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;
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),
146 explicit EigenSolver(
const MatrixType& matrix,
bool computeEigenvectors =
true)
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.");
281 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
299 void doComputeEigenvectors();
303 static void check_template_parameters()
305 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
310 EigenvalueType m_eivalues;
311 bool m_isInitialized;
312 bool m_eigenvectorsOk;
314 RealSchur<MatrixType> m_realSchur;
317 typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
318 ColumnVectorType m_tmp;
321 template<
typename MatrixType>
324 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
325 Index n = m_eivalues.rows();
327 for (
Index i=0; i<n; ++i)
329 if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i))))
330 matD.coeffRef(i,i) = numext::real(m_eivalues.coeff(i));
333 matD.template block<2,2>(i,i) << numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)),
334 -numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i));
341 template<
typename MatrixType>
344 eigen_assert(m_isInitialized &&
"EigenSolver is not initialized.");
345 eigen_assert(m_eigenvectorsOk &&
"The eigenvectors have not been computed together with the eigenvalues.");
346 Index n = m_eivec.cols();
348 for (
Index j=0; j<n; ++j)
350 if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j))) || j+1==n)
353 matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
354 matV.col(j).normalize();
359 for (
Index i=0; i<n; ++i)
361 matV.coeffRef(i,j) =
ComplexScalar(m_eivec.coeff(i,j), m_eivec.coeff(i,j+1));
362 matV.coeffRef(i,j+1) =
ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
364 matV.col(j).normalize();
365 matV.col(j+1).normalize();
372 template<
typename MatrixType>
376 check_template_parameters();
380 using numext::isfinite;
381 eigen_assert(matrix.cols() == matrix.rows());
384 m_realSchur.compute(matrix, computeEigenvectors);
386 m_info = m_realSchur.info();
390 m_matT = m_realSchur.matrixT();
391 if (computeEigenvectors)
392 m_eivec = m_realSchur.matrixU();
395 m_eivalues.resize(matrix.cols());
397 while (i < matrix.cols())
399 if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) ==
Scalar(0))
401 m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
402 if(!(isfinite)(m_eivalues.coeffRef(i)))
404 m_isInitialized =
true;
405 m_eigenvectorsOk =
false;
413 Scalar p =
Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1));
418 Scalar t0 = m_matT.coeff(i+1, i);
419 Scalar t1 = m_matT.coeff(i, i+1);
420 Scalar maxval = numext::maxi<Scalar>(abs(p),numext::maxi<Scalar>(abs(t0),abs(t1)));
424 z = maxval * sqrt(abs(p0 * p0 + t0 * t1));
427 m_eivalues.coeffRef(i) =
ComplexScalar(m_matT.coeff(i+1, i+1) + p, z);
428 m_eivalues.coeffRef(i+1) =
ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z);
429 if(!((isfinite)(m_eivalues.coeffRef(i)) && (isfinite)(m_eivalues.coeffRef(i+1))))
431 m_isInitialized =
true;
432 m_eigenvectorsOk =
false;
441 if (computeEigenvectors)
442 doComputeEigenvectors();
445 m_isInitialized =
true;
446 m_eigenvectorsOk = computeEigenvectors;
452 template<
typename Scalar>
453 std::complex<Scalar> cdiv(
const Scalar& xr,
const Scalar& xi,
const Scalar& yr,
const Scalar& yi)
457 if (abs(yr) > abs(yi))
461 return std::complex<Scalar>((xr + r*xi)/d, (xi - r*xr)/d);
467 return std::complex<Scalar>((r*xr + xi)/d, (r*xi - xr)/d);
472 template<
typename MatrixType>
473 void EigenSolver<MatrixType>::doComputeEigenvectors()
476 const Index size = m_eivec.cols();
477 const Scalar eps = NumTraits<Scalar>::epsilon();
481 for (Index j = 0; j < size; ++j)
483 norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
487 if (norm == Scalar(0))
492 for (Index n = size-1; n >= 0; n--)
494 Scalar p = m_eivalues.coeff(n).real();
495 Scalar q = m_eivalues.coeff(n).imag();
500 Scalar lastr(0), lastw(0);
503 m_matT.coeffRef(n,n) = 1.0;
504 for (Index i = n-1; i >= 0; i--)
506 Scalar w = m_matT.coeff(i,i) - p;
507 Scalar r = m_matT.row(i).segment(l,n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
509 if (m_eivalues.coeff(i).imag() < Scalar(0))
517 if (m_eivalues.coeff(i).imag() == Scalar(0))
520 m_matT.coeffRef(i,n) = -r / w;
522 m_matT.coeffRef(i,n) = -r / (eps * norm);
526 Scalar x = m_matT.coeff(i,i+1);
527 Scalar y = m_matT.coeff(i+1,i);
528 Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
529 Scalar t = (x * lastr - lastw * r) / denom;
530 m_matT.coeffRef(i,n) = t;
531 if (abs(x) > abs(lastw))
532 m_matT.coeffRef(i+1,n) = (-r - w * t) / x;
534 m_matT.coeffRef(i+1,n) = (-lastr - y * t) / lastw;
538 Scalar t = abs(m_matT.coeff(i,n));
539 if ((eps * t) * t > Scalar(1))
540 m_matT.col(n).tail(size-i) /= t;
