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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.
+// ------------------------------------------------------------------------
+
+
+//! \addtogroup op_logmat
+//! @{
+
+
+// Partly based on algorithm 11.9 (inverse scaling and squaring algorithm with Schur decomposition) in:
+// Nicholas J. Higham.
+// Functions of Matrices: Theory and Computation.
+// SIAM, 2008.
+// ISBN 978-0-89871-646-7
+
+
+template<typename T1>
+inline
+void
+op_logmat::apply(Mat< std::complex<typename T1::elem_type> >& out, const mtOp<std::complex<typename T1::elem_type>,T1,op_logmat>& in)
+ {
+ arma_extra_debug_sigprint();
+
+ const bool status = op_logmat::apply_direct(out, in.m, in.aux_uword_a);
+
+ if(status == false)
+ {
+ out.soft_reset();
+ arma_stop_runtime_error("logmat(): transformation failed");
+ }
+ }
+
+
+
+template<typename T1>
+inline
+bool
+op_logmat::apply_direct(Mat< std::complex<typename T1::elem_type> >& out, const Op<T1,op_diagmat>& expr, const uword)
+ {
+ arma_extra_debug_sigprint();
+
+ typedef typename T1::elem_type T;
+
+ const diagmat_proxy<T1> P(expr.m);
+
+ arma_debug_check( (P.n_rows != P.n_cols), "logmat(): given matrix must be square sized" );
+
+ const uword N = P.n_rows;
+
+ out.zeros(N,N); // aliasing can't happen as op_logmat is defined as cx_mat = op(mat)
+
+ for(uword i=0; i<N; ++i)
+ {
+ const T val = P[i];
+
+ if(val >= T(0))
+ {
+ out.at(i,i) = std::log(val);
+ }
+ else
+ {
+ out.at(i,i) = std::log( std::complex<T>(val) );
+ }
+ }
+
+ return true;
+ }
+
+
+
+template<typename T1>
+inline
+bool
+op_logmat::apply_direct(Mat< std::complex<typename T1::elem_type> >& out, const Base<typename T1::elem_type,T1>& expr, const uword n_iters)
+ {
+ arma_extra_debug_sigprint();
+
+ typedef typename T1::elem_type in_T;
+ typedef typename std::complex<in_T> out_T;
+
+ const quasi_unwrap<T1> expr_unwrap(expr.get_ref());
+ const Mat<in_T>& A = expr_unwrap.M;
+
+ arma_debug_check( (A.is_square() == false), "logmat(): given matrix must be square sized" );
+
+ if(A.n_elem == 0)
+ {
+ out.reset();
+ return true;
+ }
+ else
+ if(A.n_elem == 1)
+ {
+ out.set_size(1,1);
+ out[0] = std::log( std::complex<in_T>( A[0] ) );
+ return true;
+ }
+
+ if(A.is_diagmat())
+ {
+ arma_extra_debug_print("op_logmat: detected diagonal matrix");
+
+ const uword N = A.n_rows;
+
+ out.zeros(N,N); // aliasing can't happen as op_logmat is defined as cx_mat = op(mat)
+
+ for(uword i=0; i<N; ++i)
+ {
+ const in_T val = A.at(i,i);
+
+ if(val >= in_T(0))
+ {
+ out.at(i,i) = std::log(val);
+ }
+ else
+ {
+ out.at(i,i) = std::log( out_T(val) );
+ }
+ }
+
+ return true;
+ }
+
+ const bool try_sympd = arma_config::optimise_sym && sym_helper::guess_sympd(A);
+
+ if(try_sympd)
+ {
+ arma_extra_debug_print("op_logmat: attempting sympd optimisation");
+
+ // if matrix A is sympd, all its eigenvalues are positive
+
+ Col<in_T> eigval;
+ Mat<in_T> eigvec;
+
+ const bool eig_status = eig_sym_helper(eigval, eigvec, A, 'd', "logmat()");
