xref: /petsc/src/mat/impls/baij/seq/baijfact13.c (revision 3ba1676111f5c958fe6c2729b46ca4d523958bb3)

/*
    Factorization code for BAIJ format.
*/
#include <../src/mat/impls/baij/seq/baij.h>
#include <petsc/private/kernels/blockinvert.h>

/*
      Version for when blocks are 3 by 3
*/
PetscErrorCode MatLUFactorNumeric_SeqBAIJ_3_inplace(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqBAIJ    *a = (Mat_SeqBAIJ *)A->data, *b = (Mat_SeqBAIJ *)C->data;
  IS              isrow = b->row, isicol = b->icol;
  const PetscInt *r, *ic;
  PetscInt        i, j, n = a->mbs, *bi = b->i, *bj = b->j;
  PetscInt       *ajtmpold, *ajtmp, nz, row, *ai = a->i, *aj = a->j;
  PetscInt       *diag_offset = b->diag, idx, *pj;
  MatScalar      *pv, *v, *rtmp, *pc, *w, *x;
  MatScalar       p1, p2, p3, p4, m1, m2, m3, m4, m5, m6, m7, m8, m9, x1, x2, x3, x4;
  MatScalar       p5, p6, p7, p8, p9, x5, x6, x7, x8, x9;
  MatScalar      *ba = b->a, *aa = a->a;
  PetscReal       shift = info->shiftamount;
  PetscBool       allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  PetscCall(ISGetIndices(isrow, &r));
  PetscCall(ISGetIndices(isicol, &ic));
  PetscCall(PetscMalloc1(9 * (n + 1), &rtmp));
  allowzeropivot = PetscNot(A->erroriffailure);

  for (i = 0; i < n; i++) {
    nz    = bi[i + 1] - bi[i];
    ajtmp = bj + bi[i];
    for (j = 0; j < nz; j++) {
      x    = rtmp + 9 * ajtmp[j];
      x[0] = x[1] = x[2] = x[3] = x[4] = x[5] = x[6] = x[7] = x[8] = 0.0;
    }
    /* load in initial (unfactored row) */
    idx      = r[i];
    nz       = ai[idx + 1] - ai[idx];
    ajtmpold = aj + ai[idx];
    v        = aa + 9 * ai[idx];
    for (j = 0; j < nz; j++) {
      x    = rtmp + 9 * ic[ajtmpold[j]];
      x[0] = v[0];
      x[1] = v[1];
      x[2] = v[2];
      x[3] = v[3];
      x[4] = v[4];
      x[5] = v[5];
      x[6] = v[6];
      x[7] = v[7];
      x[8] = v[8];
      v += 9;
    }
    row = *ajtmp++;
    while (row < i) {
      pc = rtmp + 9 * row;
      p1 = pc[0];
      p2 = pc[1];
      p3 = pc[2];
      p4 = pc[3];
      p5 = pc[4];
      p6 = pc[5];
      p7 = pc[6];
      p8 = pc[7];
      p9 = pc[8];
      if (p1 != 0.0 || p2 != 0.0 || p3 != 0.0 || p4 != 0.0 || p5 != 0.0 || p6 != 0.0 || p7 != 0.0 || p8 != 0.0 || p9 != 0.0) {
        pv    = ba + 9 * diag_offset[row];
        pj    = bj + diag_offset[row] + 1;
        x1    = pv[0];
        x2    = pv[1];
        x3    = pv[2];
        x4    = pv[3];
        x5    = pv[4];
        x6    = pv[5];
        x7    = pv[6];
        x8    = pv[7];
        x9    = pv[8];
        pc[0] = m1 = p1 * x1 + p4 * x2 + p7 * x3;
        pc[1] = m2 = p2 * x1 + p5 * x2 + p8 * x3;
        pc[2] = m3 = p3 * x1 + p6 * x2 + p9 * x3;

        pc[3] = m4 = p1 * x4 + p4 * x5 + p7 * x6;
        pc[4] = m5 = p2 * x4 + p5 * x5 + p8 * x6;
        pc[5] = m6 = p3 * x4 + p6 * x5 + p9 * x6;

