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|
/*
* Copyright (c) 2011, 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 DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
SUBROUTINE ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
$ IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.2) --
* Univ. of Tennessee, Univ. of California Berkeley,
* Univ. of Colorado Denver and NAG Ltd..
* November 2006
*/
#ifndef FLENS_LAPACK_LA_LAQR0_TCC
#define FLENS_LAPACK_LA_LAQR0_TCC 1
#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>
namespace flens { namespace lapack {
//== generic lapack implementation =============================================
namespace generic {
//
// Workspace query (real/complex variant)
//
template <typename IndexType, typename MH>
IndexType
laqr0_wsq_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
const GeMatrix<MH> &H)
{
using std::max;
using std::min;
typedef typename GeMatrix<MH>::ElementType T;
const IndexType nTiny = 11;
const IndexType n = H.numRows();
if ((n==0) || (n<=nTiny)) {
return 1;
}
char job[3];
job[0] = (wantT) ? 'S' : 'E';
job[1] = (wantZ) ? 'V' : 'N';
job[2] = 0;
//
// ==== NWR = recommended deflation window size. At this
// . point, N .GT. NTINY = 11, so there is enough
// . subdiagonal workspace for NWR.GE.2 as required.
// . (In fact, there is enough subdiagonal space for
// . NWR.GE.3.) ====
//
IndexType nwr = ilaenv<T>(13, "LAQR0", job, n, iLo, iHi, -1);
nwr = max(IndexType(2), nwr);
nwr = min(min(IndexType(iHi-iLo+1), (n-1)/3), nwr);
//
// ==== NSR = recommended number of simultaneous shifts.
// . At this point N .GT. NTINY = 11, so there is at
// . enough subdiagonal workspace for NSR to be even
// . and greater than or equal to two as required. ====
//
IndexType nsr = ilaenv<T>(15, "LAQR0", job, n, iLo, iHi, -1);
nsr = min(min(nsr, (n+6)/9), IndexType(iHi-iLo));
nsr = max(IndexType(2), nsr-(nsr%2));
//
// ==== Estimate optimal workspace ====
//
// ==== Workspace query call to DLAQR3 ====
//
IndexType lWorkOpt = laqr3_wsq(IndexType(iLo), IndexType(iHi), nwr+1, H);
//
// ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
//
return max(3*nsr/2, lWorkOpt);
}
//
// Real variant
//
template <typename IndexType, typename MH, typename VWR, typename VWI,
typename MZ, typename VWORK>
IndexType
laqr0_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VWR> &wr,
DenseVector<VWI> &wi,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
DenseVector<VWORK> &work)
{
using flens::min;
using std::abs;
using std::max;
using std::swap;
typedef typename GeMatrix<MH>::ElementType T;
const Underscore<IndexType> _;
const IndexType n = H.numRows();
// ==== Matrices of order NTINY or smaller must be processed by
// . DLAHQR because of insufficient subdiagonal scratch space.
// . (This is a hard limit.) ====
const IndexType nTiny = 11;
// ==== Exceptional deflation windows: try to cure rare
// . slow convergence by varying the size of the
// . deflation window after KEXNW iterations. ====
const IndexType kexNw = 5;
//
// ==== Exceptional shifts: try to cure rare slow convergence
// . with ad-hoc exceptional shifts every KEXSH iterations.
// . ====
const IndexType kexSh = 6;
//
// ==== The constants WILK1 and WILK2 are used to form the
// . exceptional shifts. ====
const T wilk1 = T(0.75),
wilk2 = T(-0.4375);
const T Zero(0), One(1);
IndexType info = 0;
IndexType nDec = -1;
IndexType lWork;
IndexType lWorkOpt;
//
// ==== Perform and apply a workspace query if necessary ====
//
if (work.length()==0) {
lWorkOpt = laqr0_wsq(wantT, wantZ, iLo, iHi, H);
work.resize(lWorkOpt);
}
lWork = work.length();
//
// ==== Quick return for N = 0: nothing to do. ====
//
if (n==0) {
work(1) = One;
return info;
}
if (n<=nTiny) {
//
// ==== Tiny matrices must use DLAHQR. ====
//
lWorkOpt = 1;
info = lahqr(wantT, wantZ, iLo, iHi, H, wr, wi, iLoZ, iHiZ, Z);
} else {
//
// ==== Use small bulge multi-shift QR with aggressive early
// . deflation on larger-than-tiny matrices. ====
//
// ==== Hope for the best. ====
//
info = 0;
//
// ==== Set up job flags for ILAENV. ====
//
char job[3];
job[0] = (wantT) ? 'S' : 'E';
job[1] = (wantZ) ? 'V' : 'N';
job[2] = 0;
//
// ==== NWR = recommended deflation window size. At this
// . point, N .GT. NTINY = 11, so there is enough
// . subdiagonal workspace for NWR.GE.2 as required.
// . (In fact, there is enough subdiagonal space for
// . NWR.GE.3.) ====
//
IndexType nwr = ilaenv<T>(13, "LAQR0", job, n, iLo, iHi, lWork);
nwr = max(IndexType(2), nwr);
nwr = min(min(IndexType(iHi-iLo+1), (n-1)/3), nwr);
//
// ==== NSR = recommended number of simultaneous shifts.
