Finally Block Harmonic(?) KS working

Grid/algorithms/iterative/HarmonicBlockKrylovSchur.h

@@ -34,65 +34,44 @@ See the full license in the file "LICENSE" in the top level distribution directo
 NAMESPACE_BEGIN(Grid);
 
 /**
- * Block harmonic restarted Krylov-Schur eigensolver.
+ * Block shift-targeted Krylov-Schur eigensolver.
  *
- * Harmonic Ritz values
+ * Algorithm
- * --------------------
+ * ---------
- * Standard Ritz values of A in a Krylov space K_m minimise the residual
+ * Uses a block Arnoldi factorisation:
- * in a Galerkin sense; they are good approximations to eigenvalues at the
- * *exterior* of the spectrum.  For eigenvalues *near* a target shift σ
- * (e.g. the smallest eigenvalues when σ=0) harmonic Ritz values are
- * better-suited: they are obtained by a Petrov-Galerkin condition that
- * requires the residual to be orthogonal to (A-σI)K_m instead of K_m.
- *
- * Given the block Arnoldi factorisation
  *
  *   A V = V H + F B^dag                                         (1)
  *
- * with V orthonormal (Nm columns), H the Nm² block
+ * with V orthonormal (Nm columns), H the Nm×Nm block upper-Hessenberg
- * upper-Hessenberg Rayleigh quotient, F the Nblock residual vectors and B
+ * Rayleigh quotient, F the Nblock residual vectors and B the Nm×Nblock
- * the Nm×Nblock coupling matrix, the harmonic Rayleigh quotient
+ * coupling matrix.
- * relative to shift σ is
- *
- *   Hhat = H + (H - σI)^{-H} B B^H                             (2)
- *
- * Derivation: the harmonic Ritz condition (A-σI)Vy ⊥ (A-σI)V leads to
- *
- *   [ (H-σI)^H (H-σI) + B B^H ] y = μ (H-σI)^H y
- *
- * Left-multiplying by (H-σI)^{-H} and setting θ = μ + σ gives the
- * standard eigenvalue problem  Hhat y = θ y  with Hhat as in (2).
- *
- * The harmonic Ritz values θ_j are eigenvalues of Hhat; among these,
- * the ones closest to σ (smallest |θ_j - σ|) are the best approximations
- * to the eigenvalues of A near σ.
  *
  * Thick restart
  * -------------
- * The Schur decomposition  Hhat = Q^dag S Q  is computed and the
+ * To target eigenvalues near shift σ, the Schur decomposition is computed
- * leading Nk*Nblock Schur values (sorted by the RitzFilter) are kept.
+ * for the shifted Rayleigh quotient:
- * The same unitary rotation Q is applied to both the Krylov basis and
- * to the *original* Rayleigh quotient H (not Hhat) for the restart:
  *
- *   V_new = V Q^dag [first Nk*Nblock columns]
+ *   (H - σI) = Q^dag S Q                                        (2)
- *   H_new = Q H Q^dag  [truncated Nk*Nblock × Nk*Nblock]
- *   B_new = Q B         [truncated Nk*Nblock × Nblock]
  *
- * Block Arnoldi then resumes from block Nk, restoring H to full size
+ * Sorting the Schur values of (H - σI) by smallest |S(i,i)| = |λ - σ| and
- * as new columns are appended.
+ * retaining the leading Nk is equivalent to selecting the Ritz values of H
+ * closest to σ.  Since Q diagonalises H - σI (and hence H itself), the
+ * rotated Rayleigh quotient is exactly upper triangular:
+ *
+ *   H_new = Q H Q^dag = S + σI  (upper triangular)              (3)
+ *
+ * Truncation to Nk is therefore exact: the off-diagonal coupling block
+ * H_new[Nk:, :Nk] = 0 by triangularity.  The Krylov-Schur relation after
+ * restart is exact and block Arnoldi resumes cleanly from F.
  *
  * Convergence
  * -----------
- * For a harmonic Ritz pair (θ, y) the true Ritz residual bound is
+ * Convergence is declared when || B^H y_k || < Tolerance * approxLambdaMax
- *
+ * for each Ritz pair (λ_k, y_k) of the truncated H.
- *   || (A - θI) V y || ≤ || B^H y ||
- *
- * (same as for standard Ritz, because B captures the full coupling).
- * Convergence is declared when || B^H y || < Tolerance * approxLambdaMax.
  *
  * Parameters
  * ----------
- * shift    : target shift σ (default 0.0)
+ * shift    : target shift σ (default 0.0); Schur values sorted by |λ - σ|
  * Nblock   : block size p
  * Nm       : total Krylov dimension (must be divisible by Nblock)
  * Nk       : total vectors to retain after each restart (must be divisible by Nblock, Nk < Nm)
@@ -205,70 +184,38 @@ public:
         verify(lbl);
       }
 
