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Cleaning in BDCSVD (formating, handling of transpose case, remove some for loops)

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This commit is contained in:
Gael Guennebaud 2014-09-03 10:15:24 +02:00
parent a96f3d629c
commit c82dc227f1
2 changed files with 202 additions and 240 deletions
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1
Eigen/src/SVD/SVDBase.h
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@ -267,7 +267,6 @@ void SVDBase<Derived>::_solve_impl(const RhsType &rhs, DstType &dst) const
Matrix<Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime, RhsType::MaxColsAtCompileTime> tmp; Matrix<Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime, RhsType::MaxColsAtCompileTime> tmp;
Index l_rank = rank(); Index l_rank = rank();
tmp.noalias() = m_matrixU.leftCols(l_rank).adjoint() * rhs; tmp.noalias() = m_matrixU.leftCols(l_rank).adjoint() * rhs;
tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp; tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
dst = m_matrixV.leftCols(l_rank) * tmp; dst = m_matrixV.leftCols(l_rank) * tmp;

441
unsupported/Eigen/src/BDCSVD/BDCSVD.h
View File

@ -19,10 +19,6 @@
#ifndef EIGEN_BDCSVD_H #ifndef EIGEN_BDCSVD_H
#define EIGEN_BDCSVD_H #define EIGEN_BDCSVD_H
#define EPSILON 0.0000000000000001
#define ALGOSWAP 16
namespace Eigen { namespace Eigen {
template<typename _MatrixType> class BDCSVD; template<typename _MatrixType> class BDCSVD;
@ -88,7 +84,7 @@ public:
* The default constructor is useful in cases in which the user intends to * The default constructor is useful in cases in which the user intends to
* perform decompositions via BDCSVD::compute(const MatrixType&). * perform decompositions via BDCSVD::compute(const MatrixType&).
*/ */
BDCSVD() : algoswap(ALGOSWAP), m_numIters(0) BDCSVD() : m_algoswap(16), m_numIters(0)
{} {}
@ -99,7 +95,7 @@ public:
* \sa BDCSVD() * \sa BDCSVD()
*/ */
BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0) BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
: algoswap(ALGOSWAP), m_numIters(0) : m_algoswap(16), m_numIters(0)
{ {
allocate(rows, cols, computationOptions); allocate(rows, cols, computationOptions);
} }
@ -115,7 +111,7 @@ public:
* available with the (non - default) FullPivHouseholderQR preconditioner. * available with the (non - default) FullPivHouseholderQR preconditioner.
*/ */
BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0) BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
: algoswap(ALGOSWAP), m_numIters(0) : m_algoswap(16), m_numIters(0)
{ {
compute(matrix, computationOptions); compute(matrix, computationOptions);
} }
@ -150,35 +146,7 @@ public:
void setSwitchSize(int s) void setSwitchSize(int s)
{ {
eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3"); eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
algoswap = s; m_algoswap = s;
}
const MatrixUType& matrixU() const
{
eigen_assert(this->m_isInitialized && "SVD is not initialized.");
if (isTranspose){
eigen_assert(this->computeV() && "This SVD decomposition didn't compute U. Did you ask for it?");
return this->m_matrixV;
}
else
{
eigen_assert(this->computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
return this->m_matrixU;
}
}
const MatrixVType& matrixV() const
{
eigen_assert(this->m_isInitialized && "SVD is not initialized.");
if (isTranspose){
eigen_assert(this->computeU() && "This SVD decomposition didn't compute V. Did you ask for it?");
return this->m_matrixU;
}
else
{
eigen_assert(this->computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
return this->m_matrixV;
}
} }
private: private:
@ -194,15 +162,26 @@ private:
void deflation43(Index firstCol, Index shift, Index i, Index size); void deflation43(Index firstCol, Index shift, Index i, Index size);
void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size); void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift); void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
void copyUV(const typename internal::UpperBidiagonalization<MatrixX>::HouseholderUSequenceType& householderU, template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
const typename internal::UpperBidiagonalization<MatrixX>::HouseholderVSequenceType& householderV); void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev);
protected: protected:
MatrixXr m_naiveU, m_naiveV; MatrixXr m_naiveU, m_naiveV;
MatrixXr m_computed; MatrixXr m_computed;
Index nRec; Index m_nRec;
int algoswap; int m_algoswap;
bool isTranspose, compU, compV; bool m_isTranspose, m_compU, m_compV;
using Base::m_singularValues;
using Base::m_diagSize;
using Base::m_computeFullU;