+
+ if(eig_status)
+ {
+ // ensure each eigenvalue is > 0
+
+ const uword N = eigval.n_elem;
+ const in_T* eigval_mem = eigval.memptr();
+
+ bool all_pos = true;
+
+ for(uword i=0; i<N; ++i) { all_pos = (eigval_mem[i] <= in_T(0)) ? false : all_pos; }
+
+ if(all_pos)
+ {
+ eigval = log(eigval);
+
+ out = conv_to< Mat<out_T> >::from( eigvec * diagmat(eigval) * eigvec.t() );
+
+ return true;
+ }
+ }
+
+ arma_extra_debug_print("op_logmat: sympd optimisation failed");
+
+ // fallthrough if eigen decomposition failed or an eigenvalue is <= 0
+ }
+
+
+ Mat<out_T> S(A.n_rows, A.n_cols, arma_nozeros_indicator());
+
+ const in_T* Amem = A.memptr();
+ out_T* Smem = S.memptr();
+
+ const uword n_elem = A.n_elem;
+
+ for(uword i=0; i<n_elem; ++i)
+ {
+ Smem[i] = std::complex<in_T>( Amem[i] );
+ }
+
+ return op_logmat_cx::apply_common(out, S, n_iters);
+ }
+
+
+
+template<typename T1>
+inline
+void
+op_logmat_cx::apply(Mat<typename T1::elem_type>& out, const Op<T1,op_logmat_cx>& in)
+ {
+ arma_extra_debug_sigprint();
+
+ const bool status = op_logmat_cx::apply_direct(out, in.m, in.aux_uword_a);
+
+ if(status == false)
+ {
+ out.soft_reset();
+ arma_stop_runtime_error("logmat(): transformation failed");
+ }
+ }
+
+
+
+template<typename T1>
+inline
+bool
+op_logmat_cx::apply_direct(Mat<typename T1::elem_type>& out, const Op<T1,op_diagmat>& expr, const uword)
+ {
+ arma_extra_debug_sigprint();
+
+ typedef typename T1::elem_type eT;
+
+ const diagmat_proxy<T1> P(expr.m);
+
+ bool status = false;
+
+ if(P.is_alias(out))
+ {
+ Mat<eT> tmp;
+
+ status = op_logmat_cx::apply_direct_noalias(tmp, P);
+
+ out.steal_mem(tmp);
+ }
+ else
+ {
+ status = op_logmat_cx::apply_direct_noalias(out, P);
+ }
+
+ return status;
+ }
+
+
+
+template<typename T1>
+inline
+bool
+op_logmat_cx::apply_direct_noalias(Mat<typename T1::elem_type>& out, const diagmat_proxy<T1>& P)
+ {
+ arma_extra_debug_sigprint();
+
+ arma_debug_check( (P.n_rows != P.n_cols), "logmat(): given matrix must be square sized" );
+
+ const uword N = P.n_rows;
+
+ out.zeros(N,N);
+
+ for(uword i=0; i<N; ++i)
+ {
+ out.at(i,i) = std::log(P[i]);
+ }
+
+ return true;
+ }
+
+
+
+template<typename T1>
+inline
+bool
+op_logmat_cx::apply_direct(Mat<typename T1::elem_type>& out, const Base<typename T1::elem_type,T1>& expr, const uword n_iters)
+ {
+ arma_extra_debug_sigprint();
+
+ typedef typename T1::pod_type T;
+ typedef typename T1::elem_type eT;
+
+ Mat<eT> S = expr.get_ref();
+
+ arma_debug_check( (S.n_rows != S.n_cols), "logmat(): given matrix must be square sized" );
+
+ if(S.n_elem == 0)
+ {
+ out.reset();
+ return true;
+ }
+ else
+ if(S.n_elem == 1)
+ {
+ out.set_size(1,1);
+ out[0] = std::log(S[0]);
+ return true;
+ }
+
+ if(S.is_diagmat())
+ {
+ arma_extra_debug_print("op_logmat_cx: detected diagonal matrix");
+
+ const uword N = S.n_rows;
+
+ out.zeros(N,N); // aliasing can't happen as S is generated
+
+ for(uword i=0; i<N; ++i) { out.at(i,i) = std::log( S.at(i,i) ); }
+
+ return true;
+ }
+
