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/*
* Copyright (c) 2011, Michael Lehn * * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1) Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2) Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * 3) Neither the name of the FLENS development group nor the names of * its contributors may be used to endorse or promote products derived * from this software without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ /* Based on * SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, $ X, LDX, FERR, BERR, WORK, IWORK, INFO ) SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, $ X, LDX, FERR, BERR, WORK, RWORK, INFO ) * * -- LAPACK routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 */ #ifndef FLENS_LAPACK_GE_RFS_TCC #define FLENS_LAPACK_GE_RFS_TCC 1 #include <flens/blas/blas.h> #include <flens/lapack/lapack.h> namespace flens { namespace lapack { //== generic lapack implementation ============================================= namespace generic { //-- (ge)rfs [real variant] ---------------------------------------------------- template <typename MA, typename MAF, typename VPIV, typename MB, typename MX, typename VFERR, typename VBERR, typename VWORK, typename VIWORK> void rfs_impl(Transpose trans, const GeMatrix<MA> &A, const GeMatrix<MAF> &AF, const DenseVector<VPIV> &piv, const GeMatrix<MB> &B, GeMatrix<MX> &X, DenseVector<VFERR> &fErr, DenseVector<VBERR> &bErr, DenseVector<VWORK> &work, DenseVector<VIWORK> &iwork) { using std::abs; typedef typename GeMatrix<MA>::ElementType ElementType; typedef typename GeMatrix<MA>::IndexType IndexType; const IndexType itMax = 5; const ElementType Zero(0), One(1), Two(2), Three(3); const Underscore<IndexType> _; const IndexType n = B.numRows(); const IndexType nRhs = B.numCols(); // // Local Arrays // IndexType iSaveData[3] = {0, 0, 0}; DenseVectorView<IndexType> iSave = typename DenseVectorView<IndexType>::Engine(3, iSaveData, 1); // // Quick return if possible // if (n==0 || nRhs==0) { fErr = Zero; bErr = Zero; return; } const Transpose transT = (trans==NoTrans) ? Trans : NoTrans; // // NZ = maximum number of nonzero elements in each row of A, plus 1 // const IndexType nz = n + 1; const ElementType eps = lamch<ElementType>(Eps); const ElementType safeMin = lamch<ElementType>(SafeMin); const ElementType safe1 = nz * safeMin; const ElementType safe2 = safe1 / eps; auto work1 = work(_(1,n)); auto work2 = work(_(n+1,2*n)); auto work3 = work(_(2*n+1,3*n)); // // Do for each right hand side // for (IndexType j=1; j<=nRhs; ++j) { IndexType count = 1; ElementType lastRes = Three; RETRY: // // Loop until stopping criterion is satisfied. // // Compute residual R = B - op(A) * X, // where op(A) = A, A**T, or A**H, depending on TRANS. // work2 = B(_,j); blas::mv(trans, -One, A, X(_,j), One, work2); // // Compute componentwise relative backward error from formula // // max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) // // where abs(Z) is the componentwise absolute value of the matrix // or vector Z. If the i-th component of the denominator is less // than SAFE2, then SAFE1 is added to the i-th components of the // numerator and denominator before dividing. // for (IndexType i=1; i<=n; ++i) { work(i) = abs(B(i,j)); } // // Compute abs(op(A))*abs(X) + abs(B). // if (trans==NoTrans) { for (IndexType k=1; k<=n; ++k) { const ElementType xk = abs(X(k,j)); for (IndexType i=1; i<=n; ++i) { work(i) += abs(A(i,k)) * xk; } } } else { for (IndexType k=1; k<=n; ++k) { ElementType s = Zero; for (IndexType i=1; i<=n; ++i) { s += abs(A(i,k)) * abs(X(i,j)); } work1(k) += s; } } ElementType s = Zero; for (IndexType i=1; i<=n; ++i) { if (work1(i)>safe2) { s = max(s, abs(work2(i))/work1(i)); } else { s = max(s, (abs(work2(i))+safe1)/(work1(i)+safe1)); } } bErr(j) = s; // // Test stopping criterion. Continue iterating if // 1) The residual BERR(J) is larger than machine epsilon, and // 2) BERR(J) decreased by at least a factor of 2 during the // last iteration, and // 3) At most ITMAX iterations tried. // if (bErr(j)>eps && Two*bErr(j)<=lastRes && count<=itMax) { // // Update solution and try again. // trs(trans, AF, piv, work2); X(_,j) += work2; lastRes = bErr(j); ++count; goto RETRY; } // // Bound error from formula // // norm(X - XTRUE) / norm(X) .le. FERR = // norm( abs(inv(op(A)))* // ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) // // where // norm(Z) is the magnitude of the largest component of Z // inv(op(A)) is the inverse of op(A) // abs(Z) is the componentwise absolute value of the matrix or // vector Z // NZ is the maximum number of nonzeros in any row of A, plus 1 // EPS is machine epsilon // // The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) // is incremented by SAFE1 if the i-th component of // abs(op(A))*abs(X) + abs(B) is less than SAFE2. // // Use DLACN2 to estimate the infinity-norm of the matrix // inv(op(A)) * diag(W), // where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) // for (IndexType i=1; i<=n; ++i) { if (work(i)>safe2) { work(i) = abs(work2(i)) + nz*eps*work1(i); } else { work(i) = abs(work2(i)) + nz*eps*work1(i) + safe1; } } IndexType kase = 0; while (true) { lacn2(work3, work2, iwork, fErr(j), kase, iSave); if (kase==0) { break; } if (kase==1) { // // Multiply by diag(W)*inv(op(A)**T). // trs(transT, AF, piv, work2); for (IndexType i=1; i<=n; ++i) { work2(i) *= work1(i); } } else { // // Multiply by inv(op(A))*diag(W). // for (IndexType i=1; i<=n; ++i) { work2(i) *= work1(i); } trs(trans, AF, piv, work2); } } // // Normalize error. // lastRes = Zero; for (IndexType i=1; i<=n; ++i) { lastRes = max(lastRes, abs(X(i,j))); } if (lastRes!=Zero) { fErr(j) /= lastRes; } } } } // namespace generic //== interface for native lapack =============================================== #ifdef USE_CXXLAPACK namespace external { //-- (ge)rfs [real and complex variant] ---------------------------------------- template <typename MA, typename MAF, typename VPIV, typename MB, typename MX, typename VFERR, typename VBERR, typename VWORK, typename VWORK2> void rfs_impl(Transpose trans, const GeMatrix<MA> &A, const GeMatrix<MAF> &AF, const DenseVector<VPIV> &piv, const GeMatrix<MB> &B, GeMatrix<MX> &X, DenseVector<VFERR> &fErr, DenseVector<VBERR> &bErr, DenseVector<VWORK> &work, DenseVector<VWORK2> &work2) { typedef typename GeMatrix<MA>::IndexType IndexType; IndexType info = cxxlapack::gerfs<IndexType>(getF77Char(trans), B.numRows(), B.numCols(), A.data(), A.leadingDimension(), AF.data(), AF.leadingDimension(), piv.data(), B.data(), B.leadingDimension(), X.data(), X.leadingDimension(), fErr.data(), bErr.data(), work.data(), work2.data()); ASSERT(info==0); } } // namespace external #endif // USE_CXXLAPACK //== public interface ========================================================== //-- (ge)rfs [real variant] ---------------------------------------------------- template <typename MA, typename MAF, typename VPIV, typename MB, typename MX, typename VFERR, typename VBERR, typename VWORK, typename VIWORK> typename RestrictTo<IsRealGeMatrix<MA>::value && IsRealGeMatrix<MAF>::value && IsIntegerDenseVector<VPIV>::value && IsRealGeMatrix<MB>::value && IsRealGeMatrix<MX>::value && IsRealDenseVector<VFERR>::value && IsRealDenseVector<VBERR>::value && IsRealDenseVector<VWORK>::value && IsIntegerDenseVector<VIWORK>::value, void>::Type rfs(Transpose trans, const MA &A, const MAF &AF, const VPIV &piv, const MB &B, MX &&X, VFERR &&fErr, VBERR &&bErr, VWORK &&work, VIWORK &&iwork) { LAPACK_DEBUG_OUT("(ge)rfs [real]"); // // Remove references from rvalue types // typedef typename RemoveRef<MA>::Type MatrixA; typedef typename MatrixA::IndexType IndexType; typedef typename RemoveRef<MAF>::Type MatrixAF; typedef typename RemoveRef<VPIV>::Type VectorPiv; typedef typename RemoveRef<MB>::Type MatrixB; typedef typename RemoveRef<MX>::Type MatrixX; typedef typename RemoveRef<VFERR>::Type VectorFErr; typedef typename RemoveRef<VBERR>::Type VectorBErr; typedef typename RemoveRef<VWORK>::Type VectorWork; typedef typename RemoveRef<VIWORK>::Type VectorIWork; // // Test the input parameters // # ifndef NDEBUG ASSERT(A.firstRow()==1); ASSERT(A.firstCol()==1); ASSERT(A.numRows()==A.numCols()); const IndexType n = A.numRows(); ASSERT(AF.firstRow()==1); ASSERT(AF.firstCol()==1); ASSERT(AF.numRows()==n); ASSERT(AF.numCols()==n); ASSERT(piv.firstIndex()==1); ASSERT(piv.length()==n); ASSERT(B.firstRow()==1); ASSERT(B.firstCol()==1); ASSERT(B.numRows()==n); const IndexType nRhs = B.numCols(); ASSERT(X.firstRow()==1); ASSERT(X.firstCol()==1); ASSERT(X.numRows()==n); ASSERT(X.numCols()==nRhs); ASSERT(fErr.firstIndex()==1); ASSERT(fErr.length()==nRhs); ASSERT(bErr.firstIndex()==1); ASSERT(bErr.length()==nRhs); ASSERT(work.firstIndex()==1); ASSERT(work.length()==3*n); ASSERT(iwork.firstIndex()==1); ASSERT(iwork.length()==n); # endif // // Make copies of output arguments // typename MatrixX::NoView X_org = X; typename VectorFErr::NoView fErr_org = fErr; typename VectorBErr::NoView bErr_org = bErr; typename VectorWork::NoView work_org = work; typename VectorIWork::NoView iwork_org = iwork; // // Call implementation // LAPACK_SELECT::rfs_impl(trans, A, AF, piv, B, X, fErr, bErr, work, iwork); # ifdef CHECK_CXXLAPACK // // Compare results // typename MatrixX::NoView X_generic = X; typename VectorFErr::NoView fErr_generic = fErr; typename VectorBErr::NoView bErr_generic = bErr; typename VectorWork::NoView work_generic = work; typename VectorIWork::NoView iwork_generic = iwork; X = X_org; fErr = fErr_org; bErr = bErr_org; work = work_org; iwork = iwork_org; external::rfs_impl(trans, A, AF, piv, B, X, fErr, bErr, work, iwork); bool failed = false; if (! isIdentical(X_generic, X, "X_generic", "X")) { std::cerr << "CXXLAPACK: X_generic = " << X_generic << std::endl; std::cerr << "F77LAPACK: X = " << X << std::endl; failed = true; } if (! isIdentical(fErr_generic, fErr, "fErr_generic", "fErr")) { std::cerr << "CXXLAPACK: fErr_generic = " << fErr_generic << std::endl; std::cerr << "F77LAPACK: fErr = " << fErr << std::endl; failed = true; } if (! isIdentical(bErr_generic, bErr, "bErr_generic", "bErr")) { std::cerr << "CXXLAPACK: bErr_generic = " << bErr_generic << std::endl; std::cerr << "F77LAPACK: bErr = " << bErr << 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(iwork_generic, iwork, "iwork_generic", "iwork")) { std::cerr << "CXXLAPACK: iwork_generic = " << iwork_generic << std::endl; std::cerr << "F77LAPACK: iwork = " << iwork << std::endl; failed = true; } if (failed) { ASSERT(0); } # endif } //-- (ge)rfs [complex variant] ------------------------------------------------- #ifdef USE_CXXLAPACK template <typename MA, typename MAF, typename VPIV, typename MB, typename MX, typename VFERR, typename VBERR, typename VWORK, typename VRWORK> typename RestrictTo<IsComplexGeMatrix<MA>::value && IsComplexGeMatrix<MAF>::value && IsIntegerDenseVector<VPIV>::value && IsComplexGeMatrix<MB>::value && IsComplexGeMatrix<MX>::value && IsRealDenseVector<VFERR>::value && IsRealDenseVector<VBERR>::value && IsComplexDenseVector<VWORK>::value && IsRealDenseVector<VRWORK>::value, void>::Type rfs(Transpose trans, const MA &A, const MAF &AF, const VPIV &piv, const MB &B, MX &&X, VFERR &&fErr, VBERR &&bErr, VWORK &&work, VRWORK &&rwork) { LAPACK_DEBUG_OUT("(ge)rfs [complex]"); // // Remove references from rvalue types // typedef typename RemoveRef<MA>::Type MatrixA; typedef typename MatrixA::IndexType IndexType; typedef typename RemoveRef<MAF>::Type MatrixAF; typedef typename RemoveRef<VPIV>::Type VectorPiv; typedef typename RemoveRef<MB>::Type MatrixB; typedef typename RemoveRef<MX>::Type MatrixX; typedef typename RemoveRef<VFERR>::Type VectorFErr; typedef typename RemoveRef<VBERR>::Type VectorBErr; typedef typename RemoveRef<VWORK>::Type VectorWork; typedef typename RemoveRef<VRWORK>::Type VectorRWork; // // Test the input parameters // # ifndef NDEBUG ASSERT(A.firstRow()==1); ASSERT(A.firstCol()==1); ASSERT(A.numRows()==A.numCols()); const IndexType n = A.numRows(); ASSERT(AF.firstRow()==1); ASSERT(AF.firstCol()==1); ASSERT(AF.numRows()==n); ASSERT(AF.numCols()==n); ASSERT(piv.firstIndex()==1); ASSERT(piv.length()==n); ASSERT(B.firstRow()==1); ASSERT(B.firstCol()==1); ASSERT(B.numRows()==n); const IndexType nRhs = B.numCols(); ASSERT(X.firstRow()==1); ASSERT(X.firstCol()==1); ASSERT(X.numRows()==n); ASSERT(X.numCols()==nRhs); ASSERT(fErr.firstIndex()==1); ASSERT(fErr.length()==nRhs); ASSERT(bErr.firstIndex()==1); ASSERT(bErr.length()==nRhs); ASSERT(work.firstIndex()==1); ASSERT(work.length()==2*n); ASSERT(rwork.firstIndex()==1); ASSERT(rwork.length()==n); # endif // // Call implementation // external::rfs_impl(trans, A, AF, piv, B, X, fErr, bErr, work, rwork); } #endif // USE_CXXLAPACK } } // namespace lapack, flens #endif // FLENS_LAPACK_GE_RFS_TCC |