#include <petsc/private/kspimpl.h>
#include <petsc/private/vecimpl.h>

#define offset(j)       PetscMax(((j) - (2 * l)), 0)
#define shift(i, j)     ((i) - offset(j))
#define G(i, j)         (plcg->G[((j) * (2 * l + 1)) + (shift((i), (j)))])
#define G_noshift(i, j) (plcg->G[((j) * (2 * l + 1)) + (i)])
#define alpha(i)        (plcg->alpha[i])
#define gamma(i)        (plcg->gamma[i])
#define delta(i)        (plcg->delta[i])
#define sigma(i)        (plcg->sigma[i])
#define req(i)          (plcg->req[i])

typedef struct KSP_CG_PIPE_L_s KSP_CG_PIPE_L;
struct KSP_CG_PIPE_L_s {
  PetscInt     l; /* pipeline depth */
  Vec         *Z; /* Z vectors (shifted base) */
  Vec         *U; /* U vectors (unpreconditioned shifted base) */
  Vec         *V; /* V vectors (original base) */
  Vec         *Q; /* Q vectors (auxiliary bases) */
  Vec          p; /* work vector */
  PetscScalar *G; /* such that Z = VG (band matrix)*/
  PetscScalar *gamma, *delta, *alpha;
  PetscReal    lmin, lmax; /* min and max eigen values estimates to compute base shifts */
  PetscReal   *sigma;      /* base shifts */
  MPI_Request *req;        /* request array for asynchronous global collective */
  PetscBool    show_rstrt; /* flag to show restart information in output (default: not shown) */
};

/*
  KSPSetUp_PIPELCG - Sets up the workspace needed by the PIPELCG method.

  This is called once, usually automatically by KSPSolve() or KSPSetUp()
  but can be called directly by KSPSetUp()
*/
static PetscErrorCode KSPSetUp_PIPELCG(KSP ksp)
{
  KSP_CG_PIPE_L *plcg = (KSP_CG_PIPE_L *)ksp->data;
  PetscInt       l = plcg->l, max_it = ksp->max_it;
  MPI_Comm       comm;

  PetscFunctionBegin;
  comm = PetscObjectComm((PetscObject)ksp);
  PetscCheck(max_it >= 1, comm, PETSC_ERR_ARG_OUTOFRANGE, "%s: max_it argument must be positive.", ((PetscObject)ksp)->type_name);
  PetscCheck(l >= 1, comm, PETSC_ERR_ARG_OUTOFRANGE, "%s: pipel argument must be positive.", ((PetscObject)ksp)->type_name);
  PetscCheck(l <= max_it, comm, PETSC_ERR_ARG_OUTOFRANGE, "%s: pipel argument must be less than max_it.", ((PetscObject)ksp)->type_name);

  PetscCall(KSPSetWorkVecs(ksp, 1)); /* get work vectors needed by PIPELCG */
  plcg->p = ksp->work[0];

  PetscCall(VecDuplicateVecs(plcg->p, PetscMax(3, l + 1), &plcg->Z));
  PetscCall(VecDuplicateVecs(plcg->p, 3, &plcg->U));
  PetscCall(VecDuplicateVecs(plcg->p, 3, &plcg->V));
  PetscCall(VecDuplicateVecs(plcg->p, 3 * (l - 1) + 1, &plcg->Q));
  PetscCall(PetscCalloc1(2, &plcg->alpha));
  PetscCall(PetscCalloc1(l, &plcg->sigma));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode KSPReset_PIPELCG(KSP ksp)
{
  KSP_CG_PIPE_L *plcg = (KSP_CG_PIPE_L *)ksp->data;
  PetscInt       l    = plcg->l;

