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+// SPDX-License-Identifier: Apache-2.0
+//
+// Copyright 2008-2016 Conrad Sanderson (http://conradsanderson.id.au)
+// Copyright 2008-2016 National ICT Australia (NICTA)
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+// http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+// ------------------------------------------------------------------------
+
+
+namespace newarp
+{
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::fill_rand(eT* dest, const uword N, const uword seed_val)
+ {
+ arma_extra_debug_sigprint();
+
+ typedef typename std::mt19937_64::result_type seed_type;
+
+ local_rng.seed( seed_type(seed_val) );
+
+ std::uniform_real_distribution<double> dist(-1.0, +1.0);
+
+ for(uword i=0; i < N; ++i) { dest[i] = eT(dist(local_rng)); }
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::factorise_from(uword from_k, uword to_m, const Col<eT>& fk)
+ {
+ arma_extra_debug_sigprint();
+
+ if(to_m <= from_k) { return; }
+
+ fac_f = fk;
+
+ Col<eT> w(dim_n, arma_zeros_indicator());
+ // Norm of f
+ eT beta = norm(fac_f);
+ // Used to test beta~=0
+ const eT beta_thresh = eps * eop_aux::sqrt(dim_n);
+ // Keep the upperleft k x k submatrix of H and set other elements to 0
+ fac_H.tail_cols(ncv - from_k).zeros();
+ fac_H.submat(span(from_k, ncv - 1), span(0, from_k - 1)).zeros();
+ for(uword i = from_k; i <= to_m - 1; i++)
+ {
+ bool restart = false;
+ // If beta = 0, then the next V is not full rank
+ // We need to generate a new residual vector that is orthogonal
+ // to the current V, which we call a restart
+ if(beta < near0)
+ {
+ // // Generate new random vector for fac_f
+ // blas_int idist = 2;
+ // blas_int iseed[4] = {1, 3, 5, 7};
+ // iseed[0] = (i + 100) % 4095;
+ // blas_int n = dim_n;
+ // lapack::larnv(&idist, &iseed[0], &n, fac_f.memptr());
+
+ // Generate new random vector for fac_f
+ fill_rand(fac_f.memptr(), dim_n, i+1);
+
+ // f <- f - V * V' * f, so that f is orthogonal to V
+ Mat<eT> Vs(fac_V.memptr(), dim_n, i, false); // First i columns
+ Col<eT> Vf = Vs.t() * fac_f;
+ fac_f -= Vs * Vf;
+ // beta <- ||f||
+ beta = norm(fac_f);
+
+ restart = true;
+ }
+
+ // v <- f / ||f||
+ Col<eT> v(fac_V.colptr(i), dim_n, false); // The (i+1)-th column
+ v = fac_f / beta;
+
+ // Note that H[i+1, i] equals to the unrestarted beta
+ fac_H(i, i - 1) = restart ? eT(0) : beta;
+
+ // w <- A * v, v = fac_V.col(i)
+ op.perform_op(v.memptr(), w.memptr());
+ nmatop++;
+
+ fac_H(i - 1, i) = fac_H(i, i - 1); // Due to symmetry
+ eT Hii = dot(v, w);
+ fac_H(i, i) = Hii;
+
+ // f <- w - V * V' * w = w - H[i+1, i] * V{i} - H[i+1, i+1] * V{i+1}
+ // If restarting, we know that H[i+1, i] = 0
+ if(restart)
+ {
+ fac_f = w - Hii * v;
+ }
+ else
+ {
+ fac_f = w - fac_H(i, i - 1) * fac_V.col(i - 1) - Hii * v;
+ }
+
+ beta = norm(fac_f);
+
+ // f/||f|| is going to be the next column of V, so we need to test
+ // whether V' * (f/||f||) ~= 0
+ Mat<eT> Vs(fac_V.memptr(), dim_n, i + 1, false); // First i+1 columns
+ Col<eT> Vf = Vs.t() * fac_f;
+ eT ortho_err = abs(Vf).max();
+ // If not, iteratively correct the residual
+ uword count = 0;
+ while(count < 5 && ortho_err > eps * beta)
+ {
+ // There is an edge case: when beta=||f|| is close to zero, f mostly consists
+ // of rounding errors, so the test [ortho_err < eps * beta] is very
+ // likely to fail. In particular, if beta=0, then the test is ensured to fail.