using Base::m_computeFullV;
using Base::m_computeThinU;
using Base::m_computeThinV;
using Base::m_matrixU;
using Base::m_matrixV;
using Base::m_isInitialized;
using Base::m_nonzeroSingularValues;
public: public:
int m_numIters; int m_numIters;
@ -213,50 +192,35 @@ public:
template<typename MatrixType> template<typename MatrixType>
void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions) void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
{ {
isTranspose = (cols > rows); m_isTranspose = (cols > rows);
if (Base::allocate(rows, cols, computationOptions)) return; if (Base::allocate(rows, cols, computationOptions))
m_computed = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize ); return;
if (isTranspose){
compU = this->computeU();
compV = this->computeV();
}
else
{
compV = this->computeU();
compU = this->computeV();
}
if (compU) m_naiveU = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize + 1 );
else m_naiveU = MatrixXr::Zero(2, this->m_diagSize + 1 );
if (compV) m_naiveV = MatrixXr::Zero(this->m_diagSize, this->m_diagSize); m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize );
m_compU = computeV();
m_compV = computeU();
if (m_isTranspose)
std::swap(m_compU, m_compV);
if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 );
//should be changed for a cleaner implementation else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 );
if (isTranspose){
bool aux; if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize);
if (this->computeU()||this->computeV()){
aux = this->m_computeFullU;
this->m_computeFullU = this->m_computeFullV;
this->m_computeFullV = aux;
aux = this->m_computeThinU;
this->m_computeThinU = this->m_computeThinV;
this->m_computeThinV = aux;
}
}
}// end allocate }// end allocate
// Methode which compute the BDCSVD for the int // Methode which compute the BDCSVD for the int
template<> template<>
BDCSVD<Matrix<int, Dynamic, Dynamic> >& BDCSVD<Matrix<int, Dynamic, Dynamic> >::compute(const MatrixType& matrix, unsigned int computationOptions) { BDCSVD<Matrix<int, Dynamic, Dynamic> >& BDCSVD<Matrix<int, Dynamic, Dynamic> >::compute(const MatrixType& matrix, unsigned int computationOptions)
{
allocate(matrix.rows(), matrix.cols(), computationOptions); allocate(matrix.rows(), matrix.cols(), computationOptions);
this->m_nonzeroSingularValues = 0; m_nonzeroSingularValues = 0;
m_computed = Matrix<int, Dynamic, Dynamic>::Zero(rows(), cols()); m_computed = Matrix<int, Dynamic, Dynamic>::Zero(rows(), cols());
for (int i=0; i<this->m_diagSize; i++) {
this->m_singularValues.coeffRef(i) = 0; m_singularValues.head(m_diagSize).setZero();
}
if (this->m_computeFullU) this->m_matrixU = Matrix<int, Dynamic, Dynamic>::Zero(rows(), rows()); if (m_computeFullU) m_matrixU.setZero(rows(), rows());
if (this->m_computeFullV) this->m_matrixV = Matrix<int, Dynamic, Dynamic>::Zero(cols(), cols()); if (m_computeFullV) m_matrixV.setZero(cols(), cols());
this->m_isInitialized = true; m_isInitialized = true;
return *this; return *this;
} }
@ -268,59 +232,62 @@ BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsign
allocate(matrix.rows(), matrix.cols(), computationOptions); allocate(matrix.rows(), matrix.cols(), computationOptions);
using std::abs; using std::abs;
//**** step 1 Bidiagonalization isTranspose = (matrix.cols()>matrix.rows()) ; //**** step 1 Bidiagonalization m_isTranspose = (matrix.cols()>matrix.rows()) ;
MatrixType copy; MatrixType copy;
if (isTranspose) copy = matrix.adjoint(); if (m_isTranspose) copy = matrix.adjoint();
else copy = matrix; else copy = matrix;
internal::UpperBidiagonalization<MatrixX> bid(copy); internal::UpperBidiagonalization<MatrixX> bid(copy);
//**** step 2 Divide //**** step 2 Divide
m_computed.topRows(this->m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose(); m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
m_computed.template bottomRows<1>().setZero(); m_computed.template bottomRows<1>().setZero();
divide(0, this->m_diagSize - 1, 0, 0, 0); divide(0, m_diagSize - 1, 0, 0, 0);
//**** step 3 copy //**** step 3 copy
for (int i=0; i<this->m_diagSize; i++) { for (int i=0; i<m_diagSize; i++)
{
RealScalar a = abs(m_computed.coeff(i, i)); RealScalar a = abs(m_computed.coeff(i, i));
this->m_singularValues.coeffRef(i) = a; m_singularValues.coeffRef(i) = a;
if (a == 0){ if (a == 0)
this->m_nonzeroSingularValues = i; {
this->m_singularValues.tail(this->m_diagSize - i - 1).setZero(); m_nonzeroSingularValues = i;
m_singularValues.tail(m_diagSize - i - 1).setZero();
break; break;
} }
else if (i == this->m_diagSize - 1) else if (i == m_diagSize - 1)
{ {
