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* Clean a bit the eigenvalue solver: if the matrix is known to be

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selfadjoint at compile time, then it returns real eigenvalues.
* Fix a couple of bugs with the new product.
This commit is contained in:
Gael Guennebaud 2008-05-13 07:40:25 +00:00
parent 3eccfd1a78
commit fd2e9e5c3c
2 changed files with 257 additions and 234 deletions
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5
Eigen/src/Core/ProductWIP.h
View File

@ -143,6 +143,7 @@ template<typename Lhs, typename Rhs> struct ei_product_eval_mode
{ {
enum{ value = Lhs::MaxRowsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD enum{ value = Lhs::MaxRowsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD
&& Rhs::MaxColsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD && Rhs::MaxColsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD
&& Lhs::MaxColsAtCompileTime >= EIGEN_CACHEFRIENDLY_PRODUCT_THRESHOLD
? CacheFriendlyProduct : NormalProduct }; ? CacheFriendlyProduct : NormalProduct };
}; };
@ -188,7 +189,7 @@ template<typename T, int n=1> struct ei_product_nested_lhs
(ei_traits<T>::Flags & EvalBeforeNestingBit) (ei_traits<T>::Flags & EvalBeforeNestingBit)
|| (!(ei_traits<T>::Flags & DirectAccessBit)) || (!(ei_traits<T>::Flags & DirectAccessBit))
|| (n+1) * NumTraits<typename ei_traits<T>::Scalar>::ReadCost < (n-1) * T::CoeffReadCost, || (n+1) * NumTraits<typename ei_traits<T>::Scalar>::ReadCost < (n-1) * T::CoeffReadCost,
typename ei_product_eval_to_column_major<T>::type, typename ei_eval<T>::type,
const T& const T&
>::ret >::ret
>::ret type; >::ret type;
@ -201,7 +202,7 @@ struct ei_traits<Product<Lhs, Rhs, EvalMode> >
// the cache friendly product evals lhs once only // the cache friendly product evals lhs once only
// FIXME what to do if we chose to dynamically call the normal product from the cache friendly one for small matrices ? // FIXME what to do if we chose to dynamically call the normal product from the cache friendly one for small matrices ?
typedef typename ei_meta_if<EvalMode==CacheFriendlyProduct, typedef typename ei_meta_if<EvalMode==CacheFriendlyProduct,
typename ei_product_nested_lhs<Rhs,0>::type, typename ei_product_nested_lhs<Lhs,0>::type,
typename ei_nested<Lhs,Rhs::ColsAtCompileTime>::type>::ret LhsNested; typename ei_nested<Lhs,Rhs::ColsAtCompileTime>::type>::ret LhsNested;
// NOTE that rhs must be ColumnMajor, so we might need a special nested type calculation // NOTE that rhs must be ColumnMajor, so we might need a special nested type calculation

482
Eigen/src/QR/EigenSolver.h
View File

@ -30,94 +30,117 @@
* \brief Eigen values/vectors solver * \brief Eigen values/vectors solver
* *
* \param MatrixType the type of the matrix of which we are computing the eigen decomposition * \param MatrixType the type of the matrix of which we are computing the eigen decomposition
* \param IsSelfadjoint tells the input matrix is guaranteed to be selfadjoint (hermitian). In that case the
* return type of eigenvalues() is a real vector.
*
* Currently it only support real matrices.
* *
* \note this code was adapted from JAMA (public domain) * \note this code was adapted from JAMA (public domain)
* *
* \sa MatrixBase::eigenvalues() * \sa MatrixBase::eigenvalues()
*/ */
template<typename _MatrixType> class EigenSolver template<typename _MatrixType, bool IsSelfadjoint=false> class EigenSolver
{ {
public: public:
typedef _MatrixType MatrixType; typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::Scalar Scalar;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType; typedef typename NumTraits<Scalar>::Real RealScalar;
typedef std::complex<RealScalar> Complex;
typedef Matrix<typename ei_meta_if<IsSelfadjoint, Scalar, Complex>::ret, MatrixType::ColsAtCompileTime, 1> EigenvalueType;
typedef Matrix<RealScalar, MatrixType::ColsAtCompileTime, 1> RealVectorType;
EigenSolver(const MatrixType& matrix) EigenSolver(const MatrixType& matrix)
: m_eivec(matrix.rows(), matrix.cols()), : m_eivec(matrix.rows(), matrix.cols()),
m_eivalr(matrix.cols()), m_eivali(matrix.cols()), m_eivalues(matrix.cols())
m_H(matrix.rows(), matrix.cols()),
m_ort(matrix.cols())
{ {
_compute(matrix); _compute(matrix);
} }
MatrixType eigenvectors(void) const { return m_eivec; } MatrixType eigenvectors(void) const { return m_eivec; }
VectorType eigenvalues(void) const { return m_eivalr; } EigenvalueType eigenvalues(void) const { return m_eivalues; }
private: private:
void _compute(const MatrixType& matrix); void _compute(const MatrixType& matrix)
{
computeImpl(matrix, typename ei_meta_if<IsSelfadjoint, ei_meta_true, ei_meta_false>::ret());
}
void computeImpl(const MatrixType& matrix, ei_meta_true isSelfadjoint);
void computeImpl(const MatrixType& matrix, ei_meta_false isNotSelfadjoint);
void tridiagonalization(void); void tridiagonalization(RealVectorType& eivalr, RealVectorType& eivali);
void tql2(void); void tql2(RealVectorType& eivalr, RealVectorType& eivali);
void orthes(void); void orthes(MatrixType& matH, RealVectorType& ort);
void hqr2(void); void hqr2(MatrixType& matH, RealVectorType& ort);
protected: protected:
MatrixType m_eivec; MatrixType m_eivec;
VectorType m_eivalr, m_eivali; EigenvalueType m_eivalues;
MatrixType m_H;
VectorType m_ort;
bool m_isSymmetric;
}; };
template<typename MatrixType> template<typename MatrixType, bool IsSelfadjoint>
void EigenSolver<MatrixType>::_compute(const MatrixType& matrix) void EigenSolver<MatrixType,IsSelfadjoint>::computeImpl(const MatrixType& matrix, ei_meta_true)
{ {
assert(matrix.cols() == matrix.rows()); assert(matrix.cols() == matrix.rows());
m_isSymmetric = true;
int n = matrix.cols(); int n = matrix.cols();
for (int j = 0; (j < n) && m_isSymmetric; j++) { m_eivalues.resize(n,1);
for (int i = 0; (i < j) && m_isSymmetric; i++) {
m_isSymmetric = (matrix(i,j) == matrix(j,i));
}
}
m_eivalr.resize(n,1); RealVectorType eivali(n);
m_eivali.resize(n,1);
if (m_isSymmetric)
{
m_eivec = matrix; m_eivec = matrix;
// Tridiagonalize. // Tridiagonalize.