+ // Hence when this happens, we force f to be zero, and then restart in the
+ // next iteration.
+ if(beta < beta_thresh)
+ {
+ fac_f.zeros();
+ beta = eT(0);
+ break;
+ }
+
+ // f <- f - V * Vf
+ fac_f -= Vs * Vf;
+ // h <- h + Vf
+ fac_H(i - 1, i) += Vf[i - 1];
+ fac_H(i, i - 1) = fac_H(i - 1, i);
+ fac_H(i, i) += Vf[i];
+ // beta <- ||f||
+ beta = norm(fac_f);
+
+ Vf = Vs.t() * fac_f;
+ ortho_err = abs(Vf).max();
+ count++;
+ }
+ }
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::restart(uword k)
+ {
+ arma_extra_debug_sigprint();
+
+ if(k >= ncv) { return; }
+
+ TridiagQR<eT> decomp;
+ Mat<eT> Q(ncv, ncv, fill::eye);
+
+ for(uword i = k; i < ncv; i++)
+ {
+ // QR decomposition of H-mu*I, mu is the shift
+ fac_H.diag() -= ritz_val(i);
+ decomp.compute(fac_H);
+
+ // Q -> Q * Qi
+ decomp.apply_YQ(Q);
+
+ // H -> Q'HQ
+ // Since QR = H - mu * I, we have H = QR + mu * I
+ // and therefore Q'HQ = RQ + mu * I
+ fac_H = decomp.matrix_RQ();
+ fac_H.diag() += ritz_val(i);
+ }
+
+ // V -> VQ, only need to update the first k+1 columns
+ // Q has some elements being zero
+ // The first (ncv - k + i) elements of the i-th column of Q are non-zero
+ Mat<eT> Vs(dim_n, k + 1, arma_nozeros_indicator());
+ uword nnz;
+ for(uword i = 0; i < k; i++)
+ {
+ nnz = ncv - k + i + 1;
+ Mat<eT> V(fac_V.memptr(), dim_n, nnz, false);
+ Col<eT> q(Q.colptr(i), nnz, false);
+ // OLD CODE:
+ // Vs.col(i) = V * q;
+ // NEW CODE:
+ Col<eT> v(Vs.colptr(i), dim_n, false, true);
+ v = V * q;
+ }
+
+ Vs.col(k) = fac_V * Q.col(k);
+ fac_V.head_cols(k + 1) = Vs;
+
+ Col<eT> fk = fac_f * Q(ncv - 1, k - 1) + fac_V.col(k) * fac_H(k, k - 1);
+ factorise_from(k, ncv, fk);
+ retrieve_ritzpair();
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+uword
+SymEigsSolver<eT, SelectionRule, OpType>::num_converged(eT tol)
+ {
+ arma_extra_debug_sigprint();
+
+ // thresh = tol * max(approx0, abs(theta)), theta for ritz value
+ const eT f_norm = norm(fac_f);
+ for(uword i = 0; i < nev; i++)
+ {
+ eT thresh = tol * (std::max)(eps23, std::abs(ritz_val(i)));
+ eT resid = std::abs(ritz_est(i)) * f_norm;
+ ritz_conv[i] = (resid < thresh);
+ }
+
+ return std::count(ritz_conv.begin(), ritz_conv.end(), true);
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+uword
+SymEigsSolver<eT, SelectionRule, OpType>::nev_adjusted(uword nconv)
+ {
+ arma_extra_debug_sigprint();
+
+ uword nev_new = nev;
+ for(uword i = nev; i < ncv; i++)
+ {
+ if(std::abs(ritz_est(i)) < near0) { nev_new++; }
+ }
+
+ // Adjust nev_new, according to dsaup2.f line 677~684 in ARPACK
+ nev_new += (std::min)(nconv, (ncv - nev_new) / 2);
+
+ if(nev_new >= ncv) { nev_new = ncv - 1; }
+
+ if(nev_new == 1)
+ {
+ if(ncv >= 6) { nev_new = ncv / 2; }
+ else if(ncv > 2) { nev_new = 2; }
+ }
+
+ return nev_new;
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::retrieve_ritzpair()
+ {
+ arma_extra_debug_sigprint();
+
+ TridiagEigen<eT> decomp(fac_H);
+ Col<eT> evals = decomp.eigenvalues();
+ Mat<eT> evecs = decomp.eigenvectors();
+
+ SortEigenvalue<eT, SelectionRule> sorting(evals.memptr(), evals.n_elem);
+ std::vector<uword> ind = sorting.index();
+
+ // For BOTH_ENDS, the eigenvalues are sorted according
+ // to the LARGEST_ALGE rule, so we need to move those smallest
+ // values to the left
+ // The order would be
+ // Largest => Smallest => 2nd largest => 2nd smallest => ...
