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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 DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
$ SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
$ LDU, NV, WV, LDWV, NH, WH, LDWH )
SUBROUTINE ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S,
$ H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV,
$ WV, LDWV, NH, WH, LDWH )
*
* -- LAPACK auxiliary routine (version 3.3.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2010
*
*/
#ifndef FLENS_LAPACK_LA_LAQR5_TCC
#define FLENS_LAPACK_LA_LAQR5_TCC 1
#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>
namespace flens { namespace lapack {
//== generic lapack implementation =============================================
namespace generic {
//
// Real variant
//
template <typename IndexType, typename VSR, typename VSI, typename MH,
typename MZ, typename MV, typename MU, typename MWV, typename MWH>
void
laqr5_impl(bool wantT,
bool wantZ,
IndexType kacc22,
IndexType kTop,
IndexType kBot,
IndexType nShifts,
DenseVector<VSR> &sr,
DenseVector<VSI> &si,
GeMatrix<MH> &H,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
GeMatrix<MV> &V,
GeMatrix<MU> &U,
GeMatrix<MWV> &WV,
GeMatrix<MWH> &WH)
{
using std::abs;
using std::max;
using std::min;
typedef typename GeMatrix<MH>::ElementType T;
const T Zero(0), One(1);
const Underscore<IndexType> _;
const IndexType n = H.numRows();
const IndexType nv = WV.numRows();
const IndexType nh = WH.numCols();
typedef typename GeMatrix<MH>::VectorView VectorView;
T vtBuffer[3];
VectorView vt = typename VectorView::Engine(3, vtBuffer);
//
// ==== If there are no shifts, then there is nothing to do. ====
//
if (nShifts<2) {
return;
}
//
// ==== If the active block is empty or 1-by-1, then there
// . is nothing to do. ====
//
if (kTop>=kBot) {
return;
}
//
// ==== Shuffle shifts into pairs of real shifts and pairs
// . of complex conjugate shifts assuming complex
// . conjugate shifts are already adjacent to one
// . another. ====
//
for (IndexType i=1; i<=nShifts-2; i+=2) {
if (si(i)!=-si(i+1)) {
T tmp = sr(i);
sr(i) = sr(i+1);
sr(i+1) = sr(i+2);
sr(i+2) = tmp;
tmp = si(i);
si(i) = si(i+1);
si(i+1) = si(i+2);
si(i+2) = tmp;
}
}
//
// ==== NSHFTS is supposed to be even, but if it is odd,
// . then simply reduce it by one. The shuffle above
// . ensures that the dropped shift is real and that
// . the remaining shifts are paired. ====
//
const IndexType ns = nShifts - (nShifts % 2);
//
// ==== Machine constants for deflation ====
//
T safeMin = lamch<T>(SafeMin);
T safeMax = One/safeMin;
labad(safeMin, safeMax);
const T ulp = lamch<T>(Precision);
const T smallNum = safeMin*(T(n)/ulp);
//
// ==== Use accumulated reflections to update far-from-diagonal
// . entries ? ====
//
const bool accum = (kacc22==1) || (kacc22==2);
//
// ==== If so, exploit the 2-by-2 block structure? ====
//
const bool blk22 = (ns>2) && (kacc22==2);
//
// ==== clear trash ====
//
if (kTop+2<=kBot) {
H(kTop+2,kTop) = Zero;
}
//
// ==== NBMPS = number of 2-shift bulges in the chain ====
//
const IndexType nBmps = ns/2;
//
// ==== KDU = width of slab ====
//
const IndexType kdu = 6*nBmps - 3;
//
// ==== Create and chase chains of NBMPS bulges ====
//
for (IndexType inCol=3*(1-nBmps)+kTop-1; inCol<=kBot-2; inCol+=3*nBmps-2) {
IndexType ndCol = inCol + kdu;
if (accum) {
auto U_ = U(_(1,kdu),_(1,kdu));
U_ = Zero;
U_.diag(0) = One;
}
//
// ==== Near-the-diagonal bulge chase. The following loop
// . performs the near-the-diagonal part of a small bulge
// . multi-shift QR sweep. Each 6*NBMPS-2 column diagonal
// . chunk extends from column INCOL to column NDCOL
// . (including both column INCOL and column NDCOL). The
// . following loop chases a 3*NBMPS column long chain of
// . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
// . may be less than KTOP and and NDCOL may be greater than
// . KBOT indicating phantom columns from which to chase
// . bulges before they are actually introduced or to which
// . to chase bulges beyond column KBOT.) ====
//
const IndexType krColMax = min(inCol+3*nBmps-3, kBot-2);
for (IndexType krCol=inCol; krCol<=krColMax; ++krCol) {
//
// ==== Bulges number MTOP to MBOT are active double implicit
// . shift bulges. There may or may not also be small
// . 2-by-2 bulge, if there is room. The inactive bulges
// . (if any) must wait until the active bulges have moved
// . down the diagonal to make room. The phantom matrix
// . paradigm described above helps keep track. ====
//
const IndexType mTop = max(IndexType(1), ((kTop-1)-krCol+2)/3+1);
const IndexType mBot = min(nBmps, (kBot-krCol)/3);
const IndexType m22 = mBot + 1;
const bool bmp22 = (mBot<nBmps ) && (krCol+3*(m22-1))==(kBot-2);
//
// ==== Generate reflections to chase the chain right
// . one column. (The minimum value of K is KTOP-1.) ====
//
IndexType k;
T alpha, beta;
for (IndexType m=mTop; m<=mBot; ++m) {
k = krCol + 3*(m-1);
if (k==kTop-1) {
laqr1(H(_(kTop,kTop+2),_(kTop,kTop+2)),
sr(2*m-1), si(2*m-1), sr(2*m), si(2*m),
V(_(1,3),m));
alpha = V(1,m);
larfg(IndexType(3), alpha, V(_(2,3),m), V(1,m));
} else {
beta = H(k+1,k);
V(2,m) = H(k+2,k);
V(3,m) = H(k+3,k);
larfg(IndexType(3), beta, V(_(2,3),m), V(1,m));
//
// ==== A Bulge may collapse because of vigilant
// . deflation or destructive underflow. In the
// . underflow case, try the two-small-subdiagonals
// . trick to try to reinflate the bulge. ====
//
if (H(k+3,k)!=Zero
|| H(k+3,k+1)!=Zero
|| H(k+3,k+2)==Zero)
{
//
// ==== Typical case: not collapsed (yet). ====
//
H(k+1, k) = beta;
H(k+2, k) = Zero;
H(k+3, k) = Zero;
} else {
//
// ==== Atypical case: collapsed. Attempt to
// . reintroduce ignoring H(K+1,K) and H(K+2,K).
// . If the fill resulting from the new
// . reflector is too large, then abandon it.
// . Otherwise, use the new one. ====
//
laqr1(H(_(k+1,k+3),_(k+1,k+3)),
sr(2*m-1), si(2*m-1), sr(2*m), si(2*m),
vt);
alpha = vt(1);
larfg(3, alpha, vt(_(2,3)), vt(1));
const T refSum = vt(1)*(H(k+1,k) + vt(2)*H(k+2,k));
if (abs(H(k+2,k)-refSum*vt(2)) + abs(refSum*vt(3))
> ulp*(abs(H(k,k))+abs(H(k+1,k+1))+abs(H(k+2,k+2))))
{
//
// ==== Starting a new bulge here would
// . create non-negligible fill. Use
// . the old one with trepidation. ====
//
H(k+1, k) = beta;
H(k+2, k) = Zero;
H(k+3, k) = Zero;
} else {
//
// ==== Stating a new bulge here would
// . create only negligible fill.
