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/*
* Copyright (c) 2014, Michael Lehn
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1) Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2) Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
* 3) Neither the name of the FLENS development group nor the names of
* its contributors may be used to endorse or promote products derived
* from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/* Based on
*
SUBROUTINE DSTERF( N, D, E, INFO )
*
* -- LAPACK routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
*/
#ifndef FLENS_LAPACK_IMPL_STERF_TCC
#define FLENS_LAPACK_IMPL_STERF_TCC 1
#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>
namespace flens { namespace lapack {
//== generic lapack implementation =============================================
namespace generic {
template <typename VD, typename VE>
typename VD::IndexType
sterf_impl(DenseVector<VD> &d,
DenseVector<VE> &e)
{
using std::abs;
using std::sqrt;
typedef typename VD::ElementType T;
typedef typename VD::IndexType IndexType;
const T Zero(0), One(1), Two(2), Three(3);
const Underscore<IndexType> _;
const IndexType n = d.length();
const IndexType maxIt = 30;
IndexType info = 0;
//
// Quick return if possible
//
if (n<=1) {
return info;
}
//
// Determine the unit roundoff for this environment.
//
const T eps = lamch<T>(Precision);
const T eps2 = pow(eps, 2);
const T safeMin = lamch<T>(SafeMin);
const T safeMax = One / SafeMin;
const T sSaveMax = sqrt(safeMax) / Three;
const T sSaveMin = sqrt(safeMin) / eps2;
//const T rMax = lamch<T>(OverflowThreshold);
//
// Compute the eigenvalues of the tridiagonal matrix.
//
const IndexType nMaxIt = n*maxIt;
T sigma = Zero;
IndexType jTot = 0;
//
// Determine where the matrix splits and choose QL or QR iteration
// for each block, according to whether top or bottom diagonal
// element is smaller.
//
IndexType l1 = 1;
IndexType l, lSv, lEnd, lEndSv, m, iScale;
T normA, p, rte, rt1, rt2, r, c, s, alpha, gamma, bb, oldC, oldGamma;
START:
do {
if (l1>n) {
goto SORT;
}
if (l1>1) {
e(l1-1) = Zero;
}
for (m=l1; m<=n-1; ++m) {
if (abs(e(m))<=(sqrt(abs(d(m)))*sqrt(abs(d(m+1))))*eps) {
e(m) = Zero;
break;
}
}
l = l1;
lSv = l;
lEnd = m;
lEndSv = lEnd;
l1 = m + 1;
} while (lEnd==l);
//
// Scale submatrix in rows and columns L to LEND
//
normA = lanst(MaximumNorm, d(_(l,lEnd)), e(_(l,lEnd-1)));
iScale = 0;
if (normA==Zero) {
goto START;
}
if (normA>sSaveMax) {
iScale = 1;
lascl(LASCL::FullMatrix, 0, 0, normA, sSaveMax, d(_(l,lEnd)));
lascl(LASCL::FullMatrix, 0, 0, normA, sSaveMax, e(_(l,lEnd-1)));
} else if (normA<sSaveMin) {
iScale = 2;
lascl(LASCL::FullMatrix, 0, 0, normA, sSaveMin, d(_(l,lEnd)));
lascl(LASCL::FullMatrix, 0, 0, normA, sSaveMin, e(_(l,lEnd-1)));
}
for (IndexType i=l; i<=lEnd-1; ++i) {
e(i) = pow(e(i), 2);
}
//
// Choose between QL and QR iteration
//
if (abs(d(lEnd))<abs(d(l))) {
lEnd = lSv;
l = lEndSv;
}
if (lEnd>=l) {
//
// QL Iteration
//
// Look for small subdiagonal element.
//
QL_Start:
while (true) {
if (l!=lEnd) {
for (IndexType m=l; l<=lEnd-1; ++l) {
if (abs(e(m))<=eps2*abs(d(m)*d(m+1))) {
break;
}
}
}
if (m<lEnd) {
e(m) = Zero;
}
p = d(l);
if (m==l) {
break;
}
//
// If remaining matrix is 2 by 2, use DLAE2 to compute its
// eigenvalues.
//
if (m==l+1) {
rte = sqrt(e(l));
lae2(d(l), rte, d(l+1), rt1, rt2);
d(l) = rt1;
d(l+1) = rt2;
e(l) = Zero;
l += 2;
if (l<=lEnd) {
continue;
}
goto UNDO_SCALING;
}
if (jTot==nMaxIt) {
goto UNDO_SCALING;
}
++jTot;
//
// Form shift.
//
T rte = sqrt(e(l));
sigma = (d(l+1)-p) / (Two*rte);
r = lapy2(sigma, One);
sigma = p - (rte / (sigma+sign(r,sigma)));
c = One;
s = Zero;
gamma = d(m) - sigma;
p = gamma*gamma;
//
// Inner loop
//
for (IndexType i=m-1; i>=l; --i) {
bb = e(i);
r = p + bb;
if (i!=m-1) {
e(i+1) = s*r;
}
oldC = c;
c = p / r;
s = bb / r;
oldGamma = gamma;
alpha = d(i);
gamma = c*(alpha-sigma) - s*oldGamma;
d(i+1) = oldGamma + (alpha-gamma);
if (c!=Zero) {
p = (gamma*gamma)/c;
} else {
p = oldC*bb;
}
}
e(l) = s*p;
d(l) = sigma + gamma;
}
//
// Eigenvalue found.
