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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 DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, $ M, N, A, LDA, SVA, U, LDU, V, LDV, $ WORK, LWORK, IWORK, 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_JSV_TCC #define FLENS_LAPACK_GE_JSV_TCC 1 #include <flens/blas/blas.h> #include <flens/lapack/lapack.h> namespace flens { namespace lapack { //== generic lapack implementation ============================================= /* template <typename MA, typename VSVA, typename MU, typename MV, typename VWORK, typename VIWORK> typename GeMatrix<MA>::IndexType jsv_generic(JSV::Accuracy accuracy, JSV::JobU jobU, JSV::JobV jobV, bool restrictedRange, bool considerTransA, bool perturb, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MU> &U, GeMatrix<MV> &V, DenseVector<VWORK> &work, DenseVector<VIWORK> &iwork) { using std::abs; using std::max; using std::min; using std::sqrt; typedef typename GeMatrix<MA>::ElementType ElementType; typedef typename GeMatrix<MA>::IndexType IndexType; const ElementType Zero(0), One(1); const Underscore<IndexType> _; const bool lsvec = (jobU==ComputeU) || (jobU==FullsetU); const bool jracc = (jobV=='J'); const bool rsvec = (jobV=='V') || jracc; const bool rowpiv = (jobA=='F') || (jobA=='G'); const bool l2rank = (jobA=='R'); const bool l2aber = (jobA=='A'); const bool errest = (jobA=='E') || (jobA=='G'); const bool l2tran = (jobT=='T'); const bool l2kill = (jobR=='R'); const bool defr = (jobR=='N'); const bool l2pert = (jobP=='P'); IndexType info = 0; // // Quick return for void matrix (Y3K safe) //#:) if (m==0 || n==0) { return info; } // // Set numerical parameters // //! NOTE: Make sure DLAMCH() does not fail on the target architecture. // const ElementType eps = lamch<ElementType>(Eps); const ElementType safeMin = lamch<ElementType>(SafeMin); const ElementType small = safeMin / eps; const ElementType big = lamch<ElementType>(OverflowThreshold); // // Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N // //! If necessary, scale SVA() to protect the largest norm from // overflow. It is possible that this scaling pushes the smallest // column norm left from the underflow threshold (extreme case). // ElementType scaleM = One / sqrt(ElementType(m)*ElementType(n)); bool noScale = true; bool goScale = true; for (IndexType p=1; p<=n; ++p) { aapp = Zero; aaqq = One; lassq(A(_,p), aapp, aaqq); if (aapp>big) { return -9; } aaqq = sqrt(aaqq); if (aapp<(big/aaqq) && noscal) { sva(p) = aapp * aaqq; } else { noscal = false; sva(p) = aapp * ( aaqq * scalem ) if (goScale) { goScale = false; sva(_(1,p)) *= scaleM; } } } if (noScale) { scaleM = One; } aapp = Zero; aaqq = big; for (IndexType p=1; p<=n) { aapp = max(aapp, sva(p)); if (sva(p)!=Zero) { aaqq = min(aaqq, sva(p)); } } // // Quick return for zero M x N matrix //#:) if (aapp==Zero) { if (lsvec) { U = Zero; U.diag(0) = One; } if (rsvec) { V = Zero; V.diag(0) = One; } work(1) = One; work(2) = One; if (errest) { work(3) = One; } if (lsvec && rsvec) { work(4) = One; work(5) = One; } if (l2tran) { work(6) = Zero; work(7) = Zero; } iwork(1) = 0; iwork(2) = 0; iwork(3) = 0; return info; } // // Issue warning if denormalized column norms detected. Override the // high relative accuracy request. Issue licence to kill columns // (set them to zero) whose norm is less than sigma_max / BIG (roughly). //#:( warning = 0 if (aaqq<=SFMIN) { l2rank = true; l2kill = true; warning = 1; } // // Quick return for one-column matrix //#:) auto U1 = U(_,_(1,n)); if (n==1) { if (lsvec) { lascl(LASCL::FullMatrix, 0, 0, sva(1), scaleM, A); U1 = A; // computing all M left singular vectors of the M x 1 matrix if (nu!=n) { auto tau = work(_(1,n)); auto _work = work(_(n+1,lWork)); qrf(U1, tau, _work); orgqr(1, U, tau, _work); U1 = A; } } if (rsvec) { V(1,1) = One } if (sva(1)<big*scaleM) { sva(1) /= scaleM; scalem = One; } work(1) = One / scaleM; work(2) = One; if (sva(1)!=Zero) { iwork(1) = 1; if (sva(1)/scaleM>=safeMin) { iwork(2) = 1; } else { iwork(2) = 0; } } else { iwork(1) = 0; iwork(2) = 0; } if (errest) { work(3) = One; } if (lsvec && rsvec) { work(4) = One; work(5) = One; } if (l2tran) { work(6) = Zero; work(7) = Zero; } return info; } bool transp = false; bool l2tran = l2tran && m==n; ElementType aatMax = -One; ElementType aatMin = big; if (rowpiv || l2tran) { // // Compute the row norms, needed to determine row pivoting sequence // (in the case of heavily row weighted A, row pivoting is strongly // advised) and to collect information needed to compare the // structures of A * A^t and A^t * A (in the case L2TRAN.EQ.true). // if (l2tran) { for (IndexType p=1; p<=m; ++p) { xsc = Zero; tmp = One; lassq(A(p,_), xsc, tmp); // DLASSQ gets both the ell_2 and the ell_infinity norm // in one pass through the vector work(m+n+p) = xsc * scaleM; work(n+p) = xsc * (scaleN*sqrt(tmp)); aatMax = max(aatMax, work(n+p)); if (work(n+p)!