  PetscFunctionBegin;
  PetscCall(PetscFree(plcg->sigma));
  PetscCall(PetscFree(plcg->alpha));
  PetscCall(VecDestroyVecs(PetscMax(3, l + 1), &plcg->Z));
  PetscCall(VecDestroyVecs(3, &plcg->U));
  PetscCall(VecDestroyVecs(3, &plcg->V));
  PetscCall(VecDestroyVecs(3 * (l - 1) + 1, &plcg->Q));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode KSPDestroy_PIPELCG(KSP ksp)
{
  PetscFunctionBegin;
  PetscCall(KSPReset_PIPELCG(ksp));
  PetscCall(KSPDestroyDefault(ksp));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode KSPSetFromOptions_PIPELCG(KSP ksp, PetscOptionItems PetscOptionsObject)
{
  KSP_CG_PIPE_L *plcg = (KSP_CG_PIPE_L *)ksp->data;
  PetscBool      flag = PETSC_FALSE;

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "KSP PIPELCG options");
  PetscCall(PetscOptionsInt("-ksp_pipelcg_pipel", "Pipeline length", "", plcg->l, &plcg->l, &flag));
  if (!flag) plcg->l = 1;
  PetscCall(PetscOptionsReal("-ksp_pipelcg_lmin", "Estimate for smallest eigenvalue", "", plcg->lmin, &plcg->lmin, &flag));
  if (!flag) plcg->lmin = 0.0;
  PetscCall(PetscOptionsReal("-ksp_pipelcg_lmax", "Estimate for largest eigenvalue", "", plcg->lmax, &plcg->lmax, &flag));
  if (!flag) plcg->lmax = 0.0;
  PetscCall(PetscOptionsBool("-ksp_pipelcg_monitor", "Output information on restarts when they occur? (default: 0)", "", plcg->show_rstrt, &plcg->show_rstrt, &flag));
  if (!flag) plcg->show_rstrt = PETSC_FALSE;
  PetscOptionsHeadEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscMPIInt MPIU_Iallreduce(void *sendbuf, void *recvbuf, PetscMPIInt count, MPI_Datatype datatype, MPI_Op op, MPI_Comm comm, MPI_Request *request)
{
  PetscMPIInt err;
#if defined(PETSC_HAVE_MPI_NONBLOCKING_COLLECTIVES)
  err = MPI_Iallreduce(sendbuf, recvbuf, count, datatype, op, comm, request);
#else
  err      = MPI_Allreduce(sendbuf, recvbuf, count, datatype, op, comm);
  *request = MPI_REQUEST_NULL;
#endif
  return err;
}

static PetscErrorCode KSPView_PIPELCG(KSP ksp, PetscViewer viewer)
{
  KSP_CG_PIPE_L *plcg    = (KSP_CG_PIPE_L *)ksp->data;
  PetscBool      isascii = PETSC_FALSE, isstring = PETSC_FALSE;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERSTRING, &isstring));
  if (isascii) {
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Pipeline depth: %" PetscInt_FMT "\n", plcg->l));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Minimal eigenvalue estimate %g\n", (double)plcg->lmin));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Maximal eigenvalue estimate %g\n", (double)plcg->lmax));
  } else if (isstring) {
    PetscCall(PetscViewerStringSPrintf(viewer, "  Pipeline depth: %" PetscInt_FMT "\n", plcg->l));
    PetscCall(PetscViewerStringSPrintf(viewer, "  Minimal eigenvalue estimate %g\n", (double)plcg->lmin));
    PetscCall(PetscViewerStringSPrintf(viewer, "  Maximal eigenvalue estimate %g\n", (double)plcg->lmax));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode KSPSolve_InnerLoop_PIPELCG(KSP ksp)
{
  KSP_CG_PIPE_L *plcg = (KSP_CG_PIPE_L *)ksp->data;
  Mat            A = NULL, Pmat = NULL;
  PetscInt       it = 0, max_it = ksp->max_it, l = plcg->l, i = 0, j = 0, k = 0;
  PetscInt       start = 0, middle = 0, end = 0;
  Vec           *Z = plcg->Z, *U = plcg->U, *V = plcg->V, *Q = plcg->Q;
  Vec            x = NULL, p = NULL, temp = NULL;
  PetscScalar    sum_dummy = 0.0, eta = 0.0, zeta = 0.0, lambda = 0.0;
  PetscReal      dp = 0.0, tmp = 0.0, beta = 0.0, invbeta2 = 0.0;
  MPI_Comm       comm;
  PetscMPIInt    mpin;

