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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 DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, $ ILOZ, IHIZ, Z, LDZ, INFO ) * * -- LAPACK auxiliary routine (version 3.2) -- * Univ. of Tennessee, Univ. of California Berkeley, * Univ. of Colorado Denver and NAG Ltd.. * November 2006 */ #ifndef FLENS_LAPACK_LA_LAHQR_TCC #define FLENS_LAPACK_LA_LAHQR_TCC 1 #include <cmath> #include <flens/blas/blas.h> #include <flens/lapack/lapack.h> namespace flens { namespace lapack { //== generic lapack implementation ============================================= namespace generic { template <typename IndexType, typename MH, typename VWR, typename VWI, typename MZ> IndexType lahqr_impl(bool wantT, bool wantZ, IndexType iLo, IndexType iHi, GeMatrix<MH> &H, DenseVector<VWR> &wr, DenseVector<VWI> &wi, IndexType iLoZ, IndexType iHiZ, GeMatrix<MZ> &Z) { using std::abs; using std::max; using std::min; typedef typename GeMatrix<MH>::ElementType T; const Underscore<IndexType> _; const T Zero(0), One(1), Two(2); const T Dat1 = T(3)/T(4), Dat2 = T(-0.4375); const IndexType itMax = 30; const IndexType n = H.numRows(); typedef typename GeMatrix<MH>::VectorView VectorView; T vBuffer[3]; VectorView v = typename VectorView::Engine(3, vBuffer); // // Quick return if possible // if (n==0) { return 0; } if (iLo==iHi) { wr(iLo) = H(iLo, iLo); wi(iLo) = Zero; return 0; } // // ==== clear out the trash ==== // for (IndexType j=iLo; j<=iHi-3; ++j) { H(j+2, j) = Zero; H(j+3, j) = Zero; } if (iLo<=iHi-2) { H(iHi, iHi-2) = Zero; } const IndexType nh = iHi - iLo + 1; // // Set machine-dependent constants for the stopping criterion. // T safeMin = lamch<T>(SafeMin); T safeMax = One / safeMin; labad(safeMin, safeMax); const T ulp = lamch<T>(Precision); const T smallNum = safeMin*(T(nh)/ulp); // // i1 and i2 are the indices of the first row and last column of H // to which transformations must be applied. If eigenvalues only are // being computed, i1 and i2 are set inside the main loop. // IndexType i1 = -1, i2 = -1; if (wantT) { i1 = 1; i2 = n; } // // The main loop begins here. Variable i is the loop index and decreases from // iHi to iLo in steps of 1 or 2. Each iteration of the loop works // with the active submatrix in rows and columns l to i. // Eigenvalues i+1 to iHi have already converged. Either l = iLo or // H(l,l-1) is negligible so that the matrix splits. // IndexType i = iHi; while (true) { IndexType l = iLo; if (i<iLo) { break; } // // Perform QR iterations on rows and columns iLo to i until a // submatrix of order 1 or 2 splits off at the bottom because a // subdiagonal element has become negligible. // IndexType its; for (its=0; its<=itMax; ++its) { // // Look for a single small subdiagonal element. // IndexType k; for (k=i; k>=l+1; --k) { if (abs(H(k,k-1))<=smallNum) { break; } T test = abs(H(k-1,k-1)) + abs(H(k,k)); if (test==Zero) { if (k-2>=iLo) { test += abs(H(k-1,k-2)); } if (k+1<=iHi) { test += abs(H(k+1,k)); } } // ==== The following is a conservative small subdiagonal // . deflation criterion due to Ahues & Tisseur (LAWN 122, // . 