=Zero) { aatMin = min(aatMin, work(n+p)); } } } else { for (IndexType p=1; p<=m; ++p) { const IndexType jp = blas::iamax(A(p,_)); work(m+n+p) = scaleM*abs(A(p,jp)); aatMax = max(aatMax, work(m+n+p)); aatMin = min(aatMin, work(m+n+p)); } } } // // For square matrix A try to determine whether A^t would be better // input for the preconditioned Jacobi SVD, with faster convergence. // The decision is based on an O(N) function of the vector of column // and row norms of A, based on the Shannon entropy. This should give // the right choice in most cases when the difference actually matters. // It may fail and pick the slower converging side. // entra = Zero entrat = Zero if (l2tran) { xsc = Zero; tmp = One; lassq(sva, xsc, tmp); tmp = One / tmp; entra = Zero for (IndexType p=1; p<=n; ++p) { const ElementType _big = pow(sva(p)/xsc, 2) * tmp; if (_big!=Zero) { entra += _big * log(_big); } } entra = - entra / log(ElementType(n)); // // Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex. // It is derived from the diagonal of A^t * A. Do the same with the // diagonal of A * A^t, compute the entropy of the corresponding // probability distribution. Note that A * A^t and A^t * A have the // same trace. // entrat = Zero for (IndexType p=n+1; p<=n+m; ++p) { const ElementType _big = pow(work(p)/xsc, 2) * tmp; if (_big!=Zero) { entrat += _big * log(_big); } } entrat = -entrat / log(ElementType(m)); // // Analyze the entropies and decide A or A^t. Smaller entropy // usually means better input for the algorithm. // transp = entrat<entra; // // If A^t is better than A, transpose A. // if (transp) { // In an optimal implementation, this trivial transpose // should be replaced with faster transpose. // TODO: in-place transpose: // transpose(A); for (IndexType p=1; p<=n-1; ++p) { for (IndexType q=p+1; q<=n; ++q) { swap(A(q,p), A(p,q)); } } for (IndexType p=1; p<=n; ++p) { work(m+n+p) = sva(p); sva(p) = work(n+p); } swap(aapp, aatMax); swap(aaqq, aatMin); swap(lsvec, rsvec); if (lsvec) { // Lehn: transposing A is only considered if A is square, // so in this case m==n and therefore nu==n. Or am I // wrong?? ASSERT(nu==n); } rowpiv = true; } } // // Scale the matrix so that its maximal singular value remains less // than DSQRT(BIG) -- the matrix is scaled so that its maximal column // has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep // DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and // BLAS routines that, in some implementations, are not capable of // working in the full interval [SFMIN,BIG] and that they may provoke // overflows in the intermediate results. If the singular values spread // from SFMIN to BIG, then DGESVJ will compute them. So, in that case, // one should use DGESVJ instead of DGEJSV. // const ElementType bigRoot = sqrt(big); tmp = sqrt(big/ElementType(n)); lascl(LASCL::FullMatrix, 0, 0, aapp, tmp, sva); if (aaqq>aapp*safeMin) { aaqq = (aaqq/aapp) * tmp; } else { aaqq = (aaqq*tmp) / aapp; } tmp *= scaleM; lascl(LASCL::FullMatrix, 0, 0, aapp, tmp, A); // // To undo scaling at the end of this procedure, multiply the // computed singular values with USCAL2 / USCAL1. // uScale1 = tmp; uScale2 = aapp; if (l2kill) { // L2KILL enforces computation of nonzero singular values in // the restricted range of condition number of the initial A, // sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN). xsc = sqrt(safeMin); } else { xsc = small; // // Now, if the condition number of A is too big, // sigma_max(A) / sigma_min(A)>DSQRT(BIG/N) * EPSLN / SFMIN, // as a precaution measure, the full SVD is computed using DGESVJ // with accumulated Jacobi rotations. This provides numerically // more robust computation, at the cost of slightly increased run // time. Depending on the concrete implementation of BLAS and LAPACK // (i.e. how they behave in presence of extreme ill-conditioning) the // implementor may decide to remove this switch. if (aaqq<sqrt(safeMin) && lsvec && rsvec ) { jracc = true; } } if (aaqq<xsc) { for (IndexType p=1; p<=n; ++p) { if (sva(p)<xsc) { A(_,p) = Zero; sva(p) = Zero; } } } // // Preconditioning using QR factorization with pivoting // if (rowpiv) { // Optional row permutation (Bjoerck row pivoting): // A result by Cox and Higham shows that the Bjoerck's // row pivoting combined with standard column pivoting // has similar effect as Powell-Reid complete pivoting. // The ell-infinity norms of A are made nonincreasing. for (IndexType p=1; p<=m-1; ++p) { const IndexType q = blas::iamax(work(_(m+n+p,2*m+n))) + p - 1; iwork(2*n+p) = q; if (p!