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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 ZSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*/
#ifndef FLENS_LAPACK_IMPL_STEQR_TCC
#define FLENS_LAPACK_IMPL_STEQR_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 MZ, typename VWORK>
typename VD::IndexType
steqr_impl(STEQR::ComputeZ compZ,
DenseVector<VD> &d,
DenseVector<VE> &e,
GeMatrix<MZ> &Z,
DenseVector<VWORK> &work)
{
using std::abs;
using std::max;
using std::sqrt;
typedef typename MZ::ElementType T;
typedef typename MZ::IndexType IndexType;
typedef typename ComplexTrait<T>::PrimitiveType PT;
const PT Zero(0), One(1), Two(2), Three(3);
const T CZero(0), COne(1);
const IndexType maxIt = 30;
const IndexType n = d.length();
const Underscore<IndexType> _;
IndexType info = 0;
//
// Quick return if possible
//
if (n==0) {
return info;
}
if (n==1) {
if (compZ==STEQR::Tri) {
Z(1,1) = One;
}
return info;
}
//
// Determine the unit roundoff and over/underflow thresholds.
//
const PT eps = lamch<PT>(Eps);
const PT eps2 = pow(eps, 2);
const PT safeMin = lamch<PT>(SafeMin);
const PT safeMax = One / safeMin;
const PT sSafeMax = sqrt(safeMax) / Three;
const PT sSafeMin = sqrt(safeMin) / eps2;
//
// Compute the eigenvalues and eigenvectors of the tridiagonal
// matrix.
//
if (compZ==STEQR::Tri) {
Z = CZero;
Z.diag(0) = COne;
}
const IndexType nMaxIt = n*maxIt;
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;
PT p, rt1, rt2, c, s, g, r, f, b, normA;
START:
do {
if (l1>n) {
goto SORT;
}
if (l1>1) {
e(l1-1) = Zero;
}
if (l1<=n-1) {
for (m=l1; m<=n-1; ++m) {
PT test = abs(e(m));
if (test==Zero) {
break;
}
if (test<=(sqrt(abs(d(m)))*sqrt(abs(d(m+1))))*eps) {
e(m) = Zero;
break;
}
}
} else {
m = n;
}
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(InfinityNorm, d(_(l,lEnd)), e(_(l,max(l-1, lEnd-1))));
iScale = 0;
if (normA==Zero) {
goto START;
}
if (normA>sSafeMax) {
iScale = 1;
lascl(LASCL::FullMatrix, 0, 0, normA, sSafeMax, d(_(l,lEnd)));
lascl(LASCL::FullMatrix, 0, 0, normA, sSafeMax, e(_(l,lEnd-1)));
} else if (normA<sSafeMin) {
iScale = 2;
lascl(LASCL::FullMatrix, 0, 0, normA, sSafeMin, d(_(l,lEnd)));
lascl(LASCL::FullMatrix, 0, 0, normA, sSafeMin, e(_(l,lEnd-1)));
}
//
// 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 (m=l; m<=lEnd-1; ++m) {
PT test = pow(abs(e(m)), 2);
if (test<=(eps2*abs(d(m)))*abs(d(m+1))+safeMin) {
break;
}
}
} else {
m = lEnd;
}
if (m<lEnd) {
e(m) = Zero;
}
p = d(l);
if (m==l) {
break;
}
//
// If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
// to compute its eigensystem.
//
if (m==l+1) {
if (compZ!=STEQR::No) {
laev2(d(l), e(l), d(l+1), rt1, rt2, c, s);
work(l) = c;
work(n-1+l) = s;
lasr(Right, LASR::VariablePivot, LASR::Backward,
work(_(l,l)), work(_(n-1+l,n-1+l)),
Z(_(1,n),_(l,l+1)));
} else {
lae2(d(l), e(l), d(l+1), rt1, rt2);
}
d(l) = rt1;
d(l+1) = rt2;
e(l) = Zero;
l+=2;
if (l<=lEnd) {
goto QL_Start;
}
goto UNDO_SCALING;
}
if (jTot==nMaxIt) {
goto UNDO_SCALING;
}
++jTot;
//
// Form shift.