// . At this point N .GT. NTINY = 11, so there is at
// . enough subdiagonal workspace for NSR to be even
// . and greater than or equal to two as required. ====
//
IndexType nsr = ilaenv<T>(15, "LAQR0", job, n, iLo, iHi, lWork);
nsr = min(min(nsr, (n+6)/9), IndexType(iHi-iLo));
nsr = max(IndexType(2), nsr - (nsr%2));
//
// ==== Estimate optimal workspace ====
//
// ==== Workspace query call to DLAQR3 ====
//
lWorkOpt = laqr3_wsq(iLo, iHi, nwr+1, H);
//
// ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ====
//
lWorkOpt = max(3*nsr/2, lWorkOpt);
//
// ==== DLAHQR/DLAQR0 crossover point ====
//
IndexType nMin = ilaenv<T>(12, "LAQR0", job, n, iLo, iHi, lWork);
nMin = max(nTiny, nMin);
//
// ==== Nibble crossover point ====
//
IndexType nibble = ilaenv<T>(14, "LAQR0", job, n, iLo, iHi, lWork);
nibble = max(IndexType(0), nibble);
//
// ==== Accumulate reflections during ttswp? Use block
// . 2-by-2 structure during matrix-matrix multiply? ====
//
IndexType kacc22 = ilaenv<T>(16, "LAQR0", job, n, iLo, iHi, lWork);
kacc22 = max(IndexType(0), kacc22);
kacc22 = min(IndexType(2), kacc22);
//
// ==== NWMAX = the largest possible deflation window for
// . which there is sufficient workspace. ====
//
IndexType nwMax = min((n-1)/3, lWork/2);
IndexType nw = nwMax;
//
// ==== NSMAX = the Largest number of simultaneous shifts
// . for which there is sufficient workspace. ====
//
IndexType nsMax = min((n+6 )/9, 2*lWork/3);
nsMax -= nsMax % 2;
//
// ==== NDFL: an iteration count restarted at deflation. ====
//
IndexType nDfl = 1;
//
// ==== ITMAX = iteration limit ====
//
IndexType itMax = max(IndexType(30), 2*kexSh)
* max(IndexType(10), iHi-iLo+1);
//
// ==== Last row and column in the active block ====
//
IndexType kBot = iHi;
//
// ==== Main Loop ====
//
IndexType it;
for (it=1; it<=itMax; ++it) {
//
// ==== Done when KBOT falls below ILO ====
//
if (kBot<iLo) {
break;
}
//
// ==== Locate active block ====
//
IndexType k;
for (k=kBot; k>=iLo+1; --k) {
if (H(k,k-1)==Zero) {
break;
}
}
ASSERT(k==iLo || H(k,k-1)==Zero);
const IndexType kTop = k;
//
// ==== Select deflation window size:
// . Typical Case:
// . If possible and advisable, nibble the entire
// . active block. If not, use size MIN(NWR,NWMAX)
// . or MIN(NWR+1,NWMAX) depending upon which has
// . the smaller corresponding subdiagonal entry
// . (a heuristic).
// .
// . Exceptional Case:
// . If there have been no deflations in KEXNW or
// . more iterations, then vary the deflation window
// . size. At first, because, larger windows are,
// . in general, more powerful than smaller ones,
// . rapidly increase the window to the maximum possible.
// . Then, gradually reduce the window size. ====
//
IndexType nh = kBot - kTop + 1;
IndexType nwUpBd = min(nh, nwMax);
if (nDfl<kexNw) {
nw = min(nwUpBd, nwr);
} else {
nw = min(nwUpBd, 2*nw);
}
if (nw<nwMax) {
if (nw>=nh-1) {
nw = nh;
} else {
const IndexType kwTop = kBot - nw + 1;
if (abs(H(kwTop,kwTop-1))>abs(H(kwTop-1, kwTop-2))) {
++nw;
}
}
}
if (nDfl<kexNw) {
nDec = -1;
} else if (nDec>=0 || nw>=nwUpBd) {
++nDec;
if (nw-nDec<2) {
nDec = 0;
}
nw -= nDec;
}
//
// ==== Aggressive early deflation:
// . split workspace under the subdiagonal into
// . - an nw-by-nw work array V in the lower
// . left-hand-corner,
// . - an NW-by-at-least-NW-but-more-is-better
// . (NW-by-NHO) horizontal work array along
// . the bottom edge,
// . - an at-least-NW-but-more-is-better (NHV-by-NW)
// . vertical work array along the left-hand-edge.
// . ====
//
auto V_ = H(_(n-nw+1, n), _( 1, nw));
auto T_ = H(_(n-nw+1, n), _(nw+1, n-nw-1));
auto WV_ = H(_( nw+2, n-nw), _( 1, nw));
//
// ==== Aggressive early deflation ====
//
IndexType ls, ld;
laqr3(wantT, wantZ, kTop, kBot, nw, H, iLoZ, iHiZ, Z, ls, ld,
wr(_(1,kBot)), wi(_(1,kBot)), V_, T_, WV_, work);
//
// ==== Adjust KBOT accounting for new deflations. ====
//
kBot -= ld;
//
// ==== KS points to the shifts. ====
//
IndexType ks = kBot - ls + 1;
//
// ==== Skip an expensive QR sweep if there is a (partly
// . heuristic) reason to expect that many eigenvalues
// . will deflate without it. Here, the QR sweep is
// . skipped if many eigenvalues have just been deflated
// . or if the remaining active block is small.