=q) { swap(work(m+n+p), work(m+n+q)); } } laswp(A, iwork(_(2*n+1, 2*n+m-1))); } // // End of the preparation phase (scaling, optional sorting and // transposing, optional flushing of small columns). // // Preconditioning // // If the full SVD is needed, the right singular vectors are computed // from a matrix equation, and for that we need theoretical analysis // of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF. // In all other cases the first RR QRF can be chosen by other criteria // (eg speed by replacing global with restricted window pivoting, such // as in SGEQPX from TOMS # 782). Good results will be obtained using // SGEQPX with properly (!) chosen numerical parameters. // Any improvement of DGEQP3 improves overal performance of DGEJSV. // // A * P1 = Q1 * [ R1^t 0]^t: auto _tau = work(_(1,n)); auto _work = work(_(n+1,lWork)); auto _iwork = iwork(_(1,n)); // .. all columns are free columns _iwork = 0; qp3(A, _iwork, _tau, _work); // // The upper triangular matrix R1 from the first QRF is inspected for // rank deficiency and possibilities for deflation, or possible // ill-conditioning. Depending on the user specified flag L2RANK, // the procedure explores possibilities to reduce the numerical // rank by inspecting the computed upper triangular factor. If // L2RANK or l2aber are up, then DGEJSV will compute the SVD of // A + dA, where ||dA|| <= f(M,N)*EPSLN. // IndexType nr = 1; if (l2aber) { // Standard absolute error bound suffices. All sigma_i with // sigma_i < N*EPSLN*||A|| are flushed to zero. This is an // agressive enforcement of lower numerical rank by introducing a // backward error of the order of N*EPSLN*||A||. tmp = sqrt(ElementType(n))*eps; for (IndexType p=2; p<=n; ++p) { if (abs(A(p,p))>=tmp*abs(A(1,1))) { ++nr; } else { break; } } } else if (l2rank) { // .. similarly as above, only slightly more gentle (less agressive). // Sudden drop on the diagonal of R1 is used as the criterion for // close-to-rank-defficient. tmp = sqrt(safeMin); for (IndexType p=2; p<=n; ++p) { if (abs(A(p,p))<eps*abs(A(p-1,p-1)) || abs(A(p,p))<small || (l2kill && abs(A(p,p))<tmp)) { break; } ++nr; } } else { // The goal is high relative accuracy. However, if the matrix // has high scaled condition number the relative accuracy is in // general not feasible. Later on, a condition number estimator // will be deployed to estimate the scaled condition number. // Here we just remove the underflowed part of the triangular // factor. This prevents the situation in which the code is // working hard to get the accuracy not warranted by the data. tmp = sqrt(safeMin); for (IndexType p=2; p<=n; ++p) { if (abs(A(p,p))<small || (l2kill && abs(A(p,p))<tmp)) { break; } ++nr; } } bool almort = false; if (nr==n) { ElementType maxprj = One; for (IndexType p=2; p<=n; ++p) { maxprj = min(maxprj, abs(A(p,p))/sva(iwork(p))); } if (pow(maxprj,2)>=One-ElementType(n)*eps) { almort = true; } } sconda = -One; condr1 = -One; condr2 = -One; if (errest) { if (n==nr) { if (rsvec) { // .. V is available as workspace V.upper() = A(_(1,n),_).upper(); for (IndexType p=1; p<=n; ++p) { tmp = sva(iwork(p)); V(_(1,p),p) *= One/tmp; } auto _work = work(_(n+1, 4*n)); auto _iwork = iwork(_(2*n+m+1, 3*n+m)); pocon(V.upper(), One, tmp, _work, _iwork); } else if (lsvec) { // .. U is available as workspace auto _U = U(_(1,n),_(1,n)); _U.upper() = A(_(1,n),_).upper(); for (IndexType p=1; p<=n; ++p) { tmp = sva(iwork(p)); U(_(1,p),p) *= One/tmp; } auto _work = work(_(n+1, 4*n)); auto _iwork = iwork(_(2*n+m+1, 3*n+m)); pocon(_U.upper(), One, tmp, _work, _iwork); } else { auto _work1 = work(_(n+1,n+n*n)); auto _work2 = work(_(n+n*n+1,n+n*n+3*n)); auto _iwork = work(_(2*n+m+1,3*n+m)); GeMatrixView<ElementType> Work(n, n, _work1, n); Work.upper() = A(_(1,n),_).upper(); for (IndexType p=1; p<=n; ++p) { tmp = sva(iwork(p)); Work(_(1,p),p) *= One/tmp; } // .. the columns of R are scaled to have unit Euclidean lengths. pocon(Work.upper(), One, tmp, _work2, _iwork); } sconda = One/sqrt(tmp); // SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). // N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA } else { sconda = -One; } } l2pert = l2pert && abs(A(1,1)/A(nr,nr))>sqrt(bigRoot); // If there is no violent scaling, artificial perturbation is not needed. // // Phase 3: // if (!