        pc[6] = m7 = p1 * x7 + p4 * x8 + p7 * x9;
        pc[7] = m8 = p2 * x7 + p5 * x8 + p8 * x9;
        pc[8] = m9 = p3 * x7 + p6 * x8 + p9 * x9;
        nz         = bi[row + 1] - diag_offset[row] - 1;
        pv += 9;
        for (j = 0; j < nz; j++) {
          x1 = pv[0];
          x2 = pv[1];
          x3 = pv[2];
          x4 = pv[3];
          x5 = pv[4];
          x6 = pv[5];
          x7 = pv[6];
          x8 = pv[7];
          x9 = pv[8];
          x  = rtmp + 9 * pj[j];
          x[0] -= m1 * x1 + m4 * x2 + m7 * x3;
          x[1] -= m2 * x1 + m5 * x2 + m8 * x3;
          x[2] -= m3 * x1 + m6 * x2 + m9 * x3;

          x[3] -= m1 * x4 + m4 * x5 + m7 * x6;
          x[4] -= m2 * x4 + m5 * x5 + m8 * x6;
          x[5] -= m3 * x4 + m6 * x5 + m9 * x6;

          x[6] -= m1 * x7 + m4 * x8 + m7 * x9;
          x[7] -= m2 * x7 + m5 * x8 + m8 * x9;
          x[8] -= m3 * x7 + m6 * x8 + m9 * x9;
          pv += 9;
        }
        PetscCall(PetscLogFlops(54.0 * nz + 36.0));
      }
      row = *ajtmp++;
    }
    /* finished row so stick it into b->a */
    pv = ba + 9 * bi[i];
    pj = bj + bi[i];
    nz = bi[i + 1] - bi[i];
    for (j = 0; j < nz; j++) {
      x     = rtmp + 9 * pj[j];
      pv[0] = x[0];
      pv[1] = x[1];
      pv[2] = x[2];
      pv[3] = x[3];
      pv[4] = x[4];
      pv[5] = x[5];
      pv[6] = x[6];
      pv[7] = x[7];
      pv[8] = x[8];
      pv += 9;
    }
    /* invert diagonal block */
    w = ba + 9 * diag_offset[i];
    PetscCall(PetscKernel_A_gets_inverse_A_3(w, shift, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) C->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;
  }

  PetscCall(PetscFree(rtmp));
  PetscCall(ISRestoreIndices(isicol, &ic));
  PetscCall(ISRestoreIndices(isrow, &r));

  C->ops->solve          = MatSolve_SeqBAIJ_3_inplace;
  C->ops->solvetranspose = MatSolveTranspose_SeqBAIJ_3_inplace;
  C->assembled           = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.333333333333 * 3 * 3 * 3 * b->mbs)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* MatLUFactorNumeric_SeqBAIJ_3 -
     copied from MatLUFactorNumeric_SeqBAIJ_N_inplace() and manually re-implemented
       PetscKernel_A_gets_A_times_B()
       PetscKernel_A_gets_A_minus_B_times_C()
       PetscKernel_A_gets_inverse_A()
*/
PetscErrorCode MatLUFactorNumeric_SeqBAIJ_3(Mat B, Mat A, const MatFactorInfo *info)
{
  Mat             C = B;
  Mat_SeqBAIJ    *a = (Mat_SeqBAIJ *)A->data, *b = (Mat_SeqBAIJ *)C->data;
  IS              isrow = b->row, isicol = b->icol;
  const PetscInt *r, *ic;
  PetscInt        i, j, k, nz, nzL, row;
  const PetscInt  n = a->mbs, *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j;
  const PetscInt *ajtmp, *bjtmp, *bdiag = b->diag, *pj, bs2 = a->bs2;
  MatScalar      *rtmp, *pc, *mwork, *v, *pv, *aa = a->a;
  PetscInt        flg;
  PetscReal       shift = info->shiftamount;
  PetscBool       allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  PetscCall(ISGetIndices(isrow, &r));
  PetscCall(ISGetIndices(isicol, &ic));
  allowzeropivot = PetscNot(A->erroriffailure);