  PetscFunctionBegin;
  x = ksp->vec_sol;
  p = plcg->p;

  comm = PetscObjectComm((PetscObject)ksp);
  PetscCall(PCGetOperators(ksp->pc, &A, &Pmat));

  for (it = 0; it < max_it + l; ++it) {
    /* Multiplication  z_{it+1} =  Az_{it} */
    /* Shift the U vector pointers */
    temp = U[2];
    for (i = 2; i > 0; i--) U[i] = U[i - 1];
    U[0] = temp;
    if (it < l) {
      /* SpMV and Sigma-shift and Prec */
      PetscCall(MatMult(A, Z[l - it], U[0]));
      PetscCall(VecAXPY(U[0], -sigma(it), U[1]));
      PetscCall(KSP_PCApply(ksp, U[0], Z[l - it - 1]));
      if (it < l - 1) PetscCall(VecCopy(Z[l - it - 1], Q[3 * it]));
    } else {
      /* Shift the Z vector pointers */
      temp = Z[PetscMax(l, 2)];
      for (i = PetscMax(l, 2); i > 0; --i) Z[i] = Z[i - 1];
      Z[0] = temp;
      /* SpMV and Prec */
      PetscCall(MatMult(A, Z[1], U[0]));
      PetscCall(KSP_PCApply(ksp, U[0], Z[0]));
    }

    /* Adjust the G matrix */
    if (it >= l) {
      if (it == l) {
        /* MPI_Wait for G(0,0),scale V0 and Z and U and Q vectors with 1/beta */
        PetscCallMPI(MPI_Wait(&req(0), MPI_STATUS_IGNORE));
        beta    = PetscSqrtReal(PetscRealPart(G(0, 0)));
        G(0, 0) = 1.0;
        PetscCall(VecAXPY(V[0], 1.0 / beta, p)); /* this assumes V[0] to be zero initially */
        for (j = 0; j <= PetscMax(l, 2); ++j) PetscCall(VecScale(Z[j], 1.0 / beta));
        for (j = 0; j <= 2; ++j) PetscCall(VecScale(U[j], 1.0 / beta));
        for (j = 0; j < l - 1; ++j) PetscCall(VecScale(Q[3 * j], 1.0 / beta));
      }

      /* MPI_Wait until the dot products,started l iterations ago,are completed */
      PetscCallMPI(MPI_Wait(&req(it - l + 1), MPI_STATUS_IGNORE));
      if (it >= 2 * l) {
        for (j = PetscMax(0, it - 3 * l + 1); j <= it - 2 * l; j++) G(j, it - l + 1) = G(it - 2 * l + 1, j + l); /* exploit symmetry in G matrix */
      }

      if (it <= 2 * l - 1) {
        invbeta2 = 1.0 / (beta * beta);
        /* Scale columns 1 up to l of G with 1/beta^2 */
        for (j = PetscMax(it - 3 * l + 1, 0); j <= it - l + 1; ++j) G(j, it - l + 1) *= invbeta2;
      }

      for (j = PetscMax(it - 2 * l + 2, 0); j <= it - l; ++j) {
        sum_dummy = 0.0;
        for (k = PetscMax(it - 3 * l + 1, 0); k <= j - 1; ++k) sum_dummy = sum_dummy + G(k, j) * G(k, it - l + 1);
        G(j, it - l + 1) = (G(j, it - l + 1) - sum_dummy) / G(j, j);
      }

      sum_dummy = 0.0;
      for (k = PetscMax(it - 3 * l + 1, 0); k <= it - l; ++k) sum_dummy = sum_dummy + G(k, it - l + 1) * G(k, it - l + 1);

