UpperHessenbergQR.h

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// Copyright (C) 2016-2025 Yixuan Qiu <yixuan.qiu@cos.name>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at https://mozilla.org/MPL/2.0/.
 
#ifndef SPECTRA_UPPER_HESSENBERG_QR_H
#define SPECTRA_UPPER_HESSENBERG_QR_H
 
#include <Eigen/Core>
#include <cmath>      // std::abs, std::sqrt, std::pow
#include <algorithm>  // std::fill
#include <stdexcept>  // std::logic_error
 
#include "../Util/TypeTraits.h"
 
namespace Spectra {
 
 
 
 
template <typename Scalar = double>
class UpperHessenbergQR
{
private:
    using Index = Eigen::Index;
    using Matrix = Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic>;
    using Vector = Eigen::Matrix<Scalar, Eigen::Dynamic, 1>;
    using RowVector = Eigen::Matrix<Scalar, 1, Eigen::Dynamic>;
    using Array = Eigen::Array<Scalar, Eigen::Dynamic, 1>;
 
    using GenericMatrix = Eigen::Ref<Matrix>;
    using ConstGenericMatrix = const Eigen::Ref<const Matrix>;
 
    Matrix m_mat_R;
 
protected:
    Index m_n;
    // Gi = [ cos[i]  sin[i]]
    //      [-sin[i]  cos[i]]
    // Q = G1 * G2 * ... * G_{n-1}
    Scalar m_shift;
    Array m_rot_cos;
    Array m_rot_sin;
    bool m_computed;
 
    // Given a >= b > 0, compute r = sqrt(a^2 + b^2), c = a / r, and s = b / r with high precision
    static void stable_scaling(const Scalar& a, const Scalar& b, Scalar& r, Scalar& c, Scalar& s)
    {
        using std::sqrt;
        using std::pow;
 
        // Let t = b / a, then 0 < t <= 1
        // c = 1 / sqrt(1 + t^2)
        // s = t * c
        // r = a * sqrt(1 + t^2)
        const Scalar t = b / a;
        // We choose a cutoff such that cutoff^4 < eps
        // If t > cutoff, use the standard way; otherwise use Taylor series expansion
        // to avoid an explicit sqrt() call that may lose precision
        const Scalar eps = TypeTraits<Scalar>::epsilon();
        // std::pow() is not constexpr, so we do not declare cutoff to be constexpr
        // But most compilers should be able to compute cutoff at compile time
        const Scalar cutoff = Scalar(0.1) * pow(eps, Scalar(0.25));
        if (t >= cutoff)
        {
            const Scalar denom = sqrt(Scalar(1) + t * t);
            c = Scalar(1) / denom;
            s = t * c;
            r = a * denom;
        }
        else
        {
            // 1 / sqrt(1 + t^2) ~=     1 - (1/2) * t^2 + (3/8) * t^4
            // 1 / sqrt(1 + l^2) ~= 1 / l - (1/2) / l^3 + (3/8) / l^5
            //                   ==     t - (1/2) * t^3 + (3/8) * t^5, where l = 1 / t
            // sqrt(1 + t^2)     ~=     1 + (1/2) * t^2 - (1/8) * t^4 + (1/16) * t^6
            //
            // c = 1 / sqrt(1 + t^2) ~= 1 - t^2 * (1/2 - (3/8) * t^2)
            // s = 1 / sqrt(1 + l^2) ~= t * (1 - t^2 * (1/2 - (3/8) * t^2))
            // r = a * sqrt(1 + t^2) ~= a + (1/2) * b * t - (1/8) * b * t^3 + (1/16) * b * t^5
            //                       == a + (b/2) * t * (1 - t^2 * (1/4 - 1/8 * t^2))
            const Scalar c1 = Scalar(1);
            const Scalar c2 = Scalar(0.5);
            const Scalar c4 = Scalar(0.25);
            const Scalar c8 = Scalar(0.125);
            const Scalar c38 = Scalar(0.375);
            const Scalar t2 = t * t;
            const Scalar tc = t2 * (c2 - c38 * t2);
            c = c1 - tc;
            s = t - t * tc;
            r = a + c2 * b * t * (c1 - t2 * (c4 - c8 * t2));
 
