;
            }
 
            using Type = matrix_element_t<Matrix1>;
 
            // Set the diagonal of L to 1.0
            for (unsigned int i = 0; i < A.rows(); ++i)
                L(i, i) = (Type) 1.0;
 
            // Compute L and U
            for(unsigned int i = 0; i < A.rows(); ++i) {
                
                for(unsigned int j = 0; j < A.rows(); ++j) {
 
                    U(i, j) = A(i, j);
                
                    for(unsigned int k = 0; k < i; ++k)
                        U(i, j) -= pair_inner_product(L(i, k), U(k, j));
                }
 
                for(unsigned int j = i + 1; j < A.rows(); ++j) {
 
                    L(j, i) = A(j, i);
 
                    for(unsigned int k = 0; k < i; ++k)
                        L(j, i) -= pair_inner_product(L(j, k), U(k, i));
 
                    L(j, i) /= U(i, i);
                }
            }
        }
 
 
        template<typename Matrix>
        inline Matrix& decompose_lu_inplace(Matrix& A) {
 
            if (!is_square(A)) {
                TH_MATH_ERROR("algebra::decompose_lu_inplace", A.rows(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            for(unsigned int j = 0; j < A.cols(); ++j) {
 
                for(unsigned int i = j + 1; i < A.rows(); ++i) {
 
                    A(i, j) /= A(j, j);
 
                    for(unsigned int k = j + 1; k < A.rows(); ++k)
                        A(i, k) -= pair_inner_product(A(i, j), A(j, k));
                }
            }
        
            return A;
        }
 
 
        template<typename Matrix>
        inline Matrix decompose_cholesky(const Matrix& A) {
 
            Matrix L;
            L.resize(A.rows(), A.cols());
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("algebra::decompose_cholesky", A.rows(), MathError::InvalidArgument);
                return mat_error(L);
            }
 
            if (!is_symmetric(A)) {
                TH_MATH_ERROR("algebra::decompose_cholesky", false, MathError::InvalidArgument);
                return mat_error(L);
            }
 
            mat_zeroes(L);
 
            for (unsigned int i = 0; i < A.cols(); ++i) {
                
                for (unsigned int j = 0; j <= i; ++j) {
                    
                    Type sum = pair_inner_product(L(i, 0), L(j, 0));
 
                    for (unsigned int k = 1; k < j; ++k)
                        sum += pair_inner_product(L(i, k), L(j, k));
 
                    if (i == j) {
 
                        const Type sqr_diag = A(j, j) - sum;
 
                        // Additional check to ensure that the matrix is positive definite
                        if (sqr_diag < MACH_EPSILON) {
                            TH_MATH_ERROR("algebra::decompose_cholesky", sqr_diag, MathError::InvalidArgument);
                            return mat_error(L);
                        }
 
                        L(i, j) = sqrt(sqr_diag);
 
                    } else {
 
                        L(i, j) = (A(i, j) - sum) / L(j, j);
                    }
                }
            }
 
            return L;
        }
 
 
        template<typename Matrix>
        inline Matrix& decompose_cholesky_inplace(Matrix& A) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("algebra::decompose_cholesky_inplace", A.rows(), MathError::InvalidArgument);
                return mat_error(A);
            }
 
            if (!is_symmetric(A)) {
                TH_MATH_ERROR("algebra::decompose_cholesky_inplace", false, MathError::InvalidArgument);
                return mat_error(A);
            }
 
            // Compute the Cholesky decomposition in-place
            for (unsigned int k = 0; k < A.rows(); ++k) {
 
                A(k, k) = sqrt(A(k, k));
 
                for (unsigned int i = k + 1; i < A.rows(); ++i)
                    A(i, k) = A(i, k) / A(k, k);
 
                for (unsigned int j = k + 1; j < A.rows(); ++j)
                    for (unsigned int i = j; i < A.cols(); ++i)
                        A(i, j) -= pair_inner_product(A(i, k), A(j, k));
            }
 
            // Zero out elements over the diagonal
            for (unsigned int i = 0; i < A.rows(); ++i)
                for (unsigned int j = i + 1; j < A.rows(); ++j)
                    A(i, j) = Type(0.0);
                    
 
            return A;
        }
 
 
        // Linear system solvers
 
 
        template<typename Matrix, typename Vector>
        inline Vector solve_triangular_lower(const Matrix& L, const Vector& b) {
 