//
if ((ld==0)
|| ((100*ld<=nw*nibble) && (kBot-kTop+1>min(nMin, nwMax))))
{
//
// ==== NS = nominal number of simultaneous shifts.
// . This may be lowered (slightly) if DLAQR3
// . did not provide that many shifts. ====
//
IndexType ns = min(nsMax, nsr, max(IndexType(2), kBot-kTop));
ns -= ns % 2;
//
// ==== If there have been no deflations
// . in a multiple of KEXSH iterations,
// . then try exceptional shifts.
// . Otherwise use shifts provided by
// . DLAQR3 above or from the eigenvalues
// . of a trailing principal submatrix. ====
//
if (nDfl%kexSh==0) {
ks = kBot - ns + 1;
for (IndexType i=kBot; i>=max(ks+1,kTop+2); i-=2) {
const T ss = abs(H(i,i-1)) + abs(H(i-1,i-2));
T aa = wilk1*ss + H(i,i);
T bb = ss;
T cc = wilk2*ss;
T dd = aa;
T cs, sn;
lanv2(aa, bb, cc, dd,
wr(i-1), wi(i-1), wr(i), wi(i),
cs, sn);
}
if (ks==kTop) {
wr(ks+1) = H(ks+1, ks+1);
wi(ks+1) = Zero;
wr(ks) = wr(ks+1);
wi(ks) = wi(ks+1);
}
} else {
//
// ==== Got NS/2 or fewer shifts? Use DLAQR4 or
// . DLAHQR on a trailing principal submatrix to
// . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
// . there is enough space below the subdiagonal
// . to fit an NS-by-NS scratch array.) ====
//
if (kBot-ks+1<=ns/2) {
ks = kBot - ns +1;
H(_(ks,kBot),_(1,ns)) = H(_(ks,kBot),_(ks,kBot));
if (ns>nMin) {
// TODO: avoid the need for ZDummy
typename GeMatrix<MZ>::NoView ZDummy;
ks += laqr4(false, false,
IndexType(1), ns,
H(_(ks,kBot),_(1,ns)),
wr(_(ks,kBot)), wi(_(ks,kBot)),
IndexType(1), IndexType(1),
ZDummy, work);
} else {
// TODO: avoid the need for ZDummy
typename GeMatrix<MZ>::NoView ZDummy;
ks += lahqr(false, false,
IndexType(1), ns,
H(_(ks,kBot),_(1,ns)),
wr(_(ks,kBot)), wi(_(ks,kBot)),
IndexType(1), IndexType(1),
ZDummy);
}
//
// ==== In case of a rare QR failure use
// . eigenvalues of the trailing 2-by-2
// . principal submatrix. ====
//
if (ks>=kBot) {
T aa = H(kBot-1,kBot-1);
T cc = H(kBot, kBot-1);
T bb = H(kBot-1,kBot);
T dd = H(kBot, kBot);
T cs, sn;
lanv2(aa, bb, cc, dd,
wr(kBot-1), wi(kBot-1), wr(kBot), wi(kBot),
cs, sn);
ks = kBot - 1;
}
}
if (kBot-ks+1>ns) {
//
// ==== Sort the shifts (Helps a little)
// . Bubble sort keeps complex conjugate
// . pairs together. ====
//
bool sorted = false;
for (IndexType k=kBot; k>=ks+1; --k) {
if (sorted) {
break;
}
sorted = true;
for (IndexType i=ks; i<=k-1; ++i) {
if (abs(wr(i))+abs(wi(i))
< abs(wr(i+1))+abs(wi(i+1)))
{
sorted = false;
swap(wr(i), wr(i+1));
swap(wi(i), wi(i+1));
}
}
}
}
//
// ==== Shuffle shifts into pairs of real shifts
// . and pairs of complex conjugate shifts
// . assuming complex conjugate shifts are
// . already adjacent to one another. (Yes,
// . they are.) ====
//
for (IndexType i=kBot; i>=ks+2; i-=2) {
if (wi(i)!=-wi(i-1)) {
T tmp = wr(i);
wr(i) = wr(i-1);
wr(i-1) = wr(i-2);
wr(i-2) = tmp;
tmp = wi(i);
wi(i) = wi(i-1);
wi(i-1) = wi(i-2);
wi(i-2) = tmp;
}
}
}
//
// ==== If there are only two shifts and both are
// . real, then use only one. ====
//
if (kBot-ks+1==2) {
if (wi(kBot)==0) {
const T H_ = H(kBot,kBot);
if (abs(wr(kBot)-H_) < abs(wr(kBot-1)-H_)) {
wr(kBot-1) = wr(kBot);
} else {
wr(kBot) = wr(kBot-1);
}
}
}
//
// ==== Use up to NS of the the smallest magnatiude
// . shifts. If there aren't NS shifts available,
// . then use them all, possibly dropping one to
// . make the number of shifts even. ====
//
ns = min(ns, kBot-ks+1);
ns -= ns % 2;
ks = kBot - ns + 1;
//
// ==== Small-bulge multi-shift QR sweep:
// . split workspace under the subdiagonal into
// . - a KDU-by-KDU work array U in the lower