this->m_nonzeroSingularValues = i + 1; m_nonzeroSingularValues = i + 1;
break; break;
} }
} }
copyUV(bid.householderU(), bid.householderV()); if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
this->m_isInitialized = true; else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
m_isInitialized = true;
return *this; return *this;
}// end compute }// end compute
template<typename MatrixType> template<typename MatrixType>
void BDCSVD<MatrixType>::copyUV(const typename internal::UpperBidiagonalization<MatrixX>::HouseholderUSequenceType& householderU, template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
const typename internal::UpperBidiagonalization<MatrixX>::HouseholderVSequenceType& householderV) void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV)
{ {
// Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
if (this->computeU()){ if (computeU())
Index Ucols = this->m_computeThinU ? this->m_nonzeroSingularValues : householderU.cols(); {
this->m_matrixU = MatrixX::Identity(householderU.cols(), Ucols); Index Ucols = m_computeThinU ? m_nonzeroSingularValues : householderU.cols();
Index blockCols = this->m_computeThinU ? this->m_nonzeroSingularValues : this->m_diagSize; m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
this->m_matrixU.block(0, 0, this->m_diagSize, blockCols) = Index blockCols = m_computeThinU ? m_nonzeroSingularValues : m_diagSize;
m_naiveV.template cast<Scalar>().block(0, 0, this->m_diagSize, blockCols); m_matrixU.topLeftCorner(m_diagSize, blockCols) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, blockCols);
this->m_matrixU = householderU * this->m_matrixU; m_matrixU = householderU * m_matrixU;
} }
if (this->computeV()){ if (computeV())
Index Vcols = this->m_computeThinV ? this->m_nonzeroSingularValues : householderV.cols(); {
this->m_matrixV = MatrixX::Identity(householderV.cols(), Vcols); Index Vcols = m_computeThinV ? m_nonzeroSingularValues : householderV.cols();
Index blockCols = this->m_computeThinV ? this->m_nonzeroSingularValues : this->m_diagSize; m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
this->m_matrixV.block(0, 0, this->m_diagSize, blockCols) = Index blockCols = m_computeThinV ? m_nonzeroSingularValues : m_diagSize;
m_naiveU.template cast<Scalar>().block(0, 0, this->m_diagSize, blockCols); m_matrixV.topLeftCorner(m_diagSize, blockCols) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, blockCols);
this->m_matrixV = householderV * this->m_matrixV; m_matrixV = householderV * m_matrixV;
} }
} }
@ -335,8 +302,7 @@ void BDCSVD<MatrixType>::copyUV(const typename internal::UpperBidiagonalization<
//@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix //@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix
// to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper. // to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
template<typename MatrixType> template<typename MatrixType>
void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift)
Index firstColW, Index shift)
{ {
// requires nbRows = nbCols + 1; // requires nbRows = nbCols + 1;
using std::pow; using std::pow;
@ -351,21 +317,19 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
MatrixXr l, f; MatrixXr l, f;
// We use the other algorithm which is more efficient for small // We use the other algorithm which is more efficient for small
// matrices. // matrices.
if (n < algoswap){ if (n < m_algoswap)
JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), {
ComputeFullU | (ComputeFullV * compV)) ; JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0)) ;
if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() << b.matrixU(); if (m_compU)
m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU();
else else
{ {
m_naiveU.row(0).segment(firstCol, n + 1).real() << b.matrixU().row(0); m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0);
m_naiveU.row(1).segment(firstCol, n + 1).real() << b.matrixU().row(n); m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n);
} }
if (compV) m_naiveV.block(firstRowW, firstColW, n, n).real() << b.matrixV(); if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV();
m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero(); m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
for (int i=0; i<n; i++) m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n);
{
m_computed(firstCol + shift + i, firstCol + shift +i) = b.singularValues().coeffRef(i);
}
return; return;
} }
// We use the divide and conquer algorithm // We use the divide and conquer algorithm
@ -376,7 +340,7 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
// right submatrix before the left one. // right submatrix before the left one.
divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift); divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1); divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
if (compU) if (m_compU)
{ {
lambda = m_naiveU(firstCol + k, firstCol + k); lambda = m_naiveU(firstCol + k, firstCol + k);
phi = m_naiveU(firstCol + k + 1, lastCol + 1); phi = m_naiveU(firstCol + k + 1, lastCol + 1);
@ -386,9 +350,8 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
lambda = m_naiveU(1, firstCol + k); lambda = m_naiveU(1, firstCol + k);
phi = m_naiveU(0, lastCol + 1); phi = m_naiveU(0, lastCol + 1);
} }
r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi));
+ abs(betaK * phi) * abs(betaK * phi)); if (m_compU)
if (compU)
{ {
l = m_naiveU.row(firstCol + k).segment(firstCol, k); l = m_naiveU.row(firstCol + k).segment(firstCol, k);
f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1); f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
@ -398,7 +361,7 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
l = m_naiveU.row(1).segment(firstCol, k); l = m_naiveU.row(1).segment(firstCol, k);
f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1); f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
} }
if (compV) m_naiveV(firstRowW+k, firstColW) = 1; if (m_compV) m_naiveV(firstRowW+k, firstColW) = 1;
if (r0 == 0) if (r0 == 0)
{ {
c0 = 1; c0 = 1;
@ -409,21 +372,18 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
c0 = alphaK * lambda / r0; c0 = alphaK * lambda / r0;
s0 = betaK * phi / r0; s0 = betaK * phi / r0;
} }
if (compU) if (m_compU)
{ {
MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1)); MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
// we shiftW Q1 to the right // we shiftW Q1 to the right
for (Index i = firstCol + k - 1; i >= firstCol; i--) for (Index i = firstCol + k - 1; i >= firstCol; i--)
{ m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1);
m_naiveU.col(i + 1).segment(firstCol, k + 1) << m_naiveU.col(i).segment(firstCol, k + 1);
}
// we shift q1 at the left with a factor c0 // we shift q1 at the left with a factor c0
m_naiveU.col(firstCol).segment( firstCol, k + 1) << (q1 * c0); m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0);
// last column = q1 * - s0 // last column = q1 * - s0
m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) << (q1 * ( - s0)); m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0));
// first column = q2 * s0 // first column = q2 * s0
m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) << m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0;
m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *s0;
// q2 *= c0 // q2 *= c0
m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0; m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
} }
@ -432,9 +392,7 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
RealScalar q1 = (m_naiveU(0, firstCol + k)); RealScalar q1 = (m_naiveU(0, firstCol + k));
// we shift Q1 to the right // we shift Q1 to the right
for (Index i = firstCol + k - 1; i >= firstCol; i--) for (Index i = firstCol + k - 1; i >= firstCol; i--)
{
m_naiveU(0, i + 1) = m_naiveU(0, i); m_naiveU(0, i + 1) = m_naiveU(0, i);
}
// we shift q1 at the left with a factor c0 // we shift q1 at the left with a factor c0
m_naiveU(0, firstCol) = (q1 * c0); m_naiveU(0, firstCol) = (q1 * c0);
// last column = q1 * - s0 // last column = q1 * - s0
@ -447,8 +405,8 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero(); m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
} }
m_computed(firstCol + shift, firstCol + shift) = r0; m_computed(firstCol + shift, firstCol + shift) = r0;
m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) << alphaK * l.transpose().real(); m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real();
m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) << betaK * f.transpose().real(); m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real();
// Second part: try to deflate singular values in combined matrix // Second part: try to deflate singular values in combined matrix
@ -458,9 +416,9 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
MatrixXr UofSVD, VofSVD; MatrixXr UofSVD, VofSVD;
VectorType singVals; VectorType singVals;
computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD); computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD);
if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1) *= UofSVD; if (m_compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1) *= UofSVD; // FIXME this requires a temporary
else m_naiveU.block(0, firstCol, 2, n + 1) *= UofSVD; else m_naiveU.block(0, firstCol, 2, n + 1) *= UofSVD; // FIXME this requires a temporary, and exploit that there are 2 rows at compile time