(rsvec || lsvec)) { // // Singular Values only // // .. transpose A(1:NR,1:N) for (IndexType p=1; p<=min(n-1, nr); ++p) { A(p,_(p+1,n)) = A(_(p+1,n),p); } // // The following two DO-loops introduce small relative perturbation // into the strict upper triangle of the lower triangular matrix. // Small entries below the main diagonal are also changed. // This modification is useful if the computing environment does not // provide/allow FLUSH TO Zero underflow, for it prevents many // annoying denormalized numbers in case of strongly scaled matrices. // The perturbation is structured so that it does not introduce any // new perturbation of the singular values, and it does not destroy // the job done by the preconditioner. // The licence for this perturbation is in the variable L2PERT, which // should be false if FLUSH TO Zero underflow is active. // if (! almort) { if (l2pert) { // XSC = DSQRT(SMALL) xsc = eps / ElementType(n); for (IndexType q=1; q<=nr; ++q) { tmp = xsc*abs(A(q,q)); for (IndexType p=1; p<=n, ++p) { if ((p>q && abs(A(p,q))<=tmp) || p<q) { A(p,q) = sign(tmp, A(p,q)); } } } } else { A(_(1,nr),_(1,nr)).strictUpper() = Zero; } // // .. second preconditioning using the QR factorization // auto _A = A(_(1,n),_(1,nr)); auto _tau = work(_(1,nr)); auto _work = work(_(n+1,lWork)); qrf(_A, _tau, _work); // // .. and transpose upper to lower triangular for (IndexType p=1; p<=nr-1; ++p) { A(p,_(p+1,nr)) = A(_(p+1,nr),p); } } // // Row-cyclic Jacobi SVD algorithm with column pivoting // // .. again some perturbation (a "background noise") is added // to drown denormals if (l2pert) { // XSC = DSQRT(SMALL) xsc = eps / ElementType(n); for (IndexType q=1; q<=nr; ++q) { ElementType tmp = xsc*abs(A(q,q)); for (IndexType p=1; p<=nr; ++p) { if (((p>q) && abs(A(p,q))<=tmp) || p<q) { A(p,q) = sign(tmp, A(p,q)); } } } } else { A(_(1,nr),_(1,nr)).strictUpper() = Zero; } // // .. and one-sided Jacobi rotations are started on a lower // triangular matrix (plus perturbation which is ignored in // the part which destroys triangular form (confusing?!)) // auto _A = A(_(1,nr),_(1,nr)); auto _sva = sva(_(1,nr)); svj(SVJ::Lower, SVJ::NoU, SVJ::NoV, _A, _sva, V, work); scaleM = work(1); numRank = nint(work(2)); } else if (rsvec && !lsvec) { // // -> Singular Values and Right Singular Vectors <- // if (almort) { // // .. in this case NR equals N ASSERT(nr==n); for (IndexType p=1; p<=nr; ++p) { V(_(p,n),p) = A(p,_(p,n)); } auto _V = V(_,_(1,nr)); _V.strictUpper() = Zero; svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, _V, sva, A, work); scaleM = work(1); numRank = nint(work(2)); } else { // // .. two more QR factorizations ( one QRF is not enough, two require // accumulated product of Jacobi rotations, three are perfect ) // A(_(1,nr),_(1,nr)).strictLower() = Zero; auto _A = A(_(1,nr),_); auto _tau1 = work(_(1,nr)); auto _work1 = work(_(n+1,lWork)); lqf(_A, _tau1, _work1); auto _V = V(_(1,nr),_(1,nr)) _V.lower() = _A(_,_(1,nr)).lower(); _V.strictUpper() = Zero; auto _tau2 = work(_(n+1,n+nr)); auto _work2 = work(_(2*n+1,lWork)); qrf(_V, _tau2, _work2); for (IndexType p=1; p<=nr; ++p) { V(_(p,nr),p) = V(p,_(p,nr)); } _V.strictUpper() = Zero; auto _sva = sva(_(1,nr)); svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, _V, _sva, U, _work1); scaleM = work(n+1); numRank = nint(work(n+2)); if (nr<n) { V(_(nr+1,n),_(1,nr)) = Zero; V(_(1,nr),_(nr+1,n)) = Zero; V(_(nr+1,n),_(nr+1,n)) = Zero; V(_(nr+1,n),_(nr+1,n)).diag(0) = One; } ormlq(Left, Trans, _A, _tau1, V, _work1); } for (IndexType p=1; p<=n; ++p) { A(iwork(p),_) = V(p,_); } V = A; if (transp) { U = V; } } else if (lsvec && !rsvec) { // // .. Singular Values and Left Singular Vectors .. // // .. second preconditioning step to avoid need to accumulate // Jacobi rotations in the Jacobi iterations. auto _U = U(_(1,nr),_(1,nr)); auto _tau = work(_(n+1,n+nr)); auto _sva = sva(_(1,nr)); auto _work1 = work(_(n+1,lWork)); auto _work2 = work(_(2*n+1,lWork)); for (IndexType p=1; p<=nr; ++p) { A(p,_(p,nr)) = U(_(p,nr),p); } _U.strictUpper() = Zero; qrf(U(_(1,n),_(1,nr)), _tau, _work2); for (IndexType p=1; p<=nr-1; ++p) { U(p,_(p+1,nr)) = U(_(p+1,nr),p); } _U.strictUpper() = Zero; svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, _U, _sva, A, _work1); scaleM = work(n+1); numRank = nint(work(n+2)); if (nr<m) { U(_(nr+1,m),_(1,nr)) = Zero; if (nr<nu) { U(_(1,nr),_(nr+1,nu)) = Zero; U(_(nr+1,m),_(nr+1,nu)) = Zero; U(_(nr+1,m),_(nr+1,nu)).diag(0) = One; } } ormqr(Left, NoTrans, A, work, U); if (rowpiv) { auto piv = iwork(_(2*n+1,2*n+m-1)); laswp(U, piv.reverse()); } for (IndexType p=1; p<=nu; ++p) { xsc = One / blas::nrm2(U(_,p)); U(_,p) *= xsc; } if (transp) { V = U; } // } else { // // .. Full SVD .. // if (!jracc) { if (!almort) { // // Second Preconditioning Step (QRF [with pivoting]) // Note that the composition of TRANSPOSE, QRF and TRANSPOSE is // equivalent to an LQF CALL. Since in many libraries the QRF // seems to be better optimized than the LQF, we do explicit // transpose and use the QRF. This is subject to changes in an // optimized implementation of DGEJSV. // for (IndexType p=1; p<=nr; ++p) { V(_(p,n),p) = A(p,_(p,n)); } // // .. the following two loops perturb small entries to avoid // denormals in the second QR factorization, where they are // as good as zeros. This is done to avoid painfully slow // computation with denormals. The relative size of the // perturbation