            /* const Scalar t_2 = Scalar(0.5) * t;
            const Scalar t2_2 = t_2 * t;
            const Scalar t3_2 = t2_2 * t;
            const Scalar t4_38 = Scalar(1.5) * t2_2 * t2_2;
            const Scalar t5_16 = Scalar(0.25) * t3_2 * t2_2;
            c = Scalar(1) - t2_2 + t4_38;
            s = t - t3_2 + Scalar(6) * t5_16;
            r = a + b * (t_2 - Scalar(0.25) * t3_2 + t5_16); */
        }
    }
 
    // Given x and y, compute 1) r = sqrt(x^2 + y^2), 2) c = x / r, 3) s = -y / r
    // If both x and y are zero, set c = 1 and s = 0
    // We must implement it in a numerically stable way
    // The implementation below is shown to be more accurate than directly computing
    //     r = std::hypot(x, y); c = x / r; s = -y / r;
    static void compute_rotation(const Scalar& x, const Scalar& y, Scalar& r, Scalar& c, Scalar& s)
    {
        using std::abs;
 
        // Only need xsign when x != 0
        const Scalar xsign = (x > Scalar(0)) ? Scalar(1) : Scalar(-1);
        const Scalar xabs = abs(x);
        if (y == Scalar(0))
        {
            c = (x == Scalar(0)) ? Scalar(1) : xsign;
            s = Scalar(0);
            r = xabs;
            return;
        }
 
        // Now we know y != 0
        const Scalar ysign = (y > Scalar(0)) ? Scalar(1) : Scalar(-1);
        const Scalar yabs = abs(y);
        if (x == Scalar(0))
        {
            c = Scalar(0);
            s = -ysign;
            r = yabs;
            return;
        }
 
        // Now we know x != 0, y != 0
        if (xabs > yabs)
        {
            stable_scaling(xabs, yabs, r, c, s);
            c = xsign * c;
            s = -ysign * s;
        }
        else
        {
            stable_scaling(yabs, xabs, r, s, c);
            c = xsign * c;
            s = -ysign * s;
        }
    }
 
public:
    UpperHessenbergQR(Index size) :
        m_n(size),
        m_shift(0),
        m_rot_cos(m_n - 1),
        m_rot_sin(m_n - 1),
        m_computed(false)
    {}
 
    UpperHessenbergQR(ConstGenericMatrix& mat, const Scalar& shift = Scalar(0)) :
        m_n(mat.rows()),
        m_shift(shift),
        m_rot_cos(m_n - 1),
        m_rot_sin(m_n - 1),
        m_computed(false)
    {
        compute(mat, shift);
    }
 
    virtual ~UpperHessenbergQR() {}
 
    virtual void compute(ConstGenericMatrix& mat, const Scalar& shift = Scalar(0))
    {
        m_n = mat.rows();
        if (m_n != mat.cols())
            throw std::invalid_argument("UpperHessenbergQR: matrix must be square");
 
        m_shift = shift;
        m_mat_R.resize(m_n, m_n);
        m_rot_cos.resize(m_n - 1);
        m_rot_sin.resize(m_n - 1);
 
        // Make a copy of mat - s * I
        m_mat_R.noalias() = mat;
        m_mat_R.diagonal().array() -= m_shift;
 
        Scalar xi, xj, r, c, s;
        Scalar *Rii, *ptr;
        const Index n1 = m_n - 1;
        for (Index i = 0; i < n1; i++)
        {
            Rii = &m_mat_R.coeffRef(i, i);
 
            // Make sure R is upper Hessenberg
            // Zero the elements below R[i + 1, i]
            std::fill(Rii + 2, Rii + m_n - i, Scalar(0));
 
            xi = Rii[0];  // R[i, i]
            xj = Rii[1];  // R[i + 1, i]
            compute_rotation(xi, xj, r, c, s);
            m_rot_cos.coeffRef(i) = c;
            m_rot_sin.coeffRef(i) = s;
 
