1
       2
       3
       4
       5
       6
       7
       8
       9
      10
      11
      12
      13
      14
      15
      16
      17
      18
      19
      20
      21
      22
      23
      24
      25
      26
      27
      28
      29
      30
      31
      32
      33
      34
      35
      36
      37
      38
      39
      40
      41
      42
      43
      44
      45
      46
      47
      48
      49
      50
      51
      52
      53
      54
      55
      56
      57
      58
      59
      60
      61
      62
      63
      64
      65
      66
      67
      68
      69
      70
      71
      72
      73
      74
      75
      76
      77
      78
      79
      80
      81
      82
      83
      84
      85
      86
      87
      88
      89
      90
      91
      92
      93
      94
      95
      96
      97
      98
      99
     100
     101
     102
     103
     104
     105
     106
     107
     108
     109
     110
     111
     112
     113
     114
     115
     116
     117
     118
     119
     120
     121
     122
     123
     124
     125
     126
     127
     128
     129
     130
     131
     132
     133
     134
     135
     136
     137
     138
     139
     140
     141
     142
     143
     144
     145
     146
     147
     148
     149
     150
     151
     152
     153
     154
     155
     156
     157
     158
     159
     160
     161
     162
     163
     164
     165
     166
     167
     168
     169
     170
     171
     172
     173
     174
     175
     176
     177
     178
     179
     180
     181
     182
     183
     184
     185
     186
     187
     188
     189
     190
     191
     192
     193
     194
     195
     196
     197
     198
     199
     200
     201
     202
     203
     204
     205
     206
     207
     208
     209
     210
     211
     212
     213
     214
     215
     216
     217
     218
     219
     220
     221
     222
     223
     224
     225
     226
     227
     228
     229
     230
     231
     232
     233
     234
     235
     236
     237
     238
     239
     240
     241
     242
     243
     244
     245
     246
     247
     248
     249
     250
     251
     252
     253
     254
     255
     256
     257
     258
     259
     260
     261
     262
     263
     264
     265
     266
     267
     268
     269
     270
     271
     272
     273
     274
     275
     276
     277
     278
     279
     280
     281
     282
     283
     284
     285
     286
     287
     288
     289
     290
     291
     292
     293
     294
     295
     296
     297
     298
     299
     300
     301
     302
     303
     304
     305
     306
     307
     308
     309
     310
     311
     312
     313
     314
     315
     316
     317
     318
     319
     320
     321
     322
     323
     324
     325
     326
     327
     328
     329
     330
     331
     332
     333
     334
     335
     336
     337
     338
     339
     340
     341
     342
     343
     344
     345
     346
     347
     348
     349
     350
     351
     352
     353
     354
     355
     356
     357
     358
     359
     360
     361
     362
     363
     364
     365
     366
     367
     368
     369
     370
     371
     372
     373
     374
     375
     376
     377
     378
     379
     380
     381
     382
     383
     384
     385
     386
     387
     388
     389
     390
     391
     392
     393
     394
     395
     396
     397
     398
     399
     400
     401
     402
     403
     404
     405
     406
     407
     408
     409
     410
     411
     412
     413
     414
     415
     416
     417
     418
     419
     420
     421
     422
     423
     424
     425
     426
     427
     428
     429
     430
     431
     432
     433
     434
     435
     436
     437
     438
     439
     440
     441
     442
     443
     444
     445
     446
     447
     448
     449
     450
     451
     452
     453
     454
     455
     456
     457
     458
     459
     460
     461
     462
     463
     464
     465
     466
     467
     468
     469
     470
     471
     472
     473
     474
     475
     476
     477
     478
     479
     480
     481
     482
     483
     484
     485
     486
     487
     488
     489
     490
     491
     492
     493
     494
     495
     496
     497
     498
     499
     500
     501
     502
     503
     504
     505
     506
     507
     508
     509
     510
     511
     512
     513
     514
     515
     516
     517
     518
     519
     520
     521
     522
     523
     524
     525
     526
     527
     528
     529
     530
     531
     532
     533
     534
     535
     536
     537
     538
     539
     540
     541
     542
     543
     544
     545
     546
     547
     548
     549
     550
     551
     552
     553
     554
     555
     556
     557
     558
     559
     560
     561
     562
     563
     564
     565
     566
     567
     568
     569
     570
     571
     572
     573
     574
     575
     576
     577
     578
     579
     580
     581
     582
     583
     584
     585
     586
     587
     588
     589
     590
     591
     592
     593
     594
     595
     596
     597
     598
     599
     600
     601
     602
     603
     604
     605
     606
     607
     608
     609
     610
     611
     612
     613
     614
     615
     616
     617
     618
     619
     620
     621
     622
     623
     624
     625
     626
     627
     628
     629
     630
     631
     632
     633
     634
     635
     636
     637
     638
     639
     640
     641
     642
     643
     644
     645
     646
     647
     648
     649
     650
     651
     652
     653
     654
     655
     656
     657
     658
     659
     660
     661
     662
     663
     664
     665
     666
     667
     668
     669
     670
     671
     672
     673
     674
     675
     676
     677
     678
     679
     680
     681
     682
     683
     684
     685
     686
     687
     688
     689
     690
     691
     692
     693
     694
     695
     696
     697
     698
     699
     700
     701
     702
     703
     704
     705
     706
     707
     708
     709
     710
     711
     712
     713
     714
     715
     716
     717
     718
     719
     720
     721
     722
     723
     724
     725
     726
     727
     728
     729
     730
     731
     732
     733
     734
     735
     736
     737
     738
     739
     740
     741
     742
     743
     744
     745
     746
     747
     748
     749
     750
     751
     752
     753
     754
     755
     756
     757
     758
     759
     760
     761
     762
     763
     764
     765
     766
     767
     768
     769
     770
     771
     772
     773
     774
     775
     776
     777
     778
     779
     780
     781
     782
     783
     784
     785
     786
     787
     788
     789
     790
     791
     792
     793
     794
     795
     796
     797
     798
     799
     800
     801
     802
     803
     804
     805
     806
     807
     808
     809
     810
     811
     812
/*
 *   Copyright (c) 2012, 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 DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
      $                   WORK, LWORK, INFO )
 *
 *  -- LAPACK driver routine (version 3.3.1) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --
 */