      tmp = PetscRealPart(G(it - l + 1, it - l + 1) - sum_dummy);
      /* Breakdown check */
      if (tmp < 0) {
        if (plcg->show_rstrt) PetscCall(PetscPrintf(comm, "Sqrt breakdown in iteration %" PetscInt_FMT ": sqrt argument is %e. Iteration was restarted.\n", ksp->its + 1, (double)tmp));
        /* End hanging dot-products in the pipeline before exiting for-loop */
        start = it - l + 2;
        end   = PetscMin(it + 1, max_it + 1); /* !warning! 'it' can actually be greater than 'max_it' */
        for (i = start; i < end; ++i) PetscCallMPI(MPI_Wait(&req(i), MPI_STATUS_IGNORE));
        break;
      }
      G(it - l + 1, it - l + 1) = PetscSqrtReal(tmp);

      if (it < 2 * l) {
        if (it == l) {
          gamma(it - l) = (G(it - l, it - l + 1) + sigma(it - l) * G(it - l, it - l)) / G(it - l, it - l);
        } else {
          gamma(it - l) = (G(it - l, it - l + 1) + sigma(it - l) * G(it - l, it - l) - delta(it - l - 1) * G(it - l - 1, it - l)) / G(it - l, it - l);
        }
        delta(it - l) = G(it - l + 1, it - l + 1) / G(it - l, it - l);
      } else {
        gamma(it - l) = (G(it - l, it - l) * gamma(it - 2 * l) + G(it - l, it - l + 1) * delta(it - 2 * l) - G(it - l - 1, it - l) * delta(it - l - 1)) / G(it - l, it - l);
        delta(it - l) = (G(it - l + 1, it - l + 1) * delta(it - 2 * l)) / G(it - l, it - l);
      }

      /* Recursively compute the next V, Q, Z and U vectors */
      /* Shift the V vector pointers */
      temp = V[2];
      for (i = 2; i > 0; i--) V[i] = V[i - 1];
      V[0] = temp;

      /* Recurrence V vectors */
      if (l == 1) PetscCall(VecCopy(Z[1], V[0]));
      else PetscCall(VecCopy(Q[0], V[0]));
      if (it == l) PetscCall(VecAXPY(V[0], sigma(0) - gamma(it - l), V[1]));
      else {
        alpha(0) = sigma(0) - gamma(it - l);
        alpha(1) = -delta(it - l - 1);
        PetscCall(VecMAXPY(V[0], 2, &alpha(0), &V[1]));
      }
      PetscCall(VecScale(V[0], 1.0 / delta(it - l)));

      /* Recurrence Q vectors */
      for (j = 0; j < l - 1; ++j) {
        /* Shift the Q vector pointers */
        temp = Q[3 * j + 2];
        for (i = 2; i > 0; i--) Q[3 * j + i] = Q[3 * j + i - 1];
        Q[3 * j] = temp;

        if (j < l - 2) PetscCall(VecCopy(Q[3 * (j + 1)], Q[3 * j]));
        else PetscCall(VecCopy(Z[1], Q[3 * j]));
        if (it == l) PetscCall(VecAXPY(Q[3 * j], sigma(j + 1) - gamma(it - l), Q[3 * j + 1]));
        else {
          alpha(0) = sigma(j + 1) - gamma(it - l);
          alpha(1) = -delta(it - l - 1);
          PetscCall(VecMAXPY(Q[3 * j], 2, &alpha(0), &Q[3 * j + 1]));
        }
        PetscCall(VecScale(Q[3 * j], 1.0 / delta(it - l)));
      }

