Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 1 | // This file is part of Eigen, a lightweight C++ template library |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 2 | // for linear algebra. |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 3 | // |
| 4 | // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr> |
| 5 | // |
| 6 | // This Source Code Form is subject to the terms of the Mozilla |
| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed |
| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| 9 | |
| 10 | #ifndef EIGEN2_SVD_H |
| 11 | #define EIGEN2_SVD_H |
| 12 | |
| 13 | namespace Eigen { |
| 14 | |
| 15 | /** \ingroup SVD_Module |
| 16 | * \nonstableyet |
| 17 | * |
| 18 | * \class SVD |
| 19 | * |
| 20 | * \brief Standard SVD decomposition of a matrix and associated features |
| 21 | * |
| 22 | * \param MatrixType the type of the matrix of which we are computing the SVD decomposition |
| 23 | * |
| 24 | * This class performs a standard SVD decomposition of a real matrix A of size \c M x \c N |
| 25 | * with \c M \>= \c N. |
| 26 | * |
| 27 | * |
| 28 | * \sa MatrixBase::SVD() |
| 29 | */ |
| 30 | template<typename MatrixType> class SVD |
| 31 | { |
| 32 | private: |
| 33 | typedef typename MatrixType::Scalar Scalar; |
| 34 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; |
| 35 | |
| 36 | enum { |
| 37 | PacketSize = internal::packet_traits<Scalar>::size, |
| 38 | AlignmentMask = int(PacketSize)-1, |
| 39 | MinSize = EIGEN_SIZE_MIN_PREFER_DYNAMIC(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime) |
| 40 | }; |
| 41 | |
| 42 | typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVector; |
| 43 | typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> RowVector; |
| 44 | |
| 45 | typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MinSize> MatrixUType; |
| 46 | typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixVType; |
| 47 | typedef Matrix<Scalar, MinSize, 1> SingularValuesType; |
| 48 | |
| 49 | public: |
| 50 | |
| 51 | SVD() {} // a user who relied on compiler-generated default compiler reported problems with MSVC in 2.0.7 |
| 52 | |
| 53 | SVD(const MatrixType& matrix) |
| 54 | : m_matU(matrix.rows(), (std::min)(matrix.rows(), matrix.cols())), |
| 55 | m_matV(matrix.cols(),matrix.cols()), |
| 56 | m_sigma((std::min)(matrix.rows(),matrix.cols())) |
| 57 | { |
| 58 | compute(matrix); |
| 59 | } |
| 60 | |
| 61 | template<typename OtherDerived, typename ResultType> |
| 62 | bool solve(const MatrixBase<OtherDerived> &b, ResultType* result) const; |
| 63 | |
| 64 | const MatrixUType& matrixU() const { return m_matU; } |
| 65 | const SingularValuesType& singularValues() const { return m_sigma; } |
| 66 | const MatrixVType& matrixV() const { return m_matV; } |
| 67 | |
| 68 | void compute(const MatrixType& matrix); |
| 69 | SVD& sort(); |
| 70 | |
| 71 | template<typename UnitaryType, typename PositiveType> |
| 72 | void computeUnitaryPositive(UnitaryType *unitary, PositiveType *positive) const; |
| 73 | template<typename PositiveType, typename UnitaryType> |
| 74 | void computePositiveUnitary(PositiveType *positive, UnitaryType *unitary) const; |
