449 lines
14 KiB
C++
449 lines
14 KiB
C++
// Copyright (C) 2018-2019 Yixuan Qiu <yixuan.qiu@cos.name>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at https://mozilla.org/MPL/2.0/.
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#ifndef SYM_EIGS_BASE_H
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#define SYM_EIGS_BASE_H
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#include <Eigen/Core>
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#include <vector> // std::vector
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#include <cmath> // std::abs, std::pow, std::sqrt
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#include <algorithm> // std::min, std::copy
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#include <stdexcept> // std::invalid_argument
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#include "Util/TypeTraits.h"
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#include "Util/SelectionRule.h"
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#include "Util/CompInfo.h"
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#include "Util/SimpleRandom.h"
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#include "MatOp/internal/ArnoldiOp.h"
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#include "LinAlg/UpperHessenbergQR.h"
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#include "LinAlg/TridiagEigen.h"
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#include "LinAlg/Lanczos.h"
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namespace Spectra {
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///
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/// \defgroup EigenSolver Eigen Solvers
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///
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/// Eigen solvers for different types of problems.
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///
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///
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/// \ingroup EigenSolver
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///
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/// This is the base class for symmetric eigen solvers, mainly for internal use.
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/// It is kept here to provide the documentation for member functions of concrete eigen solvers
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/// such as SymEigsSolver and SymEigsShiftSolver.
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///
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template <typename Scalar,
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int SelectionRule,
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typename OpType,
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typename BOpType>
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class SymEigsBase
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{
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private:
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typedef Eigen::Index Index;
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typedef Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic> Matrix;
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typedef Eigen::Matrix<Scalar, Eigen::Dynamic, 1> Vector;
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typedef Eigen::Array<Scalar, Eigen::Dynamic, 1> Array;
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typedef Eigen::Array<bool, Eigen::Dynamic, 1> BoolArray;
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typedef Eigen::Map<Matrix> MapMat;
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typedef Eigen::Map<Vector> MapVec;
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typedef Eigen::Map<const Vector> MapConstVec;
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typedef ArnoldiOp<Scalar, OpType, BOpType> ArnoldiOpType;
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typedef Lanczos<Scalar, ArnoldiOpType> LanczosFac;
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protected:
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// clang-format off
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OpType* m_op; // object to conduct matrix operation,
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// e.g. matrix-vector product
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const Index m_n; // dimension of matrix A
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const Index m_nev; // number of eigenvalues requested
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const Index m_ncv; // dimension of Krylov subspace in the Lanczos method
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Index m_nmatop; // number of matrix operations called
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Index m_niter; // number of restarting iterations
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LanczosFac m_fac; // Lanczos factorization
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Vector m_ritz_val; // Ritz values
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private:
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Matrix m_ritz_vec; // Ritz vectors
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Vector m_ritz_est; // last row of m_ritz_vec, also called the Ritz estimates
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BoolArray m_ritz_conv; // indicator of the convergence of Ritz values
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int m_info; // status of the computation
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const Scalar m_near_0; // a very small value, but 1.0 / m_near_0 does not overflow
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// ~= 1e-307 for the "double" type
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const Scalar m_eps; // the machine precision, ~= 1e-16 for the "double" type
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const Scalar m_eps23; // m_eps^(2/3), used to test the convergence
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// clang-format on
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// Implicitly restarted Lanczos factorization
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void restart(Index k)
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{
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if (k >= m_ncv)
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return;
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TridiagQR<Scalar> decomp(m_ncv);
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Matrix Q = Matrix::Identity(m_ncv, m_ncv);
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for (Index i = k; i < m_ncv; i++)
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{
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// QR decomposition of H-mu*I, mu is the shift
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decomp.compute(m_fac.matrix_H(), m_ritz_val[i]);
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// Q -> Q * Qi
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decomp.apply_YQ(Q);
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// H -> Q'HQ
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// Since QR = H - mu * I, we have H = QR + mu * I
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// and therefore Q'HQ = RQ + mu * I
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m_fac.compress_H(decomp);
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}
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m_fac.compress_V(Q);
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m_fac.factorize_from(k, m_ncv, m_nmatop);
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retrieve_ritzpair();
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}
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// Calculates the number of converged Ritz values
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Index num_converged(Scalar tol)
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{
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// thresh = tol * max(m_eps23, abs(theta)), theta for Ritz value
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Array thresh = tol * m_ritz_val.head(m_nev).array().abs().max(m_eps23);
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Array resid = m_ritz_est.head(m_nev).array().abs() * m_fac.f_norm();
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// Converged "wanted" Ritz values
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m_ritz_conv = (resid < thresh);
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return m_ritz_conv.cast<Index>().sum();
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}
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// Returns the adjusted nev for restarting
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Index nev_adjusted(Index nconv)
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{
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using std::abs;
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Index nev_new = m_nev;
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for (Index i = m_nev; i < m_ncv; i++)
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if (abs(m_ritz_est[i]) < m_near_0)
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nev_new++;
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// Adjust nev_new, according to dsaup2.f line 677~684 in ARPACK
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nev_new += std::min(nconv, (m_ncv - nev_new) / 2);
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if (nev_new == 1 && m_ncv >= 6)
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nev_new = m_ncv / 2;
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else if (nev_new == 1 && m_ncv > 2)
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nev_new = 2;
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if (nev_new > m_ncv - 1)
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nev_new = m_ncv - 1;
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return nev_new;
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}
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// Retrieves and sorts Ritz values and Ritz vectors
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void retrieve_ritzpair()
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{
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TridiagEigen<Scalar> decomp(m_fac.matrix_H());
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const Vector& evals = decomp.eigenvalues();
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const Matrix& evecs = decomp.eigenvectors();
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SortEigenvalue<Scalar, SelectionRule> sorting(evals.data(), evals.size());
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std::vector<int> ind = sorting.index();
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// For BOTH_ENDS, the eigenvalues are sorted according
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// to the LARGEST_ALGE rule, so we need to move those smallest
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// values to the left
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// The order would be
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// Largest => Smallest => 2nd largest => 2nd smallest => ...