#ifndef FLENS_LAPACK_GE_LSY_TCC
#define FLENS_LAPACK_GE_LSY_TCC 1

#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>

namespace flens { namespace lapack {

//== generic lapack implementation =============================================

namespace generic {

//-- (ge)lsy [real variant] ----------------------------------------------------

template <typename MA, typename MB, typename VJPIV, typename RCOND,
          typename RANK, typename VWORK>
void
lsy_impl(GeMatrix<MA>              &A,
         GeMatrix<MB>              &B,
         DenseVector<VJPIV>        &jPiv,
         RCOND                     rCond,
         RANK                      &rank,
         DenseVector<VWORK>        &work)
{
    using std::abs;
    using flens::max;
    using flens::min;
    using LASCL::FullMatrix;
    using LASCL::UpperTriangular;

    typedef typename GeMatrix<MA>::ElementType  ElementType;
    typedef typename GeMatrix<MA>::IndexType    IndexType;

    const ElementType Zero(0), One(1);

    const Underscore<IndexType>  _;

    const IndexType m    = A.numRows();
    const IndexType n    = A.numCols();
    const IndexType nRhs = B.numCols();
    const IndexType mn   = min(m, n);

//
//  Figure out optimal block size
//
    IndexType lWork = work.length();
    IndexType lWorkMin;
    IndexType lWorkOpt;

    if (mn==0 || nRhs==0) {
        lWorkMin1;
        lWorkOpt1;
    } else {
        IndexType nb1 = ilaenv<ElementType>(1"GEQRF""", m, n);
        IndexType nb2 = ilaenv<ElementType>(1"GERQF""", m, n);
        IndexType nb3 = ilaenv<ElementType>(1"ORMQR""", m, n, nRhs);
        IndexType nb4 = ilaenv<ElementType>(1"ORMRQ""", m, n, nRhs);
        IndexType nb = max(nb1, nb2, nb3, nb4);
        lWorkMin = mn + max(2*mn, n+1, mn+nRhs);
        lWorkOpt = max(lWorkMin, mn+2*n+nb*(n+1), 2*mn+nb*nRhs);
    }

    if (lWork==0) {
        work.resize(lWorkOpt);
        lWork = work.length();
    }
    work(1) = lWorkOpt;

    auto sMinWork = work(_(mn+1,2*mn));
    auto sMaxWork = work(_(2*mn+1,3*mn));