      /* Recurrence Z and U vectors */
      if (it == l) {
        PetscCall(VecAXPY(Z[0], -gamma(it - l), Z[1]));
        PetscCall(VecAXPY(U[0], -gamma(it - l), U[1]));
      } else {
        alpha(0) = -gamma(it - l);
        alpha(1) = -delta(it - l - 1);
        PetscCall(VecMAXPY(Z[0], 2, &alpha(0), &Z[1]));
        PetscCall(VecMAXPY(U[0], 2, &alpha(0), &U[1]));
      }
      PetscCall(VecScale(Z[0], 1.0 / delta(it - l)));
      PetscCall(VecScale(U[0], 1.0 / delta(it - l)));
    }

    /* Compute and communicate the dot products */
    if (it < l) {
      for (j = 0; j < it + 2; ++j) PetscCall((*U[0]->ops->dot_local)(U[0], Z[l - j], &G(j, it + 1))); /* dot-products (U[0],Z[j]) */
      PetscCall(PetscMPIIntCast(it + 2, &mpin));
      PetscCallMPI(MPIU_Iallreduce(MPI_IN_PLACE, &G(0, it + 1), mpin, MPIU_SCALAR, MPIU_SUM, comm, &req(it + 1)));
    } else if ((it >= l) && (it < max_it)) {
      middle = it - l + 2;
      end    = it + 2;
      PetscCall((*U[0]->ops->dot_local)(U[0], V[0], &G(it - l + 1, it + 1)));                                      /* dot-product (U[0],V[0]) */
      for (j = middle; j < end; ++j) PetscCall((*U[0]->ops->dot_local)(U[0], plcg->Z[it + 1 - j], &G(j, it + 1))); /* dot-products (U[0],Z[j]) */
      PetscCall(PetscMPIIntCast(l + 1, &mpin));
      PetscCallMPI(MPIU_Iallreduce(MPI_IN_PLACE, &G(it - l + 1, it + 1), mpin, MPIU_SCALAR, MPIU_SUM, comm, &req(it + 1)));
    }

    /* Compute solution vector and residual norm */
    if (it >= l) {
      if (it == l) {
        if (ksp->its != 0) ++ksp->its;
        eta  = gamma(0);
        zeta = beta;
        PetscCall(VecCopy(V[1], p));
        PetscCall(VecScale(p, 1.0 / eta));
        PetscCall(VecAXPY(x, zeta, p));
        dp = beta;
      } else if (it > l) {
        k = it - l;
        ++ksp->its;
        lambda = delta(k - 1) / eta;
        eta    = gamma(k) - lambda * delta(k - 1);
        zeta   = -lambda * zeta;
        PetscCall(VecScale(p, -delta(k - 1) / eta));
        PetscCall(VecAXPY(p, 1.0 / eta, V[1]));
        PetscCall(VecAXPY(x, zeta, p));
        dp = PetscAbsScalar(zeta);
      }
      ksp->rnorm = dp;
      PetscCall(KSPLogResidualHistory(ksp, dp));
      PetscCall(KSPMonitor(ksp, ksp->its, dp));
      PetscCall((*ksp->converged)(ksp, ksp->its, dp, &ksp->reason, ksp->cnvP));

      if (ksp->its >= max_it && !ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
      if (ksp->reason) {
        /* End hanging dot-products in the pipeline before exiting for-loop */
        start = it - l + 2;
        end   = PetscMin(it + 2, max_it + 1); /* !warning! 'it' can actually be greater than 'max_it' */
        for (i = start; i < end; ++i) PetscCallMPI(MPI_Wait(&req(i), MPI_STATUS_IGNORE));
        break;
      }
    }
  } /* End inner for loop */
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode KSPSolve_ReInitData_PIPELCG(KSP ksp)
{
  KSP_CG_PIPE_L *plcg = (KSP_CG_PIPE_L *)ksp->data;
  PetscInt       i = 0, j = 0, l = plcg->l, max_it = ksp->max_it;