+ // We keep this order since the first k values will always be
+ // the wanted collection, no matter k is nev_updated (used in restart())
+ // or is nev (used in sort_ritzpair())
+ if(SelectionRule == EigsSelect::BOTH_ENDS)
+ {
+ std::vector<uword> ind_copy(ind);
+ for(uword i = 0; i < ncv; i++)
+ {
+ // If i is even, pick values from the left (large values)
+ // If i is odd, pick values from the right (small values)
+
+ ind[i] = (i % 2 == 0) ? ind_copy[i / 2] : ind_copy[ncv - 1 - i / 2];
+ }
+ }
+
+ // Copy the ritz values and vectors to ritz_val and ritz_vec, respectively
+ for(uword i = 0; i < ncv; i++)
+ {
+ ritz_val(i) = evals(ind[i]);
+ ritz_est(i) = evecs(ncv - 1, ind[i]);
+ }
+ for(uword i = 0; i < nev; i++)
+ {
+ ritz_vec.col(i) = evecs.col(ind[i]);
+ }
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::sort_ritzpair()
+ {
+ arma_extra_debug_sigprint();
+
+ // SortEigenvalue<eT, EigsSelect::LARGEST_MAGN> sorting(ritz_val.memptr(), nev);
+
+ // Sort Ritz values in ascending algebraic, to be consistent with ARPACK
+ SortEigenvalue<eT, EigsSelect::SMALLEST_ALGE> sorting(ritz_val.memptr(), nev);
+
+ std::vector<uword> ind = sorting.index();
+
+ Col<eT> new_ritz_val(ncv, arma_zeros_indicator() );
+ Mat<eT> new_ritz_vec(ncv, nev, arma_nozeros_indicator());
+ std::vector<bool> new_ritz_conv(nev);
+
+ for(uword i = 0; i < nev; i++)
+ {
+ new_ritz_val(i) = ritz_val(ind[i]);
+ new_ritz_vec.col(i) = ritz_vec.col(ind[i]);
+ new_ritz_conv[i] = ritz_conv[ind[i]];
+ }
+
+ ritz_val.swap(new_ritz_val);
+ ritz_vec.swap(new_ritz_vec);
+ ritz_conv.swap(new_ritz_conv);
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+SymEigsSolver<eT, SelectionRule, OpType>::SymEigsSolver(const OpType& op_, uword nev_, uword ncv_)
+ : op(op_)
+ , nev(nev_)
+ , dim_n(op.n_rows)
+ , ncv(ncv_ > dim_n ? dim_n : ncv_)
+ , nmatop(0)
+ , niter(0)
+ , eps(std::numeric_limits<eT>::epsilon())
+ , eps23(std::pow(eps, eT(2.0) / 3))
+ , near0(std::numeric_limits<eT>::min() * eT(10))
+ {
+ arma_extra_debug_sigprint();
+
+ arma_debug_check( (nev_ < 1 || nev_ > dim_n - 1), "newarp::SymEigsSolver: nev must satisfy 1 <= nev <= n - 1, n is the size of matrix" );
+ arma_debug_check( (ncv_ <= nev_ || ncv_ > dim_n), "newarp::SymEigsSolver: ncv must satisfy nev < ncv <= n, n is the size of matrix" );
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::init(eT* init_resid)
+ {
+ arma_extra_debug_sigprint();
+
+ // Reset all matrices/vectors to zero
+ fac_V.zeros(dim_n, ncv);
+ fac_H.zeros(ncv, ncv);
+ fac_f.zeros(dim_n);
+ ritz_val.zeros(ncv);
+ ritz_vec.zeros(ncv, nev);
+ ritz_est.zeros(ncv);
+ ritz_conv.assign(nev, false);
+
+ nmatop = 0;
+ niter = 0;
+
+ Col<eT> r(init_resid, dim_n, false);