//
g = (d(l+1)-p) / (Two*e(l));
r = lapy2(g, One);
g = d(m) - p + (e(l)/(g + sign(r,g)));
s = One;
c = One;
p = Zero;
//
// Inner loop
//
for (IndexType i=m-1; i>=l; --i) {
f = s*e(i);
b = c*e(i);
lartg(g, f, c, s, r);
if (i!=m-1) {
e(i+1) = r;
}
g = d(i+1) - p;
r = (d(i)-g)*s + Two*c*b;
p = s*r;
d(i+1) = g + p;
g = c*r - b;
//
// If eigenvectors are desired, then save rotations.
//
if (compZ!=STEQR::No) {
work(i) = c;
work(n-1+i) = -s;
}
}
//
// If eigenvectors are desired, then apply saved rotations.
//
if (compZ!=STEQR::No) {
lasr(Right, LASR::VariablePivot, LASR::Backward,
work(_(l,m-1)), work(_(n-1+l,n-2+m)),
Z(_(1,n),_(l,m)));
}
d(l) -= p;
e(l) = g;
}
//
// 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) {
if (l!=lEnd) {
for (m=l; m>=lEnd+1; --m) {
PT test = pow(abs(e(m-1)), 2);
if (test<=(eps2*abs(d(m)))*abs(d(m-1))+safeMin) {
break;
}
}
} else {
m = lEnd;
}
if (m>lEnd) {
e(m-1) = Zero;
}
p = d(l);
if (m==l) {
break;
}
//
// If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
// to compute its eigensystem.
//
if (m==l-1) {
if (compZ!=STEQR::No) {
laev2(d(l-1), e(l-1), d(l), rt1, rt2, c, s);
work(m) = c;
work(n-1+m) = s;
lasr(Right, LASR::VariablePivot, LASR::Forward,
work(_(m,m)), work(_(n-1+m,n-1+m)),
Z(_(1,n),_(l-1,l)));
} else {
lae2(d(l-1), e(l-1), d(l), rt1, rt2);
}
d(l-1) = rt1;
d(l) = 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.
//
g = (d(l-1)-p) / (Two*e(l-1));
r = lapy2(g, One);
g = d(m) - p + (e(l-1)/(g + sign(r,g)));
s = One;
c = One;
p = Zero;
//
// Inner loop
//
for (IndexType i=m; i<=l-1; ++i) {
f = s*e(i);
b = c*e(i);
lartg(g, f, c, s, r);
if (i!=m) {
e(i-1) = r;
}
g = d(i) - p;
r = (d(i+1)-g)*s + Two*c*b;
p = s*r;
d(i) = g + p;
g = c*r - b;
//
// If eigenvectors are desired, then save rotations.
//
if (compZ!=STEQR::No) {
work(i) = c;
work(n-1+i) = s;
}
}
//
// If eigenvectors are desired, then apply saved rotations.
//
if (compZ!=STEQR::No) {
lasr(Right, LASR::VariablePivot, LASR::Forward,
work(_(m,l-1)), work(_(n-1+m,n-2+l)),
Z(_(1,n),_(m,l)));
}
d(l) -= p;
e(l-1) = g;
}
//
// 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, sSafeMax, normA, d(_(lSv,lEndSv)));
lascl(LASCL::FullMatrix, 0, 0, sSafeMax, normA, e(_(lSv,lEndSv-1)));
} else if (iScale==2) {
lascl(LASCL::FullMatrix, 0, 0, sSafeMin, normA, d(_(lSv,lEndSv)));
lascl(LASCL::FullMatrix, 0, 0, sSafeMin, normA, e(_(lSv,lEndSv-1)));
}
//
// Check for no convergence to an eigenvalue after a total
// of N*MAXIT iterations.
//
if (jTot==nMaxIt) {
for (IndexType i=1; i<=n-1; ++i) {
if (e(i)!=Zero) {
++info;
}
}
return info;
}
goto START;
//
// Order eigenvalues and eigenvectors.