            // For a complete QR decomposition,
            // we first obtain the rotation matrix
            // G = [ cos  sin]
            //     [-sin  cos]
            // and then do R[i:(i + 1), i:(n - 1)] = G' * R[i:(i + 1), i:(n - 1)]
 
            // Gt << c, -s, s, c;
            // m_mat_R.block(i, i, 2, m_n - i) = Gt * m_mat_R.block(i, i, 2, m_n - i);
            Rii[0] = r;       // R[i, i]     => r
            Rii[1] = 0;       // R[i + 1, i] => 0
            ptr = Rii + m_n;  // R[i, k], k = i+1, i+2, ..., n-1
            for (Index j = i + 1; j < m_n; j++, ptr += m_n)
            {
                const Scalar tmp = ptr[0];
                ptr[0] = c * tmp - s * ptr[1];
                ptr[1] = s * tmp + c * ptr[1];
            }
 
            // If we do not need to calculate the R matrix, then
            // only the cos and sin sequences are required.
            // In such case we only update R[i + 1, (i + 1):(n - 1)]
            // m_mat_R.block(i + 1, i + 1, 1, m_n - i - 1) *= c;
            // m_mat_R.block(i + 1, i + 1, 1, m_n - i - 1) += s * m_mat_R.block(i, i + 1, 1, m_n - i - 1);
        }
 
        m_computed = true;
    }
 
    virtual Matrix matrix_R() const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        return m_mat_R;
    }
 
    virtual void matrix_QtHQ(Matrix& dest) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        // Make a copy of the R matrix
        dest.resize(m_n, m_n);
        dest.noalias() = m_mat_R;
 
        // Compute the RQ matrix
        const Index n1 = m_n - 1;
        for (Index i = 0; i < n1; i++)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // RQ[, i:(i + 1)] = RQ[, i:(i + 1)] * Gi
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            Scalar *Yi, *Yi1;
            Yi = &dest.coeffRef(0, i);
            Yi1 = Yi + m_n;  // RQ[0, i + 1]
            const Index i2 = i + 2;
            for (Index j = 0; j < i2; j++)
            {
                const Scalar tmp = Yi[j];
                Yi[j] = c * tmp - s * Yi1[j];
                Yi1[j] = s * tmp + c * Yi1[j];
            }
 
            /* Vector dest = RQ.block(0, i, i + 2, 1);
            dest.block(0, i, i + 2, 1)     = c * Yi - s * dest.block(0, i + 1, i + 2, 1);
            dest.block(0, i + 1, i + 2, 1) = s * Yi + c * dest.block(0, i + 1, i + 2, 1); */
        }
 
        // Add the shift to the diagonal
        dest.diagonal().array() += m_shift;
    }
 
    // Y -> QY = G1 * G2 * ... * Y
    void apply_QY(Vector& Y) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        for (Index i = m_n - 2; i >= 0; i--)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // Y[i:(i + 1)] = Gi * Y[i:(i + 1)]
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            const Scalar tmp = Y[i];
            Y[i] = c * tmp + s * Y[i + 1];
            Y[i + 1] = -s * tmp + c * Y[i + 1];
        }
    }
 
    // Y -> Q'Y = G_{n-1}' * ... * G2' * G1' * Y
    void apply_QtY(Vector& Y) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        const Index n1 = m_n - 1;
        for (Index i = 0; i < n1; i++)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // Y[i:(i + 1)] = Gi' * Y[i:(i + 1)]
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            const Scalar tmp = Y[i];
            Y[i] = c * tmp - s * Y[i + 1];
            Y[i + 1] = s * tmp + c * Y[i + 1];
        }
    }
 
    // Y -> QY = G1 * G2 * ... * Y
    void apply_QY(GenericMatrix Y) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        RowVector Yi(Y.cols()), Yi1(Y.cols());
        for (Index i = m_n - 2; i >= 0; i--)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // Y[i:(i + 1), ] = Gi * Y[i:(i + 1), ]
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            Yi.noalias() = Y.row(i);
            Yi1.noalias() = Y.row(i + 1);
            Y.row(i) = c * Yi + s * Yi1;
            Y.row(i + 1) = -s * Yi + c * Yi1;
        }
    }
 