// . left-hand-corner,
// . - a KDU-by-at-least-KDU-but-more-is-better
// . (KDU-by-NHo) horizontal work array WH along
// . the bottom edge,
// . - and an at-least-KDU-but-more-is-better-by-KDU
// . (NVE-by-KDU) vertical work WV arrow along
// . the left-hand-edge. ====
//
IndexType kdu = 3*ns - 3;
IndexType ku = n - kdu + 1;
IndexType kwv = kdu + 4;
IndexType nho = (n-kdu+1-4) - (kdu+1) + 1;
typedef typename GeMatrix<MH>::View GeMatrixView;
GeMatrixView V_(IndexType(3), ns/2, work(_(1,3*ns/2)));
auto U_ = H(_( ku, n), _( 1, kdu));
auto WV_ = H(_(kwv,n-kdu), _( 1, kdu));
auto WH_ = H(_( ku, n), _(kdu+1,kdu+nho));
//
// ==== Small-bulge multi-shift QR sweep ====
//
laqr5(wantT, wantZ, kacc22, kTop, kBot, ns,
wr(_(ks,kBot)), wi(_(ks,kBot)), H,
iLoZ, iHiZ, Z, V_, U_, WV_, WH_);
}
//
// ==== Note progress (or the lack of it). ====
//
if (ld>0) {
nDfl = 1;
} else {
++nDfl;
}
//
// ==== End of main loop ====
//
}
//
// ==== Iteration limit exceeded. Set INFO to show where
// . the problem occurred and exit. ====
//
if (it>itMax) {
info = kBot;
}
}
work(1) = lWorkOpt;
return info;
}
//
// Complex variant
//
template <typename IndexType, typename MH, typename VW, typename MZ,
typename VWORK>
IndexType
laqr0_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VW> &w,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
DenseVector<VWORK> &work)
{
using std::abs;
using std::max;
using std::swap;
using flens::min;
typedef typename GeMatrix<MH>::ElementType T;
typedef typename ComplexTrait<T>::PrimitiveType PT;
const Underscore<IndexType> _;
const IndexType n = H.numRows();
//
// ==== Matrices of order NTINY or smaller must be processed by
// . ZLAHQR because of insufficient subdiagonal scratch space.
// . (This is a hard limit.) ====
const IndexType nTiny = 11;
//
// ==== Exceptional deflation windows: try to cure rare
// . slow convergence by varying the size of the
// . deflation window after KEXNW iterations. ====
const IndexType kexNw = 5;
//
// ==== Exceptional shifts: try to cure rare slow convergence
// . with ad-hoc exceptional shifts every KEXSH iterations.
// . ====
const IndexType kexSh = 6;
//
// ==== The constant WILK1 is used to form the exceptional
// . shifts. ====
const PT wilk1 = PT(0.75);
const T Zero(0), One(1);
const PT Two(2);
IndexType info = 0;
IndexType lWork;
IndexType lWorkOpt;
IndexType nDec = -1;
//
// ==== Perform and apply a workspace query if necessary ====
//
if (work.length()==0) {
lWorkOpt = laqr0_wsq(wantT, wantZ, iLo, iHi, H);
work.resize(lWorkOpt);
}
lWork = work.length();
//
// ==== Quick return for N = 0: nothing to do. ====
//
if (n==0) {
work(1) = One;
return info;
}
if (n<=nTiny) {
//
// ==== Tiny matrices must use ZLAHQR. ====
//
lWorkOpt = 1;
info = lahqr(wantT, wantZ, iLo, iHi, H, w, iLoZ, iHiZ, Z);
} else {
//
// ==== Use small bulge multi-shift QR with aggressive early
// . deflation on larger-than-tiny matrices. ====
//
// ==== Hope for the best. ====
//
info = 0;
//
// ==== Set up job flags for ILAENV. ====
//
char job[3];
job[0] = (wantT) ? 'S' : 'E';
job[1] = (wantZ) ? 'V' : 'N';
job[2] = 0;
//
// ==== NWR = recommended deflation window size. At this
// . point, N .GT. NTINY = 11, so there is enough
// . subdiagonal workspace for NWR.GE.2 as required.
// . (In fact, there is enough subdiagonal space for
// . NWR.GE.3.) ====
//
IndexType nwr = ilaenv<T>(13, "LAQR0", job, n, iLo, iHi, lWork);
nwr = max(IndexType(2), nwr);
nwr = min(iHi-iLo+1, (n-1)/3, nwr);
//
// ==== NSR = recommended number of simultaneous shifts.