is a parameter that can be changed by the // implementer. This perturbation device will be obsolete on // machines with properly implemented arithmetic. // To switch it off, set L2PERT=false To remove it from the // code, remove the action under L2PERT=true, leave the ELSE // part. The following two loops should be blocked and fused with // the transposed copy above. // if (l2pert) { xsc = sqrt(small); for (IndexType q=1; q<=nr; ++q) { tmp = xsc*abs(V(q,q)); for (IndexType p=1; p<=n; ++p) { if (p>q && abs(V(p,q))<=tmp || p<q) { V(p,q) = sign(tmp, V(p,q)); } if (p<q) { V(p,q) = -V(p,q); } } } } else { V(_(1,nr),_(1,nr)).strictUpper() = Zero; } // // Estimate the row scaled condition number of R1 // (If R1 is rectangular, N > NR, then the condition number // of the leading NR x NR submatrix is estimated.) // auto _work1 = work(_(2*n+1,2*n+nr*nr)); auto _work2 = work(_(2*n+nr*nr+1, 2*n+nr*nr+3*nr)); auto _iwork = iwork(_(m+2*n+1,m+2*n+nr)); GeMatrixView<ElementType> Work(nr, nr, _work1, nr); Work.lower() = V.lower(); for (IndexType p=1; p<=nr; ++p) { tmp = blas::nrm2(Work(_(p,nr),p)); Work(_(p,nr),p) *= One/tmp; } pocon(Work, One, tmp, _work2, _iwork); condr1 = One / sqrt(tmp); // .. here need a second oppinion on the condition number // .. then assume worst case scenario // R1 is OK for inverse <=> condr1 < DBLE(N) // more conservative <=> condr1 < DSQRT(DBLE(N)) // cond_ok = sqrt(ElementType(nr)); //[TP] COND_OK is a tuning parameter. if (condr1<cond_ok) { // .. the second QRF without pivoting. Note: in an optimized // implementation, this QRF should be implemented as the QRF // of a lower triangular matrix. // R1^t = Q2 * R2 auto tau = work(_(n+1,n+nr)); auto _work = work(_(2*n+1,lWork)); qrf(V(_,_(1,nr)), tau, _work); if (l2pert) { xsc = sqrt(small) / Eps; for (IndexType p=2; p<=nr; ++p) { for (IndexType q=1; q<=p-1; ++q) { tmp = xsc*min(abs(V(p,p)), abs(V(q,q))); if (abs(V(q,p))<=tmp) { V(q,p) = sign(tmp, V(q,p)); } } } } // if (nr!=n) { auto _work = work(_(2*n+1, 2*n+n*nr)); GeMatrixView<ElementType> Work(n, nr, _work, n); Work = V(_,_(1,nr)); } // .. save ... // // .. this transposed copy should be better than naive // TODO: auto _V = V(_(1,nr),_(1,nr)); // _V.lower() = transpose(_V.upper()); // for (IndexType p=1; p<=nr-1; ++p) { V(_(p+1,nr),p) = V(p,_(p+1,nr)); } condr2 = condr1; } else { // // .. ill-conditioned case: second QRF with pivoting // Note that windowed pivoting would be equaly good // numerically, and more run-time efficient. So, in // an optimal implementation, the next call to DGEQP3 // should be replaced with eg. CALL SGEQPX (ACM TOMS #782) // with properly (carefully) chosen parameters. // // R1^t * P2 = Q2 * R2 auto _V = V(_,_(1,nr)); auto piv = iwork(_(n+1,n+nr)); auto tau = work(_(n+1,,n+nr)); auto _work = work(_(2*n+1, lWork)); piv = 0; qp3(_V, piv, tau, _work); if (l2pert) { xsc = sqrt(small); for (IndexType p=2; p<=nr; ++p) { for (IndexType q=1; q<=p-1; ++q) { tmp = xsc*min(abs(V(p,p)), abs(V(q,q))); if (abs(V(q,p))<=tmp) { V(q,p) = sign(tmp, V(q,p)); } } } } auto _work1 = work(_(2*n+1, 2*n+n*nr)); GeMatrixView<ElementType> Work1(n, nr, _work1, n); Work1 = V(_,_(1,nr)); if (l2pert) { xsc = sqrt(small); for (IndexType p=2; p<=nr; ++p) { for (IndexType q=1; q<=p-1; ++q) { tmp = xsc * min(abs(V(p,p)), abs(V(q,q))); V(p,q) = -sign(tmp, V(q,p)); } } } else { V(_(1,nr),_(1,nr)).strictLower() = Zero; } // Now, compute R2 = L3 * Q3, the LQ factorization. auto _V = V(_(1,nr),_(1,nr)); auto tau = work(_(2*n+n*nr+1,2*n+n*nr+nr)); auto _work2 = work(_(2*n+n*nr+nr+1,lWork)); lqf(_V, tau, _work2); // .. and estimate the condition number auto _work3 = work(_(2*n+n*nr+nr+1,2*n+n*nr+nr+nr*nr)); GeMatrixView<ElementType> Work3(nr, nr, _work3, nr); Work3.lower() = _V.lower(); for (IndexType p=1; p<=nr; ++p) { tmp = blas::nrm2(Work3(p,_(1,p))); Work3(p,_(1,p)) *= One/tmp; } auto _work4 = work(_(2*n+n*nr+nr+nr*nr+1, 2*n+n*nr+nr+nr*nr+3*nr)); auto _iwork = iwork(_(m+2*n+1, m+2*n+nr)); pocon(Work3.lower(), One, tmp, _work4, _iwork); condr2 = One / sqrt(tmp); if (condr2>=cond_ok) { // .. save the Householder vectors used for Q3 // (this overwrittes the copy of R2, as it will not be // needed in this branch, but it does not overwritte the // Huseholder vectors of Q2.). Work1(_(1,nr),_(1,nr)).upper() = V.upper(); // .. and the rest of the information on Q3 is in // WORK(2*N+N*NR+1:2*N+N*NR+N) } } if (l2pert) { xsc = sqrt(small); for (IndexType q=2; q<=nr; ++q) { tmp = xsc * V(q,q); for (IndexType p=1; p<=q-1; ++p) { // V(p,q) = - DSIGN( TEMP1, V(q,p) ) V(p,q) = -sign(tmp, V(p,q)); } } } else { V(_(1,nr),_(1,nr)).strictLower(); } // // Second preconditioning finished; continue with Jacobi SVD // The input matrix is lower triangular. // // Recover the right singular vectors as solution of a well // conditioned