//
//  Quick return if possible
//
    if (mn==0 || nRhs==0) {
        rank0;
        return;
    }
//
//  Get machine parameters
//
    ElementType smallNum = lamch<ElementType>(SafeMin)
                         / lamch<ElementType>(Precision);
    ElementType bigNum = One / smallNum;
    labad(smallNum, bigNum);

//
//  Scale A, B if max entries outside range [SMLNUM,BIGNUM]
//
    ElementType normA = lan(MaximumNorm, A);

    IndexType scaleA = 0;

    if (normA>Zero && normA<smallNum) {
//
//      Scale matrix norm up to SMLNUM
//
        lascl(FullMatrix00, normA, smallNum, A);
        scaleA1;
    } else if (normA>bigNum) {
//
//      Scale matrix norm down to BIGNUM
//
        lascl(FullMatrix00, normA, bigNum, A);
        scaleA2;
    } else if (normA==Zero) {
//
//      Matrix all zero. Return zero solution.
//
        B = Zero;
        rank0;
        work(1) = lWorkOpt;
        return;
    }

    auto B_ = B(_(1,m),_);
    ElementType normB = lan(MaximumNorm, B_);

    IndexType scaleB = 0;

    if (normB>Zero && normB<smallNum) {
//
//      Scale matrix norm up to SMLNUM
//
        lascl(FullMatrix00, normB, smallNum, B_);
        scaleB1;
    } else if (normB>bigNum) {
//
//      Scale matrix norm down to BIGNUM
//
        lascl(FullMatrix00, normB, bigNum, B_);
        scaleB2;
    }
//
//  Compute QR factorization with column pivoting of A:
//      A * P = Q * R
//
    auto tau1    = work(_(1,mn));
    auto qp3Work = work(_(mn+1,lWork));
    qp3(A, jPiv, tau1, qp3Work);
//
//  workspace: MN+2*N+NB*(N+1).
//  Details of Householder rotations stored in WORK(1:MN).
//
//  Determine RANK using incremental condition estimation
//
    sMinWork(1) = One;
    sMaxWork(1) = One;

    ElementType sMax = abs(A(1,1));
    ElementType sMin = sMax;

    if (abs(A(1,1))==Zero) {
        rank0;
        B = Zero;
        work(1) = lWorkOpt;
        return;
    } else {
        rank1;
    }

    while (rank<mn) {
        IndexType i = rank+1;
        const auto sMinWork_ = sMinWork(_(1,rank));
        const auto sMaxWork_ = sMaxWork(_(1,rank));
        const auto A_i       = A(_(1,rank),i);

        ElementType sMinPr, sMaxPr, s1, s2, c1, c2;

        laic1(LAIC1::Min, sMinWork_, sMin, A_i, A(i,i), sMinPr, s1, c1);
        laic1(LAIC1::Max, sMaxWork_, sMax, A_i, A(i,i), sMaxPr, s2, c2);

        if (sMaxPr*rCond<=sMinPr) {
            for (IndexType i=1; i<=rank; ++i) {
                sMinWork(i) *= s1;
                sMaxWork(i) *= s2;
            }
            sMinWork(rank+1) = c1;
            sMaxWork(rank+1) = c2;
            sMin = sMinPr;
            sMax = sMaxPr;
            ++rank;
        } else {
            break;
        }
    }

//
//  workspace: 3*MN.
//
//  Logically partition R = [ R11 R12 ]
//                          [  0  R22 ]
//  where R11 = R(1:RANK,1:RANK)
//
//  [R11,R12] = [ T11, 0 ] * Y
//
    auto tau2 = work(_(mn+1,mn+rank));
    auto work_ = work(_(2*mn+1,lWork));
    if (rank<n) {
        auto A_ = A(_(1,rank),_);
        tzrzf(A_, tau2, work_);
    }
    auto T11 = A(_(1,rank),_(1,rank)).upper();
//
//  workspace: 2*MN.
//  Details of Householder rotations stored in WORK(MN+1:2*MN)
//
//  B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
//
    ormqr(Left, Trans, A(_,_(1,mn)), tau1, B_, work_);
//
//  workspace: 2*MN+NB*NRHS.
//
//  B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
//
    blas::sm(Left, NoTrans, One, T11, B(_(1,rank),_));

