486 lines
12 KiB
C++
486 lines
12 KiB
C++
// SPDX-License-Identifier: Apache-2.0
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//
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// Copyright 2008-2016 Conrad Sanderson (http://conradsanderson.id.au)
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// Copyright 2008-2016 National ICT Australia (NICTA)
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//
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// Licensed under the Apache License, Version 2.0 (the "License");
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// you may not use this file except in compliance with the License.
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// You may obtain a copy of the License at
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// http://www.apache.org/licenses/LICENSE-2.0
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//
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// Unless required by applicable law or agreed to in writing, software
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// distributed under the License is distributed on an "AS IS" BASIS,
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// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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// See the License for the specific language governing permissions and
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// limitations under the License.
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// ------------------------------------------------------------------------
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//! \addtogroup sym_helper
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//! @{
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namespace sym_helper
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{
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// computationally inexpensive algorithm to guess whether a matrix is positive definite:
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// (1) ensure the matrix is symmetric/hermitian (within a tolerance)
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// (2) ensure the diagonal entries are real and greater than zero
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// (3) ensure that the value with largest modulus is on the main diagonal
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// (4) ensure rudimentary diagonal dominance: (real(A_ii) + real(A_jj)) > 2*abs(real(A_ij))
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// the above conditions are necessary, but not sufficient;
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// doing it properly would be too computationally expensive for our purposes
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// more info:
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// http://mathworld.wolfram.com/PositiveDefiniteMatrix.html
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// http://mathworld.wolfram.com/DiagonallyDominantMatrix.html
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template<typename eT>
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inline
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typename enable_if2<is_cx<eT>::no, bool>::result
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guess_sympd_worker(const Mat<eT>& A)
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{
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arma_debug_sigprint();
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// NOTE: assuming A is square-sized
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const eT tol = eT(100) * std::numeric_limits<eT>::epsilon(); // allow some leeway
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const uword N = A.n_rows;
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const eT* A_mem = A.memptr();
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const eT* A_col = A_mem;
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eT max_diag = eT(0);
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for(uword j=0; j < N; ++j)
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{
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const eT A_jj = A_col[j];
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if(A_jj <= eT(0)) { return false; }
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max_diag = (A_jj > max_diag) ? A_jj : max_diag;
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A_col += N;
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}
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A_col = A_mem;
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const uword Nm1 = N-1;
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const uword Np1 = N+1;
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for(uword j=0; j < Nm1; ++j)
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{
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const eT A_jj = A_col[j];
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const uword jp1 = j+1;
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const eT* A_ji_ptr = &(A_mem[j + jp1*N]); // &(A.at(j,jp1));
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const eT* A_ii_ptr = &(A_mem[jp1 + jp1*N]);
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for(uword i=jp1; i < N; ++i)
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{
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const eT A_ij = A_col[i];
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const eT A_ji = (*A_ji_ptr);
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const eT A_ij_abs = (std::abs)(A_ij);
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const eT A_ji_abs = (std::abs)(A_ji);
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// if( (A_ij_abs >= max_diag) || (A_ji_abs >= max_diag) ) { return false; }
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if(A_ij_abs >= max_diag) { return false; }
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const eT A_delta = (std::abs)(A_ij - A_ji);
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const eT A_abs_max = (std::max)(A_ij_abs, A_ji_abs);
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if( (A_delta > tol) && (A_delta > (A_abs_max*tol)) ) { return false; }
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const eT A_ii = (*A_ii_ptr);
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if( (A_ij_abs + A_ij_abs) >= (A_ii + A_jj) ) { return false; }
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A_ji_ptr += N;
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A_ii_ptr += Np1;
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}
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A_col += N;
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}
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return true;
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}
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template<typename eT>
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inline
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typename enable_if2<is_cx<eT>::yes, bool>::result
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guess_sympd_worker(const Mat<eT>& A)
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{
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arma_debug_sigprint();
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// NOTE: assuming A is square-sized
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typedef typename get_pod_type<eT>::result T;
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const T tol = T(100) * std::numeric_limits<T>::epsilon(); // allow some leeway
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const uword N = A.n_rows;
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const eT* A_mem = A.memptr();
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const eT* A_col = A_mem;
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T max_diag = T(0);
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for(uword j=0; j < N; ++j)
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{
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const eT& A_jj = A_col[j];
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const T A_jj_real = std::real(A_jj);
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const T A_jj_imag = std::imag(A_jj);
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if( (A_jj_real <= T(0)) || (std::abs(A_jj_imag) > tol) ) { return false; }
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max_diag = (A_jj_real > max_diag) ? A_jj_real : max_diag;
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A_col += N;
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}
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const T square_max_diag = max_diag * max_diag;
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if(arma_isfinite(square_max_diag) == false) { return false; }
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A_col = A_mem;
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const uword Nm1 = N-1;
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const uword Np1 = N+1;