// . At this point N .GT. NTINY = 11, so there is at
// . enough subdiagonal workspace for NSR to be even
// . and greater than or equal to two as required. ====
//
IndexType nsr = ilaenv<T>(15, "LAQR0", job, n, iLo, iHi, lWork);
nsr = min(nsr, (n+6)/9, iHi-iLo);
nsr = max(IndexType(2), nsr - (nsr%2));
//
// ==== Estimate optimal workspace ====
//
// ==== Workspace query call to ZLAQR3 ====
//
lWorkOpt = laqr3_wsq(iLo, iHi, nwr+1, H);
//
// ==== Optimal workspace = MAX(ZLAQR5, ZLAQR3) ====
//
lWorkOpt = max(3*nsr/2, lWorkOpt);
//
// ==== ZLAHQR/ZLAQR0 crossover point ====
//
IndexType nMin = ilaenv<T>(12, "laqr0", job, n, iLo, iHi, lWork);
nMin = max(nTiny, nMin);
//
// ==== Nibble crossover point ====
//
IndexType nibble = ilaenv<T>(14, "laqr0", job, n, iLo, iHi, lWork);
nibble = max(IndexType(0), nibble);
//
// ==== Accumulate reflections during ttswp? Use block
// . 2-by-2 structure during matrix-matrix multiply? ====
//
IndexType kacc22 = ilaenv<T>(16, "laqr0", job, n, iLo, iHi, lWork);
kacc22 = max(IndexType(0), kacc22);
kacc22 = min(IndexType(2), kacc22);
//
// ==== NWMAX = the largest possible deflation window for
// . which there is sufficient workspace. ====
//
IndexType nwMax = min((n-1)/3, lWork/2);
IndexType nw = nwMax;
//
// ==== NSMAX = the Largest number of simultaneous shifts
// . for which there is sufficient workspace. ====
//
IndexType nsMax = min((n+6)/9, 2*lWork/3);
nsMax -= nsMax % 2;
//
// ==== NDFL: an iteration count restarted at deflation. ====
//
IndexType nDfl = 1;
//
// ==== ITMAX = iteration limit ====
//
IndexType itMax = max(IndexType(30), 2*kexSh)
* max(IndexType(10), iHi-iLo+1);
//
// ==== Last row and column in the active block ====
//
IndexType kBot = iHi;
//
// ==== Main Loop ====
//
IndexType it;
for (it=1; it<=itMax; ++it) {
//
// ==== Done when KBOT falls below ILO ====
//
if (kBot<iLo) {
break;
}
//
// ==== Locate active block ====
//
IndexType k;
for (k=kBot; k>=iLo+1; --k) {
if (H(k,k-1)==Zero) {
break;
}
}
ASSERT(k==iLo || H(k,k-1)==Zero);
const IndexType kTop = k;
//
// ==== Select deflation window size:
// . Typical Case:
// . If possible and advisable, nibble the entire
// . active block. If not, use size MIN(NWR,NWMAX)
// . or MIN(NWR+1,NWMAX) depending upon which has
// . the smaller corresponding subdiagonal entry
// . (a heuristic).
// .
// . Exceptional Case:
// . If there have been no deflations in KEXNW or
// . more iterations, then vary the deflation window
// . size. At first, because, larger windows are,
// . in general, more powerful than smaller ones,
// . rapidly increase the window to the maximum possible.
// . Then, gradually reduce the window size. ====
//
IndexType nh = kBot - kTop + 1;
IndexType nwUpBd = min(nh, nwMax);
if (nDfl<kexNw) {
nw = min(nwUpBd, nwr);
} else {
nw = min(nwUpBd, 2*nw);
}
if (nw<nwMax) {
if (nw>=nh-1) {
nw = nh;
} else {
const IndexType kwTop = kBot - nw + 1;
if (abs1(H(kwTop,kwTop-1)) > abs1(H(kwTop-1,kwTop-2))) {
++nw;
}
}
}
if (nDfl<kexNw) {
nDec = -1;
} else if (nDec>=0 || nw>=nwUpBd) {
++nDec;
if (nw-nDec<2) {
nDec = 0;
}
nw -= nDec;
}
//
// ==== Aggressive early deflation:
// . split workspace under the subdiagonal into
// . - an nw-by-nw work array V in the lower
// . left-hand-corner,
// . - an NW-by-at-least-NW-but-more-is-better
// . (NW-by-NHO) horizontal work array along
// . the bottom edge,
// . - an at-least-NW-but-more-is-better (NHV-by-NW)
// . vertical work array along the left-hand-edge.
// . ====
//
auto V_ = H(_(n-nw+1, n), _( 1, nw));
auto T_ = H(_(n-nw+1, n), _(nw+1, n-nw-1));
auto WV_ = H(_( nw+2, n-nw), _( 1, nw));
//
// ==== Aggressive early deflation ====
//
IndexType ls, ld;
laqr3(wantT, wantZ, kTop, kBot, nw, H, iLoZ, iHiZ, Z, ls, ld,
w(_(1,kBot)), V_, T_, WV_, work);
//
// ==== Adjust KBOT accounting for new deflations. ====
//
kBot -= ld;
//
// ==== KS points to the shifts. ====
//
IndexType ks = kBot - ls + 1;
//
// ==== Skip an expensive QR sweep if there is a (partly
// . heuristic) reason to expect that many eigenvalues
// . will deflate without it. Here, the QR sweep is
// . skipped if many eigenvalues have just been deflated
// . or if the remaining active block is small.
//
if ((ld==0)
|| ((100*ld <= nw*nibble) && (kBot-kTop+1 > min(nMin, nwMax))))
{
//
// ==== NS = nominal number of simultaneous shifts.
// . This may be lowered (slightly) if ZLAQR3
// . did not provide that many shifts. ====
//
IndexType ns = min(nsMax, nsr, max(IndexType(2), kBot-kTop));
ns -= ns % 2;
//
// ==== If there have been no deflations
// . in a multiple of KEXSH iterations,
// . then try exceptional shifts.