-      // ---- Form harmonic Rayleigh quotient ----
+      // ---- Schur decompose (H - σI) to select Schur vectors closest to σ ----
-      // Hhat = H + (H - σI)^{-H} * B * B^H
+      // Sorting the Schur values of (H - σI) by |S(i,i)| = |λ - σ| gives the
-      CMat Hhat = harmonicRayleigh(H, B, N);
+      // Ritz values of H nearest the target shift without any matrix inversion.
-
+      // Because Q diagonalises (H - σI), it also diagonalises H:
-      // ---- Schur decompose Hhat ----
+      //   H_new = Q H Q^dag = S + σI  (upper triangular)
-      ComplexSchurDecomposition schur(Hhat, false, ritzFilter);
+      // Truncation is therefore exact.
+      CMat Hshift = H - shift * CMat::Identity(N, N);
+      ComplexSchurDecomposition schur(Hshift, false, ritzFilter);
       schur.schurReorder(Nk);
 
       std::cout << GridLogMessage
-                << "HarmonicBlockKrylovSchur: harmonic Ritz values (first Nk):" << std::endl;
+                << "HarmonicBlockKrylovSchur: Ritz values nearest shift (first Nk):" << std::endl;
       CMat S = schur.getMatrixS();
       for (int i = 0; i < Nk; i++)
-        std::cout << GridLogMessage << "  [" << i << "] " << S(i, i) << std::endl;
+        std::cout << GridLogMessage << "  [" << i << "] " << S(i, i) + shift << std::endl;
 
       CMat Q  = schur.getMatrixQ();
       CMat Qt = Q.adjoint();
 
-      // ---- Rotate Krylov basis using Q from Hhat ----
+      // ---- Rotate Krylov basis ----
       std::vector<Field> basis2;
       constructUR(basis2, basis, Qt, N);
       basis = basis2;
 
-      // ---- Update H and B (rotate H, not Hhat) ----
+      // ---- Update H and B ----
-      H = Q * H * Qt;
+      // H_new = S + σI is upper triangular; off-diagonal block H_new[Nk:,:Nk] = 0
+      H = S + shift * CMat::Identity(N, N);
       B = Q * B;
 
-      // ---- Truncate to Nk ----
+      // ---- Truncate to Nk (exact: H upper triangular) ----
       int Nkeep = Nk;
 
-      // ---- Option B: corrected restart starting block -------------------------
-      //
-      // In standard Krylov-Schur, Q diagonalises H itself, so Q H Q^H = S is
-      // upper triangular and the off-diagonal coupling block H_dk = S[Nkeep:,:Nkeep] = 0.
-      // Truncation is therefore exact.
-      //
-      // Here Q diagonalises Hhat (not H), so H_new = Q H Q^H is generally dense.
-      // The off-diagonal block
-      //
-      //   H_dk = H_new[Nkeep:, :Nkeep]   (size (N-Nkeep) x Nkeep)
-      //
-      // is non-zero, and the true KS relation after truncation is:
-      //
-      //   A V_k = V_k H_k  +  V_disc H_dk  +  F B_k^H
-      //
-      // If we restart from F only, the V_disc H_dk term is lost.
-      //
-      // Fix: include the V_disc coupling in the new starting block:
-      //
-      //   G[t] = F[t] + sum_{s=Nkeep}^{N-1} basis[s] * H(s, t),  t = 0..Nblock-1
-      //
-      // G[:,t] is the t-th column of  (V_disc H_dk + F B_k^H) restricted to the
-      // first Nblock columns of H_dk.  Since F ⊥ V_k and V_disc ⊥ V_k, G is
-      // automatically orthogonal to the retained subspace.
-      //
-      // This must be computed BEFORE basis[Nkeep:] and H[Nkeep:, :] are discarded.
-
-      std::vector<Field> G(Nblock, Field(Grid_));
-      for (int t = 0; t < Nblock; t++) {
-        G[t] = F[t];
-        for (int s = Nkeep; s < N; s++)
-          G[t] += basis[s] * H(s, t);
-      }
-      blockQR(G);   // orthonormalise within G (G is already ⊥ to basis[0:Nkeep])
-
       CMat Htmp = H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep));
       H = CMat::Zero(N, N);
       H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep)) = Htmp;
@@ -284,10 +231,8 @@ public:
       std::cout << GridLogMessage
                 << "HarmonicBlockKrylovSchur: beta_k = " << beta_k << std::endl;
 