1997). It has better mathematical foundation and // . improves accuracy in some cases. ==== if (abs(H(k,k-1))<=ulp*test) { const T ab = max(abs(H(k, k-1)), abs(H(k-1, k))); const T ba = min(abs(H(k, k-1)), abs(H(k-1, k))); const T aa = max(abs(H(k, k)), abs(H(k-1, k-1)-H(k, k))); const T bb = min(abs(H(k, k)), abs(H(k-1, k-1)-H(k, k))); const T s = aa + ab; if (ba*(ab/s)<=max(smallNum, ulp*(bb*(aa/s)))) { break; } } } l = k; if (l>iLo) { // // H(l,l-1) is negligible // H(l, l-1) = Zero; } // // Exit from loop if a submatrix of order 1 or 2 has split off. // if (l>=i-1) { break; } // // Now the active submatrix is in rows and columns l to i. If // eigenvalues only are being computed, only the active submatrix // need be transformed. // if (!wantT) { i1 = l; i2 = i; } T H11, H12, H21, H22; T rt1r, rt2r, rt1i, rt2i; if (its==10) { // // Exceptional shift. // const T s = abs(H(l+1,l)) + abs(H(l+2, l+1)); H11 = Dat1*s + H(l,l); H12 = Dat2*s; H21 = s; H22 = H11; } else if (its==20) { // // Exceptional shift. // const T s = abs(H(i,i-1)) + abs(H(i-1,i-2)); H11 = Dat1*s + H(i,i); H12 = Dat2*s; H21 = s; H22 = H11; } else { // // Prepare to use Francis' double shift // (i.e. 2nd degree generalized Rayleigh quotient) // H11 = H(i-1, i-1); H21 = H( i, i-1); H12 = H(i-1, i); H22 = H( i, i); } const T s = abs(H11) + abs(H12) + abs(H21) + abs(H22); if (s==Zero) { rt1r = Zero; rt1i = Zero; rt2r = Zero; rt2i = Zero; } else { H11 /= s; H21 /= s; H12 /= s; H22 /= s; const T tr = (H11+H22) / Two; const T det = (H11-tr)*(H22-tr) - H12*H21; const T rtDisc = sqrt(abs(det)); if (det>=Zero) { // // ==== complex conjugate shifts ==== // rt1r = tr*s; rt2r = rt1r; rt1i = rtDisc*s; rt2i = -rt1i; } else { // // ==== real shifts (use only one of them) ==== // rt1r = tr + rtDisc; rt2r = tr - rtDisc; if (abs(rt1r-H22)<=abs(rt2r-H22)) { rt1r *= s; rt2r = rt1r; } else { rt2r *= s; rt1r = rt2r; } rt1i = Zero; rt2i = Zero; } } // // Look for two consecutive small subdiagonal elements. // IndexType m; for (m=i-2; m>=l; --m) { // Determine the effect of starting the double-shift QR // iteration at row m, and see if this would make H(m,m-1) // negligible. (The following uses scaling to avoid // overflows and most underflows.) // T H21S = H(m+1,m); T s = abs(H(m,m)-rt2r) + abs(rt2i) + abs(H21S); H21S = H(m+1,m)/s; v(1) = H21S*H(m,m+1) + (H(m,m)-rt1r)*((H(m,m)-rt2r)/s) - rt1i*(rt2i/s); v(2) = H21S*(H(m,m) + H(m+1,m+1)-rt1r-rt2r); v(3) = H21S*H(m+2,m+1); s = abs(v(1)) + abs(v(2)) + abs(v(3)); v(1) /= s; v(2) /= s; v(3) /= s; if (m==l) { break; } const T value1 = abs(H(m,m-1))*(abs(v(2))+abs(v(3))); const T value2 = ulp*abs(v(1)) *(abs(H(m-1,m-1))+abs(H(m,m))+abs(H(m+1,m+1))); if (value1<=value2) { break; } } // // Double-shift QR step // for (k=m; k<=i-1; ++k) { // // The first iteration of this loop determines a reflection G // from the vector v and applies it from left and right to H, // thus creating a nonzero bulge below the subdiagonal. // // Each subsequent iteration determines a reflection G to // restore the Hessenberg form in the (k-1)th column, and thus // chases the bulge one step toward the bottom of the active // submatrix. 