// . Replace the old reflector with
// . the new one. ====
//
H(k+1, k) -= refSum;
H(k+2, k) = Zero;
H(k+3, k) = Zero;
V(1, m) = vt(1);
V(2, m) = vt(2);
V(3, m) = vt(3);
}
}
}
}
//
// ==== Generate a 2-by-2 reflection, if needed. ====
//
k = krCol + 3*(m22-1);
if (bmp22) {
if (k==kTop-1) {
laqr1(H(_(k+1,k+2),_(k+1,k+2)),
sr(2*m22-1), si(2*m22-1), sr(2*m22), si(2*m22),
V(_(1,2),m22));
beta = V(1, m22);
larfg(IndexType(2), beta, V(_(2,2),m22), V(1,m22));
} else {
beta = H(k+1, k);
V(2, m22) = H(k+2,k);
larfg(IndexType(2), beta, V(_(2,2),m22), V(1,m22));
H(k+1, k) = beta;
H(k+2, k) = Zero;
}
}
//
// ==== Multiply H by reflections from the left ====
//
IndexType jBot;
if (accum) {
jBot = min(ndCol, kBot);
} else if (wantT) {
jBot = n;
} else {
jBot = kBot;
}
for (IndexType j=max(kTop,krCol); j<=jBot; ++j) {
IndexType mEnd = min(mBot, (j-krCol+2)/3);
for (IndexType m=mTop; m<=mEnd; ++m) {
k = krCol + 3*(m-1);
const T refSum = V(1,m)*(H(k+1,j) + V(2,m)*H(k+2,j)
+ V(3,m)*H(k+3,j));
H(k+1,j) -= refSum;
H(k+2,j) -= refSum*V(2,m);
H(k+3,j) -= refSum*V(3,m);
}
}
if (bmp22) {
k = krCol + 3*(m22-1);
for (IndexType j=max(k+1,kTop); j<=jBot; ++j) {
const T refSum = V(1,m22)*(H(k+1,j)+V(2,m22)*H(k+2,j));
H(k+1,j) -= refSum;
H(k+2,j) -= refSum*V(2,m22);
}
}
//
// ==== Multiply H by reflections from the right.
// . Delay filling in the last row until the
// . vigilant deflation check is complete. ====
//
IndexType jTop;
if (accum) {
jTop = max(kTop, inCol);
} else if (wantT) {
jTop = 1;
} else {
jTop = kTop;
}
for (IndexType m=mTop; m<=mBot; ++m) {
if (V(1,m)!=Zero) {
k = krCol + 3*(m-1);
for (IndexType j=jTop; j<=min(kBot,k+3); ++j) {
const T refSum = V(1,m)*(H(j,k+1) + V(2,m)*H(j,k+2)
+ V(3,m)*H(j,k+3));
H(j, k+1) -= refSum;
H(j, k+2) -= refSum*V(2, m);
H(j, k+3) -= refSum*V(3, m);
}
if (accum) {
//
// ==== Accumulate U. (If necessary, update Z later
// . with with an efficient matrix-matrix
// . multiply.) ====
//
IndexType kms = k - inCol;
IndexType j1 = max(IndexType(1),kTop-inCol);
for (IndexType j=j1; j<=kdu; ++j) {
const T refSum = V(1,m)*(U(j,kms+1)
+ V(2,m)*U(j,kms+2)
+ V(3,m)*U(j,kms+3));
U(j, kms+1) -= refSum;
U(j, kms+2) -= refSum*V(2, m);
U(j, kms+3) -= refSum*V(3, m);
}
} else if (wantZ) {
//
// ==== U is not accumulated, so update Z
// . now by multiplying by reflections
// . from the right. ====
//
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
const T refSum = V(1,m)*(Z(j,k+1)
+V(2,m)*Z(j,k+2)
+V(3,m)*Z(j,k+3));
Z(j,k+1) -= refSum;
Z(j,k+2) -= refSum*V(2,m);
Z(j,k+3) -= refSum*V(3,m);
}
}
}
}
//
// ==== Special case: 2-by-2 reflection (if needed) ====
//
k = krCol + 3*(m22-1);
if (bmp22) {
if (V(1,m22)!=Zero) {
for (IndexType j=jTop; j<=min(kBot,k+3); ++j) {
const T refSum = V(1,m22)*(H(j,k+1)+V(2,m22)*H(j,k+2));
H(j,k+1) -= refSum;
H(j,k+2) -= refSum*V(2,m22);
}
if (accum) {
IndexType kms = k - inCol;
IndexType j1 = max(IndexType(1),kTop-inCol);
for (IndexType j=j1; j<=kdu; ++j) {
const T refSum = V(1,m22)*(U(j,kms+1)
+ V(2,m22)*U(j,kms+2));
U(j,kms+1 ) -= refSum;
U(j,kms+2 ) -= refSum*V(2,m22);
}
} else if (wantZ) {
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
const T refSum = V(1,m22)*(Z(j,k+1)
+ V(2,m22)*Z(j,k+2));
Z(j,k+1) -= refSum;
Z(j,k+2) -= refSum*V(2,m22);
}
}
}
}
//
// ==== Vigilant deflation check ====
//
IndexType mStart = mTop;
if (krCol+3*(mStart-1)<kTop) {
++mStart;
}
IndexType mEnd = mBot;
if (bmp22) {
++mEnd;
}
if (krCol==kBot-2) {
++mEnd;
}
for (IndexType m=mStart; m<=mEnd; ++m) {
k = min(kBot-1, krCol+3*(m-1));
//
// ==== The following convergence test requires that
// . the tradition small-compared-to-nearby-diagonals
// . criterion and the Ahues & Tisseur (LAWN 122, 1997)
// . criteria both be satisfied. The latter improves
// . accuracy in some examples. Falling back on an
// . alternate convergence criterion when TST1 or TST2
// . is zero (as done here) is traditional but probably
// . unnecessary. ====
//
if (H(k+1,k)!=Zero) {
T test1 = abs(H(k,k)) + abs(H(k+1,k+1));
if (test1==Zero) {
if (k>=kTop+1) {
test1 += abs(H(k,k-1));
}
if (k>=kTop+2) {
test1 += abs(H(k,k-2));
}
if (k>=kTop+3) {
test1 += abs(H(k,k-3));
}
if (k<=kBot-2) {
test1 += abs(H(k+2,k+1));
}
if (k<=kBot-3) {
test1 += abs(H(k+3,k+1));
}
if (k<=kBot-4) {
test1 += abs(H(k+4,k+1));
}
}
if (abs(H(k+1,k))<=max(smallNum, ulp*test1)) {
const T H12 = max(abs(H(k+1,k)), abs(H(k,k+1)));
const T H21 = min(abs(H(k+1,k)), abs(H(k,k+1)));
const T H11 = max(abs(H(k+1,k+1)),
abs(H(k,k)-H(k+1,k+1)));
const T H22 = min(abs(H(k+1,k+1)),
abs(H(k,k)-H(k+1,k+1)));
const T scal = H11 + H12;
const T test2 = H22*(H11/scal);
if (test2==Zero
|| H21*(H12/scal)<=max(smallNum,ulp*test2))
{
H(k+1,k) = Zero;
}
}
}
}
//
// ==== Fill in the last row of each bulge. ====
//
mEnd = min(nBmps, (kBot-krCol-1)/3);
for (IndexType m=mTop; m<=mEnd; ++m) {
k = krCol + 3*(m-1);
const T refSum = V(1,m)*V(3,m)*H(k+4,k+3);
H(k+4,k+1) = -refSum;
H(k+4,k+2) = -refSum*V(2,m);
H(k+4,k+3) -= refSum*V(3,m);
}
//
// ==== End of near-the-diagonal bulge chase. ====
//
}
//
// ==== Use U (if accumulated) to update far-from-diagonal
// . entries in H. If required, use U to update Z as
// . well. ====
//
if (accum) {
IndexType jTop, jBot;
if (wantT) {
jTop = 1;
jBot = n;
} else {
jTop = kTop;
jBot = kBot;
}
if ((!blk22) || (inCol<kTop) || (ndCol>kBot) || (ns<=2)) {
//
// ==== Updates not exploiting the 2-by-2 block
// . structure of U. K1 and NU keep track of
// . the location and size of U in the special
// . cases of introducing bulges and chasing
// . bulges off the bottom. In these special
// . cases and in case the number of shifts
// . is NS = 2, there is no 2-by-2 block
// . structure to exploit. ====
//
const IndexType k1 = max(IndexType(1), kTop-inCol);
const IndexType nu = (kdu-max(IndexType(0), ndCol-kBot)) -k1+1;
const IndexType nu_ = (kdu-max(IndexType(0), ndCol-kBot));
//
// ==== Horizontal Multiply ====
//
for (IndexType jCol=min(ndCol,kBot)+1; jCol<=jBot; jCol+=nh) {
const IndexType jLen = min(nh, jBot-jCol+1);
auto U_ = U(_(k1, nu_), _(k1, nu_));
auto H_ = H(_(inCol+k1,inCol+nu_),_(jCol,jCol+jLen-1));
auto WH_ = WH(_(1,nu),_(1,jLen));
blas::mm(ConjTrans, NoTrans, One, U_, H_, Zero, WH_);
H_ = WH_;
}
//
// ==== Vertical multiply ====
//
for (IndexType jRow=jTop; jRow<=max(kTop,inCol)-1; jRow+=nv) {
const IndexType jLen = min(nv, max(kTop,inCol)-jRow);
auto H_ = H(_(jRow,jRow+jLen-1),_(inCol+k1,inCol+nu_));
auto U_ = U(_(k1,nu_),_(k1,nu_));
auto WV_ = WV(_(1,jLen),_(1,nu));
blas::mm(NoTrans, NoTrans, One, H_, U_, Zero, WV_);
H_ = WV_;
}
//
// ==== Z multiply (also vertical) ====
//
if (wantZ) {
for (IndexType jRow=iLoZ; jRow<=iHiZ; jRow+=nv) {
const IndexType jLen = min(nv, iHiZ-jRow+1);
auto Z_ = Z(_(jRow,jRow+jLen-1),_(inCol+k1,inCol+nu_));
auto U_ = U(_(k1,nu_),_(k1,nu_));
auto WV_ = WV(_(1,jLen),_(1,nu));
blas::mm(NoTrans, NoTrans, One, Z_, U_, Zero, WV_);
Z_ = WV_;
}
}
} else {
//
// ==== Updates exploiting U's 2-by-2 block structure.
