HermEigsBase.h

// Copyright (C) 2018-2025 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_HERM_EIGS_BASE_H
#define SPECTRA_HERM_EIGS_BASE_H
 
#include <Eigen/Core>
#include <vector>     // std::vector
#include <cmath>      // std::abs, std::pow
#include <algorithm>  // std::min
#include <stdexcept>  // std::invalid_argument
#include <utility>    // std::move
 
#include "Util/Version.h"
#include "Util/TypeTraits.h"
#include "Util/SelectionRule.h"
#include "Util/CompInfo.h"
#include "Util/SimpleRandom.h"
#include "MatOp/internal/ArnoldiOp.h"
#include "LinAlg/UpperHessenbergQR.h"
#include "LinAlg/TridiagEigen.h"
#include "LinAlg/Lanczos.h"
 
namespace Spectra {


template <typename OpType, typename BOpType>
class HermEigsBase
{
private:
    // Scalar is the type of the matrix element
    // Can be real or complex
    using Scalar = typename OpType::Scalar;
    // The real part type of the matrix element, e.g.,
    //     Scalar = double               => RealScalar = double
    //     Scalar = std::complex<double> => RealScalar = double
    // The eigenvalues are known to be real numbers
    using RealScalar = typename Eigen::NumTraits<Scalar>::Real;
    using Index = Eigen::Index;
    using Matrix = Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic>;
    using Vector = Eigen::Matrix<Scalar, Eigen::Dynamic, 1>;
    using RealMatrix = Eigen::Matrix<RealScalar, Eigen::Dynamic, Eigen::Dynamic>;
    using RealVector = Eigen::Matrix<RealScalar, Eigen::Dynamic, 1>;
    using RealArray = Eigen::Array<RealScalar, Eigen::Dynamic, 1>;
    using BoolArray = Eigen::Array<bool, Eigen::Dynamic, 1>;
    using MapMat = Eigen::Map<Matrix>;
    using MapVec = Eigen::Map<Vector>;
    using MapConstVec = Eigen::Map<const Vector>;
 
    using ArnoldiOpType = ArnoldiOp<OpType, BOpType>;
    using LanczosFac = Lanczos<ArnoldiOpType>;
 
protected:
    // clang-format off
 
    // In SymEigsSolver and SymEigsShiftSolver, the A operator is an lvalue provided by
    // the user. In SymGEigsSolver, the A operator is an rvalue. To avoid copying objects,
    // we use the following scheme:
    // 1. If the op parameter in the constructor is an lvalue, make m_op a const reference to op
    // 2. If op is an rvalue, move op to m_op_container, and then make m_op a const
    //    reference to m_op_container[0]
    std::vector<OpType> m_op_container;
    const OpType& m_op;         // matrix operator for A
    const Index   m_n;          // dimension of matrix A
    const Index   m_nev;        // number of eigenvalues requested
    const Index   m_ncv;        // dimension of Krylov subspace in the Lanczos method
    Index         m_nmatop;     // number of matrix operations called
    Index         m_niter;      // number of restarting iterations
 
    LanczosFac    m_fac;        // Lanczos factorization
    RealVector    m_ritz_val;   // Ritz values
 
private:
    RealMatrix    m_ritz_vec;   // Ritz vectors
    RealVector    m_ritz_est;   // last row of m_ritz_vec, also called the Ritz estimates
    BoolArray     m_ritz_conv;  // indicator of the convergence of Ritz values
    CompInfo      m_info;       // status of the computation
    // clang-format on
 
    // Move rvalue object to the container
    static std::vector<OpType> create_op_container(OpType&& rval)
    {
        std::vector<OpType> container;
        container.emplace_back(std::move(rval));
        return container;
    }
 
    // Implicitly restarted Lanczos factorization
    void restart(Index k, SortRule selection)
    {
        using std::abs;
 
        if (k >= m_ncv)
            return;
 
        // QR decomposition on a real symmetric matrix
        TridiagQR<RealScalar> decomp(m_ncv);
        // Q is a real orthogonal matrix
        RealMatrix Q = RealMatrix::Identity(m_ncv, m_ncv);
 
        // Apply large shifts first
        const int nshift = m_ncv - k;
        RealVector shifts = m_ritz_val.tail(nshift);
        std::sort(shifts.data(), shifts.data() + nshift,
                  [](const RealScalar& v1, const RealScalar& v2) { return abs(v1) > abs(v2); });
 
        // Apply shifts and update H and Q
        for (Index i = 0; i < nshift; i++)
        {
            // QR decomposition of H - mu[i] * I, mu[i] is the shift
            // H is known to be a real symmetric matrix
            decomp.compute(m_fac.matrix_H().real(), shifts[i]);
 
