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/*
* 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 DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, $ LDV, WORK, LWORK, INFO ) * * -- LAPACK routine (version 3.3.1) -- * * -- Contributed by Zlatko Drmac of the University of Zagreb and -- * -- Kresimir Veselic of the Fernuniversitaet Hagen -- * -- April 2011 -- * * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * This routine is also part of SIGMA (version 1.23, October 23. 2008.) * SIGMA is a library of algorithms for highly accurate algorithms for * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0. * */ #ifndef FLENS_LAPACK_GE_SVJ_TCC #define FLENS_LAPACK_GE_SVJ_TCC 1 #include <flens/blas/blas.h> #include <flens/lapack/lapack.h> namespace flens { namespace lapack { //== generic lapack implementation ============================================= namespace generic { template <typename MA, typename VSVA, typename MV, typename VWORK> typename GeMatrix<MA>::IndexType svj_impl(SVJ::TypeA typeA, SVJ::JobU jobU, SVJ::JobV jobV, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MV> &V, DenseVector<VWORK> &work) { using std::abs; using std::max; using std::min; using flens::pow; using std::sqrt; using std::swap; typedef typename GeMatrix<MA>::ElementType ElementType; typedef typename GeMatrix<MA>::IndexType IndexType; const ElementType Zero(0), Half(0.5), One(1); const IndexType nSweep = 30; const Underscore<IndexType> _; ElementType fastr_data[5]; DenseVectorView<ElementType> fastr = typename DenseVectorView<ElementType>::Engine(5, fastr_data); const IndexType m = A.numRows(); const IndexType n = A.numCols(); const IndexType mv = V.numRows(); const IndexType lWork = work.length(); const bool lower = (typeA==SVJ::Lower); const bool upper = (typeA==SVJ::Upper); const bool lhsVec = (jobU==SVJ::ComputeU); const bool controlU = (jobU==SVJ::ControlU); const bool applyV = (jobV==SVJ::ApplyV); const bool rhsVec = (jobV==SVJ::ComputeV) || applyV; IndexType info = 0; // //#:) Quick return for void matrix // if ((m==0) || (n==0)) { return info; } // // Set numerical parameters // The stopping criterion for Jacobi rotations is // // max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS // // where EPS is the round-off and CTOL is defined as follows: // ElementType cTol; if (controlU) { // ... user controlled cTol = work(1); } else { // ... default if (lhsVec || rhsVec) { cTol = sqrt(ElementType(m)); } else { cTol = ElementType(m); } } // ... and the machine dependent parameters are //[!] (Make sure that DLAMCH() works properly on the target machine.) // const ElementType eps = lamch<ElementType>(Eps); const ElementType rootEps = sqrt(eps); const ElementType safeMin = lamch<ElementType>(SafeMin); const ElementType rootSafeMin = sqrt(safeMin); const ElementType small = safeMin / eps; const ElementType big = lamch<ElementType>(OverflowThreshold); // const ElementType big = One / safeMin; const ElementType rootBig = One / rootSafeMin; const ElementType bigTheta = One / rootEps; const ElementType tol = cTol*eps; const ElementType rootTol = sqrt(tol); if (ElementType(m)*eps>=One) { // we return -4 to keep it compatible with the LAPACK implementation return -4; } // // Initialize the right singular vector matrix. // if (rhsVec && (!applyV)) { V = Zero; V.diag(0) = One; } // // Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N ) //(!) If necessary, scale A to protect the largest singular value // from overflow. It is possible that saving the largest singular // value destroys the information about the small ones. // This initial scaling is almost minimal in the sense that the // goal is to make sure that no column norm overflows, and that // DSQRT(N)*max_i SVA(i) does not