// . (I2, I4, J2, J4 are the last rows and columns
// . of the blocks.) ====
//
const IndexType i2 = (kdu+1 ) / 2;
const IndexType i4 = kdu;
const IndexType j2 = i4 - i2;
const IndexType j4 = kdu;
//
// ==== KZS and KNZ deal with the band of zeros
// . along the diagonal of one of the triangular
// . blocks. ====
//
const IndexType kZs = (j4-j2) - (ns+1);
const IndexType kNz = ns + 1;
//
// ==== Horizontal multiply ====
//
for (IndexType jCol=min(ndCol, kBot)+1; jCol<=jBot; jCol+=nh) {
const IndexType jLen = min(nh, jBot-jCol+1);
//
// ==== Copy bottom of H to top+KZS of scratch ====
// (The first KZS rows get multiplied by zero.) ====
//
WH(_(kZs+1,kZs+kNz),_(1,jLen))
= H(_(inCol+j2+1,inCol+j2+kNz),_(jCol,jCol+jLen-1));
//
// ==== Multiply by U21**T ====
//
WH(_(1,kZs),_(1,jLen)) = Zero;
blas::mm(Left, ConjTrans, One,
U(_(j2+1,j2+kNz),_(kZs+1,kZs+kNz)).upper(),
WH(_(kZs+1,kZs+kNz),_(1,jLen)));
//
// ==== Multiply top of H by U11**T ====
//
blas::mm(ConjTrans, NoTrans, One,
U(_(1,j2),_(1,i2)),
H(_(inCol+1,inCol+j2),_(jCol,jCol+jLen-1)),
One,
WH(_(1,i2),_(1,jLen)));
//
// ==== Copy top of H to bottom of WH ====
//
WH(_(i2+1,i2+j2),_(1,jLen))
= H(_(inCol+1,inCol+j2),_(jCol,jCol+jLen-1));
//
// ==== Multiply by U21**T ====
//
blas::mm(Left, ConjTrans, One,
U(_(1,j2),_(i2+1,i2+j2)).lower(),
WH(_(i2+1,i2+j2),_(1,jLen)));
//
// ==== Multiply by U22 ====
//
blas::mm(ConjTrans, NoTrans, One,
U(_(j2+1,j4),_(i2+1,i4)),
H(_(inCol+j2+1,inCol+j4),_(jCol,jCol+jLen-1)),
One,
WH(_(i2+1,i4),_(1,jLen)));
//
// ==== Copy it back ====
//
H(_(inCol+1,inCol+kdu),_(jCol,jCol+jLen-1))
= WH(_(1,kdu),_(1,jLen));
}
//
// ==== Vertical multiply ====
//
for (IndexType jRow=jTop; jRow<=max(inCol,kTop)-1; jRow+=nv) {
const IndexType jLen = min(nv, max(inCol,kTop) -jRow);
//
// ==== Copy right of H to scratch (the first KZS
// . columns get multiplied by zero) ====
//
WV(_(1,jLen),_(kZs+1,kZs+kNz))
= H(_(jRow,jRow+jLen-1),_(inCol+j2+1, inCol+j2+kNz));
//
// ==== Multiply by U21 ====
//
WV(_(1,jLen),_(1,kZs)) = Zero;
blas::mm(Right, NoTrans, One,
U(_(j2+1,j2+kNz),_(kZs+1,kZs+kNz)).upper(),
WV(_(1,jLen),_(kZs+1,kZs+kNz)));
//
// ==== Multiply by U11 ====
//
blas::mm(NoTrans, NoTrans, One,
H(_(jRow,jRow+jLen-1),_(inCol+1,inCol+j2)),
U(_(1,j2),_(1,i2)),
One,
WV(_(1,jLen),_(1,i2)));
//
// ==== Copy left of H to right of scratch ====
//
WV(_(1,jLen),_(i2+1,i2+j2))
= H(_(jRow,jRow+jLen-1), _(inCol+1,inCol+j2));
//
// ==== Multiply by U21 ====
//
blas::mm(Right, NoTrans, One,
U(_(1,i4-i2),_(i2+1,i4)).lower(),
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Multiply by U22 ====
//
blas::mm(NoTrans, NoTrans, One,
H(_(jRow,jRow+jLen-1),_(inCol+j2+1,inCol+j4)),
U(_(j2+1,j4),_(i2+1,i4)),
One,
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Copy it back ====
//
H(_(jRow,jRow+jLen-1),_(inCol+1,inCol+kdu))
= WV(_(1,jLen),_(1,kdu));
}
//
// ==== Multiply Z (also vertical) ====
//
if (wantZ) {
for (IndexType jRow=iLoZ; jRow<=iHiZ; jRow+=nv) {
const IndexType jLen = min(nv,iHiZ-jRow+1);
//
// ==== Copy right of Z to left of scratch (first
// . KZS columns get multiplied by zero) ====
//
WV(_(1,jLen),_(kZs+1,kZs+kNz))
= Z(_(jRow,jRow+jLen-1),_(inCol+j2+1,inCol+j2+kNz));
//
// ==== Multiply by U12 ====
//
WV(_(1,jLen),_(1,kZs)) = Zero;
blas::mm(Right, NoTrans, One,
U(_(j2+1,j2+kNz), _(kZs+1,kZs+kNz)).upper(),
WV(_(1,jLen),_(kZs+1,kZs+kNz)));
//
// ==== Multiply by U11 ====
//
blas::mm(NoTrans, NoTrans, One,
Z(_(jRow,jRow+jLen-1), _(inCol+1,inCol+j2)),
U(_(1,j2),_(1,i2)),
One,
WV(_(1,jLen),_(1,i2)));
//
// ==== Copy left of Z to right of scratch ====
//
WV(_(1,jLen),_(i2+1,i2+j2))
= Z(_(jRow,jRow+jLen-1), _(inCol+1,inCol+j2));
//
// ==== Multiply by U21 ====
//
blas::mm(Right, NoTrans, One,
U(_(1,i4-i2),_(i2+1,i4)).lower(),
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Multiply by U22 ====
//
blas::mm(NoTrans, NoTrans, One,
Z(_(jRow,jRow+jLen-1),_(inCol+j2+1,inCol+j4)),
U(_(j2+1,j4),_(i2+1,i4)),
One,
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Copy the result back to Z ====
//
Z(_(jRow,jRow+jLen-1),_(inCol+1,inCol+kdu))
= WV(_(1,jLen),_(1,kdu));
}
}
}
}
}
}
//
// Complex variant
//
template <typename IndexType, typename VS, typename MH, typename MZ,
typename MV, typename MU, typename MWV, typename MWH>
void
laqr5_impl(bool wantT,
bool wantZ,
IndexType kacc22,
IndexType kTop,
IndexType kBot,
IndexType nShifts,
DenseVector<VS> &s,
GeMatrix<MH> &H,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
GeMatrix<MV> &V,
GeMatrix<MU> &U,
GeMatrix<MWV> &WV,
GeMatrix<MWH> &WH)
{
using std::abs;
using std::max;
using std::min;
typedef typename GeMatrix<MH>::ElementType T;
