1 // Copyright (c) 2017-2025, Lawrence Livermore National Security, LLC and other CEED contributors. 2 // All Rights Reserved. See the top-level LICENSE and NOTICE files for details. 3 // 4 // SPDX-License-Identifier: BSD-2-Clause 5 // 6 // This file is part of CEED: http://github.com/ceed 7 8 /// @file 9 /// Eigen system solver for symmetric NxN matrices. Modified from the CC0 code provided at https://github.com/jewettaij/jacobi_pd 10 #pragma once 11 12 #include <ceed/types.h> 13 #ifndef CEED_RUNNING_JIT_PASS 14 #include <math.h> 15 #include <stdbool.h> 16 #endif 17 18 #include "utils.h" 19 20 // @typedef choose the criteria for sorting eigenvalues and eigenvectors 21 typedef enum eSortCriteria { 22 SORT_NONE, 23 SORT_DECREASING_EVALS, 24 SORT_INCREASING_EVALS, 25 SORT_DECREASING_ABS_EVALS, 26 SORT_INCREASING_ABS_EVALS 27 } SortCriteria; 28 29 ///@brief Find the off-diagonal index in row i whose absolute value is largest 30 /// 31 /// @param[in] *A matrix 32 /// @param[in] i row index 33 /// @returns Index of absolute largest off-diagonal element in row i 34 CEED_QFUNCTION_HELPER CeedInt MaxEntryRow(const CeedScalar *A, CeedInt N, CeedInt i) { 35 CeedInt j_max = i + 1; 36 for (CeedInt j = i + 2; j < N; j++) 37 if (fabs(A[i * N + j]) > fabs(A[i * N + j_max])) j_max = j; 38 return j_max; 39 } 40 41 /// @brief Find the indices (i_max, j_max) marking the location of the 42 /// entry in the matrix with the largest absolute value. This 43 /// uses the max_idx_row[] array to find the answer in O(n) time. 44 /// 45 /// @param[in] *A matrix 46 /// @param[inout] i_max row index 47 /// @param[inout] j_max column index 48 CEED_QFUNCTION_HELPER void MaxEntry(const CeedScalar *A, CeedInt N, CeedInt *max_idx_row, CeedInt *i_max, CeedInt *j_max) { 49 *i_max = 0; 50 *j_max = max_idx_row[*i_max]; 51 CeedScalar max_entry = fabs(A[*i_max * N + *j_max]); 52 for (CeedInt i = 1; i < N - 1; i++) { 53 CeedInt j = max_idx_row[i]; 54 if (fabs(A[i * N + j]) > max_entry) { 55 max_entry = fabs(A[i * N + j]); 56 *i_max = i; 57 *j_max = j; 58 } 59 } 60 } 61 62 /// @brief Calculate the components of a rotation matrix which performs a 63 /// rotation in the i,j plane by an angle (θ) that (when multiplied on 64 /// both sides) will zero the ij'th element of A, so that afterwards 65 /// A[i][j] = 0. The results will be stored in c, s, and t 66 /// (which store cos(θ), sin(θ), and tan(θ), respectively). 67 /// 68 /// @param[in] *A matrix 69 /// @param[in] i row index 70 /// @param[in] j column index 71 CEED_QFUNCTION_HELPER void CalcRot(const CeedScalar *A, CeedInt N, CeedInt i, CeedInt j, CeedScalar *rotmat_cst) { 72 rotmat_cst[2] = 1.0; // = tan(θ) 73 CeedScalar A_jj_ii = (A[j * N + j] - A[i * N + i]); 74 if (A_jj_ii != 0.0) { 75 // kappa = (A[j][j] - A[i][i]) / (2*A[i][j]) 76 CeedScalar kappa = A_jj_ii; 77 rotmat_cst[2] = 0.0; 78 CeedScalar A_ij = A[i * N + j]; 79 if (A_ij != 0.0) { 80 kappa /= (2.0 * A_ij); 81 // t satisfies: t^2 + 2*t*kappa - 1 = 0 82 // (choose the root which has the smaller absolute value) 83 rotmat_cst[2] = 1.0 / (sqrt(1 + kappa * kappa) + fabs(kappa)); 84 if (kappa < 0.0) rotmat_cst[2] = -rotmat_cst[2]; 85 } 86 } 87 rotmat_cst[0] = 1.0 / sqrt(1 + rotmat_cst[2] * rotmat_cst[2]); 88 rotmat_cst[1] = rotmat_cst[0] * rotmat_cst[2]; 89 } 90 91 /// @brief Perform a similarity transformation by multiplying matrix A on both 92 /// sides by a rotation matrix (and its transpose) to eliminate A[i][j]. 