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RealSchur: change parameter lists; minor rewrite of computeShift().
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Jitse Niesen
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b6829e1d5b
commit
7dea3a33a5
@ -96,11 +96,11 @@ template<typename _MatrixType> class RealSchur
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typedef Matrix<Scalar,3,1> Vector3s;
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Scalar computeNormOfT();
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int findSmallSubdiagEntry(int n, Scalar norm);
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void computeShift(Scalar& x, Scalar& y, Scalar& w, int iu, Scalar& exshift, int iter);
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void findTwoSmallSubdiagEntries(Scalar x, Scalar y, Scalar w, int il, int& m, int iu, Vector3s& firstHouseholderVector);
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void doFrancisStep(int il, int m, int iu, const Vector3s& firstHouseholderVector, Scalar* workspace);
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int findSmallSubdiagEntry(int iu, Scalar norm);
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void splitOffTwoRows(int iu, Scalar exshift);
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void computeShift(int iu, int iter, Scalar& exshift, Vector3s& shiftInfo);
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void initFrancisQRStep(int il, int iu, const Vector3s& shiftInfo, int& im, Vector3s& firstHouseholderVector);
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void performFrancisQRStep(int il, int im, int iu, const Vector3s& firstHouseholderVector, Scalar* workspace);
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};
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@ -125,10 +125,10 @@ void RealSchur<MatrixType>::compute(const MatrixType& matrix)
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// Rows il,...,iu is the part we are working on (the active window).
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// Rows iu+1,...,end are already brought in triangular form.
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int iu = m_matU.cols() - 1;
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Scalar exshift = 0.0;
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int iter = 0; // iteration count
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Scalar exshift = 0.0; // sum of exceptional shifts
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Scalar norm = computeNormOfT();
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int iter = 0;
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while (iu >= 0)
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{
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int il = findSmallSubdiagEntry(iu, norm);
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@ -149,33 +149,33 @@ void RealSchur<MatrixType>::compute(const MatrixType& matrix)
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}
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else // No convergence yet
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{
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Scalar x, y, w;
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Vector3s firstHouseholderVector;
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computeShift(x, y, w, iu, exshift, iter);
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Vector3s firstHouseholderVector, shiftInfo;
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computeShift(iu, iter, exshift, shiftInfo);
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iter = iter + 1; // (Could check iteration count here.)
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int m;
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findTwoSmallSubdiagEntries(x, y, w, il, m, iu, firstHouseholderVector);
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doFrancisStep(il, m, iu, firstHouseholderVector, workspace);
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} // check convergence
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} // while (iu >= 0)
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int im;
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initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
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performFrancisQRStep(il, im, iu, firstHouseholderVector, workspace);
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}
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}
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m_isInitialized = true;
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}
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// Compute matrix norm
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/** \internal Computes and returns vector L1 norm of T */
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template<typename MatrixType>
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inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
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{
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const int size = m_matU.cols();
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// FIXME to be efficient the following would requires a triangular reduxion code
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// Scalar norm = m_matT.upper().cwiseAbs().sum() + m_matT.corner(BottomLeft,size-1,size-1).diagonal().cwiseAbs().sum();
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// Scalar norm = m_matT.upper().cwiseAbs().sum()
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// + m_matT.corner(BottomLeft,size-1,size-1).diagonal().cwiseAbs().sum();
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Scalar norm = 0.0;
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for (int j = 0; j < size; ++j)
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norm += m_matT.row(j).segment(std::max(j-1,0), size-std::max(j-1,0)).cwiseAbs().sum();
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return norm;
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}
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// Look for single small sub-diagonal element
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/** \internal Look for single small sub-diagonal element and returns its index */
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template<typename MatrixType>
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inline int RealSchur<MatrixType>::findSmallSubdiagEntry(int iu, Scalar norm)
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{
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@ -192,133 +192,134 @@ inline int RealSchur<MatrixType>::findSmallSubdiagEntry(int iu, Scalar norm)
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return res;
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}
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/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::splitOffTwoRows(int iu, Scalar exshift)
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{
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const int size = m_matU.cols();
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// The eigenvalues of the 2x2 matrix [a b; c d] are
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// trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
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Scalar w = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
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Scalar p = (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu)) * Scalar(0.5);
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Scalar q = p * p + w;
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Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
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Scalar q = p * p + w; // q = tr^2 / 4 - det = discr/4
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Scalar z = ei_sqrt(ei_abs(q));
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m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
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m_matT.coeffRef(iu-1,iu-1) = m_matT.coeff(iu-1,iu-1) + exshift;
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Scalar x = m_matT.coeff(iu,iu);
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m_matT.coeffRef(iu,iu) += exshift;
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m_matT.coeffRef(iu-1,iu-1) += exshift;
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// Scalar pair
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if (q >= 0)
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if (q >= 0) // Two real eigenvalues
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{
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if (p >= 0)
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z = p + z;
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else
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z = p - z;
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m_eivalues.coeffRef(iu-1) = ComplexScalar(x + z, 0.0);
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m_eivalues.coeffRef(iu) = ComplexScalar(z!=0.0 ? x - w / z : m_eivalues.coeff(iu-1).real(), 0.0);
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PlanarRotation<Scalar> rot;
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rot.makeGivens(z, m_matT.coeff(iu, iu-1));
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if (p >= 0)
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rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
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else
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rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
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m_matT.block(0, iu-1, size, size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
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m_matT.block(0, 0, iu+1, size).applyOnTheRight(iu-1, iu, rot);
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m_matU.applyOnTheRight(iu-1, iu, rot);
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m_eivalues.coeffRef(iu-1) = ComplexScalar(m_matT.coeff(iu-1, iu-1), 0.0);
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m_eivalues.coeffRef(iu) = ComplexScalar(m_matT.coeff(iu, iu), 0.0);
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}
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else // Complex pair
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else // // Pair of complex conjugate eigenvalues
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{