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for(uword j=0; j < Nm1; ++j)
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{
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const uword jp1 = j+1;
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const eT* A_ji_ptr = &(A_mem[j + jp1*N]); // &(A.at(j,jp1));
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const eT* A_ii_ptr = &(A_mem[jp1 + jp1*N]);
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const T A_jj_real = std::real(A_col[j]);
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for(uword i=jp1; i < N; ++i)
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{
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const eT& A_ij = A_col[i];
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const T A_ij_real = std::real(A_ij);
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const T A_ij_imag = std::imag(A_ij);
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// avoid using std::abs(), as that is time consuming due to division and std::sqrt()
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const T square_A_ij_abs = (A_ij_real * A_ij_real) + (A_ij_imag * A_ij_imag);
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if(arma_isfinite(square_A_ij_abs) == false) { return false; }
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if(square_A_ij_abs >= square_max_diag) { return false; }
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const T A_ij_real_abs = (std::abs)(A_ij_real);
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const T A_ij_imag_abs = (std::abs)(A_ij_imag);
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const eT& A_ji = (*A_ji_ptr);
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const T A_ji_real = std::real(A_ji);
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const T A_ji_imag = std::imag(A_ji);
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const T A_ji_real_abs = (std::abs)(A_ji_real);
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const T A_ji_imag_abs = (std::abs)(A_ji_imag);
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const T A_real_delta = (std::abs)(A_ij_real - A_ji_real);
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const T A_real_abs_max = (std::max)(A_ij_real_abs, A_ji_real_abs);
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if( (A_real_delta > tol) && (A_real_delta > (A_real_abs_max*tol)) ) { return false; }
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const T A_imag_delta = (std::abs)(A_ij_imag + A_ji_imag); // take into account complex conjugate
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const T A_imag_abs_max = (std::max)(A_ij_imag_abs, A_ji_imag_abs);
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if( (A_imag_delta > tol) && (A_imag_delta > (A_imag_abs_max*tol)) ) { return false; }
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const T A_ii_real = std::real(*A_ii_ptr);
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if( (A_ij_real_abs + A_ij_real_abs) >= (A_ii_real + A_jj_real) ) { return false; }
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A_ji_ptr += N;
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A_ii_ptr += Np1;
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}
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A_col += N;
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}
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return true;
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}
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template<typename eT>
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inline
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bool
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guess_sympd(const Mat<eT>& A)
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{
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arma_debug_sigprint();
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// analyse matrices with size >= 4x4
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if((A.n_rows != A.n_cols) || (A.n_rows < uword(4))) { return false; }
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return guess_sympd_worker(A);
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}
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template<typename eT>
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inline
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bool
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guess_sympd(const Mat<eT>& A, const uword min_n_rows)
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{
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arma_debug_sigprint();
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if((A.n_rows != A.n_cols) || (A.n_rows < min_n_rows)) { return false; }
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return guess_sympd_worker(A);
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}
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//
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template<typename eT>
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inline
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typename enable_if2<is_cx<eT>::no, void>::result
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analyse_matrix_worker(bool& is_approx_sym, bool& is_approx_sympd, const Mat<eT>& A)
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{
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arma_debug_sigprint();
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is_approx_sym = true;
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is_approx_sympd = true;
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const eT tol = eT(100) * std::numeric_limits<eT>::epsilon(); // allow some leeway
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const uword N = A.n_rows;
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const eT* A_mem = A.memptr();
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const eT* A_col = A_mem;
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eT max_diag = eT(0);
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for(uword j=0; j < N; ++j)
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{
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const eT A_jj = A_col[j];
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if(A_jj <= eT(0)) { is_approx_sympd = false; }
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max_diag = (A_jj > max_diag) ? A_jj : max_diag;
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A_col += N;
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}
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A_col = A_mem;
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const uword Nm1 = N-1;
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const uword Np1 = N+1;
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for(uword j=0; j < Nm1; ++j)
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{
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const eT A_jj = A_col[j];
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const uword jp1 = j+1;
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const eT* A_ji_ptr = &(A_mem[j + jp1*N]); // &(A.at(j,jp1));
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const eT* A_ii_ptr = &(A_mem[jp1 + jp1*N]);
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for(uword i=jp1; i < N; ++i)
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{
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const eT A_ij = A_col[i];
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const eT A_ji = (*A_ji_ptr);
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const eT A_ij_abs = (std::abs)(A_ij);
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const eT A_ji_abs = (std::abs)(A_ji);
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const eT A_delta = (std::abs)(A_ij - A_ji);
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const eT A_abs_max = (std::max)(A_ij_abs, A_ji_abs);
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if( (A_delta > tol) && (A_delta > (A_abs_max*tol)) ) { is_approx_sym = false; return; }
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if(is_approx_sympd)
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{
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// if( (A_ij_abs >= max_diag) || (A_ji_abs >= max_diag) ) { is_approx_sympd = false; }
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if(A_ij_abs >= max_diag) { is_approx_sympd = false; }
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const eT A_ii = (*A_ii_ptr);
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if( (A_ij_abs + A_ij_abs) >= (A_ii + A_jj) ) { is_approx_sympd = false; }
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}
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A_ji_ptr += N;
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A_ii_ptr += Np1;
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}
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A_col += N;
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}
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}
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template<typename eT>