typedef typename ComplexTrait<T>::PrimitiveType PT;
const T Zero(0), One(1);
const PT RZero(0), ROne(1);
const Underscore<IndexType> _;
const IndexType n = H.numRows();
const IndexType nv = WV.numRows();
const IndexType nh = WH.numCols();
typedef typename GeMatrix<MH>::VectorView VectorView;
T vtBuffer[3];
VectorView vt = typename VectorView::Engine(3, vtBuffer);
//
// ==== If there are no shifts, then there is nothing to do. ====
//
if (nShifts<2) {
return;
}
//
// ==== If the active block is empty or 1-by-1, then there
// . is nothing to do. ====
//
if (kTop>=kBot) {
return;
}
//
// ==== NSHFTS is supposed to be even, but if it is odd,
// . then simply reduce it by one. ====
//
const IndexType ns = nShifts - (nShifts % 2);
//
// ==== Machine constants for deflation ====
//
PT safeMin = lamch<PT>(SafeMin);
PT safeMax = ROne/safeMin;
labad(safeMin, safeMax);
const PT ulp = lamch<PT>(Precision);
const PT smallNum = safeMin*(PT(n)/ulp);
//
// ==== Use accumulated reflections to update far-from-diagonal
// . entries ? ====
//
const bool accum = (kacc22==1) || (kacc22==2);
//
// ==== If so, exploit the 2-by-2 block structure? ====
//
const bool blk22 = (ns>2) && (kacc22==2);
//
// ==== clear trash ====
//
if (kTop+2<=kBot) {
H(kTop+2,kTop) = Zero;
}
//
// ==== NBMPS = number of 2-shift bulges in the chain ====
//
const IndexType nBmps = ns/2;
//
// ==== KDU = width of slab ====
//
const IndexType kdu = 6*nBmps - 3;
//
// ==== Create and chase chains of NBMPS bulges ====
//
for (IndexType inCol=3*(1-nBmps)+kTop-1; inCol<=kBot-2; inCol+=3*nBmps-2) {
IndexType ndCol = inCol + kdu;
if (accum) {
auto U_ = U(_(1,kdu),_(1,kdu));
U_ = Zero;
U_.diag(0) = One;
}
//
// ==== Near-the-diagonal bulge chase. The following loop
// . performs the near-the-diagonal part of a small bulge
// . multi-shift QR sweep. Each 6*NBMPS-2 column diagonal
// . chunk extends from column INCOL to column NDCOL
// . (including both column INCOL and column NDCOL). The
// . following loop chases a 3*NBMPS column long chain of
// . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
// . may be less than KTOP and and NDCOL may be greater than
// . KBOT indicating phantom columns from which to chase
// . bulges before they are actually introduced or to which
// . to chase bulges beyond column KBOT.) ====
//
const IndexType krColMax = min(inCol+3*nBmps-3, kBot-2);
for (IndexType krCol=inCol; krCol<=krColMax; ++krCol) {
//
// ==== Bulges number MTOP to MBOT are active double implicit
// . shift bulges. There may or may not also be small
// . 2-by-2 bulge, if there is room. The inactive bulges
// . (if any) must wait until the active bulges have moved
// . down the diagonal to make room. The phantom matrix
// . paradigm described above helps keep track. ====
//
const IndexType mTop = max(IndexType(1), ((kTop-1)-krCol+2)/3+1);
const IndexType mBot = min(nBmps, (kBot-krCol)/3);
const IndexType m22 = mBot + 1;
const bool bmp22 = (mBot<nBmps ) && (krCol+3*(m22-1))==(kBot-2);
//
// ==== Generate reflections to chase the chain right
// . one column. (The minimum value of K is KTOP-1.) ====
//
IndexType k;
T alpha, beta;
for (IndexType m=mTop; m<=mBot; ++m) {
k = krCol + 3*(m-1);
if (k==kTop-1) {
laqr1(H(_(kTop,kTop+2),_(kTop,kTop+2)),
s(2*m-1), s(2*m),
V(_(1,3),m));
alpha = V(1,m);
larfg(IndexType(3), alpha, V(_(2,3),m), V(1,m));
} else {
beta = H(k+1,k);
V(2,m) = H(k+2,k);
V(3,m) = H(k+3,k);
larfg(IndexType(3), beta, V(_(2,3),m), V(1,m));
//
// ==== A Bulge may collapse because of vigilant
// . deflation or destructive underflow. In the
// . underflow case, try the two-small-subdiagonals
// . trick to try to reinflate the bulge. ====
//
if (H(k+3,k)!=Zero
|| H(k+3,k+1)!=Zero
|| H(k+3,k+2)==Zero)
{
//
// ==== Typical case: not collapsed (yet). ====
//
H(k+1, k) = beta;
H(k+2, k) = Zero;
H(k+3, k) = Zero;
} else {
//
// ==== Atypical case: collapsed. Attempt to
// . reintroduce ignoring H(K+1,K) and H(K+2,K).
// . If the fill resulting from the new
// . reflector is too large, then abandon it.
// . Otherwise, use the new one. ====
//
laqr1(H(_(k+1,k+3),_(k+1,k+3)), s(2*m-1), s(2*m), vt);
alpha = vt(1);
larfg(3, alpha, vt(_(2,3)), vt(1));
const T refSum = conj(vt(1))*(
H(k+1,k) + conj(vt(2))*H(k+2,k)
);
if (abs1(H(k+2,k)-refSum*vt(2)) + abs1(refSum*vt(3))
> ulp*(abs1(H(k,k))+abs1(H(k+1,k+1))+abs(H(k+2,k+2))))
{
//
// ==== Starting a new bulge here would
// . create non-negligible fill. Use
// . the old one with trepidation. ====
//
H(k+1, k) = beta;
H(k+2, k) = Zero;
H(k+3, k) = Zero;
} else {
//
// ==== Stating a new bulge here would
// . create only negligible fill.