    // Y -> Q'Y = G_{n-1}' * ... * G2' * G1' * Y
    void apply_QtY(GenericMatrix Y) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        RowVector Yi(Y.cols()), Yi1(Y.cols());
        const Index n1 = m_n - 1;
        for (Index i = 0; i < n1; i++)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // Y[i:(i + 1), ] = Gi' * Y[i:(i + 1), ]
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            Yi.noalias() = Y.row(i);
            Yi1.noalias() = Y.row(i + 1);
            Y.row(i) = c * Yi - s * Yi1;
            Y.row(i + 1) = s * Yi + c * Yi1;
        }
    }
 
    // Y -> YQ = Y * G1 * G2 * ...
    void apply_YQ(GenericMatrix Y) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        /*Vector Yi(Y.rows());
        for(Index i = 0; i < m_n - 1; i++)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // Y[, i:(i + 1)] = Y[, i:(i + 1)] * Gi
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            Yi.noalias() = Y.col(i);
            Y.col(i)     = c * Yi - s * Y.col(i + 1);
            Y.col(i + 1) = s * Yi + c * Y.col(i + 1);
        }*/
        Scalar *Y_col_i, *Y_col_i1;
        const Index n1 = m_n - 1;
        const Index nrow = Y.rows();
        for (Index i = 0; i < n1; i++)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
 
            Y_col_i = &Y.coeffRef(0, i);
            Y_col_i1 = &Y.coeffRef(0, i + 1);
            for (Index j = 0; j < nrow; j++)
            {
                Scalar tmp = Y_col_i[j];
                Y_col_i[j] = c * tmp - s * Y_col_i1[j];
                Y_col_i1[j] = s * tmp + c * Y_col_i1[j];
            }
        }
    }
 
    // Y -> YQ' = Y * G_{n-1}' * ... * G2' * G1'
    void apply_YQt(GenericMatrix Y) const
    {
        if (!m_computed)
            throw std::logic_error("UpperHessenbergQR: need to call compute() first");
 
        Vector Yi(Y.rows());
        for (Index i = m_n - 2; i >= 0; i--)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            // Y[, i:(i + 1)] = Y[, i:(i + 1)] * Gi'
            // Gi = [ cos[i]  sin[i]]
            //      [-sin[i]  cos[i]]
            Yi.noalias() = Y.col(i);
            Y.col(i) = c * Yi + s * Y.col(i + 1);
            Y.col(i + 1) = -s * Yi + c * Y.col(i + 1);
        }
    }
};
 
template <typename Scalar = double>
class TridiagQR : public UpperHessenbergQR<Scalar>
{
private:
    using Index = Eigen::Index;
    using Matrix = Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic>;
    using Vector = Eigen::Matrix<Scalar, Eigen::Dynamic, 1>;
    using ConstGenericMatrix = const Eigen::Ref<const Matrix>;
    using ComplexMatrix = Eigen::Matrix<std::complex<Scalar>, Eigen::Dynamic, Eigen::Dynamic>;
 
    using UpperHessenbergQR<Scalar>::m_n;
    using UpperHessenbergQR<Scalar>::m_shift;
    using UpperHessenbergQR<Scalar>::m_rot_cos;
    using UpperHessenbergQR<Scalar>::m_rot_sin;
    using UpperHessenbergQR<Scalar>::m_computed;
 
    Vector m_T_diag;   // diagonal elements of T
    Vector m_T_subd;   // 1st subdiagonal of T
    Vector m_R_diag;   // diagonal elements of R, where T = QR
    Vector m_R_supd;   // 1st superdiagonal of R
    Vector m_R_supd2;  // 2nd superdiagonal of R
 
public:
    TridiagQR(Index size) :
        UpperHessenbergQR<Scalar>(size)
    {}
 