-      // Use corrected starting block G (not bare F).
+      // Restart from F (exact: no discarded-basis correction needed)
-      // G encodes the coupling from the discarded basis vectors V_disc through
+      startBlock = F;
-      // H_dk[:,0:Nblock], restoring the exact KS relation to first order.
-      startBlock = G;
 
       if (doVerify) {
         std::string lbl = "iter " + std::to_string(iter) + " after restart+truncation";
@@ -327,8 +272,7 @@ public:
    *   A V = V H + F B^dag                                      (KS)
    *
    * by explicit operator applications.  H here is the standard Rayleigh
-   * quotient (not Hhat), so the KS relation is the same as for
+   * quotient, so the KS relation is the same as for BlockKrylovSchur.
-   * BlockKrylovSchur.
    *
    * Prints:
    *   - H (current Rayleigh quotient, nBasis × nBasis)
@@ -445,32 +389,6 @@ public:
 
 private:
 
-  //--------------------------------------------------------------------
-  // Harmonic Rayleigh quotient
-  //--------------------------------------------------------------------
-  /**
-   * Forms the harmonic Rayleigh quotient relative to shift σ:
-   *
-   *   Hhat = H + (H - σI)^{-H} * B * B^H
-   *
-   * where H is the N×N block-Hessenberg, B is the N×Nblock coupling matrix.
-   *
-   * The N×N solve  (H - σI)^H X = B B^H  is done via Eigen's LU
-   * factorisation.  If H - σI is (nearly) singular the result is
-   * ill-conditioned; in that case σ should be perturbed slightly.
-   */
-  CMat harmonicRayleigh(const CMat& H_, const CMat& B_, int N)
-  {
-    CMat K  = H_ - shift * CMat::Identity(N, N);
-    CMat KH = K.adjoint();                         // (H - σI)^H
-
-    // Solve KH * X = B B^H  →  X = KH^{-1} B B^H = (H-σI)^{-H} B B^H
-    CMat BBH = B_ * B_.adjoint();                  // N × N
-    CMat X   = KH.lu().solve(BBH);                 // N × N
-
-    return H_ + X;
-  }
-
   //--------------------------------------------------------------------
   // Block Arnoldi iteration
   //--------------------------------------------------------------------
@@ -491,24 +409,15 @@ private:
     } else {
       // Append the new starting block to the retained basis.
       //
-      // Standard KS (startBlock = F):
+      // The exact truncated KS relation is  A V_k = V_k H_k + F B_k^dag,
-      //   The exact truncated relation is  A V_k = V_k H_k + F B_k^dag,
       // so the coupling rows are  H[Nkeep+t, j] = conj(B_k[j,t]).
-      //
+      // Since H_new = S + σI is upper triangular, the off-diagonal block
-      // Harmonic KS Option B (startBlock = G, where G = F + V_disc H_dk[:,0:Nblock]):
+      // H_new[Nkeep:, :Nkeep] = 0 and the restart from F is exact.
-      //   The exact coupling rows are  H[Nkeep+t, j] = <G[t], A V_k[:,j]>,
-      //   which differs from conj(B_k[j,t]) because G ≠ F.
-      //   For a Hermitian operator these preset rows are overwritten by the
-      //   Hermitian symmetry fill inside blockArnoldiStep (via explicit inner
-      //   products), so the approximate preset below does no harm.
-      //   For a non-Hermitian operator the preset is approximate; a more
-      //   expensive fix would compute <G[t], A V_k[:,j]> explicitly here.
       int Nkeep = startIdx * Nblock;
       for (auto& v : startBlock) basis.push_back(v);
 