tridiagonalization(); tridiagonalization(m_eivalues, eivali);
// Diagonalize. // Diagonalize.
tql2(); tql2(m_eivalues, eivali);
}
template<typename MatrixType, bool IsSelfadjoint>
void EigenSolver<MatrixType,IsSelfadjoint>::computeImpl(const MatrixType& matrix, ei_meta_false)
{
assert(matrix.cols() == matrix.rows());
int n = matrix.cols();
m_eivalues.resize(n,1);
bool isSelfadjoint = true;
for (int j = 0; (j < n) && isSelfadjoint; j++)
for (int i = 0; (i < j) && isSelfadjoint; i++)
isSelfadjoint = (matrix(i,j) == matrix(j,i));
if (isSelfadjoint)
{
RealVectorType eivalr(n);
RealVectorType eivali(n);
m_eivec = matrix;
// Tridiagonalize.
tridiagonalization(eivalr, eivali);
// Diagonalize.
tql2(eivalr, eivali);
m_eivalues = eivalr.template cast<Complex>();
} }
else else
{ {
m_H = matrix; MatrixType matH = matrix;
m_ort.resize(n, 1); RealVectorType ort(n);
// Reduce to Hessenberg form. // Reduce to Hessenberg form.
orthes(); orthes(matH, ort);
// Reduce Hessenberg to real Schur form. // Reduce Hessenberg to real Schur form.
hqr2(); hqr2(matH, ort);
} }
std::cout << m_eivali.transpose() << "\n";
} }
// Symmetric Householder reduction to tridiagonal form. // Symmetric Householder reduction to tridiagonal form.
template<typename MatrixType> template<typename MatrixType, bool IsSelfadjoint>
void EigenSolver<MatrixType>::tridiagonalization(void) void EigenSolver<MatrixType,IsSelfadjoint>::tridiagonalization(RealVectorType& eivalr, RealVectorType& eivali)
{ {
// This is derived from the Algol procedures tred2 by // This is derived from the Algol procedures tred2 by
@ -126,7 +149,7 @@ void EigenSolver<MatrixType>::tridiagonalization(void)
// Fortran subroutine in EISPACK. // Fortran subroutine in EISPACK.
int n = m_eivec.cols(); int n = m_eivec.cols();
m_eivalr = m_eivec.row(m_eivalr.size()-1); eivalr = m_eivec.row(eivalr.size()-1);
// Householder reduction to tridiagonal form. // Householder reduction to tridiagonal form.
for (int i = n-1; i > 0; i--) for (int i = n-1; i > 0; i--)
@ -134,55 +157,55 @@ void EigenSolver<MatrixType>::tridiagonalization(void)
// Scale to avoid under/overflow. // Scale to avoid under/overflow.
Scalar scale = 0.0; Scalar scale = 0.0;
Scalar h = 0.0; Scalar h = 0.0;
scale = m_eivalr.start(i).cwiseAbs().sum(); scale = eivalr.start(i).cwiseAbs().sum();
if (scale == 0.0) if (scale == 0.0)
{ {
m_eivali[i] = m_eivalr[i-1]; eivali[i] = eivalr[i-1];
m_eivalr.start(i) = m_eivec.row(i-1).start(i); eivalr.start(i) = m_eivec.row(i-1).start(i);
m_eivec.corner(TopLeft, i, i) = m_eivec.corner(TopLeft, i, i).diagonal().asDiagonal(); m_eivec.corner(TopLeft, i, i) = m_eivec.corner(TopLeft, i, i).diagonal().asDiagonal();
} }
else else
{ {
// Generate Householder vector. // Generate Householder vector.
m_eivalr.start(i) /= scale; eivalr.start(i) /= scale;
h = m_eivalr.start(i).cwiseAbs2().sum(); h = eivalr.start(i).cwiseAbs2().sum();
Scalar f = m_eivalr[i-1]; Scalar f = eivalr[i-1];
Scalar g = ei_sqrt(h); Scalar g = ei_sqrt(h);
if (f > 0) if (f > 0)
g = -g; g = -g;
m_eivali[i] = scale * g; eivali[i] = scale * g;
h = h - f * g; h = h - f * g;
m_eivalr[i-1] = f - g; eivalr[i-1] = f - g;
m_eivali.start(i).setZero(); eivali.start(i).setZero();
// Apply similarity transformation to remaining columns. // Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++) for (int j = 0; j < i; j++)
{ {
f = m_eivalr[j]; f = eivalr[j];
m_eivec(j,i) = f; m_eivec(j,i) = f;
g = m_eivali[j] + m_eivec(j,j) * f; g = eivali[j] + m_eivec(j,j) * f;
int bSize = i-j-1; int bSize = i-j-1;
if (bSize>0) if (bSize>0)
{ {
g += (m_eivec.col(j).block(j+1, bSize).transpose() * m_eivalr.block(j+1, bSize))(0,0); g += (m_eivec.col(j).block(j+1, bSize).transpose() * eivalr.block(j+1, bSize))(0,0);
m_eivali.block(j+1, bSize) += m_eivec.col(j).block(j+1, bSize) * f; eivali.block(j+1, bSize) += m_eivec.col(j).block(j+1, bSize) * f;
} }
m_eivali[j] = g; eivali[j] = g;
} }
f = (m_eivali.start(i).transpose() * m_eivalr.start(i))(0,0); f = (eivali.start(i).transpose() * eivalr.start(i))(0,0);
m_eivali.start(i) = (m_eivali.start(i) - (f / (h + h)) * m_eivalr.start(i))/h; eivali.start(i) = (eivali.start(i) - (f / (h + h)) * eivalr.start(i))/h;
m_eivec.corner(TopLeft, i, i).lower() -= m_eivec.corner(TopLeft, i, i).lower() -=
( (m_eivali.start(i) * m_eivalr.start(i).transpose()).lazy() ( (eivali.start(i) * eivalr.start(i).transpose()).lazy()
+ (m_eivalr.start(i) * m_eivali.start(i).transpose()).lazy()); + (eivalr.start(i) * eivali.start(i).transpose()).lazy());
m_eivalr.start(i) = m_eivec.row(i-1).start(i); eivalr.start(i) = m_eivec.row(i-1).start(i);
m_eivec.row(i).start(i).setZero(); m_eivec.row(i).start(i).setZero();
} }
m_eivalr[i] = h; eivalr[i] = h;
} }
// Accumulate transformations. // Accumulate transformations.