  PetscFunctionBegin;
  for (i = 0; i < PetscMax(3, l + 1); ++i) PetscCall(VecSet(plcg->Z[i], 0.0));
  for (i = 1; i < 3; ++i) PetscCall(VecSet(plcg->U[i], 0.0));
  for (i = 0; i < 3; ++i) PetscCall(VecSet(plcg->V[i], 0.0));
  for (i = 0; i < 3 * (l - 1) + 1; ++i) PetscCall(VecSet(plcg->Q[i], 0.0));
  for (j = 0; j < (max_it + 1); ++j) {
    gamma(j) = 0.0;
    delta(j) = 0.0;
    for (i = 0; i < (2 * l + 1); ++i) G_noshift(i, j) = 0.0;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  KSPSolve_PIPELCG - This routine actually applies the pipelined(l) conjugate gradient method
*/
static PetscErrorCode KSPSolve_PIPELCG(KSP ksp)
{
  KSP_CG_PIPE_L *plcg = (KSP_CG_PIPE_L *)ksp->data;
  Mat            A = NULL, Pmat = NULL;
  Vec            b = NULL, x = NULL, p = NULL;
  PetscInt       max_it = ksp->max_it, l = plcg->l;
  PetscInt       i = 0, outer_it = 0, curr_guess_zero = 0;
  PetscReal      lmin = plcg->lmin, lmax = plcg->lmax;
  PetscBool      diagonalscale = PETSC_FALSE;
  MPI_Comm       comm;

  PetscFunctionBegin;
  comm = PetscObjectComm((PetscObject)ksp);
  PetscCall(PCGetDiagonalScale(ksp->pc, &diagonalscale));
  PetscCheck(!diagonalscale, comm, PETSC_ERR_SUP, "Krylov method %s does not support diagonal scaling", ((PetscObject)ksp)->type_name);

  x = ksp->vec_sol;
  b = ksp->vec_rhs;
  p = plcg->p;

  PetscCall(PetscCalloc1((max_it + 1) * (2 * l + 1), &plcg->G));
  PetscCall(PetscCalloc1(max_it + 1, &plcg->gamma));
  PetscCall(PetscCalloc1(max_it + 1, &plcg->delta));
  PetscCall(PetscCalloc1(max_it + 1, &plcg->req));

  PetscCall(PCGetOperators(ksp->pc, &A, &Pmat));

  for (i = 0; i < l; ++i) sigma(i) = (0.5 * (lmin + lmax) + (0.5 * (lmax - lmin) * PetscCosReal(PETSC_PI * (2.0 * i + 1.0) / (2.0 * l))));

  ksp->its        = 0;
  outer_it        = 0;
  curr_guess_zero = !!ksp->guess_zero;

  while (ksp->its < max_it) { /* OUTER LOOP (gmres-like restart to handle breakdowns) */
    /* RESTART LOOP */
    if (!curr_guess_zero) {
      PetscCall(KSP_MatMult(ksp, A, x, plcg->U[0])); /* u <- b - Ax */
      PetscCall(VecAYPX(plcg->U[0], -1.0, b));
    } else {
      PetscCall(VecCopy(b, plcg->U[0])); /* u <- b (x is 0) */
    }
    PetscCall(KSP_PCApply(ksp, plcg->U[0], p)); /* p <- Bu */

    if (outer_it > 0) {
      /* Re-initialize Z,U,V,Q,gamma,delta,G after restart occurred */
      PetscCall(KSPSolve_ReInitData_PIPELCG(ksp));
    }

    PetscCall((*plcg->U[0]->ops->dot_local)(plcg->U[0], p, &G(0, 0)));
    PetscCallMPI(MPIU_Iallreduce(MPI_IN_PLACE, &G(0, 0), 1, MPIU_SCALAR, MPIU_SUM, comm, &req(0)));
    PetscCall(VecCopy(p, plcg->Z[l]));