// . Replace the old reflector with
// . the new one. ====
//
H(k+1, k) -= refSum;
H(k+2, k) = Zero;
H(k+3, k) = Zero;
V(1, m) = vt(1);
V(2, m) = vt(2);
V(3, m) = vt(3);
}
}
}
}
//
// ==== Generate a 2-by-2 reflection, if needed. ====
//
k = krCol + 3*(m22-1);
if (bmp22) {
if (k==kTop-1) {
laqr1(H(_(k+1,k+2),_(k+1,k+2)),
s(2*m22-1), s(2*m22),
V(_(1,2),m22));
beta = V(1, m22);
larfg(IndexType(2), beta, V(_(2,2),m22), V(1,m22));
} else {
beta = H(k+1, k);
V(2, m22) = H(k+2,k);
larfg(IndexType(2), beta, V(_(2,2),m22), V(1,m22));
H(k+1, k) = beta;
H(k+2, k) = Zero;
}
}
//
// ==== Multiply H by reflections from the left ====
//
IndexType jBot;
if (accum) {
jBot = min(ndCol, kBot);
} else if (wantT) {
jBot = n;
} else {
jBot = kBot;
}
for (IndexType j=max(kTop,krCol); j<=jBot; ++j) {
IndexType mEnd = min(mBot, (j-krCol+2)/3);
for (IndexType m=mTop; m<=mEnd; ++m) {
k = krCol + 3*(m-1);
const T refSum = conj(V(1,m))*(
H(k+1,j) + conj(V(2,m))*H(k+2,j)
+ conj(V(3,m))*H(k+3,j)
);
H(k+1,j) -= refSum;
H(k+2,j) -= refSum*V(2,m);
H(k+3,j) -= refSum*V(3,m);
}
}
if (bmp22) {
k = krCol + 3*(m22-1);
for (IndexType j=max(k+1,kTop); j<=jBot; ++j) {
const T refSum = conj(V(1,m22))*(
H(k+1,j) + conj(V(2,m22))*H(k+2,j)
);
H(k+1,j) -= refSum;
H(k+2,j) -= refSum*V(2,m22);
}
}
//
// ==== Multiply H by reflections from the right.
// . Delay filling in the last row until the
// . vigilant deflation check is complete. ====
//
IndexType jTop;
if (accum) {
jTop = max(kTop, inCol);
} else if (wantT) {
jTop = 1;
} else {
jTop = kTop;
}
for (IndexType m=mTop; m<=mBot; ++m) {
if (V(1,m)!=Zero) {
k = krCol + 3*(m-1);
for (IndexType j=jTop; j<=min(kBot,k+3); ++j) {
const T refSum = V(1,m)*(H(j,k+1) + V(2,m)*H(j,k+2)
+ V(3,m)*H(j,k+3));
H(j, k+1) -= refSum;
H(j, k+2) -= refSum*conj(V(2, m));
H(j, k+3) -= refSum*conj(V(3, m));
}
if (accum) {
//
// ==== Accumulate U. (If necessary, update Z later
// . with with an efficient matrix-matrix
// . multiply.) ====
//
IndexType kms = k - inCol;
IndexType j1 = max(IndexType(1),kTop-inCol);
for (IndexType j=j1; j<=kdu; ++j) {
const T refSum = V(1,m)*(U(j,kms+1)
+ V(2,m)*U(j,kms+2)
+ V(3,m)*U(j,kms+3));
U(j, kms+1) -= refSum;
U(j, kms+2) -= refSum*conj(V(2, m));
U(j, kms+3) -= refSum*conj(V(3, m));
}
} else if (wantZ) {
//
// ==== U is not accumulated, so update Z
// . now by multiplying by reflections
// . from the right. ====
//
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
const T refSum = V(1,m)*(Z(j,k+1)
+V(2,m)*Z(j,k+2)
+V(3,m)*Z(j,k+3));
Z(j,k+1) -= refSum;
Z(j,k+2) -= refSum*conj(V(2,m));
Z(j,k+3) -= refSum*conj(V(3,m));
}
}
}
}
//
// ==== Special case: 2-by-2 reflection (if needed) ====
//
k = krCol + 3*(m22-1);
if (bmp22) {
if (V(1,m22)!=Zero) {
for (IndexType j=jTop; j<=min(kBot,k+3); ++j) {
const T refSum = V(1,m22)*(H(j,k+1)+V(2,m22)*H(j,k+2));
H(j,k+1) -= refSum;
H(j,k+2) -= refSum*conj(V(2,m22));
}
if (accum) {
IndexType kms = k - inCol;
IndexType j1 = max(IndexType(1),kTop-inCol);
for (IndexType j=j1; j<=kdu; ++j) {
const T refSum = V(1,m22)*(U(j,kms+1)
+ V(2,m22)*U(j,kms+2));
U(j,kms+1 ) -= refSum;
U(j,kms+2 ) -= refSum*conj(V(2,m22));
}
} else if (wantZ) {
for (IndexType j=iLoZ; j<=iHiZ; ++j) {
const T refSum = V(1,m22)*(Z(j,k+1)
+ V(2,m22)*Z(j,k+2));
Z(j,k+1) -= refSum;
Z(j,k+2) -= refSum*conj(V(2,m22));
}
}
}
}
//
// ==== Vigilant deflation check ====
//
IndexType mStart = mTop;
if (krCol+3*(mStart-1)<kTop) {
++mStart;
}
IndexType mEnd = mBot;
if (bmp22) {
++mEnd;
}
if (krCol==kBot-2) {
++mEnd;
}
for (IndexType m=mStart; m<=mEnd; ++m) {
k = min(kBot-1, krCol+3*(m-1));
//
// ==== The following convergence test requires that
// . the tradition small-compared-to-nearby-diagonals
// . criterion and the Ahues & Tisseur (LAWN 122, 1997)
// . criteria both be satisfied. The latter improves
// . accuracy in some examples. Falling back on an
// . alternate convergence criterion when TST1 or TST2
// . is zero (as done here) is traditional but probably
// . unnecessary. ====
//
if (H(k+1,k)!=Zero) {
PT test1 = abs1(H(k,k)) + abs1(H(k+1,k+1));
if (test1==RZero) {
if (k>=kTop+1) {
test1 += abs1(H(k,k-1));
}
if (k>=kTop+2) {
test1 += abs1(H(k,k-2));
}
if (k>=kTop+3) {
test1 += abs1(H(k,k-3));
}
if (k<=kBot-2) {
test1 += abs1(H(k+2,k+1));
}
if (k<=kBot-3) {
test1 += abs1(H(k+3,k+1));
}
if (k<=kBot-4) {
test1 += abs1(H(k+4,k+1));
}
}
if (abs1(H(k+1,k))<=max(smallNum, ulp*test1)) {
const PT H12 = max(abs1(H(k+1,k)), abs1(H(k,k+1)));
const PT H21 = min(abs1(H(k+1,k)), abs1(H(k,k+1)));
const PT H11 = max(abs1(H(k+1,k+1)),
abs1(H(k,k)-H(k+1,k+1)));
const PT H22 = min(abs1(H(k+1,k+1)),
abs1(H(k,k)-H(k+1,k+1)));
const PT scal = H11 + H12;
const PT test2 = H22*(H11/scal);
if (test2==Zero
|| H21*(H12/scal)<=max(smallNum,ulp*test2))
{
H(k+1,k) = Zero;
}
}
}
}
//
// ==== Fill in the last row of each bulge. ====
//
mEnd = min(nBmps, (kBot-krCol-1)/3);
for (IndexType m=mTop; m<=mEnd; ++m) {
k = krCol + 3*(m-1);
const T refSum = V(1,m)*V(3,m)*H(k+4,k+3);
H(k+4,k+1) = -refSum;
H(k+4,k+2) = -refSum*conj(V(2,m));
H(k+4,k+3) -= refSum*conj(V(3,m));
}
//
// ==== End of near-the-diagonal bulge chase. ====
//
}
//
// ==== Use U (if accumulated) to update far-from-diagonal
// . entries in H. If required, use U to update Z as
// . well. ====
//
if (accum) {
IndexType jTop, jBot;
if (wantT) {
jTop = 1;
jBot = n;
} else {
jTop = kTop;
jBot = kBot;
}
if ((!blk22) || (inCol<kTop) || (ndCol>kBot) || (ns<=2)) {
//
// ==== Updates not exploiting the 2-by-2 block
// . structure of U. K1 and NU keep track of
// . the location and size of U in the special
// . cases of introducing bulges and chasing
// . bulges off the bottom. In these special
// . cases and in case the number of shifts
// . is NS = 2, there is no 2-by-2 block
// . structure to exploit. ====
//
const IndexType k1 = max(IndexType(1), kTop-inCol);
const IndexType nu = (kdu-max(IndexType(0), ndCol-kBot)) -k1+1;
const IndexType nu_ = (kdu-max(IndexType(0), ndCol-kBot));
//
// ==== Horizontal Multiply ====
//
for (IndexType jCol=min(ndCol,kBot)+1; jCol<=jBot; jCol+=nh) {
const IndexType jLen = min(nh, jBot-jCol+1);
auto U_ = U(_(k1, nu_), _(k1, nu_));
auto H_ = H(_(inCol+k1,inCol+nu_),_(jCol,jCol+jLen-1));
auto WH_ = WH(_(1,nu),_(1,jLen));
blas::mm(ConjTrans, NoTrans, One, U_, H_, Zero, WH_);
H_ = WH_;
}
//
// ==== Vertical multiply ====
//
for (IndexType jRow=jTop; jRow<=max(kTop,inCol)-1; jRow+=nv) {
const IndexType jLen = min(nv, max(kTop,inCol)-jRow);
auto H_ = H(_(jRow,jRow+jLen-1),_(inCol+k1,inCol+nu_));
auto U_ = U(_(k1,nu_),_(k1,nu_));
auto WV_ = WV(_(1,jLen),_(1,nu));
blas::mm(NoTrans, NoTrans, One, H_, U_, Zero, WV_);
H_ = WV_;
}
//
// ==== Z multiply (also vertical) ====
//
if (wantZ) {
for (IndexType jRow=iLoZ; jRow<=iHiZ; jRow+=nv) {
const IndexType jLen = min(nv, iHiZ-jRow+1);
auto Z_ = Z(_(jRow,jRow+jLen-1),_(inCol+k1,inCol+nu_));
auto U_ = U(_(k1,nu_),_(k1,nu_));
auto WV_ = WV(_(1,jLen),_(1,nu));
blas::mm(NoTrans, NoTrans, One, Z_, U_, Zero, WV_);
Z_ = WV_;
}
}
} else {
//
// ==== Updates exploiting U's 2-by-2 block structure.