@ -190,39 +213,38 @@ void EigenSolver<MatrixType>::tridiagonalization(void)
{ {
m_eivec(n-1,i) = m_eivec(i,i); m_eivec(n-1,i) = m_eivec(i,i);
m_eivec(i,i) = 1.0; m_eivec(i,i) = 1.0;
Scalar h = m_eivalr[i+1]; Scalar h = eivalr[i+1];
// FIXME this does not looks very stable ;) // FIXME this does not looks very stable ;)
if (h != 0.0) if (h != 0.0)
{ {
m_eivalr.start(i+1) = m_eivec.col(i+1).start(i+1) / h; eivalr.start(i+1) = m_eivec.col(i+1).start(i+1) / h;
m_eivec.corner(TopLeft, i+1, i+1) -= m_eivalr.start(i+1) m_eivec.corner(TopLeft, i+1, i+1) -= eivalr.start(i+1)
* ( m_eivec.col(i+1).start(i+1).transpose() * m_eivec.corner(TopLeft, i+1, i+1) ); * ( m_eivec.col(i+1).start(i+1).transpose() * m_eivec.corner(TopLeft, i+1, i+1) );
} }
m_eivec.col(i+1).start(i+1).setZero(); m_eivec.col(i+1).start(i+1).setZero();
} }
m_eivalr = m_eivec.row(m_eivalr.size()-1); eivalr = m_eivec.row(eivalr.size()-1);
m_eivec.row(m_eivalr.size()-1).setZero(); m_eivec.row(eivalr.size()-1).setZero();
m_eivec(n-1,n-1) = 1.0; m_eivec(n-1,n-1) = 1.0;
m_eivali[0] = 0.0; eivali[0] = 0.0;
} }
// Symmetric tridiagonal QL algorithm. // Symmetric tridiagonal QL algorithm.
template<typename MatrixType> template<typename MatrixType, bool IsSelfadjoint>
void EigenSolver<MatrixType>::tql2(void) void EigenSolver<MatrixType,IsSelfadjoint>::tql2(RealVectorType& eivalr, RealVectorType& eivali)
{ {
// This is derived from the Algol procedures tql2, by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
// This is derived from the Algol procedures tql2, by int n = eivalr.size();
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
int n = m_eivalr.size();
for (int i = 1; i < n; i++) { for (int i = 1; i < n; i++) {
m_eivali[i-1] = m_eivali[i]; eivali[i-1] = eivali[i];
} }
m_eivali[n-1] = 0.0; eivali[n-1] = 0.0;
Scalar f = 0.0; Scalar f = 0.0;
Scalar tst1 = 0.0; Scalar tst1 = 0.0;
@ -230,13 +252,13 @@ void EigenSolver<MatrixType>::tql2(void)
for (int l = 0; l < n; l++) for (int l = 0; l < n; l++)
{ {
// Find small subdiagonal element // Find small subdiagonal element
tst1 = std::max(tst1,ei_abs(m_eivalr[l]) + ei_abs(m_eivali[l])); tst1 = std::max(tst1,ei_abs(eivalr[l]) + ei_abs(eivali[l]));
int m = l; int m = l;
while ( (m < n) && (ei_abs(m_eivali[m]) > eps*tst1) ) while ( (m < n) && (ei_abs(eivali[m]) > eps*tst1) )
m++; m++;
// If m == l, m_eivalr[l] is an eigenvalue, // If m == l, eivalr[l] is an eigenvalue,
// otherwise, iterate. // otherwise, iterate.
if (m > l) if (m > l)
{ {
@ -246,26 +268,26 @@ void EigenSolver<MatrixType>::tql2(void)
iter = iter + 1; iter = iter + 1;
// Compute implicit shift // Compute implicit shift
Scalar g = m_eivalr[l]; Scalar g = eivalr[l];
Scalar p = (m_eivalr[l+1] - g) / (2.0 * m_eivali[l]); Scalar p = (eivalr[l+1] - g) / (2.0 * eivali[l]);
Scalar r = hypot(p,1.0); Scalar r = hypot(p,1.0);
if (p < 0) if (p < 0)
r = -r; r = -r;
m_eivalr[l] = m_eivali[l] / (p + r); eivalr[l] = eivali[l] / (p + r);
m_eivalr[l+1] = m_eivali[l] * (p + r); eivalr[l+1] = eivali[l] * (p + r);
Scalar dl1 = m_eivalr[l+1]; Scalar dl1 = eivalr[l+1];
Scalar h = g - m_eivalr[l]; Scalar h = g - eivalr[l];
if (l+2<n) if (l+2<n)
m_eivalr.end(n-l-2) -= VectorType::constant(n-l-2, h); eivalr.end(n-l-2) -= RealVectorType::constant(n-l-2, h);
f = f + h; f = f + h;
// Implicit QL transformation. // Implicit QL transformation.