if (compV) m_naiveV.block(firstRowW, firstColW, n, n) *= VofSVD; if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n) *= VofSVD; // FIXME this requires a temporary
m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero(); m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero();
m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals; m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals;
}// end divide }// end divide
@ -468,7 +426,7 @@ void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
// Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in // Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in
// the first column and on the diagonal and has undergone deflation, so diagonal is in increasing // the first column and on the diagonal and has undergone deflation, so diagonal is in increasing
// order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except // order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except
// that if compV is false, then V is not computed. Singular values are sorted in decreasing order. // that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order.
// //
// TODO Opportunities for optimization: better root finding algo, better stopping criterion, better // TODO Opportunities for optimization: better root finding algo, better stopping criterion, better
// handling of round-off errors, be consistent in ordering // handling of round-off errors, be consistent in ordering
@ -483,7 +441,7 @@ void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, Vec
// compute singular values and vectors (in decreasing order) // compute singular values and vectors (in decreasing order)
singVals.resize(n); singVals.resize(n);
U.resize(n+1, n+1); U.resize(n+1, n+1);
if (compV) V.resize(n, n); if (m_compV) V.resize(n, n);
if (col0.hasNaN() || diag.hasNaN()) return; if (col0.hasNaN() || diag.hasNaN()) return;
@ -495,7 +453,7 @@ void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, Vec
// Reverse order so that singular values in increased order // Reverse order so that singular values in increased order
singVals.reverseInPlace(); singVals.reverseInPlace();
U.leftCols(n) = U.leftCols(n).rowwise().reverse().eval(); U.leftCols(n) = U.leftCols(n).rowwise().reverse().eval();
if (compV) V = V.rowwise().reverse().eval(); if (m_compV) V = V.rowwise().reverse().eval();
} }
template <typename MatrixType> template <typename MatrixType>
@ -504,10 +462,13 @@ void BDCSVD<MatrixType>::computeSingVals(const ArrayXr& col0, const ArrayXr& dia
{ {
using std::abs; using std::abs;
using std::swap; using std::swap;
using std::max;
Index n = col0.size(); Index n = col0.size();
for (Index k = 0; k < n; ++k) { for (Index k = 0; k < n; ++k)
if (col0(k) == 0) { {
if (col0(k) == 0)
{
// entry is deflated, so singular value is on diagonal // entry is deflated, so singular value is on diagonal
singVals(k) = diag(k); singVals(k) = diag(k);
mus(k) = 0; mus(k) = 0;
@ -523,27 +484,29 @@ void BDCSVD<MatrixType>::computeSingVals(const ArrayXr& col0, const ArrayXr& dia
RealScalar mid = left + (right-left) / 2; RealScalar mid = left + (right-left) / 2;
RealScalar fMid = 1 + (col0.square() / ((diag + mid) * (diag - mid))).sum(); RealScalar fMid = 1 + (col0.square() / ((diag + mid) * (diag - mid))).sum();
RealScalar shift; RealScalar shift = (k == n-1 || fMid > 0) ? left : right;
if (k == n-1 || fMid > 0) shift = left;
else shift = right;
// measure everything relative to shift // measure everything relative to shift
ArrayXr diagShifted = diag - shift; ArrayXr diagShifted = diag - shift;
// initial guess // initial guess
RealScalar muPrev, muCur; RealScalar muPrev, muCur;
if (shift == left) { if (shift == left)
{
muPrev = (right - left) * 0.1; muPrev = (right - left) * 0.1;
if (k == n-1) muCur = right - left; if (k == n-1) muCur = right - left;
else muCur = (right - left) * 0.5; else muCur = (right - left) * 0.5;
} else { }
else
{
muPrev = -(right - left) * 0.1; muPrev = -(right - left) * 0.1;
muCur = -(right - left) * 0.5; muCur = -(right - left) * 0.5;
} }
RealScalar fPrev = 1 + (col0.square() / ((diagShifted - muPrev) * (diag + shift + muPrev))).sum(); RealScalar fPrev = 1 + (col0.square() / ((diagShifted - muPrev) * (diag + shift + muPrev))).sum();
RealScalar fCur = 1 + (col0.square() / ((diagShifted - muCur) * (diag + shift + muCur))).sum(); RealScalar fCur = 1 + (col0.square() / ((diagShifted - muCur) * (diag + shift + muCur))).sum();
if (abs(fPrev) < abs(fCur)) { if (abs(fPrev) < abs(fCur))
{
swap(fPrev, fCur); swap(fPrev, fCur);
swap(muPrev, muCur); swap(muPrev, muCur);
} }
@ -551,7 +514,8 @@ void BDCSVD<MatrixType>::computeSingVals(const ArrayXr& col0, const ArrayXr& dia