    B(_(rank+1,n),_) = Zero;
//
//  B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
//
    if (rank<n) {
        ormrz(Left, Trans, n-rank, A(_(1,rank),_), tau2, B(_(1,n),_), work_);
    }
//
//  workspace: 2*MN+NRHS.
//
//  B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
//
    for (IndexType j=1; j<=nRhs; ++j) {
        for (IndexType i=1; i<=n; ++i) {
            work(jPiv(i)) = B(i,j);
        }
        B(_(1,n),j) = work(_(1,n));
    }

//
//  workspace: N
//
//  Undo scaling
//
    if (scaleA==1) {
        lascl(FullMatrix00, normA, smallNum, B(_(1,n),_));
        lascl(UpperTriangular00, smallNum, normA, A(_(1,rank),_(1,rank)));
    } else if (scaleA==2) {
        lascl(FullMatrix00, normA, bigNum, B(_(1,n),_));
        lascl(UpperTriangular00, bigNum, normA, A(_(1,rank),_(1,rank)));
    }
    if (scaleB==1) {
        lascl(FullMatrix00, smallNum, normB, B(_(1,n),_));
    } else if (scaleB==2) {
        lascl(FullMatrix00, bigNum, normB, B(_(1,n),_));
    }

    work(1) = lWorkOpt;
}

// namespace generic


//== interface for native lapack ===============================================

#ifdef USE_CXXLAPACK

namespace external {

//-- (ge)lsy [real variant] ----------------------------------------------------

template <typename MA, typename MB, typename VJPIV, typename RCOND,
          typename RANK, typename VWORK>
void
lsy_impl(GeMatrix<MA>              &A,
         GeMatrix<MB>              &B,
         DenseVector<VJPIV>        &jPiv,
         RCOND                     rCond,
         RANK                      &rank,
         DenseVector<VWORK>        &work)
{
    typedef typename GeMatrix<MA>::ElementType  ElementType;
    typedef typename GeMatrix<MA>::IndexType    IndexType;

    if (work.length()==0) {
        ElementType   WORK;
        IndexType     LWORK = -1;

        cxxlapack::gelsy<IndexType>(A.numRows(),
                                    A.numCols(),
                                    B.numCols(),
                                    A.data(),
                                    A.leadingDimension(),
                                    B.data(),
                                    B.leadingDimension(),
                                    jPiv.data(),
                                    rCond,
                                    rank,
                                    &WORK,
                                    LWORK);
        work.resize(IndexType(WORK));
    }

    cxxlapack::gelsy<IndexType>(A.numRows(),
                                A.numCols(),
                                B.numCols(),
                                A.data(),
                                A.leadingDimension(),
                                B.data(),
                                B.leadingDimension(),
                                jPiv.data(),
                                rCond,
                                rank,
                                work.data(),
                                work.length());
}

//-- (ge)lsy [complex variant] -------------------------------------------------

template <typename MA, typename MB, typename VJPIV, typename RCOND,
          typename RANK, typename VWORK, typename VRWORK>
void
lsy_impl(GeMatrix<MA>              &A,
         GeMatrix<MB>              &B,
         DenseVector<VJPIV>        &jPiv,
         RCOND                     rCond,
         RANK                      &rank,
         DenseVector<VWORK>        &work,
         DenseVector<VRWORK>       &rwork)
{
    typedef typename GeMatrix<MA>::ElementType  ElementType;
    typedef typename GeMatrix<MA>::IndexType    IndexType;

    if (work.length()==0) {
        ElementType   WORK;
        IndexType     LWORK = -1;

        cxxlapack::gelsy<IndexType>(A.numRows(),
                                    A.numCols(),
                                    B.numCols(),
                                    A.data(),
                                    A.leadingDimension(),
                                    B.data(),
                                    B.leadingDimension(),
                                    jPiv.data(),
                                    rCond,
                                    rank,
                                    &WORK,
                                    LWORK,
                                    rwork.data());
        work.resize(IndexType(cxxblas::real(WORK)));
    }

    cxxlapack::gelsy<IndexType>(A.numRows(),
                                A.numCols(),
                                B.numCols(),
                                A.data(),
                                A.leadingDimension(),
                                B.data(),
                                B.leadingDimension(),
                                jPiv.data(),
                                rCond,
                                rank,
                                work.data(),
                                work.length(),
                                rwork.data());
}