| 75 | template<typename RotationType, typename ScalingType> |
| 76 | void computeRotationScaling(RotationType *unitary, ScalingType *positive) const; |
| 77 | template<typename ScalingType, typename RotationType> |
| 78 | void computeScalingRotation(ScalingType *positive, RotationType *unitary) const; |
| 79 | |
| 80 | protected: |
| 81 | /** \internal */ |
| 82 | MatrixUType m_matU; |
| 83 | /** \internal */ |
| 84 | MatrixVType m_matV; |
| 85 | /** \internal */ |
| 86 | SingularValuesType m_sigma; |
| 87 | }; |
| 88 | |
| 89 | /** Computes / recomputes the SVD decomposition A = U S V^* of \a matrix |
| 90 | * |
| 91 | * \note this code has been adapted from JAMA (public domain) |
| 92 | */ |
| 93 | template<typename MatrixType> |
| 94 | void SVD<MatrixType>::compute(const MatrixType& matrix) |
| 95 | { |
| 96 | const int m = matrix.rows(); |
| 97 | const int n = matrix.cols(); |
| 98 | const int nu = (std::min)(m,n); |
| 99 | ei_assert(m>=n && "In Eigen 2.0, SVD only works for MxN matrices with M>=N. Sorry!"); |
| 100 | ei_assert(m>1 && "In Eigen 2.0, SVD doesn't work on 1x1 matrices"); |
| 101 | |
| 102 | m_matU.resize(m, nu); |
| 103 | m_matU.setZero(); |
| 104 | m_sigma.resize((std::min)(m,n)); |
| 105 | m_matV.resize(n,n); |
| 106 | |
| 107 | RowVector e(n); |
| 108 | ColVector work(m); |
| 109 | MatrixType matA(matrix); |
| 110 | const bool wantu = true; |
| 111 | const bool wantv = true; |
| 112 | int i=0, j=0, k=0; |
| 113 | |
| 114 | // Reduce A to bidiagonal form, storing the diagonal elements |
| 115 | // in s and the super-diagonal elements in e. |
| 116 | int nct = (std::min)(m-1,n); |
| 117 | int nrt = (std::max)(0,(std::min)(n-2,m)); |
| 118 | for (k = 0; k < (std::max)(nct,nrt); ++k) |
| 119 | { |
| 120 | if (k < nct) |
| 121 | { |
| 122 | // Compute the transformation for the k-th column and |
| 123 | // place the k-th diagonal in m_sigma[k]. |
| 124 | m_sigma[k] = matA.col(k).end(m-k).norm(); |
| 125 | if (m_sigma[k] != 0.0) // FIXME |
| 126 | { |
| 127 | if (matA(k,k) < 0.0) |
| 128 | m_sigma[k] = -m_sigma[k]; |
| 129 | matA.col(k).end(m-k) /= m_sigma[k]; |
| 130 | matA(k,k) += 1.0; |
| 131 | } |
| 132 | m_sigma[k] = -m_sigma[k]; |
| 133 | } |
| 134 | |
| 135 | for (j = k+1; j < n; ++j) |
| 136 | { |
| 137 | if ((k < nct) && (m_sigma[k] != 0.0)) |
| 138 | { |
| 139 | // Apply the transformation. |
| 140 | Scalar t = matA.col(k).end(m-k).eigen2_dot(matA.col(j).end(m-k)); // FIXME dot product or cwise prod + .sum() ?? |
| 141 | t = -t/matA(k,k); |
| 142 | matA.col(j).end(m-k) += t * matA.col(k).end(m-k); |
| 143 | } |
| 144 | |
| 145 | // Place the k-th row of A into e for the |
| 146 | // subsequent calculation of the row transformation. |
| 147 | e[j] = matA(k,j); |
| 148 | } |
| 149 | |
| 150 | // Place the transformation in U for subsequent back multiplication. |
| 151 | if (wantu & (k < nct)) |
| 152 | m_matU.col(k).end(m-k) = matA.col(k).end(m-k); |
| 153 | |
| 154 | if (k < nrt) |
| 155 | { |
| 156 | // Compute the k-th row transformation and place the |
| 157 | // k-th super-diagonal in e[k]. |
| 158 | e[k] = e.end(n-k-1).norm(); |
| 159 | if (e[k] != 0.0) |