// rational interpolation: fit a function of the form a / mu + b through the two previous // rational interpolation: fit a function of the form a / mu + b through the two previous
// iterates and use its zero to compute the next iterate // iterates and use its zero to compute the next iterate
bool useBisection = false; bool useBisection = false;
while (abs(muCur - muPrev) > 8 * NumTraits<RealScalar>::epsilon() * (std::max)(abs(muCur), abs(muPrev)) && fCur != fPrev && !useBisection) { while (abs(muCur - muPrev) > 8 * NumTraits<RealScalar>::epsilon() * (max)(abs(muCur), abs(muPrev)) && fCur != fPrev && !useBisection)
{
++m_numIters; ++m_numIters;
RealScalar a = (fCur - fPrev) / (1/muCur - 1/muPrev); RealScalar a = (fCur - fPrev) / (1/muCur - 1/muPrev);
@ -567,13 +531,17 @@ void BDCSVD<MatrixType>::computeSingVals(const ArrayXr& col0, const ArrayXr& dia
} }
// fall back on bisection method if rational interpolation did not work // fall back on bisection method if rational interpolation did not work
if (useBisection) { if (useBisection)
{
RealScalar leftShifted, rightShifted; RealScalar leftShifted, rightShifted;
if (shift == left) { if (shift == left)
{
leftShifted = 1e-30; leftShifted = 1e-30;
if (k == 0) rightShifted = right - left; if (k == 0) rightShifted = right - left;
else rightShifted = (right - left) * 0.6; // theoretically we can take 0.5, but let's be safe else rightShifted = (right - left) * 0.6; // theoretically we can take 0.5, but let's be safe
} else { }
else
{
leftShifted = -(right - left) * 0.6; leftShifted = -(right - left) * 0.6;
rightShifted = -1e-30; rightShifted = -1e-30;
} }
@ -582,13 +550,17 @@ void BDCSVD<MatrixType>::computeSingVals(const ArrayXr& col0, const ArrayXr& dia
RealScalar fRight = 1 + (col0.square() / ((diagShifted - rightShifted) * (diag + shift + rightShifted))).sum(); RealScalar fRight = 1 + (col0.square() / ((diagShifted - rightShifted) * (diag + shift + rightShifted))).sum();
assert(fLeft * fRight < 0); assert(fLeft * fRight < 0);
while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * (std::max)(abs(leftShifted), abs(rightShifted))) { while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * (max)(abs(leftShifted), abs(rightShifted)))
{
RealScalar midShifted = (leftShifted + rightShifted) / 2; RealScalar midShifted = (leftShifted + rightShifted) / 2;
RealScalar fMid = 1 + (col0.square() / ((diagShifted - midShifted) * (diag + shift + midShifted))).sum(); RealScalar fMid = 1 + (col0.square() / ((diagShifted - midShifted) * (diag + shift + midShifted))).sum();
if (fLeft * fMid < 0) { if (fLeft * fMid < 0)
{
rightShifted = midShifted; rightShifted = midShifted;
fRight = fMid; fRight = fMid;
} else { }
else
{
leftShifted = midShifted; leftShifted = midShifted;
fLeft = fMid; fLeft = fMid;
} }
@ -615,13 +587,15 @@ void BDCSVD<MatrixType>::perturbCol0
(const ArrayXr& col0, const ArrayXr& diag, const VectorType& singVals, (const ArrayXr& col0, const ArrayXr& diag, const VectorType& singVals,
const ArrayXr& shifts, const ArrayXr& mus, ArrayXr& zhat) const ArrayXr& shifts, const ArrayXr& mus, ArrayXr& zhat)
{ {
using std::sqrt;
Index n = col0.size(); Index n = col0.size();
for (Index k = 0; k < n; ++k) { for (Index k = 0; k < n; ++k)
{
if (col0(k) == 0) if (col0(k) == 0)
zhat(k) = 0; zhat(k) = 0;
else { else
{
// see equation (3.6) // see equation (3.6)
using std::sqrt;
RealScalar tmp = RealScalar tmp =
sqrt( sqrt(
(singVals(n-1) + diag(k)) * (mus(n-1) + (shifts(n-1) - diag(k))) (singVals(n-1) + diag(k)) * (mus(n-1) + (shifts(n-1) - diag(k)))
@ -647,16 +621,21 @@ void BDCSVD<MatrixType>::computeSingVecs
const ArrayXr& shifts, const ArrayXr& mus, MatrixXr& U, MatrixXr& V) const ArrayXr& shifts, const ArrayXr& mus, MatrixXr& U, MatrixXr& V)
{ {
Index n = zhat.size(); Index n = zhat.size();
for (Index k = 0; k < n; ++k) { for (Index k = 0; k < n; ++k)
if (zhat(k) == 0) { {
if (zhat(k) == 0)
{
U.col(k) = VectorType::Unit(n+1, k); U.col(k) = VectorType::Unit(n+1, k);
if (compV) V.col(k) = VectorType::Unit(n, k); if (m_compV) V.col(k) = VectorType::Unit(n, k);
} else { }
else
{
U.col(k).head(n) = zhat / (((diag - shifts(k)) - mus(k)) * (diag + singVals[k])); U.col(k).head(n) = zhat / (((diag - shifts(k)) - mus(k)) * (diag + singVals[k]));
U(n,k) = 0; U(n,k) = 0;
U.col(k).normalize(); U.col(k).normalize();
if (compV) { if (m_compV)
{
V.col(k).tail(n-1) = (diag * zhat / (((diag - shifts(k)) - mus(k)) * (diag + singVals[k]))).tail(n-1); V.col(k).tail(n-1) = (diag * zhat / (((diag - shifts(k)) - mus(k)) * (diag + singVals[k]))).tail(n-1);
V(0,k) = -1; V(0,k) = -1;
V.col(k).normalize(); V.col(k).normalize();
@ -671,15 +650,17 @@ void BDCSVD<MatrixType>::computeSingVecs
// i >= 1, di almost null and zi non null. // i >= 1, di almost null and zi non null.