// namespace external

#endif // USE_CXXLAPACK


//== public interface ==========================================================

//-- (ge)lsy [real variant] ----------------------------------------------------

template <typename MA, typename MB, typename VJPIV, typename RCOND,
          typename VWORK>
typename RestrictTo<IsRealGeMatrix<MA>::value
                 && IsRealGeMatrix<MB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value
                 && IsRealDenseVector<VWORK>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    MB           &&B,
    VJPIV        &&jPiv,
    RCOND        rCond,
    VWORK        &&work)
{
    using flens::max;
    using std::min;

    LAPACK_DEBUG_OUT("(ge)lsy [real]");

//
//  Remove references from rvalue types
//
    typedef typename RemoveRef<MA>::Type    MatrixA;
    typedef typename RemoveRef<MB>::Type    MatrixB;
    typedef typename RemoveRef<VJPIV>::Type VectorJPiv;
    typedef typename RemoveRef<VWORK>::Type VectorWork;

    typedef typename MatrixA::IndexType     IndexType;

//
//  Test the input parameters
//
    const IndexType m = A.numRows();
    const IndexType n = A.numCols();
    const IndexType nRhs = B.numCols();

#   ifndef NDEBUG
    ASSERT(A.firstRow()==1);
    ASSERT(A.firstCol()==1);
    ASSERT(B.firstRow()==1);
    ASSERT(B.firstCol()==1);
    ASSERT(jPiv.firstIndex()==1);
    ASSERT(work.firstIndex()==1);
    ASSERT(B.numRows()==max(m,n));
    ASSERT(jPiv.length()==0 || jPiv.length()==n);

    if (work.length()>0) {
        const IndexType mn = min(m, n);
        const IndexType lWorkMin = mn + max(2*mn, n1, mn + nRhs);
        ASSERT(work.length()>=lWorkMin);
    }
#   endif

    if (jPiv.length()==0) {
        jPiv.resize(n, jPiv.firstIndex(), IndexType(0));
    }

//
//  Make copies of output arguments
//
#   ifdef CHECK_CXXLAPACK
    typename MatrixA::NoView        A_org      = A;
    typename MatrixB::NoView        B_org      = B;
    typename VectorJPiv::NoView     jPiv_org   = jPiv;
    typename VectorWork::NoView     work_org   = work;
#   endif

//
//  Call implementation
//
    IndexType  rank;
    LAPACK_SELECT::lsy_impl(A, B, jPiv, rCond, rank, work);

#   ifdef CHECK_CXXLAPACK
//
//  Make copies of results computed by the generic implementation
//
    typename MatrixA::NoView        A_generic    = A;
    typename MatrixB::NoView        B_generic    = B;
    typename VectorJPiv::NoView     jPiv_generic = jPiv;
    IndexType                       rank_generic = rank;
    typename VectorWork::NoView     work_generic = work;

//
//  restore output arguments
//
    A    = A_org;
    B    = B_org;
    jPiv = jPiv_org;
    rank = 0;
    work = work_org;

//
//  Compare generic results with results from the native implementation
//
    external::lsy_impl(A, B, jPiv, rCond, rank, work);