overflow. If INFinite entries // in A are detected, the procedure returns with INFO=-6. // ElementType skl = One / sqrt(ElementType(m)*ElementType(n)); bool noScale = true; bool goScale = true; ElementType aapp, aaqq, tmp; if (lower) { // the input matrix is M-by-N lower triangular (trapezoidal) for (IndexType p=1; p<=n; ++p) { aapp = Zero; aaqq = One; lassq(A(_(p,m),p), aapp, aaqq); if (aapp>big) { return -6; } aaqq = sqrt(aaqq); if ((aapp<(big/aaqq)) && noScale) { sva(p) = aapp*aaqq; } else { noScale = false; sva(p) = aapp*(aaqq*skl); if (goScale) { goScale = false; sva(_(1,p-1)) *= skl; } } } } else if (upper) { // the input matrix is M-by-N upper triangular (trapezoidal) for (IndexType p=1; p<=n; ++p) { aapp = Zero; aaqq = One; lassq(A(_(1,p), p), aapp, aaqq); if (aapp>big) { return -6; } aaqq = sqrt(aaqq); if ((aapp<(big/aaqq)) && noScale) { sva(p) = aapp*aaqq; } else { noScale = false; sva(p) = aapp*(aaqq*skl); if (goScale) { goScale = false; sva(_(1,p-1)) *= skl; } } } } else { // the input matrix is M-by-N general dense for (IndexType p=1; p<=n; ++p) { aapp = Zero; aaqq = One; lassq(A(_,p), aapp, aaqq); if (aapp>big) { return -6; } aaqq = sqrt(aaqq); if ((aapp<(big/aaqq)) && noScale) { sva(p) = aapp*aaqq; } else { noScale = false; sva(p) = aapp*(aaqq*skl); if (goScale) { goScale = false; sva(_(1,p-1)) *= skl; } } } } if (noScale) { skl = One; } // // Move the smaller part of the spectrum from the underflow threshold //(!) Start by determining the position of the nonzero entries of the // array SVA() relative to ( SFMIN, BIG ). // aapp = Zero; aaqq = big; for (IndexType p=1; p<=n; ++p) { if (sva(p)!=Zero) { aaqq = min(aaqq,sva(p)); } aapp = max(aapp,sva(p)); } // //#:) Quick return for zero matrix // if (aapp==Zero) { if (lhsVec) { A = Zero; A.diag(0) = One; } work(1) = One; work(_(2,6)) = Zero; return info; } // //#:) Quick return for one-column matrix // if (n==1) { if (lhsVec) { lascl(LASCL::FullMatrix, 0, 0, sva(1), skl, A); } work(1) = One / skl; if (sva(1)>=safeMin) { work(2) = One; } else { work(2) = Zero; } work(_(3,6)) = Zero; return info; } // // Protect small singular values from underflow, and try to // avoid underflows/overflows in computing Jacobi rotations. // ElementType cs, sn; sn = sqrt(safeMin/eps); tmp = sqrt(big/ElementType(n)); if ((aapp<=sn) || (aaqq>=tmp) || (sn<=aaqq && aapp<=tmp)) { tmp = min(big, tmp/aapp); // aaqq = aaqq*tmp // aapp = aapp*tmp } else if ((aaqq<=sn) && (aapp<=tmp)) { tmp = min(sn/aaqq, big/(aapp*sqrt(ElementType(n)))); // aaqq = aaqq*tmp // aapp = aapp*tmp } else if ((aaqq>=sn) && (aapp>=tmp)) { tmp = max(sn/aaqq, tmp/aapp); // aaqq = aaqq*tmp // aapp = aapp*tmp } else if ((aaqq<=sn) && (aapp>=tmp)) { tmp = min(sn/aaqq, big/(sqrt(ElementType(n))*aapp)); // aaqq = aaqq*tmp // aapp = aapp*tmp } else { tmp = One; } // // Scale, if necessary // if (tmp!=One) { lascl(LASCL::FullMatrix, 0, 0, One, tmp, sva); } skl *= tmp; if (skl!=One) { if (upper) { lascl(LASCL::UpperTriangular, 0, 0, One, skl, A); } else if (lower) { lascl(LASCL::LowerTriangular, 0, 0, One, skl, A); } else { lascl(LASCL::FullMatrix, 0, 0, One, skl, A); } skl = One / skl; } // // Row-cyclic Jacobi SVD algorithm with column pivoting // const IndexType emptsw = n*(n-1)/2; IndexType notRot = 0; fastr(1) = Zero; // // A is represented in factored form A = A * diag(WORK), where diag(WORK) // is initialized to identity. WORK is updated during fast scaled // rotations. // work(_(1,n)) = One; // // IndexType swBand = 3; //[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective // if DGESVJ is used as a computational routine in the preconditioned // Jacobi SVD algorithm DGESVJ. For sweeps i=1:SWBAND the procedure // works on pivots inside a band-like region around the diagonal. // The boundaries are determined dynamically, based on the number of // pivots above a threshold. // const IndexType kbl = min(IndexType(8),n); //[TP] KBL is a tuning parameter that defines the tile size in the // tiling of the p-q loops of pivot pairs. In general, an optimal // value of KBL depends on the matrix dimensions and on the // parameters of the computer's memory. // IndexType nbl = n / kbl; if (nbl*kbl!