-      // Fill restart coupling rows into H (exact for standard KS; approximate
+      // Fill restart coupling rows into H (exact: H_new is upper triangular,
-      // for harmonic KS with Option-B starting block, but overwritten by
+      // so B encodes the only non-zero coupling to the new block).
-      // Hermitian symmetry fill for Hermitian operators).
       for (int t = 0; t < Nblock; t++)
         for (int j = 0; j < Nkeep; j++)
           H(Nkeep + t, j) = std::conj(B(j, t));
@@ -560,10 +469,6 @@ private:
         for (int s = 0; s < Nblock; s++)
           B(kBase + t, s) = std::conj(R(s, t));    // B_block = R^H
       beta_k = R.norm();
-      // Hermitian symmetry fill for last block (same as non-last path below)
-      for (int t = 0; t < Nblock; t++)
-        for (int j = 0; j < kBase; j++)
-          H(kBase + t, j) = std::conj(H(j, kBase + t));
       return;
     }
 
@@ -578,23 +483,6 @@ private:
     for (int t = 0; t < Nblock; t++)
       basis.push_back(F[t]);
 
-    // Hermitian symmetry fill: H[kBase+t, j] = conj(H[j, kBase+t]) for j < kBase.
-    //
-    // In a fresh block Arnoldi the Krylov structure forces H[kBase+t, j] = 0 for
-    // j < kBase-Nblock (sub-subdiagonal), so this is a no-op.
-    //
-    // After a non-Schur restart (e.g. harmonic restart where H_new = Q H Q^dag is
-    // a full matrix), A v_k_j for j < Nkeep has components in ALL new extended
-    // vectors, making these elements non-zero.  The Arnoldi step fills column
-    // kBase+t (H[j, kBase+t] for j < prevN) via inner products, but never fills
-    // the corresponding row.  For a Hermitian operator the two are related by
-    //   H[kBase+t, j] = <basis[kBase+t] | A basis[j]>
-    //                 = conj(<basis[j] | A basis[kBase+t]>) = conj(H[j, kBase+t])
-    // Filling these ensures H = H^dag and fixes the M != H discrepancy that
-    // corrupts subsequent Arnoldi steps after a harmonic restart.
-    for (int t = 0; t < Nblock; t++)
-      for (int j = 0; j < kBase; j++)
-        H(kBase + t, j) = std::conj(H(j, kBase + t));
   }
 
   //--------------------------------------------------------------------

examples/Example_krylov_schur.cc

@@ -56,6 +56,7 @@ struct LanczosParameters: Serializable {
                                 Integer, maxIter,
                                 Integer, Nblock,
                                 Integer, verify,
+		  		RealD, shift , 
 	  			RealD, resid,
 	  			RealD, ChebyLow,
 	  			RealD, ChebyHigh,
@@ -117,12 +118,9 @@ public:
   InvertNonHermitianLinearOperator(Matrix &Mat,RealD stp=1e-8): _Mat(Mat),_stp(stp){};
   // Support for coarsening to a multigrid
   void OpDiag (const Field &in, Field &out) {
-//    _Mat.Mdiag(in,out);
-//    out = out + shift*in;
     assert(0);
   }
   void OpDir  (const Field &in, Field &out,int dir,int disp) {
-//    _Mat.Mdir(in,out,dir,disp);
     assert(0);
   }
   void OpDirAll  (const Field &in, std::vector<Field> &out){
@@ -131,11 +129,6 @@ public:
   };
   void Op     (const Field &in, Field &out){
     Field tmp(in.Grid());
-//    _Mat.M(in,out);
-//  RealD mass=-shift;
-//  WilsonCloverFermionD Dw(Umu, Grid, RBGrid, mass, csw_r, csw_t);
-//  NonHermitianLinearOperator<Matrix,Field> HermOp(_Mat);
-//  BiCGSTAB<Field> CG(_stp,10000);
     _Mat.Mdag(in,tmp);
     MdagMLinearOperator<Matrix,Field> HermOp(_Mat);
     ConjugateGradient<Field> CG(_stp,10000);
@@ -341,12 +334,7 @@ int main (int argc, char ** argv)
 