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// We keep this order since the first k values will always be
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// the wanted collection, no matter k is nev_updated (used in restart())
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// or is nev (used in sort_ritzpair())
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if (SelectionRule == BOTH_ENDS)
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{
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std::vector<int> ind_copy(ind);
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for (Index i = 0; i < m_ncv; i++)
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{
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// If i is even, pick values from the left (large values)
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// If i is odd, pick values from the right (small values)
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if (i % 2 == 0)
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ind[i] = ind_copy[i / 2];
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else
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ind[i] = ind_copy[m_ncv - 1 - i / 2];
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}
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}
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// Copy the Ritz values and vectors to m_ritz_val and m_ritz_vec, respectively
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for (Index i = 0; i < m_ncv; i++)
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{
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m_ritz_val[i] = evals[ind[i]];
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m_ritz_est[i] = evecs(m_ncv - 1, ind[i]);
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}
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for (Index i = 0; i < m_nev; i++)
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{
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m_ritz_vec.col(i).noalias() = evecs.col(ind[i]);
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}
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}
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protected:
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// Sorts the first nev Ritz pairs in the specified order
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// This is used to return the final results
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virtual void sort_ritzpair(int sort_rule)
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{
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// First make sure that we have a valid index vector
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SortEigenvalue<Scalar, LARGEST_ALGE> sorting(m_ritz_val.data(), m_nev);
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std::vector<int> ind = sorting.index();
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switch (sort_rule)
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{
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case LARGEST_ALGE:
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break;
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case LARGEST_MAGN:
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{
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SortEigenvalue<Scalar, LARGEST_MAGN> sorting(m_ritz_val.data(), m_nev);
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ind = sorting.index();
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break;
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}
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case SMALLEST_ALGE:
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{
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SortEigenvalue<Scalar, SMALLEST_ALGE> sorting(m_ritz_val.data(), m_nev);
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ind = sorting.index();
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break;
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}
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case SMALLEST_MAGN:
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{
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SortEigenvalue<Scalar, SMALLEST_MAGN> sorting(m_ritz_val.data(), m_nev);
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ind = sorting.index();
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break;
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}
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default:
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throw std::invalid_argument("unsupported sorting rule");
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}
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Vector new_ritz_val(m_ncv);
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Matrix new_ritz_vec(m_ncv, m_nev);
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BoolArray new_ritz_conv(m_nev);
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for (Index i = 0; i < m_nev; i++)
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{
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new_ritz_val[i] = m_ritz_val[ind[i]];
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new_ritz_vec.col(i).noalias() = m_ritz_vec.col(ind[i]);
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new_ritz_conv[i] = m_ritz_conv[ind[i]];
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}
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m_ritz_val.swap(new_ritz_val);
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m_ritz_vec.swap(new_ritz_vec);
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m_ritz_conv.swap(new_ritz_conv);
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}
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public:
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/// \cond
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SymEigsBase(OpType* op, BOpType* Bop, Index nev, Index ncv) :
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m_op(op),
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m_n(m_op->rows()),
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m_nev(nev),
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m_ncv(ncv > m_n ? m_n : ncv),
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m_nmatop(0),
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m_niter(0),
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m_fac(ArnoldiOpType(op, Bop), m_ncv),
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m_info(NOT_COMPUTED),
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m_near_0(TypeTraits<Scalar>::min() * Scalar(10)),
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m_eps(Eigen::NumTraits<Scalar>::epsilon()),
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m_eps23(Eigen::numext::pow(m_eps, Scalar(2.0) / 3))
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{
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if (nev < 1 || nev > m_n - 1)
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throw std::invalid_argument("nev must satisfy 1 <= nev <= n - 1, n is the size of matrix");
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if (ncv <= nev || ncv > m_n)
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throw std::invalid_argument("ncv must satisfy nev < ncv <= n, n is the size of matrix");
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}
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///
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/// Virtual destructor
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///
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virtual ~SymEigsBase() {}
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/// \endcond
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///
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/// Initializes the solver by providing an initial residual vector.
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///
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/// \param init_resid Pointer to the initial residual vector.