=n) { ++nbl; } const IndexType blSkip = pow(kbl, 2); //[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. // const IndexType rowSkip = min(IndexType(5), kbl); //[TP] ROWSKIP is a tuning parameter. // const IndexType lkAhead = 1; //[TP] LKAHEAD is a tuning parameter. // // Quasi block transformations, using the lower (upper) triangular // structure of the input matrix. The quasi-block-cycling usually // invokes cubic convergence. Big part of this cycle is done inside // canonical subspaces of dimensions less than M. // if ((lower || upper) && (n>max(IndexType(64), 4*kbl))) { //[TP] The number of partition levels and the actual partition are // tuning parameters. const IndexType n4 = n / 4; const IndexType n2 = n / 2; const IndexType n34 = 3*n4; const IndexType q = (applyV) ? 0 : 1; const IndexType qm = (applyV) ? mv : 0; auto _work = work(_(n+1,lWork)); if (lower) { // // This works very well on lower triangular matrices, in particular // in the framework of the preconditioned Jacobi SVD (xGEJSV). // The idea is simple: // [+ 0 0 0] Note that Jacobi transformations of [0 0] // [+ + 0 0] [0 0] // [+ + x 0] actually work on [x 0] [x 0] // [+ + x x] [x x]. [x x] // auto A1 = A(_( 1,m),_( 1, n4)); auto A2 = A(_( n4+1,m),_( n4+1, n2)); auto A3 = A(_( n2+1,m),_( n2+1,n34)); auto A4 = A(_(n34+1,m),_(n34+1, n)); auto A12 = A(_( 1,m),_( 1,n2)); auto A34 = A(_(n2+1,m),_(n2+1, n)); auto V1 = V(_( 1, n4*q+qm),_( 1,n4)); auto V2 = V(_( n4*q+1, n2*q+qm),_( n4+1,n2)); auto V3 = V(_( n2*q+1,n34*q+qm),_( n2+1,n34)); auto V4 = V(_(n34*q+1, mv*q+qm),_(n34+1,n)); auto V12 = V(_( 1,n2*q+qm),_( 1,n2)); auto V34 = V(_( n2*q+1,mv*q+qm),_(n2+1,n)); auto d1 = work(_( 1, n4)); auto d2 = work(_( n4+1, n2)); auto d3 = work(_( n2+1,n34)); auto d4 = work(_(n34+1, n)); auto d12 = work(_( 1,n2)); auto d34 = work(_( n2+1, n)); auto sva1 = sva(_( 1, n4)); auto sva2 = sva(_( n4+1, n2)); auto sva3 = sva(_( n2+1,n34)); auto sva4 = sva(_(n34+1, n)); auto sva12 = sva(_( 1,n2)); auto sva34 = sva(_(n2+1, n)); svj0(jobV, A4, d4, sva4, V4, eps, safeMin, tol, 2, _work); svj0(jobV, A3, d3, sva3, V3, eps, safeMin, tol, 2, _work); svj1(jobV, n4, A34, d34, sva34, V34, eps, safeMin, tol, 1, _work); svj0(jobV, A2, d2, sva2, V2, eps, safeMin, tol, 1, _work); svj0(jobV, A1, d1, sva1, V1, eps, safeMin, tol, 1, _work); svj1(jobV, n4, A12, d12, sva12, V12, eps, safeMin, tol, 1, _work); } else if (upper) { auto A1 = A(_(1, n4),_( 1, n4)); auto A2 = A(_(1, n2),_( n4+1,n4+n4)); auto A3 = A(_(1,n2+n4),_( n2+1,n2+n4)); auto A12 = A(_(1,n2),_(1,n2)); auto V1 = V(_( 1, n4*q+qm),_( 1, n4)); auto V2 = V(_( n4*q+1,(n4+n4)*q+qm),_( n4+1,n4+n4)); auto V3 = V(_( n2*q+1,(n2+n4)*q+qm),_( n2+1,n2+n4)); auto V12 = V(_(1,n2*q+qm),_(1,n2)); auto d1 = work(_( 1, n4)); auto d2 = work(_( n4+1,n4+n4)); auto d3 = work(_( n2+1,n2+n4)); auto d12 = work(_(1,n2)); auto sva1 = sva(_( 1, n4)); auto sva2 = sva(_( n4+1,n4+n4)); auto sva3 = sva(_( n2+1,n2+n4)); auto sva12 = sva(_(1,n2)); svj0(jobV, A1, d1, sva1, V1, eps, safeMin, tol, 2, _work); svj0(jobV, A2, d2, sva2, V2, eps, safeMin, tol, 1, _work); svj1(jobV, n4, A12, d12, sva12, V12, eps, safeMin, tol, 1, _work); svj0(jobV, A3, d3, sva3, V3, eps, safeMin, tol, 1, _work); } } // // .. Row-cyclic pivot strategy with de Rijk's pivoting .. // ElementType max_aapq, aapq, aapp0, aqoap, apoaq; ElementType