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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 band_helper
+//! @{
+
+
+namespace band_helper
+{
+
+
+
+template<typename eT>
+inline
+bool
+is_band(uword& out_KL, uword& out_KU, const Mat<eT>& A, const uword N_min)
+ {
+ arma_extra_debug_sigprint();
+
+ // NOTE: assuming that A has a square size
+ // NOTE: assuming that N_min is >= 4
+
+ const uword N = A.n_rows;
+
+ if(N < N_min) { return false; }
+
+ // first, quickly check bottom-left and top-right corners
+
+ const eT eT_zero = eT(0);
+
+ const eT* A_col0 = A.memptr();
+ const eT* A_col1 = A_col0 + N;
+
+ if( (A_col0[N-2] != eT_zero) || (A_col0[N-1] != eT_zero) || (A_col1[N-2] != eT_zero) || (A_col1[N-1] != eT_zero) ) { return false; }
+
+ const eT* A_colNm2 = A.colptr(N-2);
+ const eT* A_colNm1 = A_colNm2 + N;
+
+ if( (A_colNm2[0] != eT_zero) || (A_colNm2[1] != eT_zero) || (A_colNm1[0] != eT_zero) || (A_colNm1[1] != eT_zero) ) { return false; }
+
+ // if we reached this point, go through the entire matrix to work out number of subdiagonals and superdiagonals
+
+ const uword n_nonzero_threshold = (N*N)/4; // empirically determined
+
+ uword KL = 0; // number of subdiagonals
+ uword KU = 0; // number of superdiagonals
+
+ const eT* A_colptr = A.memptr();
+
+ for(uword col=0; col < N; ++col)
+ {
+ uword first_nonzero_row = col;
+ uword last_nonzero_row = col;
+
+ for(uword row=0; row < col; ++row)
+ {
+ if( A_colptr[row] != eT_zero ) { first_nonzero_row = row; break; }
+ }
+
+ for(uword row=(col+1); row < N; ++row)
+ {
+ last_nonzero_row = (A_colptr[row] != eT_zero) ? row : last_nonzero_row;
+ }
+
+ const uword L_count = last_nonzero_row - col;
+ const uword U_count = col - first_nonzero_row;
+
+ if( (L_count > KL) || (U_count > KU) )
+ {
+ KL = (std::max)(KL, L_count);
+ KU = (std::max)(KU, U_count);
+
+ const uword n_nonzero = N*(KL+KU+1) - (KL*(KL+1) + KU*(KU+1))/2;
+
+ // return as soon as we know that it's not worth analysing the matrix any further
+
+ if(n_nonzero > n_nonzero_threshold) { return false; }
+ }
+
+ A_colptr += N;
+ }
+
+ out_KL = KL;
+ out_KU = KU;
+
+ return true;
+ }
+
+
+
+template<typename eT>
+inline
+bool
+is_band_lower(uword& out_KD, const Mat<eT>& A, const uword N_min)
+ {
+ arma_extra_debug_sigprint();
+
+ // NOTE: assuming that A has a square size
+ // NOTE: assuming that N_min is >= 4
+
+ const uword N = A.n_rows;
+
+ if(N < N_min) { return false; }
+
+ // first, quickly check bottom-left corner
+
+ const eT eT_zero = eT(0);
+
+ const eT* A_col0 = A.memptr();
+ const eT* A_col1 = A_col0 + N;
+
+ if( (A_col0[N-2] != eT_zero) || (A_col0[N-1] != eT_zero) || (A_col1[N-2] != eT_zero) || (A_col1[N-1] != eT_zero) ) { return false; }
+
+ // if we reached this point, go through the bottom triangle to work out number of subdiagonals
+
+ const uword n_nonzero_threshold = ( N*N - (N*(N-1))/2 ) / 4; // empirically determined
+
+ uword KL = 0; // number of subdiagonals
+
+ const eT* A_colptr = A.memptr();
+
+ for(uword col=0; col < N; ++col)
+ {
+ uword last_nonzero_row = col;
+
+ for(uword row=(col+1); row < N; ++row)
+ {
+ last_nonzero_row = (A_colptr[row] != eT_zero) ? row : last_nonzero_row;
+ }
+
+ const uword L_count = last_nonzero_row - col;
+
+ if(L_count > KL)
+ {
+ KL = L_count;
+
+ const uword n_nonzero = N*(KL+1) - (KL*(KL+1))/2;
+
+ // return as soon as we know that it's not worth analysing the matrix any further
+
+ if(n_nonzero > n_nonzero_threshold) { return false; }
+ }
+
+ A_colptr += N;
+ }
+
+ out_KD = KL;
+
+ return true;
+ }
+
+
+
+template<typename eT>
+inline
+bool
+is_band_upper(uword& out_KD, const Mat<eT>& A, const uword N_min)
+ {
+ arma_extra_debug_sigprint();
+
+ // NOTE: assuming that A has a square size
+ // NOTE: assuming that N_min is >= 4
+
+ const uword N = A.n_rows;
+
+ if(N < N_min) { return false; }
+
+ // first, quickly check top-right corner
+
+ const eT eT_zero = eT(0);
+
+ const eT* A_colNm2 = A.colptr(N-2);
+ const eT* A_colNm1 = A_colNm2 + N;
+
+ if( (A_colNm2[0] != eT_zero) || (A_colNm2[1] != eT_zero) || (A_colNm1[0] != eT_zero) || (A_colNm1[1] != eT_zero) ) { return false; }
+
+ // if we reached this point, go through the entire matrix to work out number of superdiagonals