p = m_eivalr[m]; p = eivalr[m];
Scalar c = 1.0; Scalar c = 1.0;
Scalar c2 = c; Scalar c2 = c;
Scalar c3 = c; Scalar c3 = c;
Scalar el1 = m_eivali[l+1]; Scalar el1 = eivali[l+1];
Scalar s = 0.0; Scalar s = 0.0;
Scalar s2 = 0.0; Scalar s2 = 0.0;
for (int i = m-1; i >= l; i--) for (int i = m-1; i >= l; i--)
@ -273,14 +295,14 @@ void EigenSolver<MatrixType>::tql2(void)
c3 = c2; c3 = c2;
c2 = c; c2 = c;
s2 = s; s2 = s;
g = c * m_eivali[i]; g = c * eivali[i];
h = c * p; h = c * p;
r = hypot(p,m_eivali[i]); r = hypot(p,eivali[i]);
m_eivali[i+1] = s * r; eivali[i+1] = s * r;
s = m_eivali[i] / r; s = eivali[i] / r;
c = p / r; c = p / r;
p = c * m_eivalr[i] - s * g; p = c * eivalr[i] - s * g;
m_eivalr[i+1] = h + s * (c * g + s * m_eivalr[i]); eivalr[i+1] = h + s * (c * g + s * eivalr[i]);
// Accumulate transformation. // Accumulate transformation.
for (int k = 0; k < n; k++) for (int k = 0; k < n; k++)
@ -290,15 +312,15 @@ void EigenSolver<MatrixType>::tql2(void)
m_eivec(k,i) = c * m_eivec(k,i) - s * h; m_eivec(k,i) = c * m_eivec(k,i) - s * h;
} }
} }
p = -s * s2 * c3 * el1 * m_eivali[l] / dl1; p = -s * s2 * c3 * el1 * eivali[l] / dl1;
m_eivali[l] = s * p; eivali[l] = s * p;
m_eivalr[l] = c * p; eivalr[l] = c * p;
// Check for convergence. // Check for convergence.
} while (ei_abs(m_eivali[l]) > eps*tst1); } while (ei_abs(eivali[l]) > eps*tst1);
} }
m_eivalr[l] = m_eivalr[l] + f; eivalr[l] = eivalr[l] + f;
m_eivali[l] = 0.0; eivali[l] = 0.0;
} }
// Sort eigenvalues and corresponding vectors. // Sort eigenvalues and corresponding vectors.
@ -306,18 +328,18 @@ void EigenSolver<MatrixType>::tql2(void)
for (int i = 0; i < n-1; i++) for (int i = 0; i < n-1; i++)
{ {
int k = i; int k = i;
Scalar minValue = m_eivalr[i]; Scalar minValue = eivalr[i];
for (int j = i+1; j < n; j++) for (int j = i+1; j < n; j++)
{ {
if (m_eivalr[j] < minValue) if (eivalr[j] < minValue)
{ {
k = j; k = j;
minValue = m_eivalr[j]; minValue = eivalr[j];
} }
} }
if (k != i) if (k != i)
{ {
std::swap(m_eivalr[i], m_eivalr[k]); std::swap(eivalr[i], eivalr[k]);
m_eivec.col(i).swap(m_eivec.col(k)); m_eivec.col(i).swap(m_eivec.col(k));
} }
} }
@ -325,8 +347,8 @@ void EigenSolver<MatrixType>::tql2(void)
// Nonsymmetric reduction to Hessenberg form. // Nonsymmetric reduction to Hessenberg form.
template<typename MatrixType> template<typename MatrixType, bool IsSelfadjoint>
void EigenSolver<MatrixType>::orthes(void) void EigenSolver<MatrixType,IsSelfadjoint>::orthes(MatrixType& matH, RealVectorType& ort)
{ {
// This is derived from the Algol procedures orthes and ortran, // This is derived from the Algol procedures orthes and ortran,
// by Martin and Wilkinson, Handbook for Auto. Comp., // by Martin and Wilkinson, Handbook for Auto. Comp.,
@ -340,7 +362,7 @@ void EigenSolver<MatrixType>::orthes(void)
for (int m = low+1; m <= high-1; m++) for (int m = low+1; m <= high-1; m++)
{ {
// Scale column. // Scale column.
Scalar scale = m_H.block(m, m-1, high-m+1, 1).cwiseAbs().sum(); Scalar scale = matH.block(m, m-1, high-m+1, 1).cwiseAbs().sum();
if (scale != 0.0) if (scale != 0.0)
{ {
// Compute Householder transformation. // Compute Householder transformation.