triangular matrix equation. // if (condr1<cond_ok) { auto _U = U(_(1,nr),_(1,nr)); auto _V = V(_(1,nr),_(1,nr)); auto _sva = sva(_(1,nr)); auto _work = work(_(2*n+n*nr+nr+1,lWork)); svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, _V, _sva, U, _work); scaleM = _work(1); numRank = nint(_work(2)); for (IndexType p=1; p<=nr; ++p) { _U(_,p) = _V(_,p); _V(_,p) *= sva(p); } // .. pick the right matrix equation and solve it // if (nr==n) { //:)) .. best case, R1 is inverted. The solution of this // matrix equation is Q2*V2 = the product of the Jacobi // rotations used in DGESVJ, premultiplied with the // orthogonal matrix from the second QR factorization. const auto _A = A(_(1,nr),_(1,nr)); blas::sm(Left, NoTrans, One, _A.upper(), _V); } else { // .. R1 is well conditioned, but non-square. Transpose(R2) // is inverted to get the product of the Jacobi rotations // used in DGESVJ. The Q-factor from the second QR // factorization is then built in explicitly. auto _work = work(_(2*n+1, 2*n+n*nr)); GeMatrixView<ElementType> Work1(nr, nr, _work, n); GeMatrixView<ElementType> Work(n, nr, _work, n); blas::sm(Left, NoTrans, One, Work1.upper(), _V); if (nr<n) { V(_(nr+1,n),_(1,nr)) = Zero; V(_(1,nr),_(nr+1,n)) = Zero; V(_(nr+1,n),_(nr+1,n)) = Zero; V(_(nr+1,n),_(nr+1,n)).diag(0) = One; } auto tau = work(_(n+1,n+nr)); auto _work_ormqr = work(_(2*n+n*nr+nr+1,lWork)); ormqr(Left, NoTrans, Work, tau, V, _work_ormqr); } // } else if (condr2<cond_ok) { // //:) .. the input matrix A is very likely a relative of // the Kahan matrix :) // The matrix R2 is inverted. The solution of the matrix // equation is Q3^T*V3 = the product of the Jacobi rotations // (appplied to the lower triangular L3 from the LQ // factorization of R2=L3*Q3), pre-multiplied with the // transposed Q3. CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U, $ LDU, work(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO ) SCALEM = work(2*N+N*NR+NR+1) numRank = IDNINT(work(2*N+N*NR+NR+2)) DO 3870 p = 1, NR CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 ) CALL DSCAL( NR, SVA(p), U(1,p), 1 ) 3870 CONTINUE CALL DTRSM('L','U','N','N',NR,NR,One,work(2*N+1),N,U,LDU) // .. apply the permutation from the second QR factorization DO 873 q = 1, NR DO 872 p = 1, NR work(2*N+N*NR+NR+iwork(N+p)) = U(p,q) 872 CONTINUE DO 874 p = 1, NR U(p,q) = work(2*N+N*NR+NR+p) 874 CONTINUE 873 CONTINUE if ( NR < N ) { CALL DLASET( 'A',N-NR,NR,Zero,Zero,V(NR+1,1),LDV ) CALL DLASET( 'A',NR,N-NR,Zero,Zero,V(1,NR+1),LDV ) CALL DLASET( 'A',N-NR,N-NR,Zero,One,V(NR+1,NR+1),LDV ) } CALL DORMQR( 'L','N',N,N,NR,work(2*N+1),N,work(N+1), $ V,LDV,work(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR ) } else { // Last line of defense. //#:( This is a rather pathological case: no scaled condition // improvement after two pivoted QR factorizations. Other // possibility is that the rank revealing QR factorization // or the condition estimator has failed, or the COND_OK // is set very close to One (which is unnecessary). Normally, // this branch should never be executed, but in rare cases of // failure of the RRQR or condition estimator, the last line of // defense ensures that DGEJSV completes the task. // Compute the full SVD of L3 using DGESVJ with explicit // accumulation of Jacobi rotations. CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U, $ LDU, work(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO ) SCALEM = work(2*N+N*NR+NR+1) numRank = IDNINT(work(2*N+N*NR+NR+2)) if ( NR < N ) { CALL DLASET( 'A',N-NR,NR,Zero,Zero,V(NR+1,1),LDV ) CALL DLASET( 'A',NR,N-NR,Zero,Zero,V(1,NR+1),LDV ) CALL DLASET( 'A',N-NR,N-NR,Zero,One,V(NR+1,NR+1),LDV ) } CALL DORMQR( 'L','N',N,N,NR,work(2*N+1),N,work(N+1), $ V,LDV,work(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR ) // CALL DORMLQ( 'L', 'T', NR, NR, NR, work(2*N+1), N, $ work(2*N+N*NR+1), U, LDU, work(2*N+N*NR+NR+1), $ LWORK-2*N-N*NR-NR, IERR ) DO 773 q = 1, NR DO 772 p = 1, NR work(2*N+N*NR+NR+iwork(N+p)) = U(p,q) 772 CONTINUE DO 774 p = 1, NR U(p,q) = work(2*N+N*NR+NR+p) 774 CONTINUE 773 CONTINUE // } // // Permute the rows of V using the (column) permutation from the // first QRF. Also, scale the columns to make them unit in // Euclidean norm. This applies to all cases. // TEMP1 = DSQRT(DBLE(N)) * EPSLN DO 1972 q = 1, N DO 972 p = 1, N work(2*N+N*NR+NR+iwork(p)) = V(p,q) 972 CONTINUE DO 973 p = 1, N V(p,q) = work(2*N+N*NR+NR+p) 973 CONTINUE XSC = One / DNRM2( N, V(1,q), 1 ) if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) ) $ CALL DSCAL( N, XSC, V(1,q), 1 ) 1972 CONTINUE // At this moment, V contains the right singular vectors of A. // Next, assemble the left singular vector matrix U (M x N). if ( NR < M ) { CALL DLASET( 'A', M-NR, NR, Zero, Zero, U(NR+1,1), LDU ) if ( NR < N1 ) { CALL