| 160 | { |
| 161 | if (e[k+1] < 0.0) |
| 162 | e[k] = -e[k]; |
| 163 | e.end(n-k-1) /= e[k]; |
| 164 | e[k+1] += 1.0; |
| 165 | } |
| 166 | e[k] = -e[k]; |
| 167 | if ((k+1 < m) & (e[k] != 0.0)) |
| 168 | { |
| 169 | // Apply the transformation. |
| 170 | work.end(m-k-1) = matA.corner(BottomRight,m-k-1,n-k-1) * e.end(n-k-1); |
| 171 | for (j = k+1; j < n; ++j) |
| 172 | matA.col(j).end(m-k-1) += (-e[j]/e[k+1]) * work.end(m-k-1); |
| 173 | } |
| 174 | |
| 175 | // Place the transformation in V for subsequent back multiplication. |
| 176 | if (wantv) |
| 177 | m_matV.col(k).end(n-k-1) = e.end(n-k-1); |
| 178 | } |
| 179 | } |
| 180 | |
| 181 | |
| 182 | // Set up the final bidiagonal matrix or order p. |
| 183 | int p = (std::min)(n,m+1); |
| 184 | if (nct < n) |
| 185 | m_sigma[nct] = matA(nct,nct); |
| 186 | if (m < p) |
| 187 | m_sigma[p-1] = 0.0; |
| 188 | if (nrt+1 < p) |
| 189 | e[nrt] = matA(nrt,p-1); |
| 190 | e[p-1] = 0.0; |
| 191 | |
| 192 | // If required, generate U. |
| 193 | if (wantu) |
| 194 | { |
| 195 | for (j = nct; j < nu; ++j) |
| 196 | { |
| 197 | m_matU.col(j).setZero(); |
| 198 | m_matU(j,j) = 1.0; |
| 199 | } |
| 200 | for (k = nct-1; k >= 0; k--) |
| 201 | { |
| 202 | if (m_sigma[k] != 0.0) |
| 203 | { |
| 204 | for (j = k+1; j < nu; ++j) |
| 205 | { |
| 206 | Scalar t = m_matU.col(k).end(m-k).eigen2_dot(m_matU.col(j).end(m-k)); // FIXME is it really a dot product we want ? |
| 207 | t = -t/m_matU(k,k); |
| 208 | m_matU.col(j).end(m-k) += t * m_matU.col(k).end(m-k); |
| 209 | } |
| 210 | m_matU.col(k).end(m-k) = - m_matU.col(k).end(m-k); |
| 211 | m_matU(k,k) = Scalar(1) + m_matU(k,k); |
| 212 | if (k-1>0) |
| 213 | m_matU.col(k).start(k-1).setZero(); |
| 214 | } |
| 215 | else |
| 216 | { |
| 217 | m_matU.col(k).setZero(); |
| 218 | m_matU(k,k) = 1.0; |
| 219 | } |
| 220 | } |
| 221 | } |
| 222 | |
| 223 | // If required, generate V. |
| 224 | if (wantv) |
| 225 | { |
| 226 | for (k = n-1; k >= 0; k--) |
| 227 | { |
| 228 | if ((k < nrt) & (e[k] != 0.0)) |
| 229 | { |
| 230 | for (j = k+1; j < nu; ++j) |
| 231 | { |
| 232 | Scalar t = m_matV.col(k).end(n-k-1).eigen2_dot(m_matV.col(j).end(n-k-1)); // FIXME is it really a dot product we want ? |
| 233 | t = -t/m_matV(k+1,k); |
| 234 | m_matV.col(j).end(n-k-1) += t * m_matV.col(k).end(n-k-1); |
| 235 | } |
| 236 | } |
| 237 | m_matV.col(k).setZero(); |
| 238 | m_matV(k,k) = 1.0; |
| 239 | } |
| 240 | } |
| 241 | |
| 242 | // Main iteration loop for the singular values. |
| 243 | int pp = p-1; |
| 244 | int iter = 0; |
| 245 | Scalar eps = ei_pow(Scalar(2),ei_is_same_type<Scalar,float>::ret ? Scalar(-23) : Scalar(-52)); |
| 246 | while (p > 0) |
| 247 | { |
| 248 | int k=0; |
| 249 | int kase=0; |
| 250 | |
| 251 | // Here is where a test for too many iterations would go. |
| 252 | |
| 253 | // This section of the program inspects for |
| 254 | // negligible elements in the s and e arrays. On |
| 255 | // completion the variables kase and k are set as follows. |
| 256 | |
| 257 | // kase = 1 if s(p) and e[k-1] are negligible and k<p |
| 258 | // kase = 2 if s(k) is negligible and k<p |
| 259 | // kase = 3 if e[k-1] is negligible, k<p, and |