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m_eivalues.coeffRef(iu-1) = ComplexScalar(x + p, z);
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m_eivalues.coeffRef(iu) = ComplexScalar(x + p, -z);
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m_eivalues.coeffRef(iu-1) = ComplexScalar(m_matT.coeff(iu,iu) + p, z);
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m_eivalues.coeffRef(iu) = ComplexScalar(m_matT.coeff(iu,iu) + p, -z);
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}
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}
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// Form shift
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/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::computeShift(Scalar& x, Scalar& y, Scalar& w, int iu, Scalar& exshift, int iter)
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inline void RealSchur<MatrixType>::computeShift(int iu, int iter, Scalar& exshift, Vector3s& shiftInfo)
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{
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x = m_matT.coeff(iu,iu);
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y = m_matT.coeff(iu-1,iu-1);
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w = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
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shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
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shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
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shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
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// Wilkinson's original ad hoc shift
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if (iter == 10)
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{
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exshift += x;
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exshift += shiftInfo.coeff(0);
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for (int i = 0; i <= iu; ++i)
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m_matT.coeffRef(i,i) -= x;
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m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
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Scalar s = ei_abs(m_matT.coeff(iu,iu-1)) + ei_abs(m_matT.coeff(iu-1,iu-2));
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x = y = Scalar(0.75) * s;
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w = Scalar(-0.4375) * s * s;
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shiftInfo.coeffRef(0) = Scalar(0.75) * s;
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shiftInfo.coeffRef(1) = Scalar(0.75) * s;
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shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
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}
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// MATLAB's new ad hoc shift
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if (iter == 30)
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{
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Scalar s = Scalar((y - x) / 2.0);
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s = s * s + w;
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Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
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s = s * s + shiftInfo.coeff(2);
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if (s > 0)
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{
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s = ei_sqrt(s);
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if (y < x)
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if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
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s = -s;
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s = Scalar(x - w / ((y - x) / 2.0 + s));
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s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
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s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
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exshift += s;
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for (int i = 0; i <= iu; ++i)
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m_matT.coeffRef(i,i) -= s;
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exshift += s;
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x = y = w = Scalar(0.964);
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shiftInfo.setConstant(Scalar(0.964));
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}
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}
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}
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// Look for two consecutive small sub-diagonal elements
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/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::findTwoSmallSubdiagEntries(Scalar x, Scalar y, Scalar w, int il, int& m, int iu, Vector3s& firstHouseholderVector)
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inline void RealSchur<MatrixType>::initFrancisQRStep(int il, int iu, const Vector3s& shiftInfo, int& im, Vector3s& firstHouseholderVector)
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{
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Scalar p = 0, q = 0, r = 0;
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m = iu-2;
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while (m >= il)
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for (im = iu-2; im >= il; --im)
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{
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Scalar z = m_matT.coeff(m,m);
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r = x - z;
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Scalar s = y - z;
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p = (r * s - w) / m_matT.coeff(m+1,m) + m_matT.coeff(m,m+1);
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q = m_matT.coeff(m+1,m+1) - z - r - s;
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r = m_matT.coeff(m+2,m+1);
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Scalar z = m_matT.coeff(im,im);
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r = shiftInfo.coeff(0) - z;
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Scalar s = shiftInfo.coeff(1) - z;
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p = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
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q = m_matT.coeff(im+1,im+1) - z - r - s;
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r = m_matT.coeff(im+2,im+1);
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s = ei_abs(p) + ei_abs(q) + ei_abs(r);
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p = p / s;
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q = q / s;
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r = r / s;
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if (m == il) {
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if (im == il) {
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break;
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}
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if (ei_abs(m_matT.coeff(m,m-1)) * (ei_abs(q) + ei_abs(r)) <
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NumTraits<Scalar>::epsilon() * (ei_abs(p) * (ei_abs(m_matT.coeff(m-1,m-1)) + ei_abs(z) +
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ei_abs(m_matT.coeff(m+1,m+1)))))
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if (ei_abs(m_matT.coeff(im,im-1)) * (ei_abs(q) + ei_abs(r)) <
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NumTraits<Scalar>::epsilon() * (ei_abs(p) * (ei_abs(m_matT.coeff(im-1,im-1)) + ei_abs(z) +
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ei_abs(m_matT.coeff(im+1,im+1)))))
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{
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break;
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}
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m--;
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}
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for (int i = m+2; i <= iu; ++i)
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for (int i = im+2; i <= iu; ++i)
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{
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m_matT.coeffRef(i,i-2) = 0.0;
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if (i > m+2)
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if (i > im+2)
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m_matT.coeffRef(i,i-3) = 0.0;
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}
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firstHouseholderVector << p, q, r;
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}
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// Double QR step involving rows il:iu and columns m:iu
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/** Perform a Francis QR step involving rows il:iu and columns im:iu. */
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template<typename MatrixType>
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inline void RealSchur<MatrixType>::doFrancisStep(int il, int m, int iu, const Vector3s& firstHouseholderVector, Scalar* workspace)
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inline void RealSchur<MatrixType>::performFrancisQRStep(int il, int im, int iu, const Vector3s& firstHouseholderVector, Scalar* workspace)
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{
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assert(m >= il);
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assert(m <= iu-2);
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assert(im >= il);
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assert(im <= iu-2);
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const int size = m_matU.cols();
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for (int k = m; k <= iu-2; ++k)
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for (int k = im; k <= iu-2; ++k)
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{
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bool firstIteration = (k == m);
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bool firstIteration = (k == im);
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Vector3s v;
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if (firstIteration)
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