  /* generate work space needed by the factorization */
  PetscCall(PetscMalloc2(bs2 * n, &rtmp, bs2, &mwork));
  PetscCall(PetscArrayzero(rtmp, bs2 * n));

  for (i = 0; i < n; i++) {
    /* zero rtmp */
    /* L part */
    nz    = bi[i + 1] - bi[i];
    bjtmp = bj + bi[i];
    for (j = 0; j < nz; j++) PetscCall(PetscArrayzero(rtmp + bs2 * bjtmp[j], bs2));

    /* U part */
    nz    = bdiag[i] - bdiag[i + 1];
    bjtmp = bj + bdiag[i + 1] + 1;
    for (j = 0; j < nz; j++) PetscCall(PetscArrayzero(rtmp + bs2 * bjtmp[j], bs2));

    /* load in initial (unfactored row) */
    nz    = ai[r[i] + 1] - ai[r[i]];
    ajtmp = aj + ai[r[i]];
    v     = aa + bs2 * ai[r[i]];
    for (j = 0; j < nz; j++) PetscCall(PetscArraycpy(rtmp + bs2 * ic[ajtmp[j]], v + bs2 * j, bs2));

    /* elimination */
    bjtmp = bj + bi[i];
    nzL   = bi[i + 1] - bi[i];
    for (k = 0; k < nzL; k++) {
      row = bjtmp[k];
      pc  = rtmp + bs2 * row;
      for (flg = 0, j = 0; j < bs2; j++) {
        if (pc[j] != 0.0) {
          flg = 1;
          break;
        }
      }
      if (flg) {
        pv = b->a + bs2 * bdiag[row];
        /* PetscKernel_A_gets_A_times_B(bs,pc,pv,mwork); *pc = *pc * (*pv); */
        PetscCall(PetscKernel_A_gets_A_times_B_3(pc, pv, mwork));

        pj = b->j + bdiag[row + 1] + 1; /* beginning of U(row,:) */
        pv = b->a + bs2 * (bdiag[row + 1] + 1);
        nz = bdiag[row] - bdiag[row + 1] - 1; /* num of entries in U(row,:) excluding diag */
        for (j = 0; j < nz; j++) {
          /* PetscKernel_A_gets_A_minus_B_times_C(bs,rtmp+bs2*pj[j],pc,pv+bs2*j); */
          /* rtmp+bs2*pj[j] = rtmp+bs2*pj[j] - (*pc)*(pv+bs2*j) */
          v = rtmp + bs2 * pj[j];
          PetscCall(PetscKernel_A_gets_A_minus_B_times_C_3(v, pc, pv));
          pv += bs2;
        }
        PetscCall(PetscLogFlops(54.0 * nz + 45)); /* flops = 2*bs^3*nz + 2*bs^3 - bs2) */
      }
    }

    /* finished row so stick it into b->a */
    /* L part */
    pv = b->a + bs2 * bi[i];
    pj = b->j + bi[i];
    nz = bi[i + 1] - bi[i];
    for (j = 0; j < nz; j++) PetscCall(PetscArraycpy(pv + bs2 * j, rtmp + bs2 * pj[j], bs2));

    /* Mark diagonal and invert diagonal for simpler triangular solves */
    pv = b->a + bs2 * bdiag[i];
    pj = b->j + bdiag[i];
    PetscCall(PetscArraycpy(pv, rtmp + bs2 * pj[0], bs2));
    PetscCall(PetscKernel_A_gets_inverse_A_3(pv, shift, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) B->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    /* U part */
    pj = b->j + bdiag[i + 1] + 1;
    pv = b->a + bs2 * (bdiag[i + 1] + 1);
    nz = bdiag[i] - bdiag[i + 1] - 1;
    for (j = 0; j < nz; j++) PetscCall(PetscArraycpy(pv + bs2 * j, rtmp + bs2 * pj[j], bs2));
  }

  PetscCall(PetscFree2(rtmp, mwork));
  PetscCall(ISRestoreIndices(isicol, &ic));
  PetscCall(ISRestoreIndices(isrow, &r));