//
SORT:
if (compZ==STEQR::No) {
//
// Use Quick Sort
//
lasrt(true, d);
} else {
//
// Use Selection Sort to minimize swaps of eigenvectors
//
for (IndexType ii=2; ii<=n; ++ii) {
IndexType i = ii-1;
IndexType k = i;
p = d(i);
for (IndexType j=ii; j<=n; ++j) {
if (d(j)<p) {
k = j;
p = d(j);
}
}
if (k!=i) {
d(k) = d(i);
d(i) = p;
blas::swap(Z(_,i), Z(_,k));
}
}
}
return info;
}
} // namespace generic
//== interface for native lapack ===============================================
#ifdef USE_CXXLAPACK
namespace external {
template <typename VD, typename VE, typename MZ, typename VWORK>
typename VD::IndexType
steqr_impl(STEQR::ComputeZ compZ,
DenseVector<VD> &d,
DenseVector<VE> &e,
GeMatrix<MZ> &Z,
DenseVector<VWORK> &work)
{
return cxxlapack::steqr(getF77Char(compZ),
d.length(),
d.data(),
e.data(),
Z.data(),
Z.leadingDimension(),
work.data());
}
} // namespace external
#endif
//== public interface ==========================================================
template <typename VD, typename VE, typename MZ, typename VWORK>
typename RestrictTo<IsRealDenseVector<VD>::value
&& IsRealDenseVector<VE>::value
&& IsComplexGeMatrix<MZ>::value
&& IsRealDenseVector<VWORK>::value,
typename RemoveRef<VD>::Type::IndexType>::Type
steqr(STEQR::ComputeZ compZ,
VD &&d,
VE &&e,
MZ &&Z,
VWORK &&work)
{
LAPACK_DEBUG_OUT("steqr");
//
// 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;
typedef typename RemoveRef<MZ>::Type MatrixZ;
typedef typename RemoveRef<VWORK>::Type VectorWork;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(d.firstIndex()==1);
ASSERT(e.firstIndex()==1);
ASSERT(Z.firstRow()==1);
ASSERT(Z.firstCol()==1);
const IndexType n = d.length();
ASSERT(e.length()==n-1);
if (compZ==STEQR::Orig ||compZ==STEQR::Tri) {
ASSERT(Z.numRows()==n);
ASSERT(Z.numCols()==n);
}
ASSERT(work.firstIndex()==1);
ASSERT(work.length()==std::max(1,2*n-2));
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename VectorD::NoView d_org = d;
typename VectorE::NoView e_org = e;
typename MatrixZ::NoView Z_org = Z;
typename VectorWork::NoView work_org = work;
# endif
//
// Call implementation
//
const IndexType info = LAPACK_SELECT::steqr_impl(compZ, d, e, Z, work);
# ifdef CHECK_CXXLAPACK
//
// Compare results
//
typename VectorD::NoView d_generic = d;
typename VectorE::NoView e_generic = e;
typename MatrixZ::NoView Z_generic = Z;
typename VectorWork::NoView work_generic = work;
d = d_org;
e = e_org;
Z = Z_org;
work = work_org;
const IndexType info_ = external::steqr_impl(compZ, d, e, Z, work);
bool failed = false;
if (! isIdentical(info_, info, "info_", "info")) {
std::cerr << "CXXLAPACK: info_ = " << info_ << std::endl;
std::cerr << "F77LAPACK: info = " << info << std::endl;
failed = true;
}
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(Z_generic, Z, "Z_generic", "Z")) {
std::cerr << "CXXLAPACK: Z_generic = " << Z_generic << std::endl;
std::cerr << "F77LAPACK: Z = " << Z << std::endl;
failed = true;
}
if (! isIdentical(work_generic, work, "work_generic", "work")) {
std::cerr << "CXXLAPACK: work_generic = " << work_generic << std::endl;
std::cerr << "F77LAPACK: work = " << work << std::endl;
failed = true;
}
if (failed) {
ASSERT(0);
}
# endif
return info;
}
} } // namespace lapack, flens
#endif // FLENS_LAPACK_IMPL_STEQR_TCC
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