// . (I2, I4, J2, J4 are the last rows and columns
// . of the blocks.) ====
//
const IndexType i2 = (kdu+1 ) / 2;
const IndexType i4 = kdu;
const IndexType j2 = i4 - i2;
const IndexType j4 = kdu;
//
// ==== KZS and KNZ deal with the band of zeros
// . along the diagonal of one of the triangular
// . blocks. ====
//
const IndexType kZs = (j4-j2) - (ns+1);
const IndexType kNz = ns + 1;
//
// ==== Horizontal multiply ====
//
for (IndexType jCol=min(ndCol, kBot)+1; jCol<=jBot; jCol+=nh) {
const IndexType jLen = min(nh, jBot-jCol+1);
//
// ==== Copy bottom of H to top+KZS of scratch ====
// (The first KZS rows get multiplied by zero.) ====
//
WH(_(kZs+1,kZs+kNz),_(1,jLen))
= H(_(inCol+j2+1,inCol+j2+kNz),_(jCol,jCol+jLen-1));
//
// ==== Multiply by U21**H ====
//
WH(_(1,kZs),_(1,jLen)) = Zero;
blas::mm(Left, ConjTrans, One,
U(_(j2+1,j2+kNz),_(kZs+1,kZs+kNz)).upper(),
WH(_(kZs+1,kZs+kNz),_(1,jLen)));
//
// ==== Multiply top of H by U11**H ====
//
blas::mm(ConjTrans, NoTrans, One,
U(_(1,j2),_(1,i2)),
H(_(inCol+1,inCol+j2),_(jCol,jCol+jLen-1)),
One,
WH(_(1,i2),_(1,jLen)));
//
// ==== Copy top of H to bottom of WH ====
//
WH(_(i2+1,i2+j2),_(1,jLen))
= H(_(inCol+1,inCol+j2),_(jCol,jCol+jLen-1));
//
// ==== Multiply by U21**H ====
//
blas::mm(Left, ConjTrans, One,
U(_(1,j2),_(i2+1,i2+j2)).lower(),
WH(_(i2+1,i2+j2),_(1,jLen)));
//
// ==== Multiply by U22 ====
//
blas::mm(ConjTrans, NoTrans, One,
U(_(j2+1,j4),_(i2+1,i4)),
H(_(inCol+j2+1,inCol+j4),_(jCol,jCol+jLen-1)),
One,
WH(_(i2+1,i4),_(1,jLen)));
//
// ==== Copy it back ====
//
H(_(inCol+1,inCol+kdu),_(jCol,jCol+jLen-1))
= WH(_(1,kdu),_(1,jLen));
}
//
// ==== Vertical multiply ====
//
for (IndexType jRow=jTop; jRow<=max(inCol,kTop)-1; jRow+=nv) {
const IndexType jLen = min(nv, max(inCol,kTop) -jRow);
//
// ==== Copy right of H to scratch (the first KZS
// . columns get multiplied by zero) ====
//
WV(_(1,jLen),_(kZs+1,kZs+kNz))
= H(_(jRow,jRow+jLen-1),_(inCol+j2+1, inCol+j2+kNz));
//
// ==== Multiply by U21 ====
//
WV(_(1,jLen),_(1,kZs)) = Zero;
blas::mm(Right, NoTrans, One,
U(_(j2+1,j2+kNz),_(kZs+1,kZs+kNz)).upper(),
WV(_(1,jLen),_(kZs+1,kZs+kNz)));
//
// ==== Multiply by U11 ====
//
blas::mm(NoTrans, NoTrans, One,
H(_(jRow,jRow+jLen-1),_(inCol+1,inCol+j2)),
U(_(1,j2),_(1,i2)),
One,
WV(_(1,jLen),_(1,i2)));
//
// ==== Copy left of H to right of scratch ====
//
WV(_(1,jLen),_(i2+1,i2+j2))
= H(_(jRow,jRow+jLen-1), _(inCol+1,inCol+j2));
//
// ==== Multiply by U21 ====
//
blas::mm(Right, NoTrans, One,
U(_(1,i4-i2),_(i2+1,i4)).lower(),
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Multiply by U22 ====
//
blas::mm(NoTrans, NoTrans, One,
H(_(jRow,jRow+jLen-1),_(inCol+j2+1,inCol+j4)),
U(_(j2+1,j4),_(i2+1,i4)),
One,
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Copy it back ====
//
H(_(jRow,jRow+jLen-1),_(inCol+1,inCol+kdu))
= WV(_(1,jLen),_(1,kdu));
}
//
// ==== Multiply Z (also vertical) ====
//
if (wantZ) {
for (IndexType jRow=iLoZ; jRow<=iHiZ; jRow+=nv) {
const IndexType jLen = min(nv,iHiZ-jRow+1);
//
// ==== Copy right of Z to left of scratch (first
// . KZS columns get multiplied by zero) ====
//
WV(_(1,jLen),_(kZs+1,kZs+kNz))
= Z(_(jRow,jRow+jLen-1),_(inCol+j2+1,inCol+j2+kNz));
//
// ==== Multiply by U12 ====
//
WV(_(1,jLen),_(1,kZs)) = Zero;
blas::mm(Right, NoTrans, One,
U(_(j2+1,j2+kNz), _(kZs+1,kZs+kNz)).upper(),
WV(_(1,jLen),_(kZs+1,kZs+kNz)));
//
// ==== Multiply by U11 ====
//
blas::mm(NoTrans, NoTrans, One,
Z(_(jRow,jRow+jLen-1), _(inCol+1,inCol+j2)),
U(_(1,j2),_(1,i2)),
One,
WV(_(1,jLen),_(1,i2)));
//
// ==== Copy left of Z to right of scratch ====
//
WV(_(1,jLen),_(i2+1,i2+j2))
= Z(_(jRow,jRow+jLen-1), _(inCol+1,inCol+j2));
//
// ==== Multiply by U21 ====
//
blas::mm(Right, NoTrans, One,
U(_(1,i4-i2),_(i2+1,i4)).lower(),
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Multiply by U22 ====
//
blas::mm(NoTrans, NoTrans, One,
Z(_(jRow,jRow+jLen-1),_(inCol+j2+1,inCol+j4)),
U(_(j2+1,j4),_(i2+1,i4)),
One,
WV(_(1,jLen),_(i2+1,i4)));
//
// ==== Copy the result back to Z ====
//
Z(_(jRow,jRow+jLen-1),_(inCol+1,inCol+kdu))
= WV(_(1,jLen),_(1,kdu));
}
}
}
}
}
}
} // namespace generic
//== interface for native lapack ===============================================
#ifdef USE_CXXLAPACK
namespace external {
//
// Real variant
//
template <typename IndexType, typename VSR, typename VSI, typename MH,
typename MZ, typename MV, typename MU, typename MWV, typename MWH>
void
laqr5_impl(bool wantT,
bool wantZ,
IndexType kacc22,
IndexType kTop,
IndexType kBot,
IndexType nShifts,
DenseVector<VSR> &sr,
DenseVector<VSI> &si,
GeMatrix<MH> &H,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
GeMatrix<MV> &V,
GeMatrix<MU> &U,
GeMatrix<MWV> &WV,
GeMatrix<MWH> &WH)
{
cxxlapack::laqr5<IndexType>(wantT,
wantZ,
kacc22,
H.numRows(),
kTop,
kBot,
nShifts,
sr.data(),
si.data(),
H.data(),
H.leadingDimension(),
iLoZ,
iHiZ,
Z.data(),
Z.leadingDimension(),
V.data(),
V.leadingDimension(),
U.data(),
U.leadingDimension(),
WV.numRows(),
WV.data(),
WV.leadingDimension(),
WH.numCols(),
WH.data(),
WH.leadingDimension());
}
//
// Complex variant
//
template <typename IndexType, typename VS, typename MH, typename MZ,
typename MV, typename MU, typename MWV, typename MWH>