| 260 | // s(k), ..., s(p) are not negligible (qr step). |
| 261 | // kase = 4 if e(p-1) is negligible (convergence). |
| 262 | |
| 263 | for (k = p-2; k >= -1; --k) |
| 264 | { |
| 265 | if (k == -1) |
| 266 | break; |
| 267 | if (ei_abs(e[k]) <= eps*(ei_abs(m_sigma[k]) + ei_abs(m_sigma[k+1]))) |
| 268 | { |
| 269 | e[k] = 0.0; |
| 270 | break; |
| 271 | } |
| 272 | } |
| 273 | if (k == p-2) |
| 274 | { |
| 275 | kase = 4; |
| 276 | } |
| 277 | else |
| 278 | { |
| 279 | int ks; |
| 280 | for (ks = p-1; ks >= k; --ks) |
| 281 | { |
| 282 | if (ks == k) |
| 283 | break; |
| 284 | Scalar t = (ks != p ? ei_abs(e[ks]) : Scalar(0)) + (ks != k+1 ? ei_abs(e[ks-1]) : Scalar(0)); |
| 285 | if (ei_abs(m_sigma[ks]) <= eps*t) |
| 286 | { |
| 287 | m_sigma[ks] = 0.0; |
| 288 | break; |
| 289 | } |
| 290 | } |
| 291 | if (ks == k) |
| 292 | { |
| 293 | kase = 3; |
| 294 | } |
| 295 | else if (ks == p-1) |
| 296 | { |
| 297 | kase = 1; |
| 298 | } |
| 299 | else |
| 300 | { |
| 301 | kase = 2; |
| 302 | k = ks; |
| 303 | } |
| 304 | } |
| 305 | ++k; |
| 306 | |
| 307 | // Perform the task indicated by kase. |
| 308 | switch (kase) |
| 309 | { |
| 310 | |
| 311 | // Deflate negligible s(p). |
| 312 | case 1: |
| 313 | { |
| 314 | Scalar f(e[p-2]); |
| 315 | e[p-2] = 0.0; |
| 316 | for (j = p-2; j >= k; --j) |
| 317 | { |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 318 | Scalar t(numext::hypot(m_sigma[j],f)); |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 319 | Scalar cs(m_sigma[j]/t); |
| 320 | Scalar sn(f/t); |
| 321 | m_sigma[j] = t; |
| 322 | if (j != k) |
| 323 | { |
| 324 | f = -sn*e[j-1]; |
| 325 | e[j-1] = cs*e[j-1]; |
| 326 | } |
| 327 | if (wantv) |
| 328 | { |
| 329 | for (i = 0; i < n; ++i) |
| 330 | { |
| 331 | t = cs*m_matV(i,j) + sn*m_matV(i,p-1); |
| 332 | m_matV(i,p-1) = -sn*m_matV(i,j) + cs*m_matV(i,p-1); |
| 333 | m_matV(i,j) = t; |
| 334 | } |
| 335 | } |
| 336 | } |
| 337 | } |
| 338 | break; |
| 339 | |
| 340 | // Split at negligible s(k). |
| 341 | case 2: |
| 342 | { |
| 343 | Scalar f(e[k-1]); |
| 344 | e[k-1] = 0.0; |
| 345 | for (j = k; j < p; ++j) |
| 346 | { |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 347 | Scalar t(numext::hypot(m_sigma[j],f)); |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 348 | Scalar cs( m_sigma[j]/t); |
| 349 | Scalar sn(f/t); |
| 350 | m_sigma[j] = t; |
| 351 | f = -sn*e[j]; |
| 352 | e[j] = cs*e[j]; |
| 353 | if (wantu) |
| 354 | { |
| 355 | for (i = 0; i < m; ++i) |
| 356 | { |
| 357 | t = cs*m_matU(i,j) + sn*m_matU(i,k-1); |
| 358 | m_matU(i,k-1) = -sn*m_matU(i,j) + cs*m_matU(i,k-1); |
| 359 | m_matU(i,j) = t; |
| 360 | } |
| 361 | } |
| 362 | } |
| 363 | } |
| 364 | break; |
| 365 | |
| 366 | // Perform one qr step. |
| 367 | case 3: |
| 368 | { |
| 369 | // Calculate the shift. |
| 370 | Scalar scale = (std::max)((std::max)((std::max)((std::max)( |
| 371 | ei_abs(m_sigma[p-1]),ei_abs(m_sigma[p-2])),ei_abs(e[p-2])), |
| 372 | ei_abs(m_sigma[k])),ei_abs(e[k])); |
| 373 | Scalar sp = m_sigma[p-1]/scale; |
| 374 | Scalar spm1 = m_sigma[p-2]/scale; |
| 375 | Scalar epm1 = e[p-2]/scale; |
| 376 | Scalar sk = m_sigma[k]/scale; |