93 /// @details This rotation matrix performs a rotation in the i,j plane by 94 /// angle θ. This function assumes that c=cos(θ). s=sin(θ), t=tan(θ) 95 /// have been calculated in advance (using the CalcRot() function). 96 /// It also assumes that i<j. The max_idx_row[] array is also updated. 97 /// To save time, since the matrix is symmetric, the elements 98 /// below the diagonal (ie. A[u][v] where u>v) are not computed. 99 /// @verbatim 100 /// A' = R^T * A * R 101 /// where R the rotation in the i,j plane and ^T denotes the transpose. 102 /// i j 103 /// _ _ 104 /// | 1 | 105 /// | . | 106 /// | . | 107 /// | 1 | 108 /// | c ... s | 109 /// | . . . | 110 /// R = | . 1 . | 111 /// | . . . | 112 /// | -s ... c | 113 /// | 1 | 114 /// | . | 115 /// | . | 116 /// |_ 1 _| 117 /// @endverbatim 118 /// 119 /// Let A' denote the matrix A after multiplication by R^T and R. 120 /// The components of A' are: 121 /// 122 /// @verbatim 123 /// A'_uv = Σ_w Σ_z R_wu * A_wz * R_zv 124 /// @endverbatim 125 /// 126 /// Note that a the rotation at location i,j will modify all of the matrix 127 /// elements containing at least one index which is either i or j 128 /// such as: A[w][i], A[i][w], A[w][j], A[j][w]. 129 /// Check and see whether these modified matrix elements exceed the 130 /// corresponding values in max_idx_row[] array for that row. 131 /// If so, then update max_idx_row for that row. 132 /// This is somewhat complicated by the fact that we must only consider 133 /// matrix elements in the upper-right triangle strictly above the diagonal. 134 /// (ie. matrix elements whose second index is > the first index). 135 /// The modified elements we must consider are marked with an "X" below: 136 /// 137 /// @verbatim 138 /// i j 139 /// _ _ 140 /// | . X X | 141 /// | . X X | 142 /// | . X X | 143 /// | . X X | 144 /// | X X X X X 0 X X X X | i 145 /// | . X | 146 /// | . X | 147 /// A = | . X | 148 /// | . X | 149 /// | X X X X X | j 150 /// | . | 151 /// | . | 152 /// | . | 153 /// |_ . _| 154 /// @endverbatim 155 /// 156 /// @param[in] *A matrix 157 /// @param[in] i row index 158 /// @param[in] j column index 159 CEED_QFUNCTION_HELPER void ApplyRot(CeedScalar *A, CeedInt N, CeedInt i, CeedInt j, CeedInt *max_idx_row, CeedScalar *rotmat_cst) { 160 // Compute the diagonal elements of A which have changed: 161 A[i * N + i] -= rotmat_cst[2] * A[i * N + j]; 162 A[j * N + j] += rotmat_cst[2] * A[i * N + j]; 163 // Note: This is algebraically equivalent to: 164 // A[i][i] = c*c*A[i][i] + s*s*A[j][j] - 2*s*c*A[i][j] 165 // A[j][j] = s*s*A[i][i] + c*c*A[j][j] + 2*s*c*A[i][j] 166 167 // Update the off-diagonal elements of A which will change (above the diagonal) 168 169 A[i * N + j] = 0.0; 170 171 // compute A[w][i] and A[i][w] for all w!=i,considering above-diagonal elements 172 for (CeedInt w = 0; w < i; w++) { // 0 <= w < i < j < N 173 A[i * N + w] = A[w * N + i]; // backup the previous value. store below diagonal (i>w) 174 A[w * N + i] = rotmat_cst[0] * A[w * N + i] - rotmat_cst[1] * A[w * N + j]; // A[w][i], A[w][j] from previous iteration 175 if (i == max_idx_row[w]) max_idx_row[w] = MaxEntryRow(A, N, w); 176 else if (fabs(A[w * N + i]) > fabs(A[w * N + max_idx_row[w]])) max_idx_row[w] = i; 177 } 178 for (CeedInt w = i + 1; w < j; w++) { // 0 <= i < w < j < N 179 A[w * N + i] = A[i * N + w]; // backup the previous value. store below diagonal (w>i) 180 A[i * N + w] = rotmat_cst[0] * A[i * N + w] - rotmat_cst[1] * A[w * N + j]; // A[i][w], A[w][j] from previous iteration 181 } 182 for (CeedInt w = j + 1; w < N; w++) { // 0 <= i < j+1 <= w < N 183 A[w * N + i] = A[i * N + w]; // backup the previous value. store below diagonal (w>i) 184 A[i * N + w] = rotmat_cst[0] * A[i * N + w] - rotmat_cst[1] * A[j * N + w]; // A[i][w], A[j][w] from previous iteration 185 } 186 187 // now that we're done modifying row i, we can update max_idx_row[i] 188 max_idx_row[i] = MaxEntryRow(A, N, i); 189 190 // compute A[w][j] and A[j][w] for all w!=j,considering above-diagonal elements 191 for (CeedInt w = 0; w < i; w++) { // 0 <= w < i < j < N 192 A[w * N + j] = rotmat_cst[1] * A[i * N + w] + rotmat_cst[0] * A[w * N + j]; // A[i][w], A[w][j] from previous iteration 193 if (j == max_idx_row[w]) max_idx_row[w] = MaxEntryRow(A, N, w); 194 else if (fabs(A[w * N + j]) > fabs(A[w * N + max_idx_row[w]])) max_idx_row[w] = j; 195 } 196 for (CeedInt w = i + 1; w < j; w++) { // 0 <= i+1 <= w < j < N 197 A[w * N + j] = rotmat_cst[1] * A[w * N + i] + rotmat_cst[0] * A[w * N + j]; // A[w][i], A[w][j] from previous iteration 198 if (j == max_idx_row[w]) max_idx_row[w] = MaxEntryRow(A, N, w); 199 else if (fabs(A[w * N + j]) > fabs(A[w * N + max_idx_row[w]])) max_idx_row[w] = j; 200 } 201 for (CeedInt w = j + 1; w < N; w++) { // 0 <= i < j < w < N 202 A[j * N + w] = rotmat_cst[1] * A[w * N + i] + rotmat_cst[0] * A[j * N + w]; // A[w][i], A[j][w] from previous iteration 203 } 204 // now that we're done modifying row j, we can update max_idx_row[j] 205 max_idx_row[j] = MaxEntryRow(A, N, j); 206 } 207 208 ///@brief Multiply matrix A on the LEFT side by a transposed rotation matrix R^T 209 /// This matrix performs a rotation in the i,j plane by angle θ (where 210 /// the arguments "s" and "c" refer to cos(θ) and sin(θ), respectively). 211 /// @verbatim 212 /// A'_uv = Σ_w R_wu * A_wv 213 /// @endverbatim 214 /// 215 /// @param[in] *A matrix 216 /// @param[in] i row index 217 /// @param[in] j column index 218 CEED_QFUNCTION_HELPER void ApplyRotLeft(CeedScalar *A, CeedInt N, CeedInt i, CeedInt j, CeedScalar *rotmat_cst) { 219 // Recall that c = cos(θ) and s = sin(θ) 220 for (CeedInt v = 0; v < N; v++) { 221 CeedScalar Aiv = A[i * N + v]; 222 A[i * N + v] = rotmat_cst[0] * A[i * N + v] - rotmat_cst[1] * A[j * N + v]; 223 A[j * N + v] = rotmat_cst[1] * Aiv + rotmat_cst[0] * A[j * N + v]; 224 } 225 } 226 227 /// @brief Sort the rows in evec according to the numbers in v (also sorted) 228 /// 229 /// @param[inout] *eval vector containing the keys used for sorting 230 /// @param[inout] *evec matrix whose rows will be sorted according to v 231 /// @param[in] n size of the vector and matrix 232 /// @param[in] s sort decreasing order? 233 CEED_QFUNCTION_HELPER void SortRows(CeedScalar *eval, CeedScalar *evec, CeedInt N, SortCriteria sort_criteria) { 234 if (sort_criteria == SORT_NONE) return; 235 236 for (CeedInt i = 0; i < N - 1; i++) { 237 CeedInt i_max = i; 238 for (CeedInt j = i + 1; j < N; j++) { 239 // find the "maximum" element in the array starting at position i+1 240 switch (sort_criteria) { 241 case SORT_DECREASING_EVALS: 242 if (eval[j] > eval[i_max]) i_max = j; 243 break; 244 case SORT_INCREASING_EVALS: 245 if (eval[j] < eval[i_max]) i_max = j; 246 break; 247 case SORT_DECREASING_ABS_EVALS: 248 if (fabs(eval[j]) > fabs(eval[i_max])) i_max = j; 249 break; 250 case SORT_INCREASING_ABS_EVALS: 251 if (fabs(eval[j]) < fabs(eval[i_max])) i_max = j; 252 break; 253 default: 254 break; 255 } 256 } 257 SwapScalar(&eval[i], &eval[i_max]); 258 for (CeedInt k = 0; k < N; k++) SwapScalar(&evec[i * N + k], &evec[i_max * N + k]); 259 } 260 } 261 262 /// @brief Calculate all the eigenvalues and eigevectors of a symmetric matrix 263 /// using the Jacobi eigenvalue algorithm: 264 /// https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm 265 /// @returns The number of Jacobi iterations attempted, which should be > 0. 