void
laqr5_impl(bool wantT,
bool wantZ,
IndexType kacc22,
IndexType kTop,
IndexType kBot,
IndexType nShifts,
DenseVector<VS> &s,
GeMatrix<MH> &H,
IndexType iLoZ,
IndexType iHiZ,
GeMatrix<MZ> &Z,
GeMatrix<MV> &V,
GeMatrix<MU> &U,
GeMatrix<MWV> &WV,
GeMatrix<MWH> &WH)
{
cxxlapack::laqr5<IndexType>(wantT,
wantZ,
kacc22,
H.numRows(),
kTop,
kBot,
nShifts,
s.data(),
H.data(),
H.leadingDimension(),
iLoZ,
iHiZ,
Z.data(),
Z.leadingDimension(),
V.data(),
V.leadingDimension(),
U.data(),
U.leadingDimension(),
WV.numRows(),
WV.data(),
WV.leadingDimension(),
WH.numCols(),
WH.data(),
WH.leadingDimension());
}
} // namespace external
#endif // USE_CXXLAPACK
//== public interface ==========================================================
//
// Real variant
//
template <typename IndexType, typename VSR, typename VSI, typename MH,
typename MZ, typename MV, typename MU, typename MWV, typename MWH>
typename RestrictTo<IsRealDenseVector<VSR>::value
&& IsRealDenseVector<VSI>::value
&& IsRealGeMatrix<MH>::value
&& IsRealGeMatrix<MZ>::value
&& IsRealGeMatrix<MV>::value
&& IsRealGeMatrix<MU>::value
&& IsRealGeMatrix<MWV>::value
&& IsRealGeMatrix<MWH>::value,
void>::Type
laqr5(bool wantT,
bool wantZ,
IndexType kacc22,
IndexType kTop,
IndexType kBot,
IndexType nShifts,
VSR &&sr,
VSI &&si,
MH &&H,
IndexType iLoZ,
IndexType iHiZ,
MZ &&Z,
MV &&V,
MU &&U,
MWV &&WV,
MWH &&WH)
{
LAPACK_DEBUG_OUT("laqr5");
using std::max;
//
// Remove references from rvalue types
//
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<VSR>::Type VectorSR;
typedef typename RemoveRef<VSI>::Type VectorSI;
typedef typename RemoveRef<MH>::Type MatrixH;
typedef typename RemoveRef<MZ>::Type MatrixZ;
typedef typename RemoveRef<MV>::Type MatrixV;
typedef typename RemoveRef<MU>::Type MatrixU;
typedef typename RemoveRef<MWV>::Type MatrixWV;
typedef typename RemoveRef<MWH>::Type MatrixWH;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT((kacc22==0)||(kacc22==1)||(kacc22==2));
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
const IndexType n = H.numRows();
ASSERT(1<=kTop);
ASSERT(kBot<=n);
ASSERT(nShifts>0);
ASSERT(nShifts % 2 == 0);
ASSERT(sr.length()==nShifts);
ASSERT(si.length()==nShifts);
if (wantZ) {
ASSERT(1<=iLoZ);
ASSERT(iLoZ<=iHiZ);
ASSERT(iHiZ<=n);
}
ASSERT(V.firstRow()==1);
ASSERT(V.firstCol()==1);
ASSERT(V.numRows()>=3);
ASSERT(V.numCols()==nShifts/2);
ASSERT(U.firstRow()==1);
ASSERT(U.firstCol()==1);
ASSERT(U.numRows()>=3*nShifts-3);
ASSERT(WH.firstRow()==1);
ASSERT(WH.firstCol()==1);
ASSERT(WH.numRows()>=3*nShifts-3);
ASSERT(WH.numCols()>=1);
ASSERT(WV.firstRow()==1);
ASSERT(WV.firstCol()==1);
ASSERT(WV.numRows()>=1);
ASSERT(WV.numCols()>=3*nShifts-3);
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename VectorSR::NoView sr_org = sr;
typename VectorSI::NoView si_org = si;
typename MatrixH::NoView H_org = H;
typename MatrixZ::NoView Z_org = Z;
typename MatrixV::NoView V_org = V;
typename MatrixU::NoView U_org = U;
typename MatrixWV::NoView WV_org = WV;
typename MatrixWH::NoView WH_org = WH;
# endif
//
// Call implementation
//
LAPACK_SELECT::laqr5_impl(wantT, wantZ, kacc22, kTop, kBot, nShifts,
sr, si, H, iLoZ, iHiZ, Z,
V, U, WV, WH);
# ifdef CHECK_CXXLAPACK
//
// Make copies of results computed by the generic implementation
//
typename VectorSR::NoView sr_generic = sr;
typename VectorSI::NoView si_generic = si;
typename MatrixH::NoView H_generic = H;
typename MatrixZ::NoView Z_generic = Z;
typename MatrixV::NoView V_generic = V;
typename MatrixU::NoView U_generic = U;
typename MatrixWV::NoView WV_generic = WV;
typename MatrixWH::NoView WH_generic = WH;
//
// restore output arguments
//
sr = sr_org;
si = si_org;
H = H_org;
Z = Z_org;
V = V_org;
U = U_org;
WV = WV_org;
WH = WH_org;
//
// Compare generic results with results from the native implementation
//
external::laqr5_impl(wantT, wantZ, kacc22, kTop, kBot, nShifts,
sr, si, H, iLoZ, iHiZ, Z,
V, U, WV, WH);
bool failed = false;
if (! isIdentical(sr_generic, sr, "sr_generic", "sr")) {
std::cerr << "CXXLAPACK: sr_generic = " << sr_generic << std::endl;
std::cerr << "F77LAPACK: sr = " << sr << std::endl;
failed = true;
}
if (! isIdentical(si_generic, si, "si_generic", "si")) {
std::cerr << "CXXLAPACK: si_generic = " << si_generic << std::endl;
std::cerr << "F77LAPACK: si = " << si << std::endl;
failed = true;
}
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(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(V_generic, V, "V_generic", "V")) {
std::cerr << "CXXLAPACK: V_generic = " << V_generic << std::endl;
std::cerr << "F77LAPACK: V = " << V << std::endl;
failed = true;
}
if (! isIdentical(U_generic, U, "U_generic", "U")) {
std::cerr << "CXXLAPACK: U_generic = " << U_generic << std::endl;
std::cerr << "F77LAPACK: U = " << U << std::endl;
failed = true;
}
if (! isIdentical(WV_generic, WV, "WV_generic", "WV")) {
std::cerr << "CXXLAPACK: WV_generic = " << WV_generic << std::endl;
std::cerr << "F77LAPACK: WV = " << WV << std::endl;
failed = true;
}
if (! isIdentical(WH_generic, WH, "WH_generic", "WH")) {
std::cerr << "CXXLAPACK: WH_generic = " << WH_generic << std::endl;
std::cerr << "F77LAPACK: WH = " << WH << std::endl;
failed = true;
}
if (failed) {