+ // The first column of fac_V
+ Col<eT> v(fac_V.colptr(0), dim_n, false);
+ eT rnorm = norm(r);
+ arma_check( (rnorm < near0), "newarp::SymEigsSolver::init(): initial residual vector cannot be zero" );
+ v = r / rnorm;
+
+ Col<eT> w(dim_n, arma_zeros_indicator());
+ op.perform_op(v.memptr(), w.memptr());
+ nmatop++;
+
+ fac_H(0, 0) = dot(v, w);
+ fac_f = w - v * fac_H(0, 0);
+
+ // In some cases f is zero in exact arithmetics, but due to rounding errors
+ // it may contain tiny fluctuations. When this happens, we force f to be zero
+ if(abs(fac_f).max() < eps) { fac_f.zeros(); }
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+void
+SymEigsSolver<eT, SelectionRule, OpType>::init()
+ {
+ arma_extra_debug_sigprint();
+
+ // podarray<eT> init_resid(dim_n);
+ // blas_int idist = 2; // Uniform(-1, 1)
+ // blas_int iseed[4] = {1, 3, 5, 7}; // Fixed random seed
+ // blas_int n = dim_n;
+ // lapack::larnv(&idist, &iseed[0], &n, init_resid.memptr());
+ // init(init_resid.memptr());
+
+ podarray<eT> init_resid(dim_n);
+
+ fill_rand(init_resid.memptr(), dim_n, 0);
+
+ init(init_resid.memptr());
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+uword
+SymEigsSolver<eT, SelectionRule, OpType>::compute(uword maxit, eT tol)
+ {
+ arma_extra_debug_sigprint();
+
+ // The m-step Arnoldi factorisation
+ factorise_from(1, ncv, fac_f);
+ retrieve_ritzpair();
+ // Restarting
+ uword i, nconv = 0, nev_adj;
+ for(i = 0; i < maxit; i++)
+ {
+ nconv = num_converged(tol);
+ if(nconv >= nev) { break; }
+
+ nev_adj = nev_adjusted(nconv);
+ restart(nev_adj);
+ }
+ // Sorting results
+ sort_ritzpair();
+
+ niter = i + 1;
+
+ return (std::min)(nev, nconv);
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+Col<eT>
+SymEigsSolver<eT, SelectionRule, OpType>::eigenvalues()
+ {
+ arma_extra_debug_sigprint();
+
+ uword nconv = std::count(ritz_conv.begin(), ritz_conv.end(), true);
+ Col<eT> res(nconv, arma_zeros_indicator());
+
+ if(nconv > 0)
+ {
+ uword j = 0;
+
+ for(uword i=0; i < nev; i++)
+ {
+ if(ritz_conv[i]) { res(j) = ritz_val(i); j++; }
+ }
+ }
+
+ return res;
+ }
+
+
+
+template<typename eT, int SelectionRule, typename OpType>
+inline
+Mat<eT>
+SymEigsSolver<eT, SelectionRule, OpType>::eigenvectors(uword nvec)
+ {
+ arma_extra_debug_sigprint();
+
+ uword nconv = std::count(ritz_conv.begin(), ritz_conv.end(), true);
+ nvec = (std::min)(nvec, nconv);
+ Mat<eT> res(dim_n, nvec);
+
+ if(nvec > 0)
+ {
+ Mat<eT> ritz_vec_conv(ncv, nvec, arma_zeros_indicator());
+
+ uword j = 0;
+
+ for(uword i=0; i < nev && j < nvec; i++)
+ {
+ if(ritz_conv[i]) { ritz_vec_conv.col(j) = ritz_vec.col(i); j++; }
+ }
+
+ res = fac_V * ritz_vec_conv;
+ }
+
+ return res;
+ }
+
+
+} // namespace newarp