            // Q <- Q * Qi
            decomp.apply_YQ(Q);
            // H <- Qi' * H * Qi
            // Since Qi * Ri = H - mu[i] * I, we have H = Qi * Ri + mu[i] * I,
            // and then Qi' * H * Qi = Ri * Qi + mu[i] * I
            m_fac.compress_H(decomp);
            // Note that in our setting, mu[i] is an eigenvalue of H,
            // so after applying Qi' * H * Qi, H must be of the following form
            // H = [X   0   0]
            //     [0  mu   0]
            //     [0   0   D]
            // Then we can force H[k, k-1] = H[k-1, k] = 0 and H[k, k] = mu,
            // where k is the size of X
            //
            // Currently disabled due to numerical stability
            //
            // m_fac.deflate_H(m_ncv - i - 1, shifts[i]);
        }
        // Apply orthogonal transformation to V, V <- V * Q
        m_fac.compress_V(Q);
        // It can be verified that the updated V and H admit a k-step
        // Arnoldi factorization, and we expand it to m-step
        m_fac.factorize_from(k, m_ncv, m_nmatop);
        // Retrieve the new Ritz pairs
        retrieve_ritzpair(selection);
    }
 
    // Calculates the number of converged Ritz values
    Index num_converged(const RealScalar& tol)
    {
        using std::pow;
 
        // The machine precision, ~= 1e-16 for the "double" type
        const RealScalar eps = TypeTraits<RealScalar>::epsilon();
        // std::pow() is not constexpr, so we do not declare eps23 to be constexpr
        // But most compilers should be able to compute eps23 at compile time
        const RealScalar eps23 = pow(eps, RealScalar(2) / 3);
 
        // thresh = tol * max(eps23, abs(theta)), theta for Ritz value
        RealArray thresh = tol * m_ritz_val.head(m_nev).array().abs().max(eps23);
        RealArray resid = m_ritz_est.head(m_nev).array().abs() * m_fac.f_norm();
        // Converged "wanted" Ritz values
        m_ritz_conv = (resid < thresh);
 
        return m_ritz_conv.count();
    }
 
    // Returns the adjusted nev for restarting
    Index nev_adjusted(Index nconv)
    {
        using std::abs;
 
        // A very small value, but 1.0 / near_0 does not overflow
        // ~= 1e-307 for the "double" type
        const RealScalar near_0 = TypeTraits<RealScalar>::min() * RealScalar(10);
 
        Index nev_new = m_nev;
        for (Index i = m_nev; i < m_ncv; i++)
            if (abs(m_ritz_est[i]) < near_0)
                nev_new++;
 
        // Adjust nev_new, according to dsaup2.f line 677~684 in ARPACK
        nev_new += (std::min)(nconv, (m_ncv - nev_new) / 2);
        if (nev_new == 1 && m_ncv >= 6)
            nev_new = m_ncv / 2;
        else if (nev_new == 1 && m_ncv > 2)
            nev_new = 2;
 
        if (nev_new > m_ncv - 1)
            nev_new = m_ncv - 1;
 
        return nev_new;
    }
 
    // Retrieves and sorts Ritz values and Ritz vectors
    void retrieve_ritzpair(SortRule selection)
    {
        TridiagEigen<RealScalar> decomp(m_fac.matrix_H().real());
        const RealVector& evals = decomp.eigenvalues();
        const RealMatrix& evecs = decomp.eigenvectors();
 
        // Sort Ritz values and put the wanted ones at the beginning
        std::vector<Index> ind = argsort(selection, evals, m_ncv);
 
        // Copy the Ritz values and vectors to m_ritz_val and m_ritz_vec, respectively
        for (Index i = 0; i < m_ncv; i++)
        {
            m_ritz_val[i] = evals[ind[i]];
            m_ritz_est[i] = evecs(m_ncv - 1, ind[i]);
        }
        for (Index i = 0; i < m_nev; i++)
        {
            m_ritz_vec.col(i).noalias() = evecs.col(ind[i]);
        }
    }
 
protected:
    // Sorts the first nev Ritz pairs in the specified order
    // This is used to return the final results
    virtual void sort_ritzpair(SortRule sort_rule)
    {
        if ((sort_rule != SortRule::LargestAlge) && (sort_rule != SortRule::LargestMagn) &&
            (sort_rule != SortRule::SmallestAlge) && (sort_rule != SortRule::SmallestMagn))
            throw std::invalid_argument("unsupported sorting rule");
 
        std::vector<Index> ind = argsort(sort_rule, m_ritz_val, m_nev);
 
        RealVector new_ritz_val(m_ncv);
        RealMatrix new_ritz_vec(m_ncv, m_nev);
        BoolArray new_ritz_conv(m_nev);
 
        for (Index i = 0; i < m_nev; i++)
        {
            new_ritz_val[i] = m_ritz_val[ind[i]];
            new_ritz_vec.col(i).noalias() = m_ritz_vec.col(ind[i]);
            new_ritz_conv[i] = m_ritz_conv[ind[i]];
        }
 
        m_ritz_val.swap(new_ritz_val);
        m_ritz_vec.swap(new_ritz_vec);
        m_ritz_conv.swap(new_ritz_conv);
    }
 
public:
 