    bool failed = false;
    if (! isIdentical(A_generic, A, "A_generic""A")) {
        std::cerr << "CXXLAPACK: A_generic = " << A_generic << std::endl;
        std::cerr << "F77LAPACK: A = " << A << std::endl;
        failed = true;
    }
    if (! isIdentical(B_generic, B, "B_generic""B")) {
        std::cerr << "CXXLAPACK: B_generic = " << B_generic << std::endl;
        std::cerr << "F77LAPACK: B = " << B << std::endl;
        failed = true;
    }
    if (! isIdentical(jPiv_generic, jPiv, "jPiv_generic""jPiv")) {
        std::cerr << "CXXLAPACK: jPiv_generic = " << jPiv_generic << std::endl;
        std::cerr << "F77LAPACK: jPiv = " << jPiv << std::endl;
        failed = true;
    }
    if (! isIdentical(rank_generic, rank, "rank_generic""rank")) {
        std::cerr << "CXXLAPACK: rank_generic = " << rank_generic << std::endl;
        std::cerr << "F77LAPACK: rank = " << rank << 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 << "A.numRows() = " << A.numRows() << std::endl;
        std::cerr << "A.numCols() = " << A.numCols() << std::endl;
        std::cerr << "A = " << A << std::endl;
        std::cerr << "rank_generic = " << rank_generic << std::endl;
        std::cerr << "rank = " << rank << std::endl;
        std::cerr << "error in: lsy.tcc" << std::endl;
        ASSERT(0);
    } else {
        // std::cerr << "passed: lsy.tcc" << std::endl;
    }
#   endif

    return rank;
}

//-- (ge)lsy [complex variant] -------------------------------------------------

#ifdef USE_CXXLAPACK

template <typename MA, typename MB, typename VJPIV, typename RCOND,
          typename VWORK, typename VRWORK>
typename RestrictTo<IsComplexGeMatrix<MA>::value
                 && IsComplexGeMatrix<MB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value
                 && IsComplexDenseVector<VWORK>::value
                 && IsRealDenseVector<VRWORK>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    MB           &&B,
    VJPIV        &&jPiv,
    RCOND        rCond,
    VWORK        &&work,
    VRWORK       &&rwork)
{
    using flens::max;
    using flens::min;

    LAPACK_DEBUG_OUT("(ge)lsy [complex]");

//
//  Remove references from rvalue types
//
    typedef typename RemoveRef<MA>::Type    MatrixA;
    typedef typename RemoveRef<MB>::Type    MatrixB;
    typedef typename RemoveRef<VJPIV>::Type VectorJPiv;
    typedef typename RemoveRef<VWORK>::Type VectorWork;

    typedef typename MatrixA::IndexType     IndexType;

//
//  Test the input parameters
//
    const IndexType m = A.numRows();
    const IndexType n = A.numCols();
    const IndexType nRhs = B.numCols();

#   ifndef NDEBUG
    ASSERT(A.firstRow()==1);
    ASSERT(A.firstCol()==1);
    ASSERT(B.firstRow()==1);
    ASSERT(B.firstCol()==1);
    ASSERT(jPiv.firstIndex()==1);
    ASSERT(work.firstIndex()==1);
    ASSERT(B.numRows()==max(m,n));
    ASSERT(jPiv.length()==0 || jPiv.length()==n);

    if (work.length()>0) {
        const IndexType mn = min(m, n);
        const IndexType lWorkMin = mn + max(2*mn, n1, mn + nRhs);
        ASSERT(work.length()>=lWorkMin);
    }

    ASSERT(rwork.length()==0 || rwork.length()==2*n);
#   endif

    if (jPiv.length()==0) {
        jPiv.resize(n, jPiv.firstIndex(), IndexType(0));
    }

    if (rwork.length()==0) {
        rwork.resize(2*n);
    }

//
//  Call implementation
//
    IndexType  rank;
    external::lsy_impl(A, B, jPiv, rCond, rank, work, rwork);

    return rank;
}

#endif // USE_CXXLAPACK


//-- (ge)lsy [real variant with temporary workspace] ---------------------------

template <typename MA, typename MB, typename VJPIV, typename RCOND>
typename RestrictTo<IsRealGeMatrix<MA>::value
                 && IsRealGeMatrix<MB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    MB           &&B,
    VJPIV        &&jPiv,
    RCOND        rCond)
{
    typedef typename RemoveRef<MA>::Type::Vector WorkVector;

    WorkVector  work;
    return lsy(A, B, jPiv, rCond);
}


//-- (ge)lsy [complex variant with temporary workspace] ------------------------

#ifdef USE_CXXLAPACK

template <typename MA, typename MB, typename VJPIV, typename RCOND>
typename RestrictTo<IsComplexGeMatrix<MA>::value
                 && IsComplexGeMatrix<MB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    MB           &&B,
    VJPIV        &&jPiv,
    RCOND        rCond)
{
    typedef typename RemoveRef<MA>::Type::Vector        WorkVector;
    typedef typename RemoveRef<MA>::Type::ElementType   T;
    typedef typename ComplexTrait<T>::PrimitiveType     PT;
    typedef DenseVector<Array<PT> >                     RealWorkVector;