    TridiagQR(ConstGenericMatrix& mat, const Scalar& shift = Scalar(0)) :
        UpperHessenbergQR<Scalar>(mat.rows())
    {
        this->compute(mat, shift);
    }
 
    void compute(ConstGenericMatrix& mat, const Scalar& shift = Scalar(0)) override
    {
        using std::abs;
 
        m_n = mat.rows();
        if (m_n != mat.cols())
            throw std::invalid_argument("TridiagQR: matrix must be square");
 
        m_shift = shift;
        m_rot_cos.resize(m_n - 1);
        m_rot_sin.resize(m_n - 1);
 
        // Save the diagonal and subdiagonal elements of T
        m_T_diag.resize(m_n);
        m_T_subd.resize(m_n - 1);
        m_T_diag.noalias() = mat.diagonal();
        m_T_subd.noalias() = mat.diagonal(-1);
 
        // Deflation of small sub-diagonal elements
        const Scalar eps = TypeTraits<Scalar>::epsilon();
        for (Index i = 0; i < m_n - 1; i++)
        {
            if (abs(m_T_subd[i]) <= eps * (abs(m_T_diag[i]) + abs(m_T_diag[i + 1])))
                m_T_subd[i] = Scalar(0);
        }
 
        // Apply shift and copy T to R
        m_R_diag.resize(m_n);
        m_R_supd.resize(m_n - 1);
        m_R_supd2.resize(m_n - 2);
        m_R_diag.array() = m_T_diag.array() - m_shift;
        m_R_supd.noalias() = m_T_subd;
 
        // A number of pointers to avoid repeated address calculation
        Scalar *c = m_rot_cos.data(),  // pointer to the cosine vector
            *s = m_rot_sin.data(),     // pointer to the sine vector
            r;
        const Index n1 = m_n - 1, n2 = m_n - 2;
        for (Index i = 0; i < n1; i++)
        {
            // Rdiag[i] == R[i, i]
            // Tsubd[i] == R[i + 1, i]
            // r = sqrt(R[i, i]^2 + R[i + 1, i]^2)
            // c = R[i, i] / r, s = -R[i + 1, i] / r
            this->compute_rotation(m_R_diag.coeff(i), m_T_subd.coeff(i), r, *c, *s);
 
            // For a complete QR decomposition,
            // we first obtain the rotation matrix
            // G = [ cos  sin]
            //     [-sin  cos]
            // and then do R[i:(i + 1), i:(i + 2)] = G' * R[i:(i + 1), i:(i + 2)]
 
            // Update R[i, i] and R[i + 1, i]
            // The updated value of R[i, i] is known to be r
            // The updated value of R[i + 1, i] is known to be 0
            m_R_diag.coeffRef(i) = r;
            // Update R[i, i + 1] and R[i + 1, i + 1]
            // Rsupd[i] == R[i, i + 1]
            // Rdiag[i + 1] == R[i + 1, i + 1]
            const Scalar Tii1 = m_R_supd.coeff(i);
            const Scalar Ti1i1 = m_R_diag.coeff(i + 1);
            m_R_supd.coeffRef(i) = (*c) * Tii1 - (*s) * Ti1i1;
            m_R_diag.coeffRef(i + 1) = (*s) * Tii1 + (*c) * Ti1i1;
            // Update R[i, i + 2] and R[i + 1, i + 2]
            // Rsupd2[i] == R[i, i + 2]
            // Rsupd[i + 1] == R[i + 1, i + 2]
            if (i < n2)
            {
                m_R_supd2.coeffRef(i) = -(*s) * m_R_supd.coeff(i + 1);
                m_R_supd.coeffRef(i + 1) *= (*c);
            }
 
            c++;
            s++;
 
            // If we do not need to calculate the R matrix, then
            // only the cos and sin sequences are required.
            // In such case we only update R[i + 1, (i + 1):(i + 2)]
            // R[i + 1, i + 1] = c * R[i + 1, i + 1] + s * R[i, i + 1];
            // R[i + 1, i + 2] *= c;
        }
 
        m_computed = true;
    }
 
    Matrix matrix_R() const override
    {
        if (!m_computed)
            throw std::logic_error("TridiagQR: need to call compute() first");
 
        Matrix R = Matrix::Zero(m_n, m_n);
        R.diagonal().noalias() = m_R_diag;
        R.diagonal(1).noalias() = m_R_supd;
        R.diagonal(2).noalias() = m_R_supd2;
 
        return R;
    }
 
    void matrix_QtHQ(Matrix& dest) const override
    {
        using std::abs;
 
        if (!m_computed)
            throw std::logic_error("TridiagQR: need to call compute() first");
 
        // In exact arithmetics, Q'TQ = RQ + sI, so we can just apply Q to R and add the shift.
        // However, some numerical examples show that this algorithm decreases the precision,
        // so we directly apply Q' and Q to T.
 