   // Run KrylovSchur and Arnoldi on a Hermitian matrix
   std::cout << GridLogMessage << "Running Krylov Schur" << std::endl;
-  // KrylovSchur KrySchur (Dsq, FGrid, 1e-8, EvalNormLarge);
+    RealD shift=LanParams.shift;
-//  KrylovSchur KrySchur (Dsq, FGrid, 1e-8,EvalImNormSmall);
-//  KrySchur(src, maxIter, Nm, Nk, Nstop);
-//  KrylovSchur KrySchur (HermOp2, UGrid, resid,EvalNormSmall);
-//  Hacked, really EvalImagSmall
-    RealD shift=1.5;
 #if 0
     KrylovSchur KrySchur (Dwilson, UGrid, resid,EvalImNormSmall);
 //    KrySchur(src[0], maxIter, Nm, Nk, Nstop);
@@ -357,10 +345,8 @@ int main (int argc, char ** argv)
     Nblock=LanParams.Nblock;
     bool if_verify=false;
     if(LanParams.verify) if_verify=true;
-//    KrylovSchur KrySchur (Dwilson, UGrid, resid,EvalImNormSmall);
+//    BlockKrylovSchur KrySchur (Dwilson, UGrid, resid,EvalImNormSmall);
-//    KrySchur(src, maxIter, Nm, Nk, Nstop,true,if_verify);
+    HarmonicBlockKrylovSchur KrySchur (Dwilson, UGrid, resid,shift,EvalNormSmall);
-    BlockKrylovSchur KrySchur (Dwilson, UGrid, resid,EvalImNormSmall);
-//    HarmonicBlockKrylovSchur KrySchur (Dwilson, UGrid, resid,shift,EvalImNormSmall);
     KrySchur(src, maxIter, Nm, Nk, Nstop,Nblock,true,if_verify);
     std::cout << GridLogMessage << "BlockKrylovSchur evec.size= " << KrySchur.evecs.size()<< std::endl;
 #endif

tests/lanczos/Test_harmonic_vs_krylov_schur.cc

@@ -61,11 +61,11 @@ int main(int argc, char** argv)
   //----------------------------------------------------------------------
   // Parameters (kept small so output is readable)
   //----------------------------------------------------------------------
-  const int Nblock  = 2;
+  const int Nblock  = 4;
-  const int Nm      = 12;   // total vectors (= 6 blocks * Nblock=2)
+  const int Nm      = 20;   // total vectors (= 5 blocks * Nblock=4)
-  const int Nk      = 6;    // total kept   (= 3 blocks * Nblock=2)
+  const int Nk      = 8;    // total kept   (= 2 blocks * Nblock=4)
-  const int Nstop   = 2;
+  const int Nstop   = 4;
-  const int maxIter = 4;
+  const int maxIter = 8;
   const RealD tol   = 1e-6;
 
   // Two identical starting blocks
@@ -73,7 +73,7 @@ int main(int argc, char** argv)
   std::vector<Field> v0b(Nblock, Field(grid));
   for (int t = 0; t < Nblock; t++) {
     random(RNG, v0[t]);
-    v0b[t] = v0[t];
+    v0b[t] = v0[t];   // identical start for fair comparison
   }
 
   //----------------------------------------------------------------------
@@ -87,7 +87,7 @@ int main(int argc, char** argv)
   std::cout << GridLogMessage
             << "========================================\n" << std::endl;
 
-  BlockKrylovSchur<Field> bks(op, grid, tol, EvalReSmall);
+  BlockKrylovSchur<Field> bks(op, grid, tol, EvalImNormSmall);
   bks(v0, maxIter, Nm, Nk, Nstop, Nblock,
       /*doubleOrthog=*/true, /*doVerify=*/true);
 
@@ -108,7 +108,7 @@ int main(int argc, char** argv)
   std::cout << GridLogMessage
             << "========================================\n" << std::endl;
 
-  HarmonicBlockKrylovSchur<Field> hbks(op, grid, tol, 0.0, EvalNormSmall);
+  HarmonicBlockKrylovSchur<Field> hbks(op, grid, tol, 0.0, EvalImNormSmall);
   hbks(v0b, maxIter, Nm, Nk, Nstop, Nblock,
        /*doubleOrthog=*/true, /*doVerify=*/true);