            Vector x;
            x.resize(L.cols());
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(L)) {
                TH_MATH_ERROR("algebra::solve_triangular_lower", false, MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (b.size() != L.rows()) {
                TH_MATH_ERROR("algebra::solve_triangular_lower", b.size(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            // Solve using forward substitution
            for (unsigned int i = 0; i < L.cols(); ++i) {
                
                Type sum = L(i, 0) * x[0];
 
                for (unsigned int j = 1; j < i; ++j)
                    sum += L(i, j) * x[j];
 
                if (abs(L(i, i)) < MACH_EPSILON) {
                    TH_MATH_ERROR("algebra::solve_triangular_lower", L(i, i), MathError::DivByZero);
                    return vec_error(x);
                }
 
                x[i] = (b[i] - sum) / L(i, i);
            }
 
            return x;
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector solve_triangular_upper(const Matrix& U, const Vector& b) {
 
            Vector x;
            x.resize(U.cols());
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(U)) {
                TH_MATH_ERROR("solve_triangular_upper", false, MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (b.size() != U.rows()) {
                TH_MATH_ERROR("solve_triangular_upper", b.size(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (U.rows() == 1) {
                x[0] = b[0] / U(0, 0);
                return x;
            }
 
            // Solve using backward substitution
            for (int i = U.rows() - 1; i >= 0; --i) {
                
                Type sum = U(i, i + 1) * x[i + 1];
 
                for (unsigned int j = i + 2; j < U.cols(); ++j)
                    sum += U(i, j) * x[j];
 
                if (abs(U(i, i)) < MACH_EPSILON) {
                    TH_MATH_ERROR("solve_triangular_upper", U(i, i), MathError::DivByZero);
                    return vec_error(x);
                }
 
                x[i] = (b[i] - sum) / U(i, i);
            }
 
            return x;
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector solve_triangular(const Matrix& T, const Vector& b) {
 
            // Pick the correct solver
            if (is_lower_triangular(T))
                return solve_triangular_lower(T, b);
            else if(is_upper_triangular(T))
                return solve_triangular_upper(T, b);
            else {
                Vector err;
                err.resize(b.size());
                return vec_error(err);
            }
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector& solve_lu_inplace(const Matrix& A, Vector& b) {
 
            if (!is_square(A)) {
                TH_MATH_ERROR("algebra::solve_lu_inplace", A.rows(), MathError::InvalidArgument);
                return vec_error(b);
            }
 
            if (A.rows() != b.size()) {
                TH_MATH_ERROR("algebra::solve_lu_inplace", A.rows(), MathError::InvalidArgument);
                return vec_error(b);
            }
 
            using Type = matrix_element_t<Matrix>;
 
            // Forward elimination: solves the lower system (L matrix).
            for (unsigned int i = 1; i < A.rows(); ++i) {
                
                Type sum = Type(0.0);
 
                for (unsigned int j = 0; j < i; ++j)
                    sum += A(i, j) * b[j];
 
                b[i] -= sum;
            }
 
            // Backward elimination: solves the upper system (U matrix).
            for (int i = A.rows() - 1; i >= 0; --i) {
 
                Type sum = Type(0.0);
                
                for (unsigned int j = i + 1; j < A.rows(); ++j)
                    sum += A(i, j) * b[j];
 
                b[i] = (b[i] - sum) / A(i, i);
            }
        
            return b;
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector solve_lu(Matrix A, Vector b) {
 
            // Apply in-place LU decomposition
            decompose_lu_inplace(A);
 
            // Apply forward and backward substitution
            return solve_lu_inplace(A, b);
        }
 
 
        template<typename Matrix1, typename Matrix2, typename Vector>
        inline Vector solve_lu(const Matrix1& L, const Matrix2& U, const Vector& b) {
 
            Vector x = b;
 
            if (!is_square(L)) {
                TH_MATH_ERROR("algebra::solve_lu", L.rows(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (!is_square(U)) {
                TH_MATH_ERROR("algebra::solve_lu", U.rows(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (L.rows() != U.rows()) {
                TH_MATH_ERROR("algebra::solve_lu", U.rows(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (b.size() != L.rows()) {
                TH_MATH_ERROR("algebra::solve_lu", b.size(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            using Type1 = matrix_element_t<Matrix1>;
 