| 377 | Scalar ek = e[k]/scale; |
| 378 | Scalar b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/Scalar(2); |
| 379 | Scalar c = (sp*epm1)*(sp*epm1); |
| 380 | Scalar shift(0); |
| 381 | if ((b != 0.0) || (c != 0.0)) |
| 382 | { |
| 383 | shift = ei_sqrt(b*b + c); |
| 384 | if (b < 0.0) |
| 385 | shift = -shift; |
| 386 | shift = c/(b + shift); |
| 387 | } |
| 388 | Scalar f = (sk + sp)*(sk - sp) + shift; |
| 389 | Scalar g = sk*ek; |
| 390 | |
| 391 | // Chase zeros. |
| 392 | |
| 393 | for (j = k; j < p-1; ++j) |
| 394 | { |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 395 | Scalar t = numext::hypot(f,g); |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 396 | Scalar cs = f/t; |
| 397 | Scalar sn = g/t; |
| 398 | if (j != k) |
| 399 | e[j-1] = t; |
| 400 | f = cs*m_sigma[j] + sn*e[j]; |
| 401 | e[j] = cs*e[j] - sn*m_sigma[j]; |
| 402 | g = sn*m_sigma[j+1]; |
| 403 | m_sigma[j+1] = cs*m_sigma[j+1]; |
| 404 | if (wantv) |
| 405 | { |
| 406 | for (i = 0; i < n; ++i) |
| 407 | { |
| 408 | t = cs*m_matV(i,j) + sn*m_matV(i,j+1); |
| 409 | m_matV(i,j+1) = -sn*m_matV(i,j) + cs*m_matV(i,j+1); |
| 410 | m_matV(i,j) = t; |
| 411 | } |
| 412 | } |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 413 | t = numext::hypot(f,g); |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 414 | cs = f/t; |
| 415 | sn = g/t; |
| 416 | m_sigma[j] = t; |
| 417 | f = cs*e[j] + sn*m_sigma[j+1]; |
| 418 | m_sigma[j+1] = -sn*e[j] + cs*m_sigma[j+1]; |
| 419 | g = sn*e[j+1]; |
| 420 | e[j+1] = cs*e[j+1]; |
| 421 | if (wantu && (j < m-1)) |
| 422 | { |
| 423 | for (i = 0; i < m; ++i) |
| 424 | { |
| 425 | t = cs*m_matU(i,j) + sn*m_matU(i,j+1); |
| 426 | m_matU(i,j+1) = -sn*m_matU(i,j) + cs*m_matU(i,j+1); |
| 427 | m_matU(i,j) = t; |
| 428 | } |
| 429 | } |
| 430 | } |
| 431 | e[p-2] = f; |
| 432 | iter = iter + 1; |
| 433 | } |
| 434 | break; |
| 435 | |
| 436 | // Convergence. |
| 437 | case 4: |
| 438 | { |
| 439 | // Make the singular values positive. |
| 440 | if (m_sigma[k] <= 0.0) |
| 441 | { |
| 442 | m_sigma[k] = m_sigma[k] < Scalar(0) ? -m_sigma[k] : Scalar(0); |
| 443 | if (wantv) |
| 444 | m_matV.col(k).start(pp+1) = -m_matV.col(k).start(pp+1); |
| 445 | } |
| 446 | |
| 447 | // Order the singular values. |
| 448 | while (k < pp) |
| 449 | { |
| 450 | if (m_sigma[k] >= m_sigma[k+1]) |
| 451 | break; |
| 452 | Scalar t = m_sigma[k]; |
| 453 | m_sigma[k] = m_sigma[k+1]; |
| 454 | m_sigma[k+1] = t; |
| 455 | if (wantv && (k < n-1)) |
| 456 | m_matV.col(k).swap(m_matV.col(k+1)); |
| 457 | if (wantu && (k < m-1)) |
| 458 | m_matU.col(k).swap(m_matU.col(k+1)); |
| 459 | ++k; |
| 460 | } |
| 461 | iter = 0; |
| 462 | p--; |
| 463 | } |
| 464 | break; |
| 465 | } // end big switch |
| 466 | } // end iterations |
| 467 | } |
| 468 | |
| 469 | template<typename MatrixType> |
| 470 | SVD<MatrixType>& SVD<MatrixType>::sort() |
| 471 | { |
| 472 | int mu = m_matU.rows(); |
| 473 | int mv = m_matV.rows(); |
| 474 | int n = m_matU.cols(); |
| 475 | |
| 476 | for (int i=0; i<n; ++i) |
| 477 | { |
| 478 | int k = i; |
| 479 | Scalar p = m_sigma.coeff(i); |
| 480 | |
| 481 | for (int j=i+1; j<n; ++j) |