        // Copy the saved diagonal and subdiagonal elements of T to `dest`
        dest.resize(m_n, m_n);
        dest.setZero();
        dest.diagonal().noalias() = m_T_diag;
        dest.diagonal(-1).noalias() = m_T_subd;
 
        // Ti = [x  y  0],  Gi = [ cos[i]  sin[i]  0],  Gi' * Ti * Gi = [x'  y'  o']
        //      [y  z  w]        [-sin[i]  cos[i]  0]                   [y'  z'  w']
        //      [0  w  u]        [      0       0  1]                   [o'  w'  u']
        //
        // x' = c2*x - 2*c*s*y + s2*z
        // y' = c*s*(x-z) + (c2-s2)*y
        // z' = s2*x + 2*c*s*y + c2*z
        // o' = -s*w, w' = c*w, u' = u
        //
        // In iteration (i+1), (y', o') will be further updated to (y'', o''),
        // where o'' = 0, y'' = cos[i+1]*y' - sin[i+1]*o'
        const Index n1 = m_n - 1, n2 = m_n - 2;
        for (Index i = 0; i < n1; i++)
        {
            const Scalar c = m_rot_cos.coeff(i);
            const Scalar s = m_rot_sin.coeff(i);
            const Scalar cs = c * s, c2 = c * c, s2 = s * s;
            const Scalar x = dest.coeff(i, i),
                         y = dest.coeff(i + 1, i),
                         z = dest.coeff(i + 1, i + 1);
            const Scalar c2x = c2 * x, s2x = s2 * x, c2z = c2 * z, s2z = s2 * z;
            const Scalar csy2 = Scalar(2) * c * s * y;
 
            // Update the diagonal and the lower subdiagonal of dest
            dest.coeffRef(i, i) = c2x - csy2 + s2z;                  // x'
            dest.coeffRef(i + 1, i) = cs * (x - z) + (c2 - s2) * y;  // y'
            dest.coeffRef(i + 1, i + 1) = s2x + csy2 + c2z;          // z'
 
            if (i < n2)
            {
                const Scalar ci1 = m_rot_cos.coeff(i + 1);
                const Scalar si1 = m_rot_sin.coeff(i + 1);
                const Scalar o = -s * m_T_subd.coeff(i + 1);                     // o'
                dest.coeffRef(i + 2, i + 1) *= c;                                // w'
                dest.coeffRef(i + 1, i) = ci1 * dest.coeff(i + 1, i) - si1 * o;  // y''
            }
        }
 
        // Deflation of small sub-diagonal elements
        const Scalar eps = TypeTraits<Scalar>::epsilon();
        for (Index i = 0; i < n1; i++)
        {
            const Scalar diag = abs(dest.coeff(i, i)) + abs(dest.coeff(i + 1, i + 1));
            if (abs(dest.coeff(i + 1, i)) <= eps * diag)
                dest.coeffRef(i + 1, i) = Scalar(0);
        }
 
        // Copy the lower subdiagonal to upper subdiagonal
        dest.diagonal(1).noalias() = dest.diagonal(-1);
    }
 
    void matrix_QtHQ(ComplexMatrix& dest) const
    {
        // Simply compute the real-typed result and copy to the complex one
        Matrix dest_real;
        this->matrix_QtHQ(dest_real);
        dest.resize(m_n, m_n);
        dest.noalias() = dest_real.template cast<std::complex<Scalar>>();
    }
};
 
 
}  // namespace Spectra
 
#endif  // SPECTRA_UPPER_HESSENBERG_QR_H