'nr' is the order of G. // T t1, t2, t3, v2, v3; IndexType nr = min(IndexType(3), i-k+1); if (k>m) { v(_(1,nr)) = H(_(k,k+nr-1),k-1); } larfg(nr, v(1), v(_(2,nr)), t1); if (k>m) { H(k, k-1) = v(1); H(k+1,k-1) = Zero; if (k<i-1) { H(k+2,k-1) = Zero; } } else if (m>l) { // ==== Use the following instead of // . H( K, K-1 ) = -H( K, K-1 ) to // . avoid a bug when v(2) and v(3) // . underflow. ==== H(k, k-1) *= (One-t1); } v2 = v(2); t2 = t1*v2; if (nr==3) { v3 = v(3); t3 = t1*v3; // // Apply G from the left to transform the rows of the matrix // in columns k to i2. // for (IndexType j=k; j<=i2; ++j) { const T sum = H(k,j) + v2*H(k+1,j) + v3*H(k+2,j); H(k, j) -= sum*t1; H(k+1, j) -= sum*t2; H(k+2, j) -= sum*t3; } // // Apply G from the right to transform the columns of the // matrix in rows i1 to min(k+3,i). // for (IndexType j=i1; j<=min(k+3,i); ++j) { const T sum = H(j, k) + v2*H(j,k+1) + v3*H(j,k+2); H(j, k ) -= sum*t1; H(j, k+1) -= sum*t2; H(j, k+2) -= sum*t3; } if (wantZ) { // // Accumulate transformations in the matrix Z // for (IndexType j=iLoZ; j<=iHiZ; ++j) { const T sum = Z(j, k) + v2*Z(j, k+1) + v3*Z(j, k+2); Z(j, k ) -= sum*t1; Z(j, k+1) -= sum*t2; Z(j, k+2) -= sum*t3; } } } else if (nr==2) { // // Apply G from the left to transform the rows of the matrix // in columns K to I2. // for (IndexType j=k; j<=i2; ++j) { const T sum = H(k, j) + v2*H(k+1, j); H(k, j) -= sum*t1; H(k+1, j) -= sum*t2; } // // Apply G from the right to transform the columns of the // matrix in rows i1 to min(k+3,i). // for (IndexType j=i1; j<=i; ++j) { const T sum = H(j, k) + v2*H(j, k+1); H(j, k ) -= sum*t1; H(j, k+1) -= sum*t2; } if (wantZ) { // // Accumulate transformations in the matrix Z // for (IndexType j=iLoZ; j<=iHiZ; ++j) { const T sum = Z(j, k) + v2*Z(j, k+1); Z(j, k ) -= sum*t1; Z(j, k+1) -= sum*t2; } } } } } // // Failure to converge in remaining number of iterations // if (its>itMax) { return i; } if (l==i) { // // H(I,I-1) is negligible: one eigenvalue has converged. // wr(i) = H(i, i); wi(i) = Zero; } else if (l==i-1) { // // H(I-1,I-2) is negligible: a pair of eigenvalues have converged. // // Transform the 2-by-2 submatrix to standard Schur form, // and compute and store the eigenvalues. // T cs, sn; lanv2(H(i-1,i-1), H(i-1,i), H(i,i-1), H(i,i), wr(i-1), wi(i-1), wr(i), wi(i), cs, sn); if (wantT) { // // Apply the transformation to the rest of H. // if (i2>i) { const auto cols = _(i+1,i2); blas::rot(H(i-1,cols), H(i,cols), cs, sn); } const auto rows = _(i1,i-2); blas::rot(H(rows, i-1), H(rows, i), cs, sn); } if (wantZ) { // // Apply the transformation to Z. // const auto rows = _(iLoZ, iHiZ); blas::rot(Z(rows, i-1), Z(rows, i), cs, sn); } } // // return to start of the main loop with new value of I. // i = l - 1; } return 0; } } // namespace generic //== interface for native lapack =============================================== #ifdef USE_CXXLAPACK namespace external { template <typename IndexType, typename MH, typename VWR, typename VWI, typename MZ> IndexType lahqr_impl(bool wantT, bool wantZ, IndexType iLo, IndexType iHi, GeMatrix<MH> &H, DenseVector<VWR> &wr, DenseVector<VWI> &wi, IndexType iLoZ, IndexType iHiZ, GeMatrix<MZ> &Z) { IndexType info; info = cxxlapack::lahqr<IndexType>(wantT, wantZ, H.numRows(), iLo, iHi, H.data(), H.leadingDimension(), wr.data(), wi.data(), iLoZ, iHiZ, Z.data(), Z.leadingDimension()); return info; } } // namespace external #endif // USE_CXXLAPACK //== public interface ========================================================== template <typename IndexType, typename MH, typename VWR, typename VWI, typename MZ> IndexType lahqr(bool wantT, bool wantZ, IndexType iLo, IndexType iHi, GeMatrix<MH> &H, DenseVector<VWR> &wr, DenseVector<VWI> &wi, IndexType iLoZ, IndexType iHiZ, GeMatrix<MZ> &Z) { LAPACK_DEBUG_OUT("lahqr"); // // Test the input parameters // using std::max; ASSERT(H.firstRow()==1); ASSERT(H.firstCol()==1); ASSERT(H.numRows()==H.numCols()); ASSERT(wr.firstIndex()==1); ASSERT(wr.length()==H.numRows()); ASSERT(wi.firstIndex()==1); ASSERT(wi.length()==H.numRows()); ASSERT(wantZ || (Z.numRows()==H.numCols())); ASSERT(wantZ || (Z.numCols()==H.numCols())); // 1 <= ILO <= max(1,IHI); IHI <= N. ASSERT(1<=iLo); ASSERT(iLo<=max(IndexType(1), iHi)); ASSERT(iHi<=H.numRows()); // 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. ASSERT(1<=iLoZ); ASSERT(iLoZ<=iLo); ASSERT(iHi<=iHiZ); ASSERT(iHiZ<=H.numRows()); // // Make copies of output arguments // # ifdef CHECK_CXXLAPACK typename GeMatrix<MH>::NoView H_org = H; typename GeMatrix<MH>::NoView Z_org = Z; typename GeMatrix<MH>::NoView _H = H; typename DenseVector<VWR>::NoView _wr = wr; typename DenseVector<VWI>::NoView _wi = wi; typename GeMatrix<MZ>::NoView _Z = Z; # endif // // Call implementation // IndexType info = LAPACK_SELECT::lahqr_impl(wantT, wantZ, iLo, iHi, H, wr, wi, iLoZ, iHiZ, Z); // // Compare results // # ifdef CHECK_CXXLAPACK IndexType _info = external::lahqr_impl(wantT, wantZ, iLo, iHi, _H, _wr, _wi, iLoZ, iHiZ, _Z); bool failed = false; if (! isIdentical(H, _H, " H", "_H")) { std::cerr << "CXXLAPACK: H = " << H << std::endl; std::cerr << "F77LAPACK: _H = " << _H << std::endl; failed = true; } if (! isIdentical(wr, _wr, " wr", "_wr")) { std::cerr << "CXXLAPACK: wr = " << wr << std::endl; std::cerr << "F77LAPACK: _wr = " << _wr << std::endl; failed = true; } if (! isIdentical(wi, _wi, " wi", "_wi")) { std::cerr << "CXXLAPACK: wi = " << wi << std::endl; std::cerr << "F77LAPACK: _wi = " << _wi << std::endl; failed = true; } if (! isIdentical(Z, _Z, " Z", "_Z")) { std::cerr << "CXXLAPACK: Z = " << Z << std::endl; std::cerr << "F77LAPACK: _Z = " << _Z << 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 << "H_org = " << H_org << std::endl; std::cerr << "Z_org = " << Z_org << std::endl; std::cerr << "wantT = " << wantT << ", wantZ = " << wantZ << ", iLo = " << iLo << ", iHi = " << iHi << std::endl; std::cerr << "error in: lahqr.tcc" << std::endl; ASSERT(0); } else { // std::cerr << "passed: lahqr.tcc" << std::endl; } # endif return info; } //-- forwarding ---------------------------------------------------------------- template <typename IndexType, typename MH, typename VWR, typename VWI, typename MZ> IndexType lahqr(bool wantT, bool wantZ, IndexType iLo, IndexType iHi, MH &&H, VWR &&wr, VWI &&wi, IndexType iLoZ, IndexType iHiZ, MZ &&Z) { return lahqr(wantT, wantZ, iLo, iHi, H, wr, wi, iLoZ, iHiZ, Z); } } } // namespace lapack, flens #endif // FLENS_LAPACK_LA_LAHQR_TCC |