  C->ops->solve          = MatSolve_SeqBAIJ_3;
  C->ops->solvetranspose = MatSolveTranspose_SeqBAIJ_3;
  C->assembled           = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.333333333333 * 3 * 3 * 3 * n)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode MatLUFactorNumeric_SeqBAIJ_3_NaturalOrdering_inplace(Mat C, Mat A, const MatFactorInfo *info)
{
  Mat_SeqBAIJ *a = (Mat_SeqBAIJ *)A->data, *b = (Mat_SeqBAIJ *)C->data;
  PetscInt     i, j, n = a->mbs, *bi = b->i, *bj = b->j;
  PetscInt    *ajtmpold, *ajtmp, nz, row;
  PetscInt    *diag_offset = b->diag, *ai = a->i, *aj = a->j, *pj;
  MatScalar   *pv, *v, *rtmp, *pc, *w, *x;
  MatScalar    p1, p2, p3, p4, m1, m2, m3, m4, m5, m6, m7, m8, m9, x1, x2, x3, x4;
  MatScalar    p5, p6, p7, p8, p9, x5, x6, x7, x8, x9;
  MatScalar   *ba = b->a, *aa = a->a;
  PetscReal    shift = info->shiftamount;
  PetscBool    allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  PetscCall(PetscMalloc1(9 * (n + 1), &rtmp));
  allowzeropivot = PetscNot(A->erroriffailure);

  for (i = 0; i < n; i++) {
    nz    = bi[i + 1] - bi[i];
    ajtmp = bj + bi[i];
    for (j = 0; j < nz; j++) {
      x    = rtmp + 9 * ajtmp[j];
      x[0] = x[1] = x[2] = x[3] = x[4] = x[5] = x[6] = x[7] = x[8] = 0.0;
    }
    /* load in initial (unfactored row) */
    nz       = ai[i + 1] - ai[i];
    ajtmpold = aj + ai[i];
    v        = aa + 9 * ai[i];
    for (j = 0; j < nz; j++) {
      x    = rtmp + 9 * ajtmpold[j];
      x[0] = v[0];
      x[1] = v[1];
      x[2] = v[2];
      x[3] = v[3];
      x[4] = v[4];
      x[5] = v[5];
      x[6] = v[6];
      x[7] = v[7];
      x[8] = v[8];
      v += 9;
    }
    row = *ajtmp++;
    while (row < i) {
      pc = rtmp + 9 * row;
      p1 = pc[0];
      p2 = pc[1];
      p3 = pc[2];
      p4 = pc[3];
      p5 = pc[4];
      p6 = pc[5];
      p7 = pc[6];
      p8 = pc[7];
      p9 = pc[8];
      if (p1 != 0.0 || p2 != 0.0 || p3 != 0.0 || p4 != 0.0 || p5 != 0.0 || p6 != 0.0 || p7 != 0.0 || p8 != 0.0 || p9 != 0.0) {
        pv    = ba + 9 * diag_offset[row];
        pj    = bj + diag_offset[row] + 1;
        x1    = pv[0];
        x2    = pv[1];
        x3    = pv[2];
        x4    = pv[3];
        x5    = pv[4];
        x6    = pv[5];
        x7    = pv[6];
        x8    = pv[7];
        x9    = pv[8];
        pc[0] = m1 = p1 * x1 + p4 * x2 + p7 * x3;
        pc[1] = m2 = p2 * x1 + p5 * x2 + p8 * x3;
        pc[2] = m3 = p3 * x1 + p6 * x2 + p9 * x3;

        pc[3] = m4 = p1 * x4 + p4 * x5 + p7 * x6;
        pc[4] = m5 = p2 * x4 + p5 * x5 + p8 * x6;
        pc[5] = m6 = p3 * x4 + p6 * x5 + p9 * x6;

        pc[6] = m7 = p1 * x7 + p4 * x8 + p7 * x9;
        pc[7] = m8 = p2 * x7 + p5 * x8 + p8 * x9;
        pc[8] = m9 = p3 * x7 + p6 * x8 + p9 * x9;