@ -348,26 +370,26 @@ void EigenSolver<MatrixType>::orthes(void)
// FIXME could be rewritten, but this one looks better wrt cache // FIXME could be rewritten, but this one looks better wrt cache
for (int i = high; i >= m; i--) for (int i = high; i >= m; i--)
{ {
m_ort[i] = m_H(i,m-1)/scale; ort[i] = matH(i,m-1)/scale;
h += m_ort[i] * m_ort[i]; h += ort[i] * ort[i];
} }
Scalar g = ei_sqrt(h); Scalar g = ei_sqrt(h);
if (m_ort[m] > 0) if (ort[m] > 0)
g = -g; g = -g;
h = h - m_ort[m] * g; h = h - ort[m] * g;
m_ort[m] = m_ort[m] - g; ort[m] = ort[m] - g;
// Apply Householder similarity transformation // Apply Householder similarity transformation
// H = (I-u*u'/h)*H*(I-u*u')/h) // H = (I-u*u'/h)*H*(I-u*u')/h)
int bSize = high-m+1; int bSize = high-m+1;
m_H.block(m, m, bSize, n-m) -= ((m_ort.block(m, bSize)/h) matH.block(m, m, bSize, n-m) -= ((ort.block(m, bSize)/h)
* (m_ort.block(m, bSize).transpose() * m_H.block(m, m, bSize, n-m)).lazy()).lazy(); * (ort.block(m, bSize).transpose() * matH.block(m, m, bSize, n-m)).lazy()).lazy();
m_H.block(0, m, high+1, bSize) -= ((m_H.block(0, m, high+1, bSize) * m_ort.block(m, bSize)).lazy() matH.block(0, m, high+1, bSize) -= ((matH.block(0, m, high+1, bSize) * ort.block(m, bSize)).lazy()
* (m_ort.block(m, bSize)/h).transpose()).lazy(); * (ort.block(m, bSize)/h).transpose()).lazy();
m_ort[m] = scale*m_ort[m]; ort[m] = scale*ort[m];
m_H(m,m-1) = scale*g; matH(m,m-1) = scale*g;
} }
} }
@ -376,13 +398,13 @@ void EigenSolver<MatrixType>::orthes(void)
for (int m = high-1; m >= low+1; m--) for (int m = high-1; m >= low+1; m--)
{ {
if (m_H(m,m-1) != 0.0) if (matH(m,m-1) != 0.0)
{ {
m_ort.block(m+1, high-m) = m_H.col(m-1).block(m+1, high-m); ort.block(m+1, high-m) = matH.col(m-1).block(m+1, high-m);
int bSize = high-m+1; int bSize = high-m+1;
m_eivec.block(m, m, bSize, bSize) += ( (m_ort.block(m, bSize) / (m_H(m,m-1) * m_ort[m] ) ) m_eivec.block(m, m, bSize, bSize) += ( (ort.block(m, bSize) / (matH(m,m-1) * ort[m] ) )
* (m_ort.block(m, bSize).transpose() * m_eivec.block(m, m, bSize, bSize)).lazy()); * (ort.block(m, bSize).transpose() * m_eivec.block(m, m, bSize, bSize)).lazy());
} }
} }
} }
@ -409,8 +431,8 @@ std::complex<Scalar> cdiv(Scalar xr, Scalar xi, Scalar yr, Scalar yi)
// Nonsymmetric reduction from Hessenberg to real Schur form. // Nonsymmetric reduction from Hessenberg to real Schur form.
template<typename MatrixType> template<typename MatrixType, bool IsSelfadjoint>
void EigenSolver<MatrixType>::hqr2(void) void EigenSolver<MatrixType,IsSelfadjoint>::hqr2(MatrixType& matH, RealVectorType& ort)
{ {
// This is derived from the Algol procedure hqr2, // This is derived from the Algol procedure hqr2,
// by Martin and Wilkinson, Handbook for Auto. Comp., // by Martin and Wilkinson, Handbook for Auto. Comp.,
@ -428,17 +450,17 @@ void EigenSolver<MatrixType>::hqr2(void)
// Store roots isolated by balanc and compute matrix norm // Store roots isolated by balanc and compute matrix norm
// FIXME to be efficient the following would requires a triangular reduxion code // FIXME to be efficient the following would requires a triangular reduxion code
// Scalar norm = m_H.upper().cwiseAbs().sum() + m_H.corner(BottomLeft,n,n).diagonal().cwiseAbs().sum(); // Scalar norm = matH.upper().cwiseAbs().sum() + matH.corner(BottomLeft,n,n).diagonal().cwiseAbs().sum();
Scalar norm = 0.0; Scalar norm = 0.0;
for (int j = 0; j < nn; j++) for (int j = 0; j < nn; j++)
{ {
// FIXME what's the purpose of the following since the condition is always false // FIXME what's the purpose of the following since the condition is always false
if ((j < low) || (j > high)) if ((j < low) || (j > high))
{ {
m_eivalr[j] = m_H(j,j); m_eivalues[j].real() = matH(j,j);
m_eivali[j] = 0.0; m_eivalues[j].imag() = 0.0;
} }
norm += m_H.col(j).start(std::min(j+1,nn)).cwiseAbs().sum(); norm += matH.col(j).start(std::min(j+1,nn)).cwiseAbs().sum();
} }
// Outer loop over eigenvalue index // Outer loop over eigenvalue index
@ -449,10 +471,10 @@ void EigenSolver<MatrixType>::hqr2(void)
int l = n; int l = n;
while (l > low) while (l > low)
{ {