+ const bool try_sympd = arma_config::optimise_sym && sym_helper::guess_sympd(S);
+
+ if(try_sympd)
+ {
+ arma_extra_debug_print("op_logmat_cx: attempting sympd optimisation");
+
+ // if matrix S is sympd, all its eigenvalues are positive
+
+ Col< T> eigval;
+ Mat<eT> eigvec;
+
+ const bool eig_status = eig_sym_helper(eigval, eigvec, S, 'd', "logmat()");
+
+ if(eig_status)
+ {
+ // ensure each eigenvalue is > 0
+
+ const uword N = eigval.n_elem;
+ const T* eigval_mem = eigval.memptr();
+
+ bool all_pos = true;
+
+ for(uword i=0; i<N; ++i) { all_pos = (eigval_mem[i] <= T(0)) ? false : all_pos; }
+
+ if(all_pos)
+ {
+ eigval = log(eigval);
+
+ out = eigvec * diagmat(eigval) * eigvec.t();
+
+ return true;
+ }
+ }
+
+ arma_extra_debug_print("op_logmat_cx: sympd optimisation failed");
+
+ // fallthrough if eigen decomposition failed or an eigenvalue is <= 0
+ }
+
+ return op_logmat_cx::apply_common(out, S, n_iters);
+ }
+
+
+
+template<typename T>
+inline
+bool
+op_logmat_cx::apply_common(Mat< std::complex<T> >& out, Mat< std::complex<T> >& S, const uword n_iters)
+ {
+ arma_extra_debug_sigprint();
+
+ typedef typename std::complex<T> eT;
+
+ Mat<eT> U;
+
+ const bool schur_ok = auxlib::schur(U,S);
+
+ if(schur_ok == false) { arma_extra_debug_print("logmat(): schur decomposition failed"); return false; }
+
+ // NOTE: theta[0] and theta[1] not really used
+ double theta[] = { 1.10e-5, 1.82e-3, 1.6206284795015624e-2, 5.3873532631381171e-2, 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
+
+ const uword N = S.n_rows;
+
+ uword p = 0;
+ uword m = 6;
+
+ uword iter = 0;
+
+ while(iter < n_iters)
+ {
+ const T tau = norm( (S - eye< Mat<eT> >(N,N)), 1 );
+
+ if(tau <= theta[6])
+ {
+ p++;
+
+ uword j1 = 0;
+ uword j2 = 0;
+
+ for(uword i=2; i<=6; ++i) { if( tau <= theta[i]) { j1 = i; break; } }
+ for(uword i=2; i<=6; ++i) { if((tau/2.0) <= theta[i]) { j2 = i; break; } }
+
+ // sanity check, for development purposes only
+ arma_debug_check( (j2 > j1), "internal error: op_logmat::apply_direct(): j2 > j1" );
+
+ if( ((j1 - j2) <= 1) || (p == 2) ) { m = j1; break; }
+ }
+
+ const bool sqrtmat_ok = op_sqrtmat_cx::apply_direct(S,S);
+
+ if(sqrtmat_ok == false) { arma_extra_debug_print("logmat(): sqrtmat() failed"); return false; }
+
+ iter++;
+ }
+
+ if(iter >= n_iters) { arma_debug_warn_level(2, "logmat(): reached max iterations without full convergence"); }
+
+ S.diag() -= eT(1);
+
+ if(m >= 1)
+ {
+ const bool helper_ok = op_logmat_cx::helper(S,m);
+
+ if(helper_ok == false) { return false; }
+ }
+
+ out = U * S * U.t();
+
+ out *= eT(eop_aux::pow(double(2), double(iter)));
+
+ return true;
+ }
+
+
+
+template<typename eT>
+inline
+bool
+op_logmat_cx::helper(Mat<eT>& A, const uword m)
+ {
+ arma_extra_debug_sigprint();
+
+ if(A.internal_has_nonfinite()) { return false; }
+
+ const vec indices = regspace<vec>(1,m-1);
+
+ mat tmp(m, m, arma_zeros_indicator());
+
+ tmp.diag(-1) = indices / sqrt(square(2.0*indices) - 1.0);
+ tmp.diag(+1) = indices / sqrt(square(2.0*indices) - 1.0);