//
d(l) = p;
++l;
if (l<=lEnd) {
goto QL_Start;
}
goto UNDO_SCALING;
} else {
//
// QR Iteration
//
// Look for small superdiagonal element.
//
QR_Start:
while (true) {
for (m=l; m>=lEnd+1; --m) {
if (abs(e(m-1))<=eps2*abs(d(m)*d(m-1))) {
break;
}
}
if (m>lEnd) {
e(m-1) = Zero;
}
p = d(l);
if (m==l) {
break;
}
//
// If remaining matrix is 2 by 2, use DLAE2 to compute its
// eigenvalues.
//
if (m==l-1) {
rte = sqrt(e(l-1));
lae2(d(l), rte, d(l-1), rt1, rt2);
d(l) = rt1;
d(l-1) = rt2;
e(l-1) = Zero;
l -= 2;
if (l>=lEnd) {
goto QR_Start;
}
goto UNDO_SCALING;
}
if (jTot==nMaxIt) {
goto UNDO_SCALING;
}
++jTot;
//
// Form shift.
//
rte = sqrt(e(l-1));
sigma = (d(l-1)-p) / (Two*rte);
r = lapy2(sigma, One);
sigma = p - (rte / (sigma+sign(r, sigma)));
c = One;
s = Zero;
gamma = d(m) - sigma;
p = gamma*gamma;
//
// Inner loop
//
for (IndexType i=m; i<=l-1; ++i) {
bb = e(i);
r = p + bb;
if (i!=m) {
e(i-1) = s*r;
}
oldC = c;
c = p / r;
s = bb / r;
oldGamma = gamma;
alpha = d(i+1);
gamma = c*(alpha-sigma) - s*oldGamma;
d(i) = oldGamma + (alpha-gamma);
if (c!=Zero) {
p = (gamma*gamma) / c;
} else {
p = oldC*bb;
}
}
e(l-1) = s*p;
d(l) = sigma + gamma;
}
//
// Eigenvalue found.
//
d(l) = p;
--l;
if (l>=lEnd) {
goto QR_Start;
}
goto UNDO_SCALING;
}
//
// Undo scaling if necessary
//
UNDO_SCALING:
if (iScale==1) {
lascl(LASCL::FullMatrix, 0, 0, sSaveMax, normA, d(_(lSv,lEndSv)));
}
if (iScale==2) {
lascl(LASCL::FullMatrix, 0, 0, sSaveMin, normA, d(_(lSv,lEndSv)));
}
//
// Check for no convergence to an eigenvalue after a total
// of N*MAXIT iterations.
//
if (jTot<nMaxIt) {
goto START;
}
for (IndexType i=1; i<=n-1; ++i) {
if (e(i)!=Zero) {
++info;
}
}
goto RETURN;
//
// Sort eigenvalues in increasing order.
//
SORT:
lasrt(true, d);
RETURN:
return info;
}
} // namespace generic
//== interface for native lapack ===============================================
#ifdef USE_CXXLAPACK
namespace external {
template <typename VD, typename VE>
typename VD::IndexType
sterf_impl(DenseVector<VD> &d,
DenseVector<VE> &e)
{
return cxxlapack::sterf(d.length(),
d.data(),
e.data());
}
} // namespace external
#endif
//== public interface ==========================================================
template <typename VD, typename VE>
typename RestrictTo<IsRealDenseVector<VD>::value
&& IsRealDenseVector<VE>::value,
typename RemoveRef<VD>::Type::IndexType>::Type
sterf(VD &&d,
VE &&e)
{
LAPACK_DEBUG_OUT("sterf");
//
// Remove references from rvalue types
//
typedef typename RemoveRef<VD>::Type VectorD;
typedef typename VectorD::IndexType IndexType;
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<VE>::Type VectorE;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(d.firstIndex()==1);
ASSERT(e.firstIndex()==1);
const IndexType n = d.length();
ASSERT(e.length()==n-1);
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename VectorD::NoView d_org = d;
typename VectorE::NoView e_org = e;
# endif
//
// Call implementation
//
const IndexType info = LAPACK_SELECT::sterf_impl(d, e);
# ifdef CHECK_CXXLAPACK
//
// Compare results
//
typename VectorD::NoView d_generic = d;
typename VectorE::NoView e_generic = e;
d = d_org;
e = e_org;
const IndexType info_ = external::sterf_impl(d, e);
bool failed = false;
if (! isIdentical(d_generic, d, "d_generic", "d")) {
std::cerr << "CXXLAPACK: d_generic = " << d_generic << std::endl;
std::cerr << "F77LAPACK: d = " << d << std::endl;
failed = true;
}
if (! isIdentical(e_generic, e, "e_generic", "e")) {
std::cerr << "CXXLAPACK: e_generic = " << e_generic << std::endl;
std::cerr << "F77LAPACK: e = " << e << std::endl;
failed = true;
}
if (! isIdentical(info, info_, "info", "info_")) {
std::cerr << "CXXLAPACK: info= " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << std::endl;
failed = true;
}
if (failed) {
ASSERT(0);
}
# endif
return info;
}
} } // namespace lapack, flens
#endif // FLENS_LAPACK_IMPL_STERF_TCC
|