s = ei_abs(m_H(l-1,l-1)) + ei_abs(m_H(l,l)); s = ei_abs(matH(l-1,l-1)) + ei_abs(matH(l,l));
if (s == 0.0) if (s == 0.0)
s = norm; s = norm;
if (ei_abs(m_H(l,l-1)) < eps * s) if (ei_abs(matH(l,l-1)) < eps * s)
break; break;
l--; l--;
} }
@ -461,21 +483,21 @@ void EigenSolver<MatrixType>::hqr2(void)
// One root found // One root found
if (l == n) if (l == n)
{ {
m_H(n,n) = m_H(n,n) + exshift; matH(n,n) = matH(n,n) + exshift;
m_eivalr[n] = m_H(n,n); m_eivalues[n].real() = matH(n,n);
m_eivali[n] = 0.0; m_eivalues[n].imag() = 0.0;
n--; n--;
iter = 0; iter = 0;
} }
else if (l == n-1) // Two roots found else if (l == n-1) // Two roots found
{ {
w = m_H(n,n-1) * m_H(n-1,n); w = matH(n,n-1) * matH(n-1,n);
p = (m_H(n-1,n-1) - m_H(n,n)) / 2.0; p = (matH(n-1,n-1) - matH(n,n)) / 2.0;
q = p * p + w; q = p * p + w;
z = ei_sqrt(ei_abs(q)); z = ei_sqrt(ei_abs(q));
m_H(n,n) = m_H(n,n) + exshift; matH(n,n) = matH(n,n) + exshift;
m_H(n-1,n-1) = m_H(n-1,n-1) + exshift; matH(n-1,n-1) = matH(n-1,n-1) + exshift;
x = m_H(n,n); x = matH(n,n);
// Scalar pair // Scalar pair
if (q >= 0) if (q >= 0)
@ -485,14 +507,14 @@ void EigenSolver<MatrixType>::hqr2(void)
else else
z = p - z; z = p - z;
m_eivalr[n-1] = x + z; m_eivalues[n-1].real() = x + z;
m_eivalr[n] = m_eivalr[n-1]; m_eivalues[n].real() = m_eivalues[n-1].real();
if (z != 0.0) if (z != 0.0)
m_eivalr[n] = x - w / z; m_eivalues[n].real() = x - w / z;
m_eivali[n-1] = 0.0; m_eivalues[n-1].imag() = 0.0;
m_eivali[n] = 0.0; m_eivalues[n].imag() = 0.0;
x = m_H(n,n-1); x = matH(n,n-1);
s = ei_abs(x) + ei_abs(z); s = ei_abs(x) + ei_abs(z);
p = x / s; p = x / s;
q = z / s; q = z / s;
@ -503,17 +525,17 @@ void EigenSolver<MatrixType>::hqr2(void)
// Row modification // Row modification
for (int j = n-1; j < nn; j++) for (int j = n-1; j < nn; j++)
{ {
z = m_H(n-1,j); z = matH(n-1,j);
m_H(n-1,j) = q * z + p * m_H(n,j); matH(n-1,j) = q * z + p * matH(n,j);
m_H(n,j) = q * m_H(n,j) - p * z; matH(n,j) = q * matH(n,j) - p * z;
} }
// Column modification // Column modification
for (int i = 0; i <= n; i++) for (int i = 0; i <= n; i++)
{ {
z = m_H(i,n-1); z = matH(i,n-1);
m_H(i,n-1) = q * z + p * m_H(i,n); matH(i,n-1) = q * z + p * matH(i,n);
m_H(i,n) = q * m_H(i,n) - p * z; matH(i,n) = q * matH(i,n) - p * z;
} }
// Accumulate transformations // Accumulate transformations
@ -526,10 +548,10 @@ void EigenSolver<MatrixType>::hqr2(void)
} }
else // Complex pair else // Complex pair
{ {
m_eivalr[n-1] = x + p; m_eivalues[n-1].real() = x + p;
m_eivalr[n] = x + p; m_eivalues[n].real() = x + p;
m_eivali[n-1] = z; m_eivalues[n-1].imag() = z;
m_eivali[n] = -z; m_eivalues[n].imag() = -z;
} }
n = n - 2; n = n - 2;
iter = 0; iter = 0;
@ -537,13 +559,13 @@ void EigenSolver<MatrixType>::hqr2(void)
else // No convergence yet else // No convergence yet
{ {
// Form shift // Form shift
x = m_H(n,n); x = matH(n,n);
y = 0.0; y = 0.0;
w = 0.0; w = 0.0;
if (l < n) if (l < n)
{ {
y = m_H(n-1,n-1); y = matH(n-1,n-1);
w = m_H(n,n-1) * m_H(n-1,n); w = matH(n,n-1) * matH(n-1,n);
} }
// Wilkinson's original ad hoc shift // Wilkinson's original ad hoc shift
@ -551,8 +573,8 @@ void EigenSolver<MatrixType>::hqr2(void)
{ {
exshift += x; exshift += x;
for (int i = low; i <= n; i++) for (int i = low; i <= n; i++)
m_H(i,i) -= x; matH(i,i) -= x;
s = ei_abs(m_H(n,n-1)) + ei_abs(m_H(n-1,n-2)); s = ei_abs(matH(n,n-1)) + ei_abs(matH(n-1,n-2));
x = y = 0.75 * s; x = y = 0.75 * s;
w = -0.4375 * s * s; w = -0.4375 * s * s;
} }
@ -569,7 +591,7 @@ void EigenSolver<MatrixType>::hqr2(void)
s = -s; s = -s;
s = x - w / ((y - x) / 2.0 + s); s = x - w / ((y - x) / 2.0 + s);
for (int i = low; i <= n; i++) for (int i = low; i <= n; i++)
m_H(i,i) -= s; matH(i,i) -= s;
exshift += s; exshift += s;
x = y = w = 0.964; x = y = w = 0.964;
} }
@ -581,12 +603,12 @@ void EigenSolver<MatrixType>::hqr2(void)
int m = n-2; int m = n-2;
while (m >= l) while (m >= l)
{ {
z = m_H(m,m); z = matH(m,m);
r = x - z; r = x - z;
s = y - z; s = y - z;
p = (r * s - w) / m_H(m+1,m) + m_H(m,m+1); p = (r * s - w) / matH(m+1,m) + matH(m,m+1);
q = m_H(m+1,m+1) - z - r - s; q = matH(m+1,m+1) - z - r - s;
r = m_H(m+2,m+1); r = matH(m+2,m+1);