        nz = bi[row + 1] - diag_offset[row] - 1;
        pv += 9;
        for (j = 0; j < nz; j++) {
          x1 = pv[0];
          x2 = pv[1];
          x3 = pv[2];
          x4 = pv[3];
          x5 = pv[4];
          x6 = pv[5];
          x7 = pv[6];
          x8 = pv[7];
          x9 = pv[8];
          x  = rtmp + 9 * pj[j];
          x[0] -= m1 * x1 + m4 * x2 + m7 * x3;
          x[1] -= m2 * x1 + m5 * x2 + m8 * x3;
          x[2] -= m3 * x1 + m6 * x2 + m9 * x3;

          x[3] -= m1 * x4 + m4 * x5 + m7 * x6;
          x[4] -= m2 * x4 + m5 * x5 + m8 * x6;
          x[5] -= m3 * x4 + m6 * x5 + m9 * x6;

          x[6] -= m1 * x7 + m4 * x8 + m7 * x9;
          x[7] -= m2 * x7 + m5 * x8 + m8 * x9;
          x[8] -= m3 * x7 + m6 * x8 + m9 * x9;
          pv += 9;
        }
        PetscCall(PetscLogFlops(54.0 * nz + 36.0));
      }
      row = *ajtmp++;
    }
    /* finished row so stick it into b->a */
    pv = ba + 9 * bi[i];
    pj = bj + bi[i];
    nz = bi[i + 1] - bi[i];
    for (j = 0; j < nz; j++) {
      x     = rtmp + 9 * pj[j];
      pv[0] = x[0];
      pv[1] = x[1];
      pv[2] = x[2];
      pv[3] = x[3];
      pv[4] = x[4];
      pv[5] = x[5];
      pv[6] = x[6];
      pv[7] = x[7];
      pv[8] = x[8];
      pv += 9;
    }
    /* invert diagonal block */
    w = ba + 9 * diag_offset[i];
    PetscCall(PetscKernel_A_gets_inverse_A_3(w, shift, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) C->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;
  }

  PetscCall(PetscFree(rtmp));

  C->ops->solve          = MatSolve_SeqBAIJ_3_NaturalOrdering_inplace;
  C->ops->solvetranspose = MatSolveTranspose_SeqBAIJ_3_NaturalOrdering_inplace;
  C->assembled           = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.333333333333 * 3 * 3 * 3 * b->mbs)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  MatLUFactorNumeric_SeqBAIJ_3_NaturalOrdering -
    copied from MatLUFactorNumeric_SeqBAIJ_2_NaturalOrdering_inplace()
*/
PetscErrorCode MatLUFactorNumeric_SeqBAIJ_3_NaturalOrdering(Mat B, Mat A, const MatFactorInfo *info)
{
  Mat             C = B;
  Mat_SeqBAIJ    *a = (Mat_SeqBAIJ *)A->data, *b = (Mat_SeqBAIJ *)C->data;
  PetscInt        i, j, k, nz, nzL, row;
  const PetscInt  n = a->mbs, *ai = a->i, *aj = a->j, *bi = b->i, *bj = b->j;
  const PetscInt *ajtmp, *bjtmp, *bdiag = b->diag, *pj, bs2 = a->bs2;
  MatScalar      *rtmp, *pc, *mwork, *v, *pv, *aa = a->a;
  PetscInt        flg;
  PetscReal       shift = info->shiftamount;
  PetscBool       allowzeropivot, zeropivotdetected;

  PetscFunctionBegin;
  allowzeropivot = PetscNot(A->erroriffailure);

  /* generate work space needed by the factorization */
  PetscCall(PetscMalloc2(bs2 * n, &rtmp, bs2, &mwork));
  PetscCall(PetscArrayzero(rtmp, bs2 * n));

  for (i = 0; i < n; i++) {
    /* zero rtmp */
    /* L part */
    nz    = bi[i + 1] - bi[i];
    bjtmp = bj + bi[i];
    for (j = 0; j < nz; j++) PetscCall(PetscArrayzero(rtmp + bs2 * bjtmp[j], bs2));