    // If op is an lvalue
    HermEigsBase(OpType& op, const BOpType& Bop, Index nev, Index ncv) :
        m_op(op),
        m_n(op.rows()),
        m_nev(nev),
        m_ncv(ncv > m_n ? m_n : ncv),
        m_nmatop(0),
        m_niter(0),
        m_fac(ArnoldiOpType(op, Bop), m_ncv),
        m_info(CompInfo::NotComputed)
    {
        if (nev < 1 || nev > m_n - 1)
            throw std::invalid_argument("nev must satisfy 1 <= nev <= n - 1, n is the size of matrix");
 
        if (ncv <= nev || ncv > m_n)
            throw std::invalid_argument("ncv must satisfy nev < ncv <= n, n is the size of matrix");
    }
 
    // If op is an rvalue
    HermEigsBase(OpType&& op, const BOpType& Bop, Index nev, Index ncv) :
        m_op_container(create_op_container(std::move(op))),
        m_op(m_op_container.front()),
        m_n(m_op.rows()),
        m_nev(nev),
        m_ncv(ncv > m_n ? m_n : ncv),
        m_nmatop(0),
        m_niter(0),
        m_fac(ArnoldiOpType(m_op, Bop), m_ncv),
        m_info(CompInfo::NotComputed)
    {
        if (nev < 1 || nev > m_n - 1)
            throw std::invalid_argument("nev must satisfy 1 <= nev <= n - 1, n is the size of matrix");
 
        if (ncv <= nev || ncv > m_n)
            throw std::invalid_argument("ncv must satisfy nev < ncv <= n, n is the size of matrix");
    }

    virtual ~HermEigsBase() {}


    void init(const Scalar* init_resid)
    {
        // Reset all matrices/vectors to zero
        m_ritz_val.resize(m_ncv);
        m_ritz_vec.resize(m_ncv, m_nev);
        m_ritz_est.resize(m_ncv);
        m_ritz_conv.resize(m_nev);
 
        m_ritz_val.setZero();
        m_ritz_vec.setZero();
        m_ritz_est.setZero();
        m_ritz_conv.setZero();
 
        m_nmatop = 0;
        m_niter = 0;
 
        // Initialize the Lanczos factorization
        MapConstVec v0(init_resid, m_n);
        m_fac.init(v0, m_nmatop);
    }
    void init(const Scalar* init_resid) {…}

    void init()
    {
        SimpleRandom<Scalar> rng(0);
        Vector init_resid = rng.random_vec(m_n);
        init(init_resid.data());
    }
    void init() {…}

    Index compute(SortRule selection = SortRule::LargestMagn, Index maxit = 1000,
                  RealScalar tol = 1e-10, SortRule sorting = SortRule::LargestAlge)
    {
        // The m-step Lanczos factorization
        m_fac.factorize_from(1, m_ncv, m_nmatop);
        retrieve_ritzpair(selection);
        // Restarting
        Index i, nconv = 0, nev_adj;
        for (i = 0; i < maxit; i++)
        {
            nconv = num_converged(tol);
            if (nconv >= m_nev)
                break;
 
            nev_adj = nev_adjusted(nconv);
            restart(nev_adj, selection);
        }
        // Sorting results
        sort_ritzpair(sorting);
 
        m_niter += (i + 1);
        m_info = (nconv >= m_nev) ? CompInfo::Successful : CompInfo::NotConverging;
 
        return (std::min)(m_nev, nconv);
    }
    Index compute(SortRule selection = SortRule::LargestMagn, Index maxit = 1000, {…}

    CompInfo info() const { return m_info; }

    Index num_iterations() const { return m_niter; }

    Index num_operations() const { return m_nmatop; }

    RealVector eigenvalues() const
    {
        const Index nconv = m_ritz_conv.count();
        RealVector res(nconv);
 
        if (!nconv)
            return res;
 
        Index j = 0;
        for (Index i = 0; i < m_nev; i++)
        {
            if (m_ritz_conv[i])
            {
                res[j] = m_ritz_val[i];
                j++;
            }
        }
 
        return res;
    }
    RealVector eigenvalues() const {…}

    virtual Matrix eigenvectors(Index nvec) const
    {
        const Index nconv = m_ritz_conv.count();
        nvec = (std::min)(nvec, nconv);
        Matrix res(m_n, nvec);
 
        if (!nvec)
            return res;
 
        RealMatrix ritz_vec_conv(m_ncv, nvec);
        Index j = 0;
        for (Index i = 0; i < m_nev && j < nvec; i++)
        {
            if (m_ritz_conv[i])
            {
                ritz_vec_conv.col(j).noalias() = m_ritz_vec.col(i);
                j++;
            }
        }
 
        res.noalias() = m_fac.matrix_V() * ritz_vec_conv;
 
        return res;
    }
    virtual Matrix eigenvectors(Index nvec) const {…}

    virtual Matrix eigenvectors() const
    {
        return eigenvectors(m_nev);
    }
    virtual Matrix eigenvectors() const {…}
};
class HermEigsBase {…};
 
}  // namespace Spectra
 
#endif  // SPECTRA_HERM_EIGS_BASE_H