            // Forward elimination for L
            for (unsigned int i = 1; i < L.rows(); ++i) {
                
                Type1 sum = Type1(0.0);
 
                for (unsigned int j = 0; j < i; ++j)
                    sum += L(i, j) * x[j];
 
                x[i] -= sum;
            }
 
            using Type2 = matrix_element_t<Matrix2>;
 
            // Backward elimination for U
            for (int i = U.rows() - 1; i >= 0; --i) {
 
                Type2 sum = Type2(0.0);
                
                for (unsigned int j = i + 1; j < U.rows(); ++j)
                    sum += U(i, j) * x[j];
 
                if (abs(U(i, i)) < MACH_EPSILON) {
                    TH_MATH_ERROR("algebra::solve_lu", U(i, i), MathError::DivByZero);
                    return vec_error(x);
                }
 
                x[i] = (x[i] - sum) / U(i, i);
            }
 
            return x;
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector solve_cholesky(const Matrix& L, const Vector& b) {
 
            Vector x = b;
 
            if (!is_square(L)) {
                TH_MATH_ERROR("algebra::solve_cholesky", L.rows(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            if (L.rows() != b.size()) {
                TH_MATH_ERROR("algebra::solve_cholesky", b.size(), MathError::InvalidArgument);
                return vec_error(x);
            }
 
            using Type = matrix_element_t<Matrix>;
 
            // Forward elimination for L
            for (unsigned int i = 0; i < L.rows(); ++i) {
                
                Type sum = Type(0.0);
 
                for (unsigned int j = 0; j < i; ++j)
                    sum += L(i, j) * x[j];
 
                x[i] = (x[i] - sum) / L(i, i);
            }
 
            // Backward elimination for L transpose
            for (int i = L.rows() - 1; i >= 0; --i) {
 
                Type sum = Type(0.0);
                
                for (unsigned int j = i + 1; j < L.rows(); ++j)
                    sum += L(j, i) * x[j];
 
                x[i] = (x[i] - sum) / L(i, i);
            }
 
            return x;
        }
 
 
        template<typename Matrix, typename Vector>
        inline Vector solve(const Matrix& A, const Vector& b) {
 
            // Use LU decomposition to solve the linear system
            return solve_lu(A, b);
        }
 
 
        // Other composite operations
 
 
        template<typename Matrix1, typename Matrix2>
        inline Matrix1& inverse(Matrix1& dest, const Matrix2& src) {
 
            if(src.rows() != src.cols()) {
                TH_MATH_ERROR("algebra::inverse", src.rows(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            if(dest.rows() != src.rows()) {
                TH_MATH_ERROR("algebra::inverse", dest.rows(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            if(dest.cols() != src.cols()) {
                TH_MATH_ERROR("algebra::inverse", dest.cols(), MathError::InvalidArgument);
                return mat_error(dest);
            }
 
            using Type = matrix_element_t<Matrix2>;
 
            // Prepare extended matrix (A|B)
            Matrix1 A;
            A.resize(src.rows(), src.cols());
            dest.resize(src.rows(), src.cols());
            mat_copy(A, src);
            make_identity(dest);
 
            // Iterate on all columns
            for (unsigned int i = 0; i < src.rows(); ++i) {
                
                // Make sure the element on the diagonal
                // is non-zero by adding the first non-zero row
                if(A(i, i) == (Type) 0) {
 
                    bool flag = false;
 
                    // Iterate on all rows
                    for (unsigned int j = i + 1; j < src.rows(); ++j) {
 
                        // Add the j-th row to the i-th row
                        // if Aji is non-zero
                        if(A(j, i) != (Type) 0) {
 
                            for (unsigned int k = 0; k < src.rows(); ++k) {
                                A(i, k) += A(j, k);
                                dest(i, k) += dest(j, k);
                            }
 
                            flag = true;
                            break;
                        }
                    }
 
                    if(!flag) {
                        TH_MATH_ERROR("algebra::inverse", flag, MathError::ImpossibleOperation);
                        return mat_error(dest);
                    }
                }
 
                auto inv_pivot = ((Type) 1.0) / A(i, i);
 
                // Use the current row to make all other
                // elements of the column equal to zero
                for (unsigned int j = 0; j < src.rows(); ++j) {
 
                    // Skip the current row
                    if(j == i)
                        continue;
 