std::cerr << "error in: laqr5.tcc" << std::endl;
std::cerr << "N = H.numRows() = " << H.numRows() << std::endl;
std::cerr << "H.numCols() = " << H.numCols() << std::endl;
std::cerr << "NV = WV.numRows() = " << WV.numRows() << std::endl;
std::cerr << "WV.numCols() = " << WV.numCols() << std::endl;
std::cerr << "NH = WH.numRows() = " << WH.numRows() << std::endl;
std::cerr << "WH.numCols() = " << WH.numCols() << std::endl;
ASSERT(0);
} else {
// std::cerr << "passed: laqr5.tcc" << std::endl;
}
# endif
}
//
// Complex variant
//
template <typename IndexType, typename VS, typename MH, typename MZ,
typename MV, typename MU, typename MWV, typename MWH>
typename RestrictTo<IsComplexDenseVector<VS>::value
&& IsComplexGeMatrix<MH>::value
&& IsComplexGeMatrix<MZ>::value
&& IsComplexGeMatrix<MV>::value
&& IsComplexGeMatrix<MU>::value
&& IsComplexGeMatrix<MWV>::value
&& IsComplexGeMatrix<MWH>::value,
void>::Type
laqr5(bool wantT,
bool wantZ,
IndexType kacc22,
IndexType kTop,
IndexType kBot,
IndexType nShifts,
VS &&s,
MH &&H,
IndexType iLoZ,
IndexType iHiZ,
MZ &&Z,
MV &&V,
MU &&U,
MWV &&WV,
MWH &&WH)
{
LAPACK_DEBUG_OUT("laqr5");
using std::max;
//
// Remove references from rvalue types
//
# ifdef CHECK_CXXLAPACK
typedef typename RemoveRef<VS>::Type VectorS;
typedef typename RemoveRef<MH>::Type MatrixH;
typedef typename RemoveRef<MZ>::Type MatrixZ;
typedef typename RemoveRef<MV>::Type MatrixV;
typedef typename RemoveRef<MU>::Type MatrixU;
typedef typename RemoveRef<MWV>::Type MatrixWV;
typedef typename RemoveRef<MWH>::Type MatrixWH;
# endif
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT((kacc22==0)||(kacc22==1)||(kacc22==2));
ASSERT(H.firstRow()==1);
ASSERT(H.firstCol()==1);
ASSERT(H.numRows()==H.numCols());
const IndexType n = H.numRows();
ASSERT(1<=kTop);
ASSERT(kBot<=n);
ASSERT(nShifts>0);
ASSERT(nShifts % 2 == 0);
ASSERT(s.length()==nShifts);
if (wantZ) {
ASSERT(1<=iLoZ);
ASSERT(iLoZ<=iHiZ);
ASSERT(iHiZ<=n);
}
ASSERT(V.firstRow()==1);
ASSERT(V.firstCol()==1);
ASSERT(V.numRows()>=3);
ASSERT(V.numCols()==nShifts/2);
ASSERT(U.firstRow()==1);
ASSERT(U.firstCol()==1);
ASSERT(U.numRows()>=3*nShifts-3);
ASSERT(WH.firstRow()==1);
ASSERT(WH.firstCol()==1);
ASSERT(WH.numRows()>=3*nShifts-3);
ASSERT(WH.numCols()>=1);
ASSERT(WV.firstRow()==1);
ASSERT(WV.firstCol()==1);
ASSERT(WV.numRows()>=1);
ASSERT(WV.numCols()>=3*nShifts-3);
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename VectorS::NoView s_org = s;
typename MatrixH::NoView H_org = H;
typename MatrixZ::NoView Z_org = Z;
typename MatrixV::NoView V_org = V;
typename MatrixU::NoView U_org = U;
typename MatrixWV::NoView WV_org = WV;
typename MatrixWH::NoView WH_org = WH;
# endif
//
// Call implementation
//
LAPACK_SELECT::laqr5_impl(wantT, wantZ, kacc22, kTop, kBot, nShifts, s, H,
iLoZ, iHiZ, Z, V, U, WV, WH);
# ifdef CHECK_CXXLAPACK
//
// Make copies of results computed by the generic implementation
//
typename VectorS::NoView s_generic = s;
typename MatrixH::NoView H_generic = H;
typename MatrixZ::NoView Z_generic = Z;
typename MatrixV::NoView V_generic = V;
typename MatrixU::NoView U_generic = U;
typename MatrixWV::NoView WV_generic = WV;
typename MatrixWH::NoView WH_generic = WH;
//
// restore output arguments
//
s = s_org;
H = H_org;
Z = Z_org;
V = V_org;
U = U_org;
WV = WV_org;
WH = WH_org;
//
// Compare generic results with results from the native implementation
//
external::laqr5_impl(wantT, wantZ, kacc22, kTop, kBot, nShifts, s, H,
iLoZ, iHiZ, Z, V, U, WV, WH);
bool failed = false;
if (! isIdentical(s_generic, s, "s_generic", "s")) {
std::cerr << "CXXLAPACK: s_generic = " << s_generic << std::endl;
std::cerr << "F77LAPACK: s = " << s << std::endl;
failed = true;
}
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(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(V_generic, V, "V_generic", "V")) {
std::cerr << "CXXLAPACK: V_generic = " << V_generic << std::endl;
std::cerr << "F77LAPACK: V = " << V << std::endl;
failed = true;
}
if (! isIdentical(U_generic, U, "U_generic", "U")) {
std::cerr << "CXXLAPACK: U_generic = " << U_generic << std::endl;
std::cerr << "F77LAPACK: U = " << U << std::endl;
failed = true;
}
if (! isIdentical(WV_generic, WV, "WV_generic", "WV")) {
std::cerr << "CXXLAPACK: WV_generic = " << WV_generic << std::endl;
std::cerr << "F77LAPACK: WV = " << WV << std::endl;
failed = true;
}
if (! isIdentical(WH_generic, WH, "WH_generic", "WH")) {
std::cerr << "CXXLAPACK: WH_generic = " << WH_generic << std::endl;
std::cerr << "F77LAPACK: WH = " << WH << std::endl;
failed = true;
}
if (failed) {
std::cerr << "error in: laqr5.tcc" << std::endl;
std::cerr << "N = H.numRows() = " << H.numRows() << std::endl;
std::cerr << "H.numCols() = " << H.numCols() << std::endl;
std::cerr << "NV = WV.numRows() = " << WV.numRows() << std::endl;
std::cerr << "WV.numCols() = " << WV.numCols() << std::endl;
std::cerr << "NH = WH.numRows() = " << WH.numRows() << std::endl;
std::cerr << "WH.numCols() = " << WH.numCols() << std::endl;
ASSERT(0);
} else {
// std::cerr << "passed: laqr5.tcc" << std::endl;
}
# endif
}
} } // namespace lapack, flens
#endif // FLENS_LAPACK_LA_LAQR5_TCC
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