// We use a rotation to zero out zi applied to the left of M // We use a rotation to zero out zi applied to the left of M
template <typename MatrixType> template <typename MatrixType>
void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size){ void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size)
{
using std::abs; using std::abs;
using std::sqrt; using std::sqrt;
using std::pow; using std::pow;
RealScalar c = m_computed(firstCol + shift, firstCol + shift); RealScalar c = m_computed(firstCol + shift, firstCol + shift);
RealScalar s = m_computed(i, firstCol + shift); RealScalar s = m_computed(i, firstCol + shift);
RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2)); RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
if (r == 0){ if (r == 0)
m_computed(i, i)=0; {
m_computed(i, i) = 0;
return; return;
} }
c/=r; c/=r;
@ -687,7 +668,8 @@ void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index
m_computed(firstCol + shift, firstCol + shift) = r; m_computed(firstCol + shift, firstCol + shift) = r;
m_computed(i, firstCol + shift) = 0; m_computed(i, firstCol + shift) = 0;
m_computed(i, i) = 0; m_computed(i, i) = 0;
if (compU){ if (m_compU)
{
m_naiveU.col(firstCol).segment(firstCol,size) = m_naiveU.col(firstCol).segment(firstCol,size) =
c * m_naiveU.col(firstCol).segment(firstCol, size) - c * m_naiveU.col(firstCol).segment(firstCol, size) -
s * m_naiveU.col(i).segment(firstCol, size) ; s * m_naiveU.col(i).segment(firstCol, size) ;
@ -703,7 +685,8 @@ void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index
// i,j >= 1, i != j and |di - dj| < epsilon * norm2(M) // i,j >= 1, i != j and |di - dj| < epsilon * norm2(M)
// We apply two rotations to have zj = 0; // We apply two rotations to have zj = 0;
template <typename MatrixType> template <typename MatrixType>
void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size){ void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size)
{
using std::abs; using std::abs;
using std::sqrt; using std::sqrt;
using std::conj; using std::conj;
@ -711,7 +694,8 @@ void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index fi
RealScalar c = m_computed(firstColm, firstColm + j - 1); RealScalar c = m_computed(firstColm, firstColm + j - 1);
RealScalar s = m_computed(firstColm, firstColm + i - 1); RealScalar s = m_computed(firstColm, firstColm + i - 1);
RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2)); RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
if (r==0){ if (r==0)
{
m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j); m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
return; return;
} }
@ -720,7 +704,8 @@ void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index fi
m_computed(firstColm + i, firstColm) = r; m_computed(firstColm + i, firstColm) = r;
m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j); m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
m_computed(firstColm + j, firstColm) = 0; m_computed(firstColm + j, firstColm) = 0;
if (compU){ if (m_compU)
{
m_naiveU.col(firstColu + i).segment(firstColu, size) = m_naiveU.col(firstColu + i).segment(firstColu, size) =
c * m_naiveU.col(firstColu + i).segment(firstColu, size) - c * m_naiveU.col(firstColu + i).segment(firstColu, size) -
s * m_naiveU.col(firstColu + j).segment(firstColu, size) ; s * m_naiveU.col(firstColu + j).segment(firstColu, size) ;
@ -729,7 +714,8 @@ void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index fi
(c + s*s/c) * m_naiveU.col(firstColu + j).segment(firstColu, size) + (c + s*s/c) * m_naiveU.col(firstColu + j).segment(firstColu, size) +
(s/c) * m_naiveU.col(firstColu + i).segment(firstColu, size); (s/c) * m_naiveU.col(firstColu + i).segment(firstColu, size);
} }
if (compV){ if (m_compV)
{
m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) = m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) =
c * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) + c * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) +
s * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) ; s * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) ;
@ -743,72 +729,56 @@ void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index fi
// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive] // acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
template <typename MatrixType> template <typename MatrixType>
void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift){ void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift)
{
//condition 4.1 //condition 4.1
using std::sqrt; using std::sqrt;
using std::abs;
const Index length = lastCol + 1 - firstCol; const Index length = lastCol + 1 - firstCol;
RealScalar norm1 = m_computed.block(firstCol+shift, firstCol+shift, length, 1).squaredNorm(); RealScalar norm1 = m_computed.block(firstCol+shift, firstCol+shift, length, 1).squaredNorm();
RealScalar norm2 = m_computed.block(firstCol+shift, firstCol+shift, length, length).diagonal().squaredNorm(); RealScalar norm2 = m_computed.block(firstCol+shift, firstCol+shift, length, length).diagonal().squaredNorm();
RealScalar EPS = 10 * NumTraits<RealScalar>::epsilon() * sqrt(norm1 + norm2); RealScalar epsilon = 10 * NumTraits<RealScalar>::epsilon() * sqrt(norm1 + norm2);