| 482 | { |
| 483 | if (m_sigma.coeff(j) > p) |
| 484 | { |
| 485 | k = j; |
| 486 | p = m_sigma.coeff(j); |
| 487 | } |
| 488 | } |
| 489 | if (k != i) |
| 490 | { |
| 491 | m_sigma.coeffRef(k) = m_sigma.coeff(i); // i.e. |
| 492 | m_sigma.coeffRef(i) = p; // swaps the i-th and the k-th elements |
| 493 | |
| 494 | int j = mu; |
| 495 | for(int s=0; j!=0; ++s, --j) |
| 496 | std::swap(m_matU.coeffRef(s,i), m_matU.coeffRef(s,k)); |
| 497 | |
| 498 | j = mv; |
| 499 | for (int s=0; j!=0; ++s, --j) |
| 500 | std::swap(m_matV.coeffRef(s,i), m_matV.coeffRef(s,k)); |
| 501 | } |
| 502 | } |
| 503 | return *this; |
| 504 | } |
| 505 | |
| 506 | /** \returns the solution of \f$ A x = b \f$ using the current SVD decomposition of A. |
| 507 | * The parts of the solution corresponding to zero singular values are ignored. |
| 508 | * |
| 509 | * \sa MatrixBase::svd(), LU::solve(), LLT::solve() |
| 510 | */ |
| 511 | template<typename MatrixType> |
| 512 | template<typename OtherDerived, typename ResultType> |
| 513 | bool SVD<MatrixType>::solve(const MatrixBase<OtherDerived> &b, ResultType* result) const |
| 514 | { |
Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 515 | ei_assert(b.rows() == m_matU.rows()); |
Narayan Kamath | c981c48 | 2012-11-02 10:59:05 +0000 | [diff] [blame] | 516 | |
| 517 | Scalar maxVal = m_sigma.cwise().abs().maxCoeff(); |
| 518 | for (int j=0; j<b.cols(); ++j) |
| 519 | { |
| 520 | Matrix<Scalar,MatrixUType::RowsAtCompileTime,1> aux = m_matU.transpose() * b.col(j); |
| 521 | |
| 522 | for (int i = 0; i <m_matU.cols(); ++i) |
| 523 | { |
| 524 | Scalar si = m_sigma.coeff(i); |
| 525 | if (ei_isMuchSmallerThan(ei_abs(si),maxVal)) |
| 526 | aux.coeffRef(i) = 0; |
| 527 | else |
| 528 | aux.coeffRef(i) /= si; |
| 529 | } |
| 530 | |
| 531 | result->col(j) = m_matV * aux; |
| 532 | } |
| 533 | return true; |
| 534 | } |
| 535 | |
| 536 | /** Computes the polar decomposition of the matrix, as a product unitary x positive. |
| 537 | * |
| 538 | * If either pointer is zero, the corresponding computation is skipped. |
| 539 | * |
| 540 | * Only for square matrices. |
| 541 | * |
| 542 | * \sa computePositiveUnitary(), computeRotationScaling() |
| 543 | */ |
| 544 | template<typename MatrixType> |
| 545 | template<typename UnitaryType, typename PositiveType> |
| 546 | void SVD<MatrixType>::computeUnitaryPositive(UnitaryType *unitary, |
| 547 | PositiveType *positive) const |
| 548 | { |
| 549 | ei_assert(m_matU.cols() == m_matV.cols() && "Polar decomposition is only for square matrices"); |
| 550 | if(unitary) *unitary = m_matU * m_matV.adjoint(); |
| 551 | if(positive) *positive = m_matV * m_sigma.asDiagonal() * m_matV.adjoint(); |
| 552 | } |
| 553 | |
| 554 | /** Computes the polar decomposition of the matrix, as a product positive x unitary. |
| 555 | * |
| 556 | * If either pointer is zero, the corresponding computation is skipped. |
| 557 | * |
| 558 | * Only for square matrices. |
| 559 | * |
| 560 | * \sa computeUnitaryPositive(), computeRotationScaling() |
| 561 | */ |
| 562 | template<typename MatrixType> |
| 563 | template<typename UnitaryType, typename PositiveType> |