s = ei_abs(p) + ei_abs(q) + ei_abs(r); s = ei_abs(p) + ei_abs(q) + ei_abs(r);
p = p / s; p = p / s;
q = q / s; q = q / s;
@ -594,9 +616,9 @@ void EigenSolver<MatrixType>::hqr2(void)
if (m == l) { if (m == l) {
break; break;
} }
if (ei_abs(m_H(m,m-1)) * (ei_abs(q) + ei_abs(r)) < if (ei_abs(matH(m,m-1)) * (ei_abs(q) + ei_abs(r)) <
eps * (ei_abs(p) * (ei_abs(m_H(m-1,m-1)) + ei_abs(z) + eps * (ei_abs(p) * (ei_abs(matH(m-1,m-1)) + ei_abs(z) +
ei_abs(m_H(m+1,m+1))))) ei_abs(matH(m+1,m+1)))))
{ {
break; break;
} }
@ -605,9 +627,9 @@ void EigenSolver<MatrixType>::hqr2(void)
for (int i = m+2; i <= n; i++) for (int i = m+2; i <= n; i++)
{ {
m_H(i,i-2) = 0.0; matH(i,i-2) = 0.0;
if (i > m+2) if (i > m+2)
m_H(i,i-3) = 0.0; matH(i,i-3) = 0.0;
} }
// Double QR step involving rows l:n and columns m:n // Double QR step involving rows l:n and columns m:n
@ -615,9 +637,9 @@ void EigenSolver<MatrixType>::hqr2(void)
{ {
int notlast = (k != n-1); int notlast = (k != n-1);
if (k != m) { if (k != m) {
p = m_H(k,k-1); p = matH(k,k-1);
q = m_H(k+1,k-1); q = matH(k+1,k-1);
r = (notlast ? m_H(k+2,k-1) : 0.0); r = (notlast ? matH(k+2,k-1) : 0.0);
x = ei_abs(p) + ei_abs(q) + ei_abs(r); x = ei_abs(p) + ei_abs(q) + ei_abs(r);
if (x != 0.0) if (x != 0.0)
{ {
@ -638,9 +660,9 @@ void EigenSolver<MatrixType>::hqr2(void)
if (s != 0) if (s != 0)
{ {
if (k != m) if (k != m)
m_H(k,k-1) = -s * x; matH(k,k-1) = -s * x;
else if (l != m) else if (l != m)
m_H(k,k-1) = -m_H(k,k-1); matH(k,k-1) = -matH(k,k-1);
p = p + s; p = p + s;
x = p / s; x = p / s;
@ -652,27 +674,27 @@ void EigenSolver<MatrixType>::hqr2(void)
// Row modification // Row modification
for (int j = k; j < nn; j++) for (int j = k; j < nn; j++)
{ {
p = m_H(k,j) + q * m_H(k+1,j); p = matH(k,j) + q * matH(k+1,j);
if (notlast) if (notlast)
{ {
p = p + r * m_H(k+2,j); p = p + r * matH(k+2,j);
m_H(k+2,j) = m_H(k+2,j) - p * z; matH(k+2,j) = matH(k+2,j) - p * z;
} }
m_H(k,j) = m_H(k,j) - p * x; matH(k,j) = matH(k,j) - p * x;
m_H(k+1,j) = m_H(k+1,j) - p * y; matH(k+1,j) = matH(k+1,j) - p * y;
} }
// Column modification // Column modification
for (int i = 0; i <= std::min(n,k+3); i++) for (int i = 0; i <= std::min(n,k+3); i++)
{ {
p = x * m_H(i,k) + y * m_H(i,k+1); p = x * matH(i,k) + y * matH(i,k+1);
if (notlast) if (notlast)
{ {
p = p + z * m_H(i,k+2); p = p + z * matH(i,k+2);
m_H(i,k+2) = m_H(i,k+2) - p * r; matH(i,k+2) = matH(i,k+2) - p * r;
} }
m_H(i,k) = m_H(i,k) - p; matH(i,k) = matH(i,k) - p;
m_H(i,k+1) = m_H(i,k+1) - p * q; matH(i,k+1) = matH(i,k+1) - p * q;
} }
// Accumulate transformations // Accumulate transformations
@ -700,20 +722,20 @@ void EigenSolver<MatrixType>::hqr2(void)
for (n = nn-1; n >= 0; n--) for (n = nn-1; n >= 0; n--)
{ {
p = m_eivalr[n]; p = m_eivalues[n].real();
q = m_eivali[n]; q = m_eivalues[n].imag();
// Scalar vector // Scalar vector
if (q == 0) if (q == 0)
{ {
int l = n; int l = n;
m_H(n,n) = 1.0; matH(n,n) = 1.0;
for (int i = n-1; i >= 0; i--) for (int i = n-1; i >= 0; i--)
{ {
w = m_H(i,i) - p; w = matH(i,i) - p;
r = (m_H.row(i).end(nn-l) * m_H.col(n).end(nn-l))(0,0); r = (matH.row(i).end(nn-l) * matH.col(n).end(nn-l))(0,0);
if (m_eivali[i] < 0.0) if (m_eivalues[i].imag() < 0.0)
{ {
z = w; z = w;
s = r; s = r;
@ -721,30 +743,30 @@ void EigenSolver<MatrixType>::hqr2(void)
else else
{ {
l = i; l = i;
if (m_eivali[i] == 0.0) if (m_eivalues[i].imag() == 0.0)
{ {
if (w != 0.0) if (w != 0.0)
m_H(i,n) = -r / w; matH(i,n) = -r / w;
else else
m_H(i,n) = -r / (eps * norm); matH(i,n) = -r / (eps * norm);
} }
else // Solve real equations else // Solve real equations
{ {
x = m_H(i,i+1); x = matH(i,i+1);
y = m_H(i+1,i); y = matH(i+1,i);
q = (m_eivalr[i] - p) * (m_eivalr[i] - p) + m_eivali[i] * m_eivali[i]; q = (m_eivalues[i].real() - p) * (m_eivalues[i].real() - p) + m_eivalues[i].imag() * m_eivalues[i].imag();
t = (x * s - z * r) / q; t = (x * s - z * r) / q;
m_H(i,n) = t; matH(i,n) = t;
if (ei_abs(x) > ei_abs(z)) if (ei_abs(x) > ei_abs(z))
m_H(i+1,n) = (-r - w * t) / x; matH(i+1,n) = (-r - w * t) / x;
else else
m_H(i+1,n) = (-s - y * t) / z; matH(i+1,n) = (-s - y * t) / z;
} }
// Overflow control // Overflow control