if (m_computed(firstCol + shift, firstCol + shift) < EPS){ if (m_computed(firstCol + shift, firstCol + shift) < epsilon)
m_computed(firstCol + shift, firstCol + shift) = EPS; m_computed(firstCol + shift, firstCol + shift) = epsilon;
}
//condition 4.2 //condition 4.2
for (Index i=firstCol + shift + 1;i<=lastCol + shift;i++){ for (Index i=firstCol + shift + 1;i<=lastCol + shift;i++)
if (std::abs(m_computed(i, firstCol + shift)) < EPS){ if (abs(m_computed(i, firstCol + shift)) < epsilon)
m_computed(i, firstCol + shift) = 0; m_computed(i, firstCol + shift) = 0;
}
}
//condition 4.3 //condition 4.3
for (Index i=firstCol + shift + 1;i<=lastCol + shift; i++){ for (Index i=firstCol + shift + 1;i<=lastCol + shift; i++)
if (m_computed(i, i) < EPS){ if (m_computed(i, i) < epsilon)
deflation43(firstCol, shift, i, length); deflation43(firstCol, shift, i, length);
}
}
//condition 4.4 //condition 4.4
Index i=firstCol + shift + 1, j=firstCol + shift + k + 1; Index i=firstCol + shift + 1, j=firstCol + shift + k + 1;
//we stock the final place of each line //we stock the final place of each line
Index *permutation = new Index[length]; Index *permutation = new Index[length]; // FIXME avoid repeated dynamic memory allocation
for (Index p =1; p < length; p++) { for (Index p =1; p < length; p++)
if (i> firstCol + shift + k){ {
permutation[p] = j; if (i> firstCol + shift + k) permutation[p] = j++;
j++; else if (j> lastCol + shift) permutation[p] = i++;
} else if (j> lastCol + shift) else if (m_computed(i, i) < m_computed(j, j)) permutation[p] = j++;
{ else permutation[p] = i++;
permutation[p] = i;
i++;
}
else
{
if (m_computed(i, i) < m_computed(j, j)){
permutation[p] = j;
j++;
}
else
{
permutation[p] = i;
i++;
}
}
} }
//we do the permutation //we do the permutation
RealScalar aux; RealScalar aux;
//we stock the current index of each col //we stock the current index of each col
//and the column of each index //and the column of each index
Index *realInd = new Index[length]; Index *realInd = new Index[length]; // FIXME avoid repeated dynamic memory allocation
Index *realCol = new Index[length]; Index *realCol = new Index[length]; // FIXME avoid repeated dynamic memory allocation
for (int pos = 0; pos< length; pos++){ for (int pos = 0; pos< length; pos++)
{
realCol[pos] = pos + firstCol + shift; realCol[pos] = pos + firstCol + shift;
realInd[pos] = pos; realInd[pos] = pos;
} }
const Index Zero = firstCol + shift; const Index Zero = firstCol + shift;
VectorType temp; VectorType temp;
for (int i = 1; i < length - 1; i++){ for (int i = 1; i < length - 1; i++)
{
const Index I = i + Zero; const Index I = i + Zero;
const Index realI = realInd[i]; const Index realI = realInd[i];
const Index j = permutation[length - i] - Zero; const Index j = permutation[length - i] - Zero;
@ -825,25 +795,25 @@ void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index
m_computed(J, Zero) = aux; m_computed(J, Zero) = aux;
// change columns // change columns
if (compU) { if (m_compU)
{
temp = m_naiveU.col(I - shift).segment(firstCol, length + 1); temp = m_naiveU.col(I - shift).segment(firstCol, length + 1);
m_naiveU.col(I - shift).segment(firstCol, length + 1) << m_naiveU.col(I - shift).segment(firstCol, length + 1) = m_naiveU.col(J - shift).segment(firstCol, length + 1);
m_naiveU.col(J - shift).segment(firstCol, length + 1); m_naiveU.col(J - shift).segment(firstCol, length + 1) = temp;
m_naiveU.col(J - shift).segment(firstCol, length + 1) << temp;
} }
else else
{ {
temp = m_naiveU.col(I - shift).segment(0, 2); temp = m_naiveU.col(I - shift).segment(0, 2);
m_naiveU.col(I - shift).segment(0, 2) << m_naiveU.col(I - shift).template head<2>() = m_naiveU.col(J - shift).segment(0, 2);
m_naiveU.col(J - shift).segment(0, 2); m_naiveU.col(J - shift).template head<2>() = temp;
m_naiveU.col(J - shift).segment(0, 2) << temp;
} }
if (compV) { if (m_compV)
{
const Index CWI = I + firstColW - Zero; const Index CWI = I + firstColW - Zero;
const Index CWJ = J + firstColW - Zero; const Index CWJ = J + firstColW - Zero;
temp = m_naiveV.col(CWI).segment(firstRowW, length); temp = m_naiveV.col(CWI).segment(firstRowW, length);
m_naiveV.col(CWI).segment(firstRowW, length) << m_naiveV.col(CWJ).segment(firstRowW, length); m_naiveV.col(CWI).segment(firstRowW, length) = m_naiveV.col(CWJ).segment(firstRowW, length);
m_naiveV.col(CWJ).segment(firstRowW, length) << temp; m_naiveV.col(CWJ).segment(firstRowW, length) = temp;
} }
//update real pos //update real pos
@ -852,20 +822,13 @@ void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index
realInd[J - Zero] = realI; realInd[J - Zero] = realI;
realInd[I - Zero] = j; realInd[I - Zero] = j;
} }
for (Index i = firstCol + shift + 1; i<lastCol + shift;i++){ for (Index i = firstCol + shift + 1; i<lastCol + shift;i++)
if ((m_computed(i + 1, i + 1) - m_computed(i, i)) < EPS){ if ((m_computed(i + 1, i + 1) - m_computed(i, i)) < epsilon)
deflation44(firstCol , deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i - Zero, i + 1 - Zero, length);
firstCol + shift,
firstRowW, delete[] permutation;
firstColW, delete[] realInd;
i - Zero, delete[] realCol;
i + 1 - Zero,
length);
}
}
delete [] permutation;
delete [] realInd;
delete [] realCol;
}//end deflation }//end deflation