266 /// If the return value is not strictly > 0 then convergence failed. 267 /// @note To reduce the computation time further, set calc_evecs=false. 268 /// Additionally, note that the output evecs should be normalized. It 269 /// simply takes the Identity matrix and performs (isometric) rotations 270 /// on it, so divergence from normalized is due to finite-precision 271 /// arithmetic of the rotations. 272 // 273 // @param[in] A the matrix you wish to diagonalize (size NxN) 274 // @param[in] N size of the matrix 275 // @param[out] eval store the eigenvalues here (size N) 276 // @param[out] evec store the eigenvectors here (in rows, size NxN) 277 // @param[out] max_idx_row work vector of size N 278 // @param[in] sort_criteria sort results? 279 // @param[in] calc_evecs calculate the eigenvectors? 280 // @param[in] max_num_sweeps maximum number of iterations = max_num_sweeps * number of off-diagonals (N*(N-1)/2) 281 CEED_QFUNCTION_HELPER CeedInt Diagonalize(CeedScalar *A, CeedInt N, CeedScalar *eval, CeedScalar *evec, CeedInt *max_idx_row, 282 SortCriteria sort_criteria, bool calc_evec, const CeedInt max_num_sweeps) { 283 CeedScalar rotmat_cst[3] = {0.}; // cos(θ), sin(θ), and tan(θ), 284 285 if (calc_evec) 286 for (CeedInt i = 0; i < N; i++) 287 for (CeedInt j = 0; j < N; j++) evec[i * N + j] = (i == j) ? 1.0 : 0.0; // Set evec equal to the identity matrix 288 289 for (CeedInt i = 0; i < N - 1; i++) max_idx_row[i] = MaxEntryRow(A, N, i); 290 291 // -- Iteration -- 292 CeedInt n_iters; 293 CeedInt max_num_iters = max_num_sweeps * N * (N - 1) / 2; 294 for (n_iters = 1; n_iters <= max_num_iters; n_iters++) { 295 CeedInt i, j; 296 MaxEntry(A, N, max_idx_row, &i, &j); 297 298 // If A[i][j] is small compared to A[i][i] and A[j][j], set it to 0. 299 if ((A[i * N + i] + A[i * N + j] == A[i * N + i]) && (A[j * N + j] + A[i * N + j] == A[j * N + j])) { 300 A[i * N + j] = 0.0; 301 max_idx_row[i] = MaxEntryRow(A, N, i); 302 } 303 304 if (A[i * N + j] == 0.0) break; 305 306 CalcRot(A, N, i, j, rotmat_cst); // Calculate the parameters of the rotation matrix. 307 ApplyRot(A, N, i, j, max_idx_row, rotmat_cst); // Apply this rotation to the A matrix. 308 if (calc_evec) ApplyRotLeft(evec, N, i, j, rotmat_cst); 309 } 310 311 for (CeedInt i = 0; i < N; i++) eval[i] = A[i * N + i]; 312 313 // Optional: Sort results by eigenvalue. 314 SortRows(eval, evec, N, sort_criteria); 315 316 if ((n_iters > max_num_iters) && (N > 1)) // If we exceeded max_num_iters, 317 return 0; // indicate an error occured. 318 319 return n_iters; 320 } 321 322 // @brief Interface to Diagonalize for 3x3 systems 323 CEED_QFUNCTION_HELPER CeedInt Diagonalize3(CeedScalar A[3][3], CeedScalar eval[3], CeedScalar evec[3][3], CeedInt max_idx_row[3], 324 SortCriteria sort_criteria, bool calc_evec, const CeedInt max_num_sweeps) { 325 return Diagonalize((CeedScalar *)A, 3, (CeedScalar *)eval, (CeedScalar *)evec, (CeedInt *)max_idx_row, sort_criteria, calc_evec, max_num_sweeps); 326 } 327