                    // Multiplication coefficient for
                    // the elision of Ajk
                    const auto coeff = A(j, i) * inv_pivot;
                    
                    for (unsigned int k = 0; k < src.rows(); ++k) {
                        A(j, k) -= coeff * A(i, k);
                        dest(j, k) -= coeff * dest(i, k);
                    }
                }
 
                // Divide the current row by the pivot
                for (unsigned int j = 0; j < src.rows(); ++j) {
                    A(i, j) *= inv_pivot;
                    dest(i, j) *= inv_pivot;
                }
                
            }
 
            return dest;
        }
 
 
        template<typename Matrix, typename MatrixInv = Matrix>
        inline MatrixInv inverse(const Matrix& m) {
            MatrixInv res;
            res.resize(m.rows(), m.cols());
            inverse(res, m);
            return res;
        }
 
 
        template<typename Matrix>
        inline Matrix& invert(Matrix& m) {
 
            if(m.rows() != m.cols()) {
                TH_MATH_ERROR("algebra::invert", m.rows(), MathError::InvalidArgument);
                return mat_error(m);
            }
 
            // Prepare extended matrix (A|B)
            Matrix tmp;
            tmp.resize(m.rows(), m.cols());
            inverse(tmp, m);
 
            // Modify the matrix only when the inversion
            // has succeeded
            mat_copy(m, tmp);
            return m;
        }
 
 
        template<typename Matrix>
        inline auto det(const Matrix& A) {
 
            Matrix LU = A;
            decompose_lu_inplace(LU);
 
            // The determinant of a triangular matrix
            // is the product of the elements on its diagonal
            return diagonal_product(LU);
        }
 
 
        // Eigensolvers
 
 
        template<typename Matrix, typename Vector>
        inline auto rayleigh_quotient(const Matrix& A, const Vector& x) {
 
            const auto p = dot(x, x);
 
            // Check for division by zero
            if (abs(p) < MACH_EPSILON) {
                TH_MATH_ERROR("rayleigh_quotient", abs(p), MathError::DivByZero);
                return vector_element_t<Vector>(nan());
            }
 
            return dot(x, transform(A, x)) / p;
        }
 
 
        template<typename Matrix, typename Vector>
        inline auto eigenvalue_power(
            const Matrix& A, const Vector& x,
            real tolerance = ALGEBRA_EIGEN_TOL, unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenvalue_power", is_square(A), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (x.size() != A.rows()) {
                TH_MATH_ERROR("eigenvalue_power", x.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            // Apply a first iteration to initialize
            // the current and previous vectors
            Vector x_prev = x;
            Vector x_curr = normalize(A * x_prev);
 
            // Iteration counter
            unsigned int i;
 
            // Iteratively apply the matrix to the vector
            for (i = 1; i <= max_iter; ++i) {
                
                x_prev = x_curr;
                x_curr = normalize(transform(A, x_prev));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(x_curr - x_prev) <= tolerance ||
                    norm(x_curr + x_prev) <= tolerance)
                    break;
            }
 
            // The algorithm did not converge
            if (i > max_iter) {
                TH_MATH_ERROR("eigenvalue_power", i, MathError::NoConvergence);
                return Type(nan());
            }
 
            return dot(x_curr, A * x_curr);
        }
 
 
        template<typename Matrix, typename Vector1, typename Vector2 = Vector1>
        inline auto eigenpair_power(
            const Matrix& A, const Vector1& x, Vector2& v,
            real tolerance = ALGEBRA_EIGEN_TOL, unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenpair_power", is_square(A), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (x.size() != A.rows()) {
                TH_MATH_ERROR("eigenpair_power", x.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (v.size() != x.size()) {
                TH_MATH_ERROR("eigenpair_power", v.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            // Apply a first iteration to initialize
            // the current and previous vectors
            Vector1 x_prev = x;
            Vector1 x_curr = normalize(A * x_prev);
 
            // Iteration counter
            unsigned int i;
 
            // Iteratively apply the matrix to the vector
            for (i = 1; i <= max_iter; ++i) {
                
                x_prev = x_curr;
                x_curr = normalize(transform(A, x_prev));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(x_curr - x_prev) <= tolerance ||
                    norm(x_curr + x_prev) <= tolerance)
                    break;
            }
 
            // The algorithm did not converge
            if (i > max_iter) {
                TH_MATH_ERROR("eigenpair_power", i, MathError::NoConvergence);
                return Type(nan());
            }
 
