11 #ifndef EIGEN_EIGENSOLVER_H
12 #define EIGEN_EIGENSOLVER_H
14 #include "./RealSchur.h"
72 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
73 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
74 Options = MatrixType::Options,
75 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
76 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
80 typedef typename MatrixType::Scalar
Scalar;
82 typedef typename MatrixType::Index Index;
113 EigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false), m_realSchur(), m_matT(), m_tmp() {}
122 : m_eivec(size, size),
124 m_isInitialized(false),
125 m_eigenvectorsOk(false),
147 : m_eivec(matrix.rows(), matrix.cols()),
148 m_eivalues(matrix.cols()),
149 m_isInitialized(false),
150 m_eigenvectorsOk(false),
151 m_realSchur(matrix.cols()),
152 m_matT(matrix.rows(), matrix.cols()),
155 compute(matrix, computeEigenvectors);
200 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
201 eigen_assert(m_eigenvectorsOk &&
"The eigenvectors have not been computed together with the eigenvalues.");
245 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
280 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
281 return m_realSchur.
info();
285 void doComputeEigenvectors();
290 bool m_isInitialized;
291 bool m_eigenvectorsOk;
292 RealSchur<MatrixType> m_realSchur;
295 typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
296 ColumnVectorType m_tmp;
299 template<
typename MatrixType>
302 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
303 Index n = m_eivalues.rows();
305 for (Index i=0; i<n; ++i)
319 template<
typename MatrixType>
322 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
323 eigen_assert(m_eigenvectorsOk &&
"The eigenvectors have not been computed together with the eigenvalues.");
324 Index n = m_eivec.cols();
326 for (Index j=0; j<n; ++j)
331 matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
332 matV.col(j).normalize();
337 for (Index i=0; i<n; ++i)
339 matV.coeffRef(i,j) =
ComplexScalar(m_eivec.coeff(i,j), m_eivec.coeff(i,j+1));
340 matV.coeffRef(i,j+1) =
ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
342 matV.col(j).normalize();
343 matV.col(j+1).normalize();
350 template<
typename MatrixType>
353 assert(matrix.cols() == matrix.rows());
356 m_realSchur.
compute(matrix, computeEigenvectors);
357 if (m_realSchur.info() ==
Success)
359 m_matT = m_realSchur.matrixT();
360 if (computeEigenvectors)
361 m_eivec = m_realSchur.matrixU();
364 m_eivalues.resize(matrix.cols());
366 while (i < matrix.cols())
368 if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) ==
Scalar(0))
370 m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
375 Scalar p =
Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1));
376 Scalar z = internal::sqrt(
internal::abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
377 m_eivalues.coeffRef(i) =
ComplexScalar(m_matT.coeff(i+1, i+1) + p, z);
378 m_eivalues.coeffRef(i+1) =
ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z);
384 if (computeEigenvectors)
385 doComputeEigenvectors();
388 m_isInitialized =
true;
389 m_eigenvectorsOk = computeEigenvectors;
395 template<
typename Scalar>
396 std::complex<Scalar> cdiv(Scalar xr, Scalar xi, Scalar yr, Scalar yi)
403 return std::complex<Scalar>((xr + r*xi)/d, (xi - r*xr)/d);
409 return std::complex<Scalar>((r*xr + xi)/d, (r*xi - xr)/d);
414 template<
typename MatrixType>
415 void EigenSolver<MatrixType>::doComputeEigenvectors()
417 const Index size = m_eivec.cols();
418 const Scalar eps = NumTraits<Scalar>::epsilon();
422 for (Index j = 0; j < size; ++j)
424 norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
433 for (Index n = size-1; n >= 0; n--)
435 Scalar p = m_eivalues.coeff(n).real();
436 Scalar q = m_eivalues.coeff(n).imag();
441 Scalar lastr(0), lastw(0);
444 m_matT.coeffRef(n,n) = 1.0;
445 for (Index i = n-1; i >= 0; i--)
447 Scalar w = m_matT.coeff(i,i) - p;
448 Scalar r = m_matT.row(i).segment(l,n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
450 if (m_eivalues.coeff(i).imag() < 0.0)
458 if (m_eivalues.coeff(i).imag() == 0.0)
461 m_matT.coeffRef(i,n) = -r / w;
463 m_matT.coeffRef(i,n) = -r / (eps * norm);
467 Scalar x = m_matT.coeff(i,i+1);
468 Scalar y = m_matT.coeff(i+1,i);
469 Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
470 Scalar t = (x * lastr - lastw * r) / denom;
471 m_matT.coeffRef(i,n) = t;
473 m_matT.coeffRef(i+1,n) = (-r - w * t) / x;
475 m_matT.coeffRef(i+1,n) = (-lastr - y * t) / lastw;
480 if ((eps * t) * t > Scalar(1))
481 m_matT.col(n).tail(size-i) /= t;
485 else if (q < Scalar(0) && n > 0)
487 Scalar lastra(0), lastsa(0), lastw(0);
493 m_matT.coeffRef(n-1,n-1) = q / m_matT.coeff(n,n-1);
494 m_matT.coeffRef(n-1,n) = -(m_matT.coeff(n,n) - p) / m_matT.coeff(n,n-1);
498 std::complex<Scalar> cc = cdiv<Scalar>(0.0,-m_matT.coeff(n-1,n),m_matT.coeff(n-1,n-1)-p,q);
502 m_matT.coeffRef(n,n-1) = 0.0;
503 m_matT.coeffRef(n,n) = 1.0;
504 for (Index i = n-2; i >= 0; i--)
506 Scalar ra = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n-1).segment(l, n-l+1));
507 Scalar sa = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
508 Scalar w = m_matT.coeff(i,i) - p;
510 if (m_eivalues.coeff(i).imag() < 0.0)
519 if (m_eivalues.coeff(i).imag() == RealScalar(0))
521 std::complex<Scalar> cc = cdiv(-ra,-sa,w,q);
528 Scalar x = m_matT.coeff(i,i+1);
529 Scalar y = m_matT.coeff(i+1,i);
530 Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
531 Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
532 if ((vr == 0.0) && (vi == 0.0))
535 std::complex<Scalar> cc = cdiv(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra,vr,vi);
540 m_matT.coeffRef(i+1,n-1) = (-ra - w * m_matT.coeff(i,n-1) + q * m_matT.coeff(i,n)) / x;
541 m_matT.coeffRef(i+1,n) = (-sa - w * m_matT.coeff(i,n) - q * m_matT.coeff(i,n-1)) / x;
545 cc = cdiv(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n),lastw,q);
554 if ((eps * t) * t > Scalar(1))
555 m_matT.block(i, n-1, size-i, 2) /= t;
565 eigen_assert(0 &&
"Internal bug in EigenSolver");
570 for (Index j = size-1; j >= 0; j--)
572 m_tmp.noalias() = m_eivec.leftCols(j+1) * m_matT.col(j).segment(0, j+1);
573 m_eivec.col(j) = m_tmp;
579 #endif // EIGEN_EIGENSOLVER_H