DLASET('A',NR,N1-NR,Zero,Zero,U(1,NR+1),LDU) CALL DLASET('A',M-NR,N1-NR,Zero,One,U(NR+1,NR+1),LDU) } } // // The Q matrix from the first QRF is built into the left singular // matrix U. This applies to all cases. // CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, work, U, $ LDU, work(N+1), LWORK-N, IERR ) // The columns of U are normalized. The cost is O(M*N) flops. TEMP1 = DSQRT(DBLE(M)) * EPSLN DO 1973 p = 1, NR XSC = One / DNRM2( M, U(1,p), 1 ) if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) ) $ CALL DSCAL( M, XSC, U(1,p), 1 ) 1973 CONTINUE // // If the initial QRF is computed with row pivoting, the left // singular vectors must be adjusted. // if ( ROWPIV ) $ CALL DLASWP( N1, U, LDU, 1, M-1, iwork(2*N+1), -1 ) // } else { // // .. the initial matrix A has almost orthogonal columns and // the second QRF is not needed // CALL DLACPY( 'Upper', N, N, A, LDA, work(N+1), N ) if ( L2PERT ) { XSC = DSQRT(SMALL) DO 5970 p = 2, N TEMP1 = XSC * work( N + (p-1)*N + p ) DO 5971 q = 1, p - 1 work(N+(q-1)*N+p)=-DSIGN(TEMP1,work(N+(p-1)*N+q)) 5971 CONTINUE 5970 CONTINUE } else { CALL DLASET( 'Lower',N-1,N-1,Zero,Zero,work(N+2),N ) } // CALL DGESVJ( 'Upper', 'U', 'N', N, N, work(N+1), N, SVA, $ N, U, LDU, work(N+N*N+1), LWORK-N-N*N, INFO ) // SCALEM = work(N+N*N+1) numRank = IDNINT(work(N+N*N+2)) DO 6970 p = 1, N CALL DCOPY( N, work(N+(p-1)*N+1), 1, U(1,p), 1 ) CALL DSCAL( N, SVA(p), work(N+(p-1)*N+1), 1 ) 6970 CONTINUE // CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N, $ One, A, LDA, work(N+1), N ) DO 6972 p = 1, N CALL DCOPY( N, work(N+p), N, V(iwork(p),1), LDV ) 6972 CONTINUE TEMP1 = DSQRT(DBLE(N))*EPSLN DO 6971 p = 1, N XSC = One / DNRM2( N, V(1,p), 1 ) if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) ) $ CALL DSCAL( N, XSC, V(1,p), 1 ) 6971 CONTINUE // // Assemble the left singular vector matrix U (M x N). // if ( N < M ) { CALL DLASET( 'A', M-N, N, Zero, Zero, U(N+1,1), LDU ) if ( N < N1 ) { CALL DLASET( 'A',N, N1-N, Zero, Zero, U(1,N+1),LDU ) CALL DLASET( 'A',M-N,N1-N, Zero, One,U(N+1,N+1),LDU ) } } CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, work, U, $ LDU, work(N+1), LWORK-N, IERR ) TEMP1 = DSQRT(DBLE(M))*EPSLN DO 6973 p = 1, N1 XSC = One / DNRM2( M, U(1,p), 1 ) if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) ) $ CALL DSCAL( M, XSC, U(1,p), 1 ) 6973 CONTINUE // if ( ROWPIV ) $ CALL DLASWP( N1, U, LDU, 1, M-1, iwork(2*N+1), -1 ) // } // // end of the >> almost orthogonal case << in the full SVD // } else { // // This branch deploys a preconditioned Jacobi SVD with explicitly // accumulated rotations. It is included as optional, mainly for // experimental purposes. It does perfom well, and can also be used. // In this implementation, this branch will be automatically activated // if the condition number sigma_max(A) / sigma_min(A) is predicted // to be greater than the overflow threshold. This is because the // a posteriori computation of the singular vectors assumes robust // implementation of BLAS and some LAPACK procedures, capable of // working in presence of extreme values. Since that is not always // the case, ... // DO 7968 p = 1, NR CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 ) 7968 CONTINUE // if ( L2PERT ) { XSC = DSQRT(SMALL/EPSLN) for (IndexType q=1; q<=nr; ++q) { tmp = xsc*abs(V(q,q)); for (IndexType p=1; p<=n; ++p) { if (p>q && abs(V(p,q))<=tmp || p<q) { V(p,q) = sign(tmp, V(p,q)); } if (p<q) { V(p,q) = - V(p,q); } } } } else { CALL DLASET( 'U', NR-1, NR-1, Zero, Zero, V(1,2), LDV ) } CALL DGEQRF( N, NR, V, LDV, work(N+1), work(2*N+1), $ LWORK-2*N, IERR ) CALL DLACPY( 'L', N, NR, V, LDV, work(2*N+1), N ) // DO 7969 p = 1, NR CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 ) 7969 CONTINUE if ( L2PERT ) { XSC = DSQRT(SMALL/EPSLN) DO 9970 q = 2, NR DO 9971 p = 1, q - 1 TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q))) U(p,q) = - DSIGN( TEMP1, U(q,p) ) 9971 CONTINUE 9970 CONTINUE } else { CALL DLASET('U', NR-1, NR-1, Zero, Zero, U(1,2), LDU ) } CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA, $ N, V, LDV, work(2*N+N*NR+1), LWORK-2*N-N*NR, INFO ) SCALEM = work(2*N+N*NR+1) numRank = IDNINT(work(2*N+N*NR+2)) if ( NR < N ) { CALL DLASET( 'A',N-NR,NR,Zero,Zero,V(NR+1,1),LDV ) CALL DLASET( 'A',NR,N-NR,Zero,Zero,V(1,NR+1),LDV ) CALL DLASET( 'A',N-NR,N-NR,Zero,One,V(NR+1,NR+1),LDV ) } CALL DORMQR( 'L','N',N,N,NR,work(2*N+1),N,work(N+1), $ V,LDV,work(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR ) // // Permute the rows of V using the (column) permutation from the // first QRF. Also, scale the columns to make them unit in // Euclidean norm. This applies to all cases. // TEMP1 = DSQRT(DBLE(N)) * EPSLN DO 7972 q = 1, N DO 8972 p = 1, N work(2*N+N*NR+NR+iwork(p)) = V(p,q) 8972 CONTINUE DO 8973 p = 1, N V(p,q) = work(2*N+N*NR+NR+p) 8973 CONTINUE