+
+ const uword n_nonzero_threshold = ( N*N - (N*(N-1))/2 ) / 4; // empirically determined
+
+ uword KU = 0; // number of superdiagonals
+
+ const eT* A_colptr = A.memptr();
+
+ for(uword col=0; col < N; ++col)
+ {
+ uword first_nonzero_row = col;
+
+ for(uword row=0; row < col; ++row)
+ {
+ if( A_colptr[row] != eT_zero ) { first_nonzero_row = row; break; }
+ }
+
+ const uword U_count = col - first_nonzero_row;
+
+ if(U_count > KU)
+ {
+ KU = U_count;
+
+ const uword n_nonzero = N*(KU+1) - (KU*(KU+1))/2;
+
+ // return as soon as we know that it's not worth analysing the matrix any further
+
+ if(n_nonzero > n_nonzero_threshold) { return false; }
+ }
+
+ A_colptr += N;
+ }
+
+ out_KD = KU;
+
+ return true;
+ }
+
+
+
+template<typename eT>
+inline
+void
+compress(Mat<eT>& AB, const Mat<eT>& A, const uword KL, const uword KU, const bool use_offset)
+ {
+ arma_extra_debug_sigprint();
+
+ // NOTE: assuming that A has a square size
+
+ // band matrix storage format
+ // http://www.netlib.org/lapack/lug/node124.html
+
+ // for ?gbsv, matrix AB size: 2*KL+KU+1 x N; band representation of A stored in rows KL+1 to 2*KL+KU+1 (note: fortran counts from 1)
+ // for ?gbsvx, matrix AB size: KL+KU+1 x N; band representaiton of A stored in rows 1 to KL+KU+1 (note: fortran counts from 1)
+ //
+ // the +1 in the above formulas is to take into account the main diagonal
+
+ const uword AB_n_rows = (use_offset) ? uword(2*KL + KU + 1) : uword(KL + KU + 1);
+ const uword N = A.n_rows;
+
+ AB.set_size(AB_n_rows, N);
+
+ if(A.is_empty()) { AB.zeros(); return; }
+
+ if(AB_n_rows == uword(1))
+ {
+ eT* AB_mem = AB.memptr();
+
+ for(uword i=0; i<N; ++i) { AB_mem[i] = A.at(i,i); }
+ }
+ else
+ {
+ AB.zeros(); // paranoia
+
+ for(uword j=0; j < N; ++j)
+ {
+ const uword A_row_start = (j > KU) ? uword(j - KU) : uword(0);
+ const uword A_row_endp1 = (std::min)(N, j+KL+1);
+
+ const uword length = A_row_endp1 - A_row_start;
+
+ const uword AB_row_start = (KU > j) ? (KU - j) : uword(0);
+
+ const eT* A_colptr = A.colptr(j) + A_row_start;
+ eT* AB_colptr = AB.colptr(j) + AB_row_start + ( (use_offset) ? KL : uword(0) );
+
+ arrayops::copy( AB_colptr, A_colptr, length );
+ }
+ }
+ }
+
+
+
+template<typename eT>
+inline
+void
+uncompress(Mat<eT>& A, const Mat<eT>& AB, const uword KL, const uword KU, const bool use_offset)
+ {
+ arma_extra_debug_sigprint();
+
+ const uword AB_n_rows = AB.n_rows;
+ const uword N = AB.n_cols;
+
+ arma_debug_check( (AB_n_rows != ((use_offset) ? uword(2*KL + KU + 1) : uword(KL + KU + 1))), "band_helper::uncompress(): detected inconsistency" );
+
+ A.zeros(N,N); // assuming there is no aliasing between A and AB
+
+ if(AB_n_rows == uword(1))
+ {
+ const eT* AB_mem = AB.memptr();
+
+ for(uword i=0; i<N; ++i) { A.at(i,i) = AB_mem[i]; }
+ }
+ else
+ {
+ for(uword j=0; j < N; ++j)
+ {
+ const uword A_row_start = (j > KU) ? uword(j - KU) : uword(0);
+ const uword A_row_endp1 = (std::min)(N, j+KL+1);
+
+ const uword length = A_row_endp1 - A_row_start;
+
+ const uword AB_row_start = (KU > j) ? (KU - j) : uword(0);
+
+ const eT* AB_colptr = AB.colptr(j) + AB_row_start + ( (use_offset) ? KL : uword(0) );
+ eT* A_colptr = A.colptr(j) + A_row_start;
+
+ arrayops::copy( A_colptr, AB_colptr, length );
+ }
+ }
+ }
+
+
+
+template<typename eT>
+inline
+void
+extract_tridiag(Mat<eT>& out, const Mat<eT>& A)
+ {
+ arma_extra_debug_sigprint();
+
+ // NOTE: assuming that A has a square size and is at least 2x2
+
+ const uword N = A.n_rows;
+
+ out.set_size(N, 3); // assuming there is no aliasing between 'out' and 'A'
+
+ if(N < 2) { return; }
+
+ eT* DL = out.colptr(0);
+ eT* DD = out.colptr(1);
+ eT* DU = out.colptr(2);
+
+ DD[0] = A[0];
+ DL[0] = A[1];
+
+ const uword Nm1 = N-1;
+ const uword Nm2 = N-2;
+
+ for(uword i=0; i < Nm2; ++i)
+ {
+ const uword ip1 = i+1;
+
+ const eT* data = &(A.at(i, ip1));
+
+ const eT tmp0 = data[0];
+ const eT tmp1 = data[1];
+ const eT tmp2 = data[2];
+
+ DL[ip1] = tmp2;
+ DD[ip1] = tmp1;
+ DU[i ] = tmp0;
+ }
+
+ const eT* data = &(A.at(Nm2, Nm1));
+
+ DL[Nm1] = 0;
+ DU[Nm2] = data[0];
+ DU[Nm1] = 0;
+ DD[Nm1] = data[1];
+ }
+
+
+
+} // end of namespace band_helper
+
+
+//! @}
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