max_sinj; ElementType theta, thetaSign, t; IndexType i, iswRot, pSkipped, ibr, igl, jgl, ir1, p, q, jbc, ijblsk; bool converged = false; bool rotOk; for (i=1; i<=nSweep; ++i) { // // .. go go go ... // max_aapq = Zero; max_sinj = Zero; iswRot = 0; pSkipped = 0; notRot = 0; // // Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs // 1 <= p < q <= N. This is the first step toward a blocked implementation // of the rotations. New implementation, based on block transformations, // is under development. // for (ibr=1; ibr<=nbl; ++ibr) { igl = (ibr-1)*kbl + 1; for (ir1=0; ir1<=min(lkAhead,nbl-ibr); ++ir1) { igl += ir1*kbl; for (p=igl; p<=min(igl+kbl-1,n-1); ++p) { // // .. de Rijk's pivoting // q = blas::iamax(sva(_(p,n))) + p - 1; if (p!=q) { blas::swap(A(_,p),A(_,q)); if (rhsVec) { blas::swap(V(_,p),V(_,q)); } swap(sva(p),sva(q)); swap(work(p),work(q)); } if (ir1==0) { // // Column norms are periodically updated by explicit // norm computation. // Caveat: // Unfortunately, some BLAS implementations compute DNRM2(M,A(1,p),1) // as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to // overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and to // underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold). // Hence, DNRM2 cannot be trusted, not even in the case when // the true norm is far from the under(over)flow boundaries. // If properly implemented DNRM2 is available, the IF-THEN-ELSE // below should read "AAPP = DNRM2( M, A(1,p), 1 ) * WORK(p)". // if ((sva(p)<rootBig) && (sva(p)>rootSafeMin)) { sva(p) = blas::nrm2(A(_,p)) * work(p); } else { tmp = Zero; aapp = One; lassq(A(_,p), tmp, aapp); sva(p) = tmp * sqrt(aapp) * work(p); } aapp = sva(p); } else { aapp = sva(p); } if (aapp>Zero) { pSkipped = 0; for (q=p+1; q<=min(igl+kbl-1,n); ++q) { aaqq = sva(q); if (aaqq>Zero) { aapp0 = aapp; if (aaqq>=One) { rotOk = (small*aapp<=aaqq); if (aapp<big/aaqq) { aapq = (A(_,p)*A(_,q)*work(p)*work(q)/aaqq) / aapp; } else { auto _work = work(_(n+1,n+m)); _work = A(_,p); lascl(LASCL::FullMatrix, 0, 0, aapp, work(p), _work); aapq = _work*A(_,q)*work(q)/aaqq; } } else { rotOk = (aapp<=aaqq/small); if (aapp>(small/aaqq)) { aapq = (A(_,p)*A(_,q)*work(p)*work(q)/aaqq) / aapp; } else { auto _work = work(_(n+1,n+m)); _work = A(_,q); lascl(LASCL::FullMatrix, 0, 0, aaqq, work(q), _work); aapq = _work*A(_,p)*work(p)/aapp; } } max_aapq = max(max_aapq, abs(aapq)); // // TO rotate or NOT to rotate, THAT is the question ... // if (abs(aapq)>tol) { // // .. rotate //[RTD] ROTATED = ROTATED + ONE // if (ir1==0) { notRot = 0; pSkipped = 0; ++iswRot; } if (rotOk) { aqoap = aaqq / aapp; apoaq = aapp / aaqq; theta = -Half*abs(aqoap-apoaq)/aapq; if (abs(theta)>bigTheta) { t = Half / theta; fastr(3) = t*work(p) / work(q); fastr(4) = -t*work(q) / work(p); blas::rotm(A(_,p), A(_,q), fastr); if (rhsVec) { blas::rotm(V(_,p), V(_,q), fastr); } sva(q) =aaqq*sqrt(max(Zero,One+t*apoaq*aapq)); aapp *= sqrt(max(Zero, One-t*aqoap*aapq)); max_sinj = max(max_sinj, abs(t)); } else { // // .. choose correct signum for THETA and rotate // thetaSign = -sign(One,aapq); t = One / (theta +thetaSign*sqrt(One+theta*theta)); cs = sqrt(One / (One+t*t)); sn = t*cs; max_sinj = max(max_sinj, abs(sn)); sva(q) = aaqq *sqrt(max(Zero, One+t*apoaq*aapq)); aapp *= sqrt(max(Zero, One-t*aqoap*aapq)); apoaq = work(p) / work(q); aqoap = work(q) / work(p); if (work(p)>=One) { if (work(q)>=One) { fastr(3) = t*apoaq; fastr(4) = -t*aqoap; work(p) *= cs; work(q) *= cs; blas::rotm(A(_,p), A(_,q), fastr); if (rhsVec) { blas::rotm(V(_,p), V(_,q), fastr); } } else { A(_,p) -= t*aqoap*A(_,q); A(_,q) += cs*sn*apoaq*A(_,p); work(p) *= cs; work(q) /= cs; if (rhsVec) { V(_,p) -= t*aqoap*V(_,q); V(_,q) += cs*sn*apoaq*V(_,p); } } } else { if (work(q)>=One) { A(_,q) += t*apoaq*A(_,p); A(_,p) -= cs*sn*aqoap*A(_,q); work(p) /= cs; work(q) *= cs; if (rhsVec) { V(_,q) += t*apoaq*V(_,p); V(_,p) -= cs*sn*aqoap*V(_,q); } } else { if (work(p)>=work(q)) { A(_,p) -= t*aqoap*A(_,q); A(_,q) += cs*sn*apoaq*A(_,p); work(p) *= cs; work(q) /= cs; if (rhsVec) { V(_,p) -= t*aqoap*V(_,q); V(_,q) += cs*sn*apoaq*V(_,p); } } else { A(_,q) += t*apoaq*A(_,p); A(_,p) -= cs*sn*aqoap*A(_,q); work(p) /= cs; work(q) *= cs; if (rhsVec) { V(_,q) += t*apoaq*V(_,p); V(_,p) -= cs*sn*aqoap*V(_,q); } } } } } } else { // .. have to use modified Gram-Schmidt like transformation auto _work = work(_(n+1,lWork)); _work = A(_,p); lascl(LASCL::FullMatrix, 0, 0, aapp, One, _work); lascl(LASCL::FullMatrix, 0, 0, aaqq, One, A(_,q)); tmp = -aapq*work(p)/work(q); A(_,q) += tmp*_work; lascl(LASCL::FullMatrix, 0, 0, One, aaqq, A(_,q)); sva(q) = aaqq*sqrt(max(Zero, One-aapq*aapq)); max_sinj = max(max_sinj, safeMin); } // END IF ROTOK THEN ... ELSE // // In the case of cancellation in updating SVA(q), SVA(p) // recompute SVA(q), SVA(p). // if (pow(sva(q)/aaqq,2)<=rootEps) { if ((aaqq<rootBig) && (aaqq>rootSafeMin)) { sva(q) = blas::nrm2(A(_,q))*work(q); } else { t = Zero; aaqq = One; lassq(A(_,q), t, aaqq); sva(q) = t*sqrt(aaqq)*work(q); } } if (aapp/aapp0<=rootEps) { if ((aapp<rootBig) && (aapp>rootSafeMin)) { aapp = blas::nrm2(A(_,p))*work(p); } else { t = Zero; aapp = One; lassq(A(_,p), t, aapp); aapp = t*sqrt(aapp)*work(p); } sva(p) = aapp; } // } else { // A(:,p) and A(:,q) already numerically orthogonal if (ir1==0) { ++notRot; } //[RTD] SKIPPED = SKIPPED + 1 ++pSkipped; } } else { // A(:,q) is zero column if (ir1==0) { ++notRot; } ++pSkipped; } // if ((i<=swBand) && (pSkipped>rowSkip)) { if (ir1==0) { aapp = -aapp; } notRot = 0; break; } } // END q-LOOP // sva(p) = aapp; } else { sva(p) = aapp; if ((ir1==0) && (aapp==Zero)) { notRot += min(igl+kbl-1,n) - p; } } } // end of the p-loop // end of doing the block ( ibr, ibr ) } // end of ir1-loop // //... go to the off diagonal blocks // igl = (ibr-1)*kbl + 1; for (jbc=ibr+1; jbc<=nbl; ++jbc) { jgl = (jbc-1)*kbl + 1; // // doing the block at ( ibr, jbc ) // ijblsk = 0; for (p=igl; p<=min(igl+kbl-1,n); ++p) { aapp = sva(p); if (aapp>Zero) { pSkipped = 0; for (q=jgl; q<=min(jgl+kbl-1,n); ++q) { aaqq = sva(q); if (aaqq>Zero) { aapp0 = aapp; // // .. M x 2 Jacobi SVD .. // // Safe Gram matrix computation // if (aaqq>=One) { if (aapp>=aaqq) { rotOk = (small*aapp)<=aaqq; } else { rotOk = (small*aaqq)<=aapp; } if (aapp<(big/aaqq)) { aapq = (A(_,p)*A(_,q)*work(p)*work(q)/aaqq) / aapp; } else { auto _work = work(_(n+1,n+m)); _work = A(_,p); lascl(LASCL::FullMatrix, 0, 0, aapp, work(p), _work); aapq = _work*A(_,q)*work(q) / aaqq; } } else { if (aapp>=aaqq) { rotOk = aapp<=(aaqq/small); } else { rotOk = aaqq<=(aapp/small); } if (aapp>(small/aaqq)) { aapq = (A(_,p)*A(_,q)*work(p)*work(q)/aaqq) / aapp; } else { auto _work = work(_(n+1,n+m)); _work = A(_,q); lascl(LASCL::FullMatrix, 0, 0, aaqq, work(q), _work); aapq = _work*A(_,p)*work(p)/aapp; } } max_aapq = max(max_aapq, abs(aapq)); // // TO rotate or NOT to rotate, THAT is the question ... // if (abs(aapq)>tol) { notRot = 0; //[RTD] ROTATED = ROTATED + 1 pSkipped = 0; ++iswRot; if (rotOk) { aqoap = aaqq / aapp; apoaq = aapp / aaqq; theta = -Half*abs(aqoap-apoaq)/aapq; if (aaqq>aapp0) { theta = -theta; } if (abs(theta)>bigTheta) { t = Half/theta; fastr(3) = t*work(p) / work(q); fastr(4) = -t*work(q) / work(p); blas::rotm(A(_,p), A(_,q), fastr); if (rhsVec) { blas::rotm(V(_,p), V(_,q), fastr); } sva(q) = aaqq * sqrt(max(Zero, One+t*apoaq*aapq)); aapp *= sqrt(max(Zero,One-t*aqoap*aapq)); max_sinj = max(max_sinj, abs(t)); } else { // // .. choose correct signum for THETA and rotate // thetaSign = -sign(One,aapq); if (aaqq>aapp0) { thetaSign = -thetaSign; } t = One / (theta+thetaSign*sqrt(One+theta*theta)); cs = sqrt(One / (One + t*t)); sn = t*cs; max_sinj = max(max_sinj, abs(sn)); sva(q) = aaqq * sqrt(max(Zero, One+t*apoaq*aapq)); aapp *= sqrt(max(Zero, One-t*aqoap*aapq)); apoaq = work(p) / work(q); aqoap = work(q) / work(p); if (work(p)>=One) { if (work(q)>=One) { fastr(3) = t*apoaq; fastr(4) = -t*aqoap; work(p) *= cs; work(q) *= cs; blas::rotm(A(_,p), A(_,q), fastr); if (rhsVec) { blas::rotm(V(_,p), V(_,q), fastr); } } else { A(_,p) -= t*aqoap*A(_,q); A(_,q) += cs*sn*apoaq*A(_,p); if (rhsVec) { V(_,p) -= t*aqoap*V(_,q); V(_,q) += cs*sn*apoaq*V(_,p); } work(p) *= cs; work(q) /= cs; } } else { if (work(q)>=One) { A(_,q) += t*apoaq*A(_,p); A(_,p) -= cs*sn*aqoap*A(_,q); if (rhsVec) { V(_,q) += t*apoaq*V(_,p); V(_,p) -= cs*sn*aqoap*V(_,q); } work(p) /= cs; work(q) *= cs; } else { if (work(p)>=work(q)) { A(_,p) -= t*aqoap*A(_,q); A(_,q) += cs*sn*apoaq*A(_,p); work(p) *= cs; work(q) /= cs; if (rhsVec) { V(_,p) -= t*aqoap*V(_,q); V(_,q) += cs*sn*apoaq*V(_,p); } } else { A(_,q) += t*apoaq*A(_,p); A(_,p) -= cs*sn*aqoap*A(_,q); work(p) /= cs; work(q) *= cs; if (rhsVec) { V(_,q) += t*apoaq*V(_,p); V(_,p) -= cs*sn*aqoap*V(_,q); } } } } } } else { auto _work = work(_(n+1,lWork)); if (aapp>aaqq) { _work = A(_,p); lascl(LASCL::FullMatrix, 0, 0, aapp, One, _work); lascl(LASCL::FullMatrix, 0, 0, aaqq, One, A(_,q)); tmp = -aapq*work(p) / work(q); A(_,q) += tmp*_work; lascl(LASCL::FullMatrix, 0, 0, One, aaqq, A(_,q)); sva(q) = aaqq*sqrt(max(Zero, One-aapq*aapq)); max_sinj = max(max_sinj, safeMin); } else { _work = A(_,q); lascl(LASCL::FullMatrix, 0, 0, aaqq, One, _work); lascl(LASCL::FullMatrix, 0, 0, aapp, One, A(_,p)); tmp = -aapq*work(q) / work(p); A(_,p) += tmp * _work; lascl(LASCL::FullMatrix, 0, 0, One, aapp, A(_,p)); sva(p) = aapp*sqrt(max(Zero, One-aapq*aapq)); max_sinj = max(max_sinj, safeMin); } } // END IF ROTOK THEN ... ELSE // // In the case of cancellation in updating SVA(q) // .. recompute SVA(q) if (pow(sva(q)/aaqq,2)<=rootEps) { if ((aaqq<rootBig) && (aaqq>rootSafeMin)) { sva(q) = blas::nrm2(A(_,q))*work(q); } else { t = Zero; aaqq = One; lassq(A(_,q), t, aaqq); sva(q) = t*sqrt(aaqq)*work(q); } } if (pow(aapp/aapp0,2)<=rootEps) { if ((aapp<rootBig) && (aapp>rootSafeMin)) { aapp = blas::nrm2(A(_,p))*work(p); } else { t = Zero; aapp = One; lassq(A(_,p), t, aapp); aapp = t*sqrt(aapp)*work(p); } sva(p) = aapp; } // end of OK rotation } else { ++notRot; //[RTD] SKIPPED = SKIPPED + 1 ++pSkipped; ++ijblsk; } } else { ++notRot; ++pSkipped; ++ijblsk; } if ((i<=swBand) && (ijblsk>=blSkip)) { sva(p) = aapp; notRot = 0; goto jbcLoopExit; } if ((i<=swBand) && (pSkipped>rowSkip)) { aapp = -aapp; notRot = 0; break; } } // end of the q-loop sva(p) = aapp; } else { if (aapp==Zero) { notRot += min(jgl+kbl-1,n) -jgl + 1; } if (aapp<Zero) { notRot = 0; } } } // end of the p-loop } // end of the jbc-loop jbcLoopExit: // bailed out of the jbc-loop for (p=igl; p<=min(igl+kbl-1,n); ++p) { sva(p) = abs(sva(p)); } //** } // end of the ibr-loop // // .. update SVA(N) if ((sva(n)<rootBig) && (sva(n)>rootSafeMin)) { sva(n) = blas::nrm2(A(_,n))*work(n); } else { t = Zero; aapp = One; lassq(A(_,n), t, aapp); sva(n) = t*sqrt(aapp)*work(n); } // // Additional steering devices // if ((i<swBand) && ((max_aapq<=rootTol) || (iswRot<=n))) { swBand = i; } if (i>swBand+1 && max_aapq<sqrt(ElementType(n))*tol && ElementType(n)*max_aapq*max_sinj<tol) { converged = true; break; } if (notRot>=emptsw) { converged = true; break; } } // end i=1:NSWEEP loop // if (converged) { //#:) INFO = 0 confirms successful iterations. info = 0; } else { //#:( Reaching this point means that the procedure has not converged. info = nSweep - 1; } // // Sort the singular values and find how many are above // the underflow threshold. // IndexType n2 = 0; IndexType n4 = 0; for (IndexType p=1; p<=n-1; ++p) { const IndexType q = blas::iamax(sva(_(p,n))) + p - 1; if (p!