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///
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/// **Spectra** (and also **ARPACK**) uses an iterative algorithm
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/// to find eigenvalues. This function allows the user to provide the initial
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/// residual vector.
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///
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void init(const Scalar* init_resid)
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{
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// Reset all matrices/vectors to zero
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m_ritz_val.resize(m_ncv);
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m_ritz_vec.resize(m_ncv, m_nev);
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m_ritz_est.resize(m_ncv);
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m_ritz_conv.resize(m_nev);
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m_ritz_val.setZero();
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m_ritz_vec.setZero();
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m_ritz_est.setZero();
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m_ritz_conv.setZero();
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m_nmatop = 0;
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m_niter = 0;
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// Initialize the Lanczos factorization
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MapConstVec v0(init_resid, m_n);
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m_fac.init(v0, m_nmatop);
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}
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///
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/// Initializes the solver by providing a random initial residual vector.
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///
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/// This overloaded function generates a random initial residual vector
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/// (with a fixed random seed) for the algorithm. Elements in the vector
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/// follow independent Uniform(-0.5, 0.5) distribution.
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///
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void init()
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{
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SimpleRandom<Scalar> rng(0);
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Vector init_resid = rng.random_vec(m_n);
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init(init_resid.data());
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}
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///
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/// Conducts the major computation procedure.
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///
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/// \param maxit Maximum number of iterations allowed in the algorithm.
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/// \param tol Precision parameter for the calculated eigenvalues.
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/// \param sort_rule Rule to sort the eigenvalues and eigenvectors.
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/// Supported values are
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/// `Spectra::LARGEST_ALGE`, `Spectra::LARGEST_MAGN`,
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/// `Spectra::SMALLEST_ALGE` and `Spectra::SMALLEST_MAGN`,
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/// for example `LARGEST_ALGE` indicates that largest eigenvalues
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/// come first. Note that this argument is only used to
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/// **sort** the final result, and the **selection** rule
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/// (e.g. selecting the largest or smallest eigenvalues in the
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/// full spectrum) is specified by the template parameter
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/// `SelectionRule` of SymEigsSolver.
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///
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/// \return Number of converged eigenvalues.
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///
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Index compute(Index maxit = 1000, Scalar tol = 1e-10, int sort_rule = LARGEST_ALGE)
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{
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// The m-step Lanczos factorization
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m_fac.factorize_from(1, m_ncv, m_nmatop);
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retrieve_ritzpair();
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// Restarting
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Index i, nconv = 0, nev_adj;
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for (i = 0; i < maxit; i++)
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{
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nconv = num_converged(tol);
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if (nconv >= m_nev)
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break;
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nev_adj = nev_adjusted(nconv);
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restart(nev_adj);
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}
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// Sorting results
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sort_ritzpair(sort_rule);
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m_niter += i + 1;
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m_info = (nconv >= m_nev) ? SUCCESSFUL : NOT_CONVERGING;
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return std::min(m_nev, nconv);
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}
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///
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/// Returns the status of the computation.
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/// The full list of enumeration values can be found in \ref Enumerations.
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///
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int info() const { return m_info; }
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///
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/// Returns the number of iterations used in the computation.
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///
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Index num_iterations() const { return m_niter; }
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///
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/// Returns the number of matrix operations used in the computation.
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///
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Index num_operations() const { return m_nmatop; }
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///
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/// Returns the converged eigenvalues.
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///
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/// \return A vector containing the eigenvalues.
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/// Returned vector type will be `Eigen::Vector<Scalar, ...>`, depending on
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/// the template parameter `Scalar` defined.
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///
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Vector eigenvalues() const
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{
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const Index nconv = m_ritz_conv.cast<Index>().sum();
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Vector res(nconv);
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if (!nconv)
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return res;
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Index j = 0;
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for (Index i = 0; i < m_nev; i++)
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{
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if (m_ritz_conv[i])
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{
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res[j] = m_ritz_val[i];
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j++;
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}
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}
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return res;
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}
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///
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/// Returns the eigenvectors associated with the converged eigenvalues.
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///
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/// \param nvec The number of eigenvectors to return.
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///
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/// \return A matrix containing the eigenvectors.
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/// Returned matrix type will be `Eigen::Matrix<Scalar, ...>`,
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/// depending on the template parameter `Scalar` defined.
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///
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virtual Matrix eigenvectors(Index nvec) const
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{
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const Index nconv = m_ritz_conv.cast<Index>().sum();
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nvec = std::min(nvec, nconv);
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Matrix res(m_n, nvec);
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if (!nvec)
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return res;
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Matrix ritz_vec_conv(m_ncv, nvec);
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Index j = 0;
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for (Index i = 0; i < m_nev && j < nvec; i++)
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{
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if (m_ritz_conv[i])
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{
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ritz_vec_conv.col(j).noalias() = m_ritz_vec.col(i);
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j++;
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}
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}
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res.noalias() = m_fac.matrix_V() * ritz_vec_conv;
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return res;
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}
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///
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/// Returns all converged eigenvectors.
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///
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virtual Matrix eigenvectors() const
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{
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return eigenvectors(m_nev);
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}
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};
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} // namespace Spectra
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#endif // SYM_EIGS_BASE_H
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