    PetscCall(KSPSolve_InnerLoop_PIPELCG(ksp));

    if (ksp->reason) break; /* convergence or divergence */
    ++outer_it;
    curr_guess_zero = 0;
  }

  if (!ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
  PetscCall(PetscFree(plcg->G));
  PetscCall(PetscFree(plcg->gamma));
  PetscCall(PetscFree(plcg->delta));
  PetscCall(PetscFree(plcg->req));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*MC
    KSPPIPELCG - Deep pipelined (length l) Conjugate Gradient method {cite}`cornelis2018communication` and {cite}`cools2019numerically`.
    This method has only a single non-blocking global
    reduction per iteration, compared to 2 blocking reductions for standard `KSPCG`. The reduction is overlapped by the
    matrix-vector product and preconditioner application of the next l iterations. The pipeline length l is a parameter
    of the method. [](sec_pipelineksp)

    Options Database Keys:
+   -ksp_pipelcg_pipel pipel - pipelined length
.   -ksp_pipelcg_lmin min    - approximation to the smallest eigenvalue of the preconditioned operator (default: 0.0)
.   -ksp_pipelcg_lmax max    - approximation to the largest eigenvalue of the preconditioned operator (default: 0.0)
-   -ksp_pipelcg_monitor     - output where/why the method restarts when a square-root breakdown occurs

    Example usage:
.vb
    KSP tutorials ex2, no preconditioner, pipel = 2, lmin = 0.0, lmax = 8.0 :
        $mpiexec -n 14 ./ex2 -m 1000 -n 1000 -ksp_type pipelcg -pc_type none -ksp_norm_type natural
           -ksp_rtol 1e-10 -ksp_max_it 1000 -ksp_pipelcg_pipel 2 -ksp_pipelcg_lmin 0.0 -ksp_pipelcg_lmax 8.0 -log_view
    SNES tutorials ex48, bjacobi preconditioner, pipel = 3, lmin = 0.0, lmax = 2.0, show restart information :
        $mpiexec -n 14 ./ex48 -M 150 -P 100 -ksp_type pipelcg -pc_type bjacobi -ksp_rtol 1e-10 -ksp_pipelcg_pipel 3
           -ksp_pipelcg_lmin 0.0 -ksp_pipelcg_lmax 2.0 -ksp_pipelcg_monitor -log_view
.ve

    Level: advanced

    Notes:
    MPI configuration may be necessary for reductions to make asynchronous progress, which is important for
    performance of pipelined methods. See [](doc_faq_pipelined)

    Contributed by:
    Siegfried Cools, University of Antwerp, Dept. Mathematics and Computer Science,
    funded by Flemish Research Foundation (FWO) grant number 12H4617N.

.seealso: [](ch_ksp), [](sec_pipelineksp), [](doc_faq_pipelined), `KSPCreate()`, `KSPSetType()`, `KSPType`, `KSPCG`, `KSPPIPECG`, `KSPPIPECGRR`, `KSPPGMRES`,
          `KSPPIPEBCGS`, `KSPSetPCSide()`, `KSPGROPPCG`
M*/
PETSC_EXTERN PetscErrorCode KSPCreate_PIPELCG(KSP ksp)
{
  KSP_CG_PIPE_L *plcg = NULL;

  PetscFunctionBegin;
  PetscCall(PetscNew(&plcg));
  ksp->data = (void *)plcg;

  PetscCall(KSPSetSupportedNorm(ksp, KSP_NORM_NONE, PC_LEFT, 1));
  PetscCall(KSPSetSupportedNorm(ksp, KSP_NORM_NATURAL, PC_LEFT, 2));

  ksp->ops->setup          = KSPSetUp_PIPELCG;
  ksp->ops->solve          = KSPSolve_PIPELCG;
  ksp->ops->reset          = KSPReset_PIPELCG;
  ksp->ops->destroy        = KSPDestroy_PIPELCG;
  ksp->ops->view           = KSPView_PIPELCG;
  ksp->ops->setfromoptions = KSPSetFromOptions_PIPELCG;
  ksp->ops->buildsolution  = KSPBuildSolutionDefault;
  ksp->ops->buildresidual  = KSPBuildResidualDefault;
  PetscFunctionReturn(PETSC_SUCCESS);
}