// . Otherwise use shifts provided by
// . ZLAQR3 above or from the eigenvalues
// . of a trailing principal submatrix. ====
//
if (nDfl%kexSh==0) {
ks = kBot - ns + 1;
for (IndexType i=kBot; i>=ks+1; i-=2) {
w(i) = H(i,i) + wilk1*abs1(H(i,i-1));
w(i-1) = w(i);
}
} else {
//
// ==== Got NS/2 or fewer shifts? Use ZLAQR4 or
// . ZLAHQR on a trailing principal submatrix to
// . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
// . there is enough space below the subdiagonal
// . to fit an NS-by-NS scratch array.) ====
//
if (kBot-ks+1<=ns/2) {
ks = kBot - ns +1;
H(_(ks,kBot),_(1,ns)) = H(_(ks,kBot),_(ks,kBot));
if (ns>nMin) {
// TODO: avoid the need for ZDummy
typename GeMatrix<MZ>::NoView ZDummy;
ks += laqr4(false, false,
IndexType(1), ns,
H(_(ks,kBot),_(1,ns)),
w(_(ks,kBot)),
IndexType(1), IndexType(1),
ZDummy, work);
} else {
// TODO: avoid the need for ZDummy
typename GeMatrix<MZ>::NoView ZDummy;
ks += lahqr(false, false,
IndexType(1), ns,
H(_(ks,kBot),_(1,ns)),
w(_(ks,kBot)),
IndexType(1), IndexType(1),
ZDummy);
}
// ==== In case of a rare QR failure use
// . eigenvalues of the trailing 2-by-2
// . principal submatrix. Scale to avoid
// . overflows, underflows and subnormals.
// . (The scale factor S can not be zero,
// . because H(KBOT,KBOT-1) is nonzero.) ====
//
if (ks>=kBot) {
PT s = abs1(H(kBot-1, kBot-1)) +
abs1(H(kBot, kBot-1)) +
abs1(H(kBot-1, kBot )) +
abs1(H(kBot, kBot ));
T aa = H(kBot-1, kBot-1) / s;
T cc = H(kBot, kBot-1) / s;
T bb = H(kBot-1, kBot ) / s;
T dd = H(kBot, kBot ) / s;
T tr2 = (aa+dd) / Two;
T det = (aa-tr2) * (dd-tr2) - bb*cc;
T rtDisc = sqrt(-det);
w(kBot-1) = (tr2+rtDisc)*s;
w(kBot) = (tr2-rtDisc)*s;
ks = kBot - 1;
}
}
if (kBot-ks+1>ns) {
//
// ==== Sort the shifts (Helps a little) ====
//
bool sorted = false;
for (IndexType k=kBot; k>=ks+1; --k) {
if (sorted) {
break;
}
sorted = true;
for (IndexType i=ks; i<=k-1; ++i) {
if (abs1(w(i))<abs1(w(i+1))) {
sorted = false;
swap(w(i), w(i+1));
}
}
}
}
}
//
// ==== If there are only two shifts, then use
// . only one. ====
//
if (kBot-ks+1==2) {
if (abs1(w(kBot)-H(kBot,kBot))
< abs1(w(kBot-1)-H(kBot,kBot)))
{
w(kBot-1) = w(kBot);
} else {
w(kBot) = w(kBot-1);
}
}
//
// ==== Use up to NS of the the smallest magnatiude
// . shifts. If there aren't NS shifts available,
// . then use them all, possibly dropping one to
// . make the number of shifts even. ====
//
ns = min(ns, kBot-ks+1);
ns -= ns % 2;
ks = kBot - ns + 1;
//
// ==== Small-bulge multi-shift QR sweep:
// . split workspace under the subdiagonal into
// . - a KDU-by-KDU work array U in the lower
// . left-hand-corner,
// . - a KDU-by-at-least-KDU-but-more-is-better
// . (KDU-by-NHo) horizontal work array WH along
// . the bottom edge,
// . - and an at-least-KDU-but-more-is-better-by-KDU
// . (NVE-by-KDU) vertical work WV arrow along
// . the left-hand-edge. ====
//
IndexType kdu = 3*ns - 3;
IndexType ku = n - kdu + 1;
IndexType kwv = kdu + 4;
IndexType nho = (n-kdu+1-4) - (kdu+1) + 1;
typedef typename GeMatrix<MH>::View GeMatrixView;
GeMatrixView V_(IndexType(3), ns/2, work(_(1,3*ns/2)));
auto U_ = H(_( ku, n), _( 1, kdu));
auto WV_ = H(_(kwv,n-kdu), _( 1, kdu));
auto WH_ = H(_( ku, n), _(kdu+1,kdu+nho));
//
// ==== Small-bulge multi-shift QR sweep ====
//
laqr5(wantT, wantZ, kacc22, kTop, kBot, ns,
w(_(ks,kBot)), H, iLoZ, iHiZ, Z,
V_, U_, WV_, WH_);
}
//
// ==== Note progress (or the lack of it). ====
//
if (ld>0) {
nDfl = 1;
} else {
++nDfl;
}
//
// ==== End of main loop ====
//
}
//
// ==== Iteration limit exceeded. Set INFO to show where
// . the problem occurred and exit. ====
//
if (it>itMax) {
info = kBot;
}
}
work(1) = T(lWorkOpt, 0);
return info;
}
} // namespace generic
//== interface for native lapack ===============================================
#ifdef USE_CXXLAPACK
namespace external {
//
// Real variant
//
template <typename IndexType, typename MH, typename VWR, typename VWI,
typename MZ, typename VWORK>
IndexType
laqr0_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VWR> &wr,
DenseVector<VWI> &wi,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
DenseVector<VWORK> &work)
{
IndexType info;
info = cxxlapack::laqr0<IndexType>(wantT,
wantZ,
H.numRows(),
iLo,
iHi,
H.data(),
H.leadingDimension(),
wr.data(),
wi.data(),
iLoZ,
iHiZ,
Z.data(),
Z.leadingDimension(),
work.data(),
work.length());
ASSERT(info>=0);