XSC = One / DNRM2( N, V(1,q), 1 ) if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) ) $ CALL DSCAL( N, XSC, V(1,q), 1 ) 7972 CONTINUE // // At this moment, V contains the right singular vectors of A. // Next, assemble the left singular vector matrix U (M x N). // if (NR<M) { CALL DLASET( 'A', M-NR, NR, Zero, Zero, U(NR+1,1), LDU ) if (NR<N1) { CALL DLASET( 'A',NR, N1-NR, Zero, Zero, U(1,NR+1),LDU ) CALL DLASET( 'A',M-NR,N1-NR, Zero, One,U(NR+1,NR+1),LDU ) } } CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, work, U, $ LDU, work(N+1), LWORK-N, IERR ) if (ROWPIV) $ CALL DLASWP( N1, U, LDU, 1, M-1, iwork(2*N+1), -1 ) } if ( TRANSP ) { // .. swap U and V because the procedure worked on A^t for (IndexType p=1; p<=n; ++p) { blas::swap(U(_,p),V(_,p)); } } } // end of the full SVD // // Undo scaling, if necessary (and possible) // if ( USCAL2 <= (BIG/SVA(1))*USCAL1 ) { CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR ) USCAL1 = One USCAL2 = One } if ( NR < N ) { sva(_(nr+1,n)) = Zero; } work(1) = USCAL2 * SCALEM work(2) = USCAL1 if ( errest ) work(3) = SCONDA if ( lsvec && rsvec ) { work(4) = condr1 work(5) = condr2 } if ( L2TRAN ) { work(6) = entra work(7) = entrat } iwork(1) = NR iwork(2) = numRank iwork(3) = warning } */ //== interface for native lapack =============================================== #ifdef USE_CXXLAPACK namespace external { template <typename MA, typename VSVA, typename MU, typename MV, typename VWORK, typename VIWORK> typename GeMatrix<MA>::IndexType jsv_impl(JSV::Accuracy accuracy, JSV::JobU jobU, JSV::JobV jobV, bool restrictedRange, bool considerTransA, bool perturb, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MU> &U, GeMatrix<MV> &V, DenseVector<VWORK> &work, DenseVector<VIWORK> &iwork) { typedef typename GeMatrix<MA>::IndexType IndexType; IndexType info; info = cxxlapack::gejsv<IndexType>(getF77Char(accuracy), getF77Char(jobU), getF77Char(jobV), restrictedRange ? 'R' : 'N', considerTransA ? 'T' : 'N', perturb ? 'P' : 'N', A.numRows(), A.numCols(), A.data(), A.leadingDimension(), sva.data(), U.data(), U.leadingDimension(), V.data(), V.leadingDimension(), work.data(), work.length(), iwork.data()); return info; } } // namespace external #endif // USE_CXXLAPACK //== public interface ========================================================== template <typename MA, typename VSVA, typename MU, typename MV, typename VWORK, typename VIWORK> typename GeMatrix<MA>::IndexType jsv(JSV::Accuracy accuracy, JSV::JobU jobU, JSV::JobV jobV, bool restrictedRange, bool considerTransA, bool perturb, GeMatrix<MA> &A, DenseVector<VSVA> &sva, GeMatrix<MU> &U, GeMatrix<MV> &V, DenseVector<VWORK> &work, DenseVector<VIWORK> &iwork) { 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(U.firstRow()==1); ASSERT(U.firstCol()==1); ASSERT(V.firstRow()==1); ASSERT(V.firstCol()==1); ASSERT(iwork.length()==m+3*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<MU>::NoView U_org = U; typename GeMatrix<MV>::NoView V_org = V; typename DenseVector<VWORK>::NoView work_org = work; # endif // // Call implementation // /* IndexType info = jsv_generic(accuracy, jobU, jobV, restrictedRange, considerTransA, perturb, A, sva, U, V, work); */ # ifdef USE_CXXLAPACK IndexType info = external::jsv_impl(accuracy, jobU, jobV, restrictedRange, considerTransA, perturb, A, sva, U, V, work, iwork); # else IndexType info = -1; # endif # 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<MU>::NoView U_generic = U; typename GeMatrix<MV>::NoView V_generic = V; typename DenseVector<VWORK>::NoView work_generic = work; // // restore output arguments // A = A_org; sva = sva_org; U = U_org; V = V_org; work = work_org; // // Compare generic results with results from the native implementation // IndexType _info = external::jsv_impl(accuracy, jobU, jobV, restrictedRange, considerTransA, perturb, A, sva, U, V, work, iwork); 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(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(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: jsv.tcc " << std::endl; ASSERT(0); } else { std::cerr << "passed: jsv.tcc " << std::endl; } # endif return info; } //-- forwarding ---------------------------------------------------------------- template <typename MA, typename VSVA, typename MU, typename MV, typename VWORK, typename VIWORK> typename MA::IndexType jsv(JSV::Accuracy accuracy, JSV::JobU jobU, JSV::JobV jobV, bool restrictedRange, bool considerTransA, bool perturb, MA &&A, VSVA &&sva, MU &&U, MV &&V, VWORK &&work, VIWORK &&iwork) { typename MA::IndexType info; CHECKPOINT_ENTER; info = jsv(accuracy, jobU, jobV, restrictedRange, considerTransA, perturb, A, sva, U, V, work, iwork); CHECKPOINT_LEAVE; return info; } } } // namespace lapack, flens #endif // FLENS_LAPACK_GE_JSV_TCC |