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inline
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typename enable_if2<is_cx<eT>::yes, void>::result
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analyse_matrix_worker(bool& is_approx_sym, bool& is_approx_sympd, const Mat<eT>& A)
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{
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arma_debug_sigprint();
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typedef typename get_pod_type<eT>::result T;
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is_approx_sym = true;
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is_approx_sympd = true;
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const T tol = T(100) * std::numeric_limits<T>::epsilon(); // allow some leeway
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const uword N = A.n_rows;
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const eT* A_mem = A.memptr();
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const eT* A_col = A_mem;
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T max_diag = T(0);
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for(uword j=0; j < N; ++j)
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{
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const eT& A_jj = A_col[j];
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const T A_jj_real = std::real(A_jj);
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const T A_jj_imag = std::imag(A_jj);
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if( (A_jj_real <= T(0)) || (std::abs(A_jj_imag) > tol) ) { is_approx_sympd = false; }
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max_diag = (A_jj_real > max_diag) ? A_jj_real : max_diag;
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A_col += N;
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}
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const T square_max_diag = max_diag * max_diag;
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if(arma_isfinite(square_max_diag) == false) { is_approx_sympd = false; }
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A_col = A_mem;
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const uword Nm1 = N-1;
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const uword Np1 = N+1;
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for(uword j=0; j < Nm1; ++j)
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{
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const uword jp1 = j+1;
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const eT* A_ji_ptr = &(A_mem[j + jp1*N]); // &(A.at(j,jp1));
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const eT* A_ii_ptr = &(A_mem[jp1 + jp1*N]);
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const T A_jj_real = std::real(A_col[j]);
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for(uword i=jp1; i < N; ++i)
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{
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const eT& A_ij = A_col[i];
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const T A_ij_real = std::real(A_ij);
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const T A_ij_imag = std::imag(A_ij);
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const T A_ij_real_abs = (std::abs)(A_ij_real);
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const T A_ij_imag_abs = (std::abs)(A_ij_imag);
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const eT& A_ji = (*A_ji_ptr);
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const T A_ji_real = std::real(A_ji);
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const T A_ji_imag = std::imag(A_ji);
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const T A_ji_real_abs = (std::abs)(A_ji_real);
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const T A_ji_imag_abs = (std::abs)(A_ji_imag);
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const T A_real_delta = (std::abs)(A_ij_real - A_ji_real);
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const T A_real_abs_max = (std::max)(A_ij_real_abs, A_ji_real_abs);
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if( (A_real_delta > tol) && (A_real_delta > (A_real_abs_max*tol)) ) { is_approx_sym = false; return; }
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const T A_imag_delta = (std::abs)(A_ij_imag + A_ji_imag); // take into account complex conjugate
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const T A_imag_abs_max = (std::max)(A_ij_imag_abs, A_ji_imag_abs);
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if( (A_imag_delta > tol) && (A_imag_delta > (A_imag_abs_max*tol)) ) { is_approx_sym = false; return; }
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if(is_approx_sympd)
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{
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// avoid using std::abs(), as that is time consuming due to division and std::sqrt()
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const T square_A_ij_abs = (A_ij_real * A_ij_real) + (A_ij_imag * A_ij_imag);
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if(arma_isfinite(square_A_ij_abs) == false)
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{
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is_approx_sympd = false;
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}
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else
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{
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const T A_ii_real = std::real(*A_ii_ptr);
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if( (A_ij_real_abs + A_ij_real_abs) >= (A_ii_real + A_jj_real) ) { is_approx_sympd = false; }
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if(square_A_ij_abs >= square_max_diag) { is_approx_sympd = false; }
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}
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}
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A_ji_ptr += N;
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A_ii_ptr += Np1;
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}
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A_col += N;
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}
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}
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template<typename eT>
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inline
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void
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analyse_matrix(bool& is_approx_sym, bool& is_approx_sympd, const Mat<eT>& A)
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{
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arma_debug_sigprint();
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if((A.n_rows != A.n_cols) || (A.n_rows < uword(4)))
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{
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is_approx_sym = false;
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is_approx_sympd = false;
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return;
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}
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analyse_matrix_worker(is_approx_sym, is_approx_sympd, A);
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if(is_approx_sym == false) { is_approx_sympd = false; }
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}
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template<typename eT>
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inline
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bool
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check_diag_imag(const Mat<eT>& A)
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{
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arma_debug_sigprint();
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// NOTE: assuming matrix A is square-sized
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typedef typename get_pod_type<eT>::result T;
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const T tol = T(10000) * std::numeric_limits<T>::epsilon(); // allow some leeway
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const eT* colmem = A.memptr();
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const uword N = A.n_rows;
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for(uword i=0; i<N; ++i)
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{
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const eT& A_ii = colmem[i];
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const T A_ii_imag = access::tmp_imag(A_ii);
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if(std::abs(A_ii_imag) > tol) { return false; }
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colmem += N;
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}
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return true;
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}
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} // end of namespace sym_helper
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//! @}
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