t = ei_abs(m_H(i,n)); t = ei_abs(matH(i,n));
if ((eps * t) * t > 1) if ((eps * t) * t > 1)
m_H.col(n).end(nn-i) /= t; matH.col(n).end(nn-i) /= t;
} }
} }
} }
@ -754,27 +776,27 @@ void EigenSolver<MatrixType>::hqr2(void)
int l = n-1; int l = n-1;
// Last vector component imaginary so matrix is triangular // Last vector component imaginary so matrix is triangular
if (ei_abs(m_H(n,n-1)) > ei_abs(m_H(n-1,n))) if (ei_abs(matH(n,n-1)) > ei_abs(matH(n-1,n)))
{ {
m_H(n-1,n-1) = q / m_H(n,n-1); matH(n-1,n-1) = q / matH(n,n-1);
m_H(n-1,n) = -(m_H(n,n) - p) / m_H(n,n-1); matH(n-1,n) = -(matH(n,n) - p) / matH(n,n-1);
} }
else else
{ {
cc = cdiv<Scalar>(0.0,-m_H(n-1,n),m_H(n-1,n-1)-p,q); cc = cdiv<Scalar>(0.0,-matH(n-1,n),matH(n-1,n-1)-p,q);
m_H(n-1,n-1) = ei_real(cc); matH(n-1,n-1) = ei_real(cc);
m_H(n-1,n) = ei_imag(cc); matH(n-1,n) = ei_imag(cc);
} }
m_H(n,n-1) = 0.0; matH(n,n-1) = 0.0;
m_H(n,n) = 1.0; matH(n,n) = 1.0;
for (int i = n-2; i >= 0; i--) for (int i = n-2; i >= 0; i--)
{ {
Scalar ra,sa,vr,vi; Scalar ra,sa,vr,vi;
ra = (m_H.row(i).end(nn-l) * m_H.col(n-1).end(nn-l)).lazy()(0,0); ra = (matH.row(i).end(nn-l) * matH.col(n-1).end(nn-l)).lazy()(0,0);
sa = (m_H.row(i).end(nn-l) * m_H.col(n).end(nn-l)).lazy()(0,0); sa = (matH.row(i).end(nn-l) * matH.col(n).end(nn-l)).lazy()(0,0);
w = m_H(i,i) - p; w = matH(i,i) - p;
if (m_eivali[i] < 0.0) if (m_eivalues[i].imag() < 0.0)
{ {
z = w; z = w;
r = ra; r = ra;
@ -783,42 +805,42 @@ void EigenSolver<MatrixType>::hqr2(void)
else else
{ {
l = i; l = i;
if (m_eivali[i] == 0) if (m_eivalues[i].imag() == 0)
{ {
cc = cdiv(-ra,-sa,w,q); cc = cdiv(-ra,-sa,w,q);
m_H(i,n-1) = ei_real(cc); matH(i,n-1) = ei_real(cc);
m_H(i,n) = ei_imag(cc); matH(i,n) = ei_imag(cc);
} }
else else
{ {
// Solve complex equations // Solve complex equations
x = m_H(i,i+1); x = matH(i,i+1);
y = m_H(i+1,i); y = matH(i+1,i);
vr = (m_eivalr[i] - p) * (m_eivalr[i] - p) + m_eivali[i] * m_eivali[i] - q * q; vr = (m_eivalues[i].real() - p) * (m_eivalues[i].real() - p) + m_eivalues[i].imag() * m_eivalues[i].imag() - q * q;
vi = (m_eivalr[i] - p) * 2.0 * q; vi = (m_eivalues[i].real() - p) * 2.0 * q;
if ((vr == 0.0) && (vi == 0.0)) if ((vr == 0.0) && (vi == 0.0))
vr = eps * norm * (ei_abs(w) + ei_abs(q) + ei_abs(x) + ei_abs(y) + ei_abs(z)); vr = eps * norm * (ei_abs(w) + ei_abs(q) + ei_abs(x) + ei_abs(y) + ei_abs(z));
cc= cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi); cc= cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
m_H(i,n-1) = ei_real(cc); matH(i,n-1) = ei_real(cc);
m_H(i,n) = ei_imag(cc); matH(i,n) = ei_imag(cc);
if (ei_abs(x) > (ei_abs(z) + ei_abs(q))) if (ei_abs(x) > (ei_abs(z) + ei_abs(q)))
{ {
m_H(i+1,n-1) = (-ra - w * m_H(i,n-1) + q * m_H(i,n)) / x; matH(i+1,n-1) = (-ra - w * matH(i,n-1) + q * matH(i,n)) / x;
m_H(i+1,n) = (-sa - w * m_H(i,n) - q * m_H(i,n-1)) / x; matH(i+1,n) = (-sa - w * matH(i,n) - q * matH(i,n-1)) / x;
} }
else else
{ {
cc = cdiv(-r-y*m_H(i,n-1),-s-y*m_H(i,n),z,q); cc = cdiv(-r-y*matH(i,n-1),-s-y*matH(i,n),z,q);
m_H(i+1,n-1) = ei_real(cc); matH(i+1,n-1) = ei_real(cc);
m_H(i+1,n) = ei_imag(cc); matH(i+1,n) = ei_imag(cc);
} }
} }
// Overflow control // Overflow control
t = std::max(ei_abs(m_H(i,n-1)),ei_abs(m_H(i,n))); t = std::max(ei_abs(matH(i,n-1)),ei_abs(matH(i,n)));
if ((eps * t) * t > 1) if ((eps * t) * t > 1)
m_H.block(i, n-1, nn-i, 2) /= t; matH.block(i, n-1, nn-i, 2) /= t;
} }
} }
@ -832,7 +854,7 @@ void EigenSolver<MatrixType>::hqr2(void)
// in this algo low==0 and high==nn-1 !! // in this algo low==0 and high==nn-1 !!
if (i < low || i > high) if (i < low || i > high)
{ {
m_eivec.row(i).end(nn-i) = m_H.row(i).end(nn-i); m_eivec.row(i).end(nn-i) = matH.row(i).end(nn-i);
} }
} }
@ -841,7 +863,7 @@ void EigenSolver<MatrixType>::hqr2(void)
for (int j = nn-1; j >= low; j--) for (int j = nn-1; j >= low; j--)
{ {
int bSize = std::min(j,high)-low+1; int bSize = std::min(j,high)-low+1;
m_eivec.col(j).block(low, bRows) = (m_eivec.block(low, low, bRows, bSize) * m_H.col(j).block(low, bSize)); m_eivec.col(j).block(low, bRows) = (m_eivec.block(low, low, bRows, bSize) * matH.col(j).block(low, bSize));
} }
} }