+
+ vec eigval;
+ mat eigvec;
+
+ const bool eig_ok = eig_sym_helper(eigval, eigvec, tmp, 'd', "logmat()");
+
+ if(eig_ok == false) { arma_extra_debug_print("logmat(): eig_sym() failed"); return false; }
+
+ const vec nodes = (eigval + 1.0) / 2.0;
+ const vec weights = square(eigvec.row(0).t());
+
+ const uword N = A.n_rows;
+
+ Mat<eT> B(N, N, arma_zeros_indicator());
+
+ Mat<eT> X;
+
+ for(uword i=0; i < m; ++i)
+ {
+ // B += weights(i) * solve( (nodes(i)*A + eye< Mat<eT> >(N,N)), A );
+
+ //const bool solve_ok = solve( X, (nodes(i)*A + eye< Mat<eT> >(N,N)), A, solve_opts::fast );
+ const bool solve_ok = solve( X, trimatu(nodes(i)*A + eye< Mat<eT> >(N,N)), A, solve_opts::no_approx );
+
+ if(solve_ok == false) { arma_extra_debug_print("logmat(): solve() failed"); return false; }
+
+ B += weights(i) * X;
+ }
+
+ A = B;
+
+ return true;
+ }
+
+
+
+template<typename T1>
+inline
+void
+op_logmat_sympd::apply(Mat<typename T1::elem_type>& out, const Op<T1,op_logmat_sympd>& in)
+ {
+ arma_extra_debug_sigprint();
+
+ const bool status = op_logmat_sympd::apply_direct(out, in.m);
+
+ if(status == false)
+ {
+ out.soft_reset();
+ arma_stop_runtime_error("logmat_sympd(): transformation failed");
+ }
+ }
+
+
+
+template<typename T1>
+inline
+bool
+op_logmat_sympd::apply_direct(Mat<typename T1::elem_type>& out, const Base<typename T1::elem_type,T1>& expr)
+ {
+ arma_extra_debug_sigprint();
+
+ #if defined(ARMA_USE_LAPACK)
+ {
+ typedef typename T1::pod_type T;
+ typedef typename T1::elem_type eT;
+
+ const unwrap<T1> U(expr.get_ref());
+ const Mat<eT>& X = U.M;
+
+ arma_debug_check( (X.is_square() == false), "logmat_sympd(): given matrix must be square sized" );
+
+ if((arma_config::debug) && (arma_config::warn_level > 0) && (is_cx<eT>::yes) && (sym_helper::check_diag_imag(X) == false))
+ {
+ arma_debug_warn_level(1, "logmat_sympd(): imaginary components on diagonal are non-zero");
+ }
+
+ if(is_op_diagmat<T1>::value || X.is_diagmat())
+ {
+ arma_extra_debug_print("op_logmat_sympd: detected diagonal matrix");
+
+ out = X;
+
+ eT* colmem = out.memptr();
+
+ const uword N = X.n_rows;
+
+ for(uword i=0; i<N; ++i)
+ {
+ eT& out_ii = colmem[i];
+ T out_ii_real = access::tmp_real(out_ii);
+
+ if(out_ii_real <= T(0)) { return false; }
+
+ out_ii = std::log(out_ii);
+
+ colmem += N;
+ }
+
+ return true;
+ }
+
+ Col< T> eigval;
+ Mat<eT> eigvec;
+
+ const bool status = eig_sym_helper(eigval, eigvec, X, 'd', "logmat_sympd()");
+
+ if(status == false) { return false; }
+
+ const uword N = eigval.n_elem;
+ const T* eigval_mem = eigval.memptr();
+
+ bool all_pos = true;
+
+ for(uword i=0; i<N; ++i) { all_pos = (eigval_mem[i] <= T(0)) ? false : all_pos; }
+
+ if(all_pos == false) { return false; }
+
+ eigval = log(eigval);
+
+ out = eigvec * diagmat(eigval) * eigvec.t();
+
+ return true;
+ }
+ #else
+ {
+ arma_ignore(out);
+ arma_ignore(expr);
+ arma_stop_logic_error("logmat_sympd(): use of LAPACK must be enabled");
+ return false;
+ }
+ #endif
+ }
+
+
+
+//! @}