return info;
}
//
// Workspace query (real variant)
//
template <typename IndexType, typename MH>
typename RestrictTo<IsReal<typename MH::ElementType>::value,
IndexType>::Type
laqr0_wsq_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
const GeMatrix<MH> &H)
{
typedef typename GeMatrix<MH>::ElementType T;
T WORK, DUMMY;
const IndexType LWORK = -1;
cxxlapack::laqr0<IndexType>(wantT,
wantZ,
H.numRows(),
iLo,
iHi,
&DUMMY,
H.leadingDimension(),
&DUMMY,
&DUMMY,
IndexType(1),
IndexType(1),
&DUMMY,
IndexType(1),
&WORK,
LWORK);
return WORK;
}
//
// Complex variant
//
template <typename IndexType, typename MH, typename VW, typename MZ,
typename VWORK>
IndexType
laqr0_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
GeMatrix<MH> &H,
DenseVector<VW> &w,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
DenseVector<VWORK> &work)
{
IndexType info;
info = cxxlapack::laqr0<IndexType>(wantT,
wantZ,
H.numRows(),
iLo,
iHi,
H.data(),
H.leadingDimension(),
w.data(),
iLoZ,
iHiZ,
Z.data(),
Z.leadingDimension(),
work.data(),
work.length());
ASSERT(info>=0);
return info;
}
//
// Workspace query (complex variant)
//
template <typename IndexType, typename MH>
typename RestrictTo<IsComplex<typename MH::ElementType>::value,
IndexType>::Type
laqr0_wsq_impl(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
const GeMatrix<MH> &H)
{
typedef typename GeMatrix<MH>::ElementType T;
T WORK, DUMMY;
const IndexType LWORK = -1;
cxxlapack::laqr0<IndexType>(wantT,
wantZ,
H.numRows(),
iLo,
iHi,
&DUMMY,
H.leadingDimension(),
&DUMMY,
IndexType(1),
IndexType(1),
&DUMMY,
IndexType(1),
&WORK,
LWORK);
return real(WORK);
}
} // namespace external
#endif // USE_CXXLAPACK
//== public interface ==========================================================
//
// Real variant
//
template <typename IndexType, typename MH, typename VWR, typename VWI,
typename MZ, typename VWORK>
typename RestrictTo<IsRealGeMatrix<MH>::value
&& IsRealDenseVector<VWR>::value
&& IsRealDenseVector<VWI>::value
&& IsRealGeMatrix<MZ>::value
&& IsRealDenseVector<VWORK>::value,
IndexType>::Type
laqr0(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
MH &&H,
VWR &&wr,
VWI &&wi,
IndexType iLoZ,
IndexType iHiZ,
MZ &&Z,
VWORK &&work)
{
LAPACK_DEBUG_OUT("laqr0");
using std::max;
//
// Remove references from rvalue types
//
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<MH>::Type MatrixH;
typedef typename RemoveRef<VWR>::Type VectorWR;
typedef typename RemoveRef<VWI>::Type VectorWI;
typedef typename RemoveRef<MZ>::Type MatrixZ;
typedef typename RemoveRef<VWORK>::Type VectorWork;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
const IndexType n = H.numRows();
if (n>0) {
ASSERT(1<=iLo);
ASSERT(iLo<=iHi);
ASSERT(iHi<=n);
} else {
ASSERT(iLo==1);
ASSERT(iHi==0);
}
ASSERT(wr.firstIndex()==1);
ASSERT(wr.length()>=iHi);
ASSERT(wi.firstIndex()==1);
ASSERT(wi.length()>=iHi);
ASSERT(1<=iLoZ);
ASSERT(iLoZ<=iLo);
ASSERT(iHi<=iHiZ);
ASSERT(iHiZ<=n);
ASSERT(Z.firstRow()==1);
ASSERT(Z.firstCol()==1);
ASSERT(Z.numRows()>=iHi);
ASSERT(Z.numCols()>=iHi);
ASSERT((work.length()==0) || (work.length()>=n));
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename MatrixH::NoView H_org = H;
typename VectorWR::NoView wr_org = wr;
typename VectorWI::NoView wi_org = wi;
typename MatrixZ::NoView Z_org = Z;
typename VectorWork::NoView work_org = work;
# endif
//
// Call implementation
//
IndexType info = LAPACK_SELECT::laqr0_impl(wantT, wantZ, iLo, iHi, H,
wr, wi, iLoZ, iHiZ, Z, work);
# ifdef CHECK_CXXLAPACK
//
// Make copies of results computed by the generic implementation
//
typename MatrixH::NoView H_generic = H;
typename VectorWR::NoView wr_generic = wr;
typename VectorWI::NoView wi_generic = wi;
typename MatrixZ::NoView Z_generic = Z;
typename VectorWork::NoView work_generic = work;
//
// restore output arguments
//
H = H_org;
wr = wr_org;
wi = wi_org;
Z = Z_org;
work = work_org;
//
// Compare generic results with results from the native implementation
//
IndexType info_ = external::laqr0_impl(wantT, wantZ, iLo, iHi, H, wr, wi,
iLoZ, iHiZ, Z, work);
bool failed = false;
if (! isIdentical(H_generic, H, "H_generic", "H")) {
std::cerr << "CXXLAPACK: H_generic = " << H_generic << std::endl;
std::cerr << "F77LAPACK: H = " << H << std::endl;
failed = true;
}
if (! isIdentical(wr_generic, wr, "wr_generic", "wr")) {