=q) { swap(sva(p), sva(q)); swap(work(p), work(q)); blas::swap(A(_,p), A(_,q)); if (rhsVec) { blas::swap(V(_,p), V(_,q)); } } if (sva(p)!=Zero) { ++n4; if (sva(p)*skl>safeMin) { ++n2; } } } if (sva(n)!=Zero) { ++n4; if (sva(n)*skl>safeMin) { ++n2; } } // // Normalize the left singular vectors. // if (lhsVec || controlU) { for (IndexType p=1; p<=n2; ++p) { A(_,p) *= work(p)/sva(p); } } // // Scale the product of Jacobi rotations (assemble the fast rotations). // if (rhsVec) { if (applyV) { for (IndexType p=1; p<=n; ++p) { V(_,p) *= work(p); } } else { for (IndexType p=1; p<=n; ++p) { V(_,p) *= One / blas::nrm2(V(_,p)); } } } // // Undo scaling, if necessary (and possible). if (((skl>One) && (sva(1)<big/skl)) || ((skl<One) && (sva(n2)>safeMin/skl))) { sva *= skl; skl = One; } // work(1) = skl; // The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE // then some of the singular values may overflow or underflow and // the spectrum is given in this factored representation. // work(2) = ElementType(n4); // N4 is the number of computed nonzero singular values of A. // work(3) = ElementType(n2); // N2 is the number of singular values of A greater than SFMIN. // If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers // that may carry some information. // work(4) = ElementType(i); // i is the index of the last sweep before declaring convergence. // work(5) = max_aapq; // MXAAPQ is the largest absolute value of scaled pivots in the // last sweep // work(6) = max_sinj; // MXSINJ is the largest absolute value of the sines of Jacobi angles // in the last sweep // return info; } } // namespace generic //== interface for native lapack =============================================== #ifdef USE_CXXLAPACK namespace external { template <typename MA, typename VSVA, typename MV, typename VWORK> typename GeMatrix<MA>::IndexType svj_impl(SVJ::TypeA typeA, SVJ::JobU jobU, SVJ::JobV jobV, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MV> &V, DenseVector<VWORK> &work) { typedef typename GeMatrix<MA>::IndexType IndexType; return cxxlapack::gesvj<IndexType>(getF77Char(typeA), getF77Char(jobU), getF77Char(jobV), A.numRows(), A.numCols(), A.data(), A.leadingDimension(), sva.data(), V.numRows(), V.data(), V.leadingDimension(), work.data(), work.length()); } } // namespace external #endif // USE_CXXLAPACK //== public interface ========================================================== template <typename MA, typename VSVA, typename MV, typename VWORK> typename GeMatrix<MA>::IndexType svj_(SVJ::TypeA typeA, SVJ::JobU jobU, SVJ::JobV jobV, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MV> &V, DenseVector<VWORK> &work) { using std::max; using std::min; typedef typename GeMatrix<MA>::IndexType IndexType; // // Test the input parameters // # ifndef NDEBUG ASSERT(A.firstRow()==1); ASSERT(A.firstCol()==1); const IndexType m = A.numRows(); const IndexType n = A.numCols(); ASSERT(m>=n); ASSERT(sva.firstIndex()==1); ASSERT(sva.length()==n); ASSERT(V.firstRow()==1); ASSERT(V.firstCol()==1); if (jobV==SVJ::ComputeV) { ASSERT(V.numCols()==n); ASSERT(V.numRows()==n); } if (jobV==SVJ::ApplyV) { ASSERT(V.numCols()==n); } if (work.length()>0) { ASSERT(work.length()>=max(IndexType(6),m+n)); } # endif // // Make copies of output arguments // # ifdef CHECK_CXXLAPACK typename GeMatrix<MA>::NoView A_org = A; typename DenseVector<VSVA>::NoView sva_org = sva; typename GeMatrix<MV>::NoView V_org = V; typename DenseVector<VWORK>::NoView work_org = work; # endif // // Call implementation // IndexType info = LAPACK_SELECT::svj_impl(typeA, jobU, jobV, A, sva, V, work); # ifdef CHECK_CXXLAPACK // // Make copies of results computed by the generic implementation // typename GeMatrix<MA>::NoView A_generic = A; typename DenseVector<VSVA>::NoView sva_generic = sva; typename GeMatrix<MV>::NoView V_generic = V; typename DenseVector<VWORK>::NoView work_generic = work; // // restore output arguments // A = A_org; sva = sva_org; V = V_org; work = work_org; // // Compare generic results with results from the native implementation // IndexType _info = external::svj_impl(typeA, jobU, jobV, A, sva, V, 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(sva_generic, sva, "sva_generic", "sva")) { std::cerr << "CXXLAPACK: sva_generic = " << sva_generic << std::endl; std::cerr << "F77LAPACK: sva = " << sva << 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(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 (! isIdentical(info, _info, "info", "_info")) { std::cerr << "CXXLAPACK: info = " << info << std::endl; std::cerr << "F77LAPACK: _info = " << _info << std::endl; failed = true; } if (failed) { std::cerr << "error in: svj.tcc (" << ", m = " << m << ", n = " << n << ", typeA = " << char(typeA) << ", jobU = " << char(jobU) << ", jobV = " << char(jobV) << ", info = " << info << ") " << std::endl; ASSERT(0); } else { /* std::cerr << "passed: svj.tcc (" << ", m = " << m << ", n = " << n << ", typeA = " << char(typeA) << ", jobU = " << char(jobU) << ", jobV = " << char(jobV) << ", info = " << info << ") " << std::endl; */ } # endif return info; } template <typename MA, typename VSVA, typename MV, typename VWORK> typename GeMatrix<MA>::IndexType svj(SVJ::TypeA typeA, SVJ::JobU jobU, SVJ::JobV jobV, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MV> &V, DenseVector<VWORK> &work) { # ifdef CHECK_CXXLAPACK typename GeMatrix<MA>::NoView A_org = A; typename DenseVector<VSVA>::NoView sva_org = sva; typename GeMatrix<MV>::NoView V_org = V; typename DenseVector<VSVA>::NoView work_org = work; svj_(SVJ::Lower, jobU, jobV, A, sva, V, work); A = A_org; sva = sva_org; V = V_org; work = work_org; svj_(SVJ::Upper, jobU, jobV, A, sva, V, work); A = A_org; sva = sva_org; V = V_org; work = work_org; svj_(SVJ::General, jobU, jobV, A, sva, V, work); A = A_org; sva = sva_org; V = V_org; work = work_org; # endif return svj_(typeA, jobU, jobV, A, sva, V, work); } template <typename MA, typename VSVA, typename MV, typename VWORK> typename GeMatrix<MA>::IndexType svj(SVJ::JobU jobU, SVJ::JobV jobV, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MV> &V, DenseVector<VWORK> &work) { return svj(SVJ::General, jobU, jobV, A, sva, V, work); } template <typename MA, typename VSVA, typename MV, typename VWORK> typename TrMatrix<MA>::IndexType svj(SVJ::JobU jobU, SVJ::JobV jobV, TrMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MV> &V, DenseVector<VWORK> &work) { SVJ::TypeA upLo = (A.upLo()==Upper) ? SVJ::Upper : SVJ::Lower; return svj(upLo, jobU, jobV, A.general(), sva, V, work); } //-- forwarding ---------------------------------------------------------------- template <typename MA, typename VSVA, typename MV, typename VWORK> typename MA::IndexType svj(SVJ::TypeA typeA, SVJ::JobU jobU, SVJ::JobV jobV, MA &&A, VSVA &&sva, MV &&V, VWORK &&work) { typename MA::IndexType info; CHECKPOINT_ENTER; info = svj(typeA, jobU, jobV, A, sva, V, work); CHECKPOINT_LEAVE; return info; } template <typename MA, typename VSVA, typename MV, typename VWORK> typename MA::IndexType svj(SVJ::JobU jobU, SVJ::JobV jobV, MA &&A, VSVA &&sva, MV &&V, VWORK &&work) { typename MA::IndexType info; CHECKPOINT_ENTER; info = svj(jobU, jobV, A, sva, V, work); CHECKPOINT_LEAVE; return info; } } } // namespace lapack, flens #endif // FLENS_LAPACK_GE_SVJ_TCC |