544 else if (q < Scalar(0) && n > 0)
546 Scalar lastra(0), lastsa(0), lastw(0);
550 if (abs(m_matT.coeff(n,n-1)) > abs(m_matT.coeff(n-1,n)))
552 m_matT.coeffRef(n-1,n-1) = q / m_matT.coeff(n,n-1);
553 m_matT.coeffRef(n-1,n) = -(m_matT.coeff(n,n) - p) / m_matT.coeff(n,n-1);
557 std::complex<Scalar> cc = cdiv<Scalar>(Scalar(0),-m_matT.coeff(n-1,n),m_matT.coeff(n-1,n-1)-p,q);
558 m_matT.coeffRef(n-1,n-1) = numext::real(cc);
559 m_matT.coeffRef(n-1,n) = numext::imag(cc);
561 m_matT.coeffRef(n,n-1) = Scalar(0);
562 m_matT.coeffRef(n,n) = Scalar(1);
563 for (Index i = n-2; i >= 0; i--)
565 Scalar ra = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n-1).segment(l, n-l+1));
566 Scalar sa = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
567 Scalar w = m_matT.coeff(i,i) - p;
569 if (m_eivalues.coeff(i).imag() < Scalar(0))
578 if (m_eivalues.coeff(i).imag() == RealScalar(0))
580 std::complex<Scalar> cc = cdiv(-ra,-sa,w,q);
581 m_matT.coeffRef(i,n-1) = numext::real(cc);
582 m_matT.coeffRef(i,n) = numext::imag(cc);
587 Scalar x = m_matT.coeff(i,i+1);
588 Scalar y = m_matT.coeff(i+1,i);
589 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;
590 Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
591 if ((vr == Scalar(0)) && (vi == Scalar(0)))
592 vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw));
594 std::complex<Scalar> cc = cdiv(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra,vr,vi);
595 m_matT.coeffRef(i,n-1) = numext::real(cc);
596 m_matT.coeffRef(i,n) = numext::imag(cc);
597 if (abs(x) > (abs(lastw) + abs(q)))
599 m_matT.coeffRef(i+1,n-1) = (-ra - w * m_matT.coeff(i,n-1) + q * m_matT.coeff(i,n)) / x;
600 m_matT.coeffRef(i+1,n) = (-sa - w * m_matT.coeff(i,n) - q * m_matT.coeff(i,n-1)) / x;
604 cc = cdiv(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n),lastw,q);
605 m_matT.coeffRef(i+1,n-1) = numext::real(cc);
606 m_matT.coeffRef(i+1,n) = numext::imag(cc);
611 Scalar t = numext::maxi<Scalar>(abs(m_matT.coeff(i,n-1)),abs(m_matT.coeff(i,n)));
612 if ((eps * t) * t > Scalar(1))
613 m_matT.block(i, n-1, size-i, 2) /= t;
623 eigen_assert(0 &&
"Internal bug in EigenSolver (INF or NaN has not been detected)");
628 for (Index j = size-1; j >= 0; j--)
630 m_tmp.noalias() = m_eivec.leftCols(j+1) * m_matT.col(j).segment(0, j+1);
631 m_eivec.col(j) = m_tmp;
637 #endif // EIGEN_EIGENSOLVER_H
Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime > EigenvectorsType
Type for matrix of eigenvectors as returned by eigenvectors().
Definition: EigenSolver.h:104
EigenvectorsType eigenvectors() const
Returns the eigenvectors of given matrix.
Definition: EigenSolver.h:342
EigenSolver(Index size)
Default constructor with memory preallocation.
Definition: EigenSolver.h:121
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:107
Index getMaxIterations()
Returns the maximum number of iterations.
Definition: EigenSolver.h:293
Eigen::Index Index
Definition: EigenSolver.h:82
ComputationInfo info() const
Definition: EigenSolver.h:279
EigenSolver(const MatrixType &matrix, bool computeEigenvectors=true)
Constructor; computes eigendecomposition of given matrix.
Definition: EigenSolver.h:146
Index getMaxIterations()
Returns the maximum number of iterations.
Definition: RealSchur.h:211
MatrixType pseudoEigenvalueMatrix() const
Returns the block-diagonal matrix in the pseudo-eigendecomposition.
Definition: EigenSolver.h:322
const MatrixType & pseudoEigenvectors() const
Returns the pseudo-eigenvectors of given matrix.
Definition: EigenSolver.h:198
Definition: Constants.h:426
EigenSolver & setMaxIterations(Index maxIters)
Sets the maximum number of iterations allowed.
Definition: EigenSolver.h:286
std::complex< RealScalar > ComplexScalar
Complex scalar type for MatrixType.
Definition: EigenSolver.h:90
Definition: Constants.h:424
_MatrixType MatrixType
Synonym for the template parameter _MatrixType.
Definition: EigenSolver.h:69
EigenSolver()
Default constructor.
Definition: EigenSolver.h:113
RealSchur & setMaxIterations(Index maxIters)
Sets the maximum number of iterations allowed.
Definition: RealSchur.h:204
MatrixType::Scalar Scalar
Scalar type for matrices of type MatrixType.
Definition: EigenSolver.h:80
Computes eigenvalues and eigenvectors of general matrices.
Definition: EigenSolver.h:64
ComputationInfo
Definition: Constants.h:422
const EigenvalueType & eigenvalues() const
Returns the eigenvalues of given matrix.
Definition: EigenSolver.h:243
Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > EigenvalueType
Type for vector of eigenvalues as returned by eigenvalues().
Definition: EigenSolver.h:97
EigenSolver & compute(const MatrixType &matrix, bool computeEigenvectors=true)
Computes eigendecomposition of given matrix.
Definition: EigenSolver.h:374