    /* U part */
    nz    = bdiag[i] - bdiag[i + 1];
    bjtmp = bj + bdiag[i + 1] + 1;
    for (j = 0; j < nz; j++) PetscCall(PetscArrayzero(rtmp + bs2 * bjtmp[j], bs2));

    /* load in initial (unfactored row) */
    nz    = ai[i + 1] - ai[i];
    ajtmp = aj + ai[i];
    v     = aa + bs2 * ai[i];
    for (j = 0; j < nz; j++) PetscCall(PetscArraycpy(rtmp + bs2 * ajtmp[j], v + bs2 * j, bs2));

    /* elimination */
    bjtmp = bj + bi[i];
    nzL   = bi[i + 1] - bi[i];
    for (k = 0; k < nzL; k++) {
      row = bjtmp[k];
      pc  = rtmp + bs2 * row;
      for (flg = 0, j = 0; j < bs2; j++) {
        if (pc[j] != 0.0) {
          flg = 1;
          break;
        }
      }
      if (flg) {
        pv = b->a + bs2 * bdiag[row];
        /* PetscKernel_A_gets_A_times_B(bs,pc,pv,mwork); *pc = *pc * (*pv); */
        PetscCall(PetscKernel_A_gets_A_times_B_3(pc, pv, mwork));

        pj = b->j + bdiag[row + 1] + 1; /* beginning of U(row,:) */
        pv = b->a + bs2 * (bdiag[row + 1] + 1);
        nz = bdiag[row] - bdiag[row + 1] - 1; /* num of entries in U(row,:) excluding diag */
        for (j = 0; j < nz; j++) {
          /* PetscKernel_A_gets_A_minus_B_times_C(bs,rtmp+bs2*pj[j],pc,pv+bs2*j); */
          /* rtmp+bs2*pj[j] = rtmp+bs2*pj[j] - (*pc)*(pv+bs2*j) */
          v = rtmp + bs2 * pj[j];
          PetscCall(PetscKernel_A_gets_A_minus_B_times_C_3(v, pc, pv));
          pv += bs2;
        }
        PetscCall(PetscLogFlops(54.0 * nz + 45)); /* flops = 2*bs^3*nz + 2*bs^3 - bs2) */
      }
    }

    /* finished row so stick it into b->a */
    /* L part */
    pv = b->a + bs2 * bi[i];
    pj = b->j + bi[i];
    nz = bi[i + 1] - bi[i];
    for (j = 0; j < nz; j++) PetscCall(PetscArraycpy(pv + bs2 * j, rtmp + bs2 * pj[j], bs2));

    /* Mark diagonal and invert diagonal for simpler triangular solves */
    pv = b->a + bs2 * bdiag[i];
    pj = b->j + bdiag[i];
    PetscCall(PetscArraycpy(pv, rtmp + bs2 * pj[0], bs2));
    PetscCall(PetscKernel_A_gets_inverse_A_3(pv, shift, allowzeropivot, &zeropivotdetected));
    if (zeropivotdetected) B->factorerrortype = MAT_FACTOR_NUMERIC_ZEROPIVOT;

    /* U part */
    pv = b->a + bs2 * (bdiag[i + 1] + 1);
    pj = b->j + bdiag[i + 1] + 1;
    nz = bdiag[i] - bdiag[i + 1] - 1;
    for (j = 0; j < nz; j++) PetscCall(PetscArraycpy(pv + bs2 * j, rtmp + bs2 * pj[j], bs2));
  }
  PetscCall(PetscFree2(rtmp, mwork));

  C->ops->solve          = MatSolve_SeqBAIJ_3_NaturalOrdering;
  C->ops->forwardsolve   = MatForwardSolve_SeqBAIJ_3_NaturalOrdering;
  C->ops->backwardsolve  = MatBackwardSolve_SeqBAIJ_3_NaturalOrdering;
  C->ops->solvetranspose = MatSolveTranspose_SeqBAIJ_3_NaturalOrdering;
  C->assembled           = PETSC_TRUE;

  PetscCall(PetscLogFlops(1.333333333333 * 3 * 3 * 3 * n)); /* from inverting diagonal blocks */
  PetscFunctionReturn(PETSC_SUCCESS);
}