    WorkVector      work;
    RealWorkVector  rwork;
    return lsy(A, B, jPiv, rCond, work, rwork);
}

#endif // USE_CXXLAPACK


//== (ge)lsy variant if B is vector ============================================


//-- (ge)lsy [real variant] ----------------------------------------------------

template <typename MA, typename VB, typename VJPIV, typename RCOND,
          typename VWORK>
typename RestrictTo<IsRealGeMatrix<MA>::value
                 && IsRealDenseVector<VB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value
                 && IsRealDenseVector<VWORK>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    VB           &&b,
    VJPIV        &&jPiv,
    RCOND        rCond,
    VWORK        &&work)
{
//
//  Remove references from rvalue types
//
    typedef typename RemoveRef<MA>::Type    MatrixA;
    typedef typename RemoveRef<VB>::Type    VectorB;

    typedef typename VectorB::ElementType  ElementType;
    typedef typename VectorB::IndexType    IndexType;

    const IndexType    n     = b.length();
    const StorageOrder order = MatrixA::Engine::order;

    GeMatrix<FullStorageView<ElementType, order> >  B(n1, b, n);
    return lsy(A, B, jPiv, rCond, work);
}


//-- (ge)lsy [complex variant] -------------------------------------------------

#ifdef USE_CXXLAPACK

template <typename MA, typename VB, typename VJPIV, typename RCOND,
          typename VWORK, typename VRWORK>
typename RestrictTo<IsComplexGeMatrix<MA>::value
                 && IsComplexDenseVector<VB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value
                 && IsComplexDenseVector<VWORK>::value
                 && IsRealDenseVector<VRWORK>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    VB           &&b,
    VJPIV        &&jPiv,
    RCOND        rCond,
    VWORK        &&work,
    VRWORK       &&rwork)
{
//
//  Remove references from rvalue types
//
    typedef typename RemoveRef<MA>::Type    MatrixA;
    typedef typename RemoveRef<VB>::Type    VectorB;

    typedef typename VectorB::ElementType  ElementType;
    typedef typename VectorB::IndexType    IndexType;

    const IndexType    n     = b.length();
    const StorageOrder order = MatrixA::Engine::order;

    GeMatrix<FullStorageView<ElementType, order> >  B(n1, b, n);
    return lsy(A, B, jPiv, rCond, work, rwork);
}

#endif // USE_CXXLAPACK

//-- (ge)lsy [real variant with temporary workspace] ---------------------------

template <typename MA, typename VB, typename VJPIV, typename RCOND>
typename RestrictTo<IsRealGeMatrix<MA>::value
                 && IsRealDenseVector<VB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    VB           &&b,
    VJPIV        &&jPiv,
    RCOND        rCond)
{
    typedef typename RemoveRef<MA>::Type::Vector WorkVector;

    WorkVector  work;
    return lsy(A, b, jPiv, rCond, work);
}


//-- (ge)lsy [complex variant with temporary workspace] ------------------------

#ifdef USE_CXXLAPACK

template <typename MA, typename VB, typename VJPIV, typename RCOND>
typename RestrictTo<IsComplexGeMatrix<MA>::value
                 && IsComplexDenseVector<VB>::value
                 && IsIntegerDenseVector<VJPIV>::value
                 && IsReal<RCOND>::value,
         typename RemoveRef<MA>::Type::IndexType>::Type
lsy(MA           &&A,
    VB           &&b,
    VJPIV        &&jPiv,
    RCOND        rCond)
{
    typedef typename RemoveRef<MA>::Type::Vector        WorkVector;
    typedef typename RemoveRef<MA>::Type::ElementType   T;
    typedef typename ComplexTrait<T>::PrimitiveType     PT;
    typedef DenseVector<Array<PT> >                     RealWorkVector;

    WorkVector      work;
    RealWorkVector  rwork;
    return lsy(A, b, jPiv, rCond, work, rwork);
}

#endif // USE_CXXLAPACK



} } // namespace lapack, flens

#endif // FLENS_LAPACK_GE_LSY_TCC