            // Overwrite with the eigenvector
            vec_copy(v, x_curr);
 
            return dot(x_curr, A * x_curr);
        }
 
 
        template<typename Matrix, typename Vector>
        inline auto eigenvalue_inverse(
            const Matrix& A, const Vector& x,
            real tolerance = ALGEBRA_EIGEN_TOL,
            unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenvalue_inverse", is_square(A), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (A.rows() != x.size()) {
                TH_MATH_ERROR("eigenvalue_inverse", x.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            // Compute the LU decomposition of A to speed up system solution
            Matrix LU = A;
            decompose_lu_inplace(LU);
 
            // Compute the first step to initialize the two vectors
            Vector x_prev = normalize(x);
            Vector x_curr = x_prev;
            solve_lu_inplace(LU, x_curr);
 
            // Iteration counter
            unsigned int i;
 
            for (i = 1; i <= max_iter; ++i) {
                
                // Solve the linear system using the LU decomposition
                x_prev = x_curr;
                make_normalized(solve_lu_inplace(LU, x_curr));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(x_curr - x_prev) <= tolerance ||
                    norm(x_curr + x_prev) <= tolerance)
                    break;
            }
 
            // The algorithm did not converge
            if (i > max_iter) {
                TH_MATH_ERROR("eigenvalue_inverse", i, MathError::NoConvergence);
                return Type(nan());
            }
 
            // A and inverse(A) have the same eigenvectors
            return dot(x_curr, A * x_curr);
        }
 
 
        template<typename Matrix, typename Vector1, typename Vector2>
        inline auto eigenpair_inverse(
            const Matrix& A, const Vector1& x, Vector2& v,
            real tolerance = ALGEBRA_EIGEN_TOL, unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenpair_inverse", is_square(A), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (A.rows() != x.size()) {
                TH_MATH_ERROR("eigenpair_inverse", x.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            // Compute the LU decomposition of A to speed up system solution
            Matrix LU = A;
            decompose_lu_inplace(LU);
 
            // Compute the first step to initialize the two vectors
            Vector1 x_prev = normalize(x);
            Vector1 x_curr = x_prev;
            solve_lu_inplace(LU, x_curr);
 
            // Iteration counter
            unsigned int i;
 
            for (i = 1; i <= max_iter; ++i) {
                
                // Solve the linear system using the LU decomposition
                x_prev = x_curr;
                make_normalized(solve_lu_inplace(LU, x_curr));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(x_curr - x_prev) <= tolerance ||
                    norm(x_curr + x_prev) <= tolerance)
                    break;
            }
 
            // The algorithm did not converge
            if (i > max_iter) {
                TH_MATH_ERROR("eigenpair_inverse", i, MathError::NoConvergence);
                return Type(nan());
            }
 
            // Overwrite with the eigenvector
            vec_copy(v, x_curr);
 
            // A and inverse(A) have the same eigenvectors
            return dot(x_curr, A * x_curr);
        }
 
 
        template<typename Matrix, typename Vector, typename T = matrix_element_t<Matrix>>
        inline Vector eigenvector_inverse(
            const Matrix& A, const T& lambda, const Vector& x,
            real tolerance = ALGEBRA_EIGEN_TOL, unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenvector_inverse", is_square(A), MathError::InvalidArgument);
                return make_error<Vector>(A.rows());
            }
 
            if (A.rows() != x.size()) {
                TH_MATH_ERROR("eigenvector_inverse", x.size(), MathError::InvalidArgument);
                return make_error<Vector>(A.rows());
            }
 
            Matrix LU = A;
            mat_shift_diagonal(LU, -lambda);
 
            // Compute the LU decomposition of A
            // to speed up system solution
            decompose_lu_inplace(LU);
 
            // Compute the first step to initialize the two vectors
            Vector v_prev = normalize(x);
            Vector v_curr = v_prev;
            solve_lu_inplace(LU, v_curr);
 
            // Iteration counter
            unsigned int i;
 
            for (i = 1; i <= max_iter; ++i) {
                
                // Solve the linear system using the LU decomposition
                v_prev = v_curr;
                make_normalized(solve_lu_inplace(LU, v_curr));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(v_curr - v_prev) <= tolerance ||
                    norm(v_curr + v_prev) <= tolerance)
                    break;
            }
 