std::cerr << "CXXLAPACK: wr_generic = " << wr_generic << std::endl;
std::cerr << "F77LAPACK: wr = " << wr << std::endl;
failed = true;
}
if (! isIdentical(wi_generic, wi, "wi_generic", "wi")) {
std::cerr << "CXXLAPACK: wi_generic = " << wi_generic << std::endl;
std::cerr << "F77LAPACK: wi = " << wi << 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(info, info_, " info", "info_")) {
std::cerr << "CXXLAPACK: info = " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << 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) {
std::cerr << "error in: laqr0.tcc" << std::endl;
ASSERT(0);
} else {
// std::cerr << "passed: laqr0.tcc" << std::endl;
}
# endif
return info;
}
//
// Complex variant
//
template <typename IndexType, typename MH, typename VW, typename MZ,
typename VWORK>
typename RestrictTo<IsComplexGeMatrix<MH>::value
&& IsComplexDenseVector<VW>::value
&& IsComplexGeMatrix<MZ>::value
&& IsComplexDenseVector<VWORK>::value,
IndexType>::Type
laqr0(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
MH &&H,
VW &&w,
IndexType iLoZ,
IndexType iHiZ,
MZ &&Z,
VWORK &&work)
{
LAPACK_DEBUG_OUT("laqr0");
using std::max;
//
// Remove references from rvalue types
//
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<MH>::Type MatrixH;
typedef typename RemoveRef<VW>::Type VectorW;
typedef typename RemoveRef<MZ>::Type MatrixZ;
typedef typename RemoveRef<VWORK>::Type VectorWork;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
const IndexType n = H.numRows();
if (n>0) {
ASSERT(1<=iLo);
ASSERT(iLo<=iHi);
ASSERT(iHi<=n);
} else {
ASSERT(iLo==1);
ASSERT(iHi==0);
}
ASSERT(w.firstIndex()==1);
ASSERT(w.length()>=iHi);
ASSERT(1<=iLoZ);
ASSERT(iLoZ<=iLo);
ASSERT(iHi<=iHiZ);
ASSERT(iHiZ<=n);
ASSERT(Z.firstRow()==1);
ASSERT(Z.firstCol()==1);
ASSERT(Z.numRows()>=iHi);
ASSERT(Z.numCols()>=iHi);
ASSERT((work.length()==0) || (work.length()>=n));
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename MatrixH::NoView H_org = H;
typename VectorW::NoView w_org = w;
typename MatrixZ::NoView Z_org = Z;
typename VectorWork::NoView work_org = work;
# endif
//
// Call implementation
//
IndexType info = LAPACK_SELECT::laqr0_impl(wantT, wantZ, iLo, iHi, H,
w, iLoZ, iHiZ, Z, work);
# ifdef CHECK_CXXLAPACK
//
// Make copies of results computed by the generic implementation
//
typename MatrixH::NoView H_generic = H;
typename VectorW::NoView w_generic = w;
typename MatrixZ::NoView Z_generic = Z;
typename VectorWork::NoView work_generic = work;
//
// restore output arguments
//
H = H_org;
w = w_org;
Z = Z_org;
work = work_org;
//
// Compare generic results with results from the native implementation
//
IndexType info_ = external::laqr0_impl(wantT, wantZ, iLo, iHi, H, w,
iLoZ, iHiZ, Z, work);
bool failed = false;
if (! isIdentical(H_generic, H, "H_generic", "H")) {
std::cerr << "CXXLAPACK: H_generic = " << H_generic << std::endl;
std::cerr << "F77LAPACK: H = " << H << std::endl;
failed = true;
}
if (! isIdentical(w_generic, w, "w_generic", "w")) {
std::cerr << "CXXLAPACK: w_generic = " << w_generic << std::endl;
std::cerr << "F77LAPACK: w = " << w << 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(info, info_, " info", "info_")) {
std::cerr << "CXXLAPACK: info = " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << 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) {
std::cerr << "error in: laqr0.tcc" << std::endl;
ASSERT(0);
} else {
// std::cerr << "passed: laqr0.tcc" << std::endl;
}
# endif
return info;
}
//
// Workspace query (real/complex variant)
//
template <typename IndexType, typename MH>
IndexType
laqr0_wsq(bool wantT,
bool wantZ,
IndexType iLo,
IndexType iHi,
const GeMatrix<MH> &H)
{
using std::max;
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
const IndexType n = H.numRows();
if (n>0) {
ASSERT(1<=iLo);
ASSERT(iLo<=iHi);
ASSERT(iHi<=n);
} else {
ASSERT(iLo==1);
ASSERT(iHi==0);
}
# endif
//
// Call implementation
//
IndexType info = LAPACK_SELECT::laqr0_wsq_impl(wantT, wantZ, iLo, iHi, H);
# ifdef CHECK_CXXLAPACK
//
// Compare results
//
IndexType info_ = external::laqr0_wsq_impl(wantT, wantZ, iLo, iHi, H);
if (info!=info_) {
std::cerr << "CXXLAPACK: info = " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << std::endl;
ASSERT(0);
}
# endif
return info;
}
} } // namespace lapack, flens
#endif // FLENS_LAPACK_LA_LAQR0_TCC
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