| 564 | void SVD<MatrixType>::computePositiveUnitary(UnitaryType *positive, |
| 565 | PositiveType *unitary) const |
| 566 | { |
| 567 | ei_assert(m_matU.rows() == m_matV.rows() && "Polar decomposition is only for square matrices"); |
| 568 | if(unitary) *unitary = m_matU * m_matV.adjoint(); |
| 569 | if(positive) *positive = m_matU * m_sigma.asDiagonal() * m_matU.adjoint(); |
| 570 | } |
| 571 | |
| 572 | /** decomposes the matrix as a product rotation x scaling, the scaling being |
| 573 | * not necessarily positive. |
| 574 | * |
| 575 | * If either pointer is zero, the corresponding computation is skipped. |
| 576 | * |
| 577 | * This method requires the Geometry module. |
| 578 | * |
| 579 | * \sa computeScalingRotation(), computeUnitaryPositive() |
| 580 | */ |
| 581 | template<typename MatrixType> |
| 582 | template<typename RotationType, typename ScalingType> |
| 583 | void SVD<MatrixType>::computeRotationScaling(RotationType *rotation, ScalingType *scaling) const |
| 584 | { |
| 585 | ei_assert(m_matU.rows() == m_matV.rows() && "Polar decomposition is only for square matrices"); |
| 586 | Scalar x = (m_matU * m_matV.adjoint()).determinant(); // so x has absolute value 1 |
| 587 | Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> sv(m_sigma); |
| 588 | sv.coeffRef(0) *= x; |
| 589 | if(scaling) scaling->lazyAssign(m_matV * sv.asDiagonal() * m_matV.adjoint()); |
| 590 | if(rotation) |
| 591 | { |
| 592 | MatrixType m(m_matU); |
| 593 | m.col(0) /= x; |
| 594 | rotation->lazyAssign(m * m_matV.adjoint()); |
| 595 | } |
| 596 | } |
| 597 | |
| 598 | /** decomposes the matrix as a product scaling x rotation, the scaling being |
| 599 | * not necessarily positive. |
| 600 | * |
| 601 | * If either pointer is zero, the corresponding computation is skipped. |
| 602 | * |
| 603 | * This method requires the Geometry module. |
| 604 | * |
| 605 | * \sa computeRotationScaling(), computeUnitaryPositive() |
| 606 | */ |
| 607 | template<typename MatrixType> |
| 608 | template<typename ScalingType, typename RotationType> |
| 609 | void SVD<MatrixType>::computeScalingRotation(ScalingType *scaling, RotationType *rotation) const |
| 610 | { |
| 611 | ei_assert(m_matU.rows() == m_matV.rows() && "Polar decomposition is only for square matrices"); |
| 612 | Scalar x = (m_matU * m_matV.adjoint()).determinant(); // so x has absolute value 1 |
| 613 | Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> sv(m_sigma); |
| 614 | sv.coeffRef(0) *= x; |
| 615 | if(scaling) scaling->lazyAssign(m_matU * sv.asDiagonal() * m_matU.adjoint()); |
| 616 | if(rotation) |
| 617 | { |
| 618 | MatrixType m(m_matU); |
| 619 | m.col(0) /= x; |
| 620 | rotation->lazyAssign(m * m_matV.adjoint()); |
| 621 | } |
| 622 | } |
| 623 | |
| 624 | |
| 625 | /** \svd_module |
| 626 | * \returns the SVD decomposition of \c *this |
| 627 | */ |
| 628 | template<typename Derived> |
| 629 | inline SVD<typename MatrixBase<Derived>::PlainObject> |
| 630 | MatrixBase<Derived>::svd() const |
| 631 | { |
| 632 | return SVD<PlainObject>(derived()); |
| 633 | } |
| 634 | |
| 635 | } // end namespace Eigen |
| 636 | |
| 637 | #endif // EIGEN2_SVD_H |