            // The algorithm did not converge
            if (i > max_iter) {
                TH_MATH_ERROR("eigenvector_inverse", i, MathError::NoConvergence);
                return vec_error(v_curr);
            }
 
            return v_curr;
        }
 
 
        template<typename Matrix, typename Vector, typename T = matrix_element_t<Matrix>>
        inline auto eigenvalue_rayleigh(
            const Matrix& A, const T& lambda, const Vector& x,
            real tolerance = ALGEBRA_EIGEN_TOL, unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenvalue_rayleigh", is_square(A), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (A.rows() != x.size()) {
                TH_MATH_ERROR("eigenvalue_rayleigh", x.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            // Keep track of the shifted matrix
            Matrix A_shift = A;
            mat_shift_diagonal(A_shift, -lambda);
 
            Type lambda_prev = lambda;
            Type lambda_curr = lambda;
 
            Vector x_prev = normalize(x);
            Vector x_curr = normalize(solve(A_shift, x));
 
            unsigned int i;
 
            for (i = 1; i <= max_iter; ++i) {
                    
                // Update the eigenvalue approximation
                // using Rayleigh quotient (avoiding normalization)
                lambda_prev = lambda_curr;
                lambda_curr = dot(x_curr, transform(A, x_curr));
 
                // Shift the diagonal by the difference between
                // subsequent eigenvalues steps, to avoid copying matrix A
                mat_shift_diagonal(A_shift, lambda_prev - lambda_curr);
 
                // Solve the linear system each time, as the shifted
                // matrix is continuously updated
                x_prev = x_curr;
                x_curr = normalize(solve(A_shift, x_curr));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(x_curr - x_prev) <= tolerance ||
                    norm(x_curr + x_prev) <= tolerance)
                    break;
            }
 
            if (i > max_iter) {
                TH_MATH_ERROR("eigenvalue_rayleigh", i, MathError::NoConvergence);
                return Type(nan());
            }
 
            return dot(x_curr, transform(A, x_curr));
        }
 
 
        template<typename Matrix, typename Vector1, typename Vector2,
        typename T = matrix_element_t<Matrix>>
        inline auto eigenpair_rayleigh(
            const Matrix& A, const T& lambda, const Vector1& x, Vector2& v,
            real tolerance = ALGEBRA_EIGEN_TOL, unsigned int max_iter = ALGEBRA_EIGEN_ITER) {
 
            using Type = matrix_element_t<Matrix>;
 
            if (!is_square(A)) {
                TH_MATH_ERROR("eigenpair_rayleigh", is_square(A), MathError::InvalidArgument);
                return Type(nan());
            }
 
            if (A.rows() != x.size()) {
                TH_MATH_ERROR("eigenpair_rayleigh", x.size(), MathError::InvalidArgument);
                return Type(nan());
            }
 
            // Keep track of the shifted matrix
            Matrix A_shift = A;
            mat_shift_diagonal(A_shift, -lambda);
 
            Type lambda_prev = lambda;
            Type lambda_curr = lambda;
 
            Vector1 x_prev = normalize(x);
            Vector1 x_curr = normalize(solve(A_shift, x));
 
            unsigned int i;
 
            for (i = 1; i <= max_iter; ++i) {
                    
                // Update the eigenvalue approximation
                // using Rayleigh quotient (avoiding normalization)
                lambda_prev = lambda_curr;
                lambda_curr = dot(x_curr, transform(A, x_curr));
 
                // Shift the diagonal by the difference between
                // subsequent eigenvalues steps, to avoid copying matrix A
                mat_shift_diagonal(A_shift, lambda_prev - lambda_curr);
 
                // Solve the linear system each time, as the shifted
                // matrix is continuously updated
                x_prev = x_curr;
                x_curr = normalize(solve(A_shift, x_curr));
 
                // Stop the algorithm when |x_k+1 +- x_k| is
                // less then the tolerance in module
                if (norm(x_curr - x_prev) <= tolerance ||
                    norm(x_curr + x_prev) <= tolerance)
                    break;
            }
 
            if (i > max_iter) {
                TH_MATH_ERROR("eigenpair_rayleigh", i, MathError::NoConvergence);
                return Type(nan());
            }
 
            // Overwrite with the eigenvector
            vec_copy(v, x_curr);
 
            return dot(x_curr, transform(A, x_curr));
        }
    }
}
 
#endif