xref: /petsc/src/mat/impls/kaij/kaij.c (revision 31a97b9a3c358588ac4428bcc7cbff2ea31f18a2)

/*
  Defines the basic matrix operations for the KAIJ  matrix storage format.
  This format is used to evaluate matrices of the form:

    [I \otimes S + A \otimes T]

  where
    S is a dense (p \times q) matrix
    T is a dense (p \times q) matrix
    A is an AIJ  (n \times n) matrix
    I is the identity matrix

  The resulting matrix is (np \times nq)

  We provide:
     MatMult()
     MatMultAdd()
     MatInvertBlockDiagonal()
  and
     MatCreateKAIJ(Mat,PetscInt,PetscInt,const PetscScalar[],const PetscScalar[],Mat*)

  This single directory handles both the sequential and parallel codes
*/

#include <../src/mat/impls/kaij/kaij.h> /*I "petscmat.h" I*/
#include <../src/mat/utils/freespace.h>
#include <petsc/private/vecimpl.h>

/*@C
   MatKAIJGetAIJ - Get the AIJ matrix describing the blockwise action of the KAIJ matrix

   Not Collective, but if the KAIJ matrix is parallel, the AIJ matrix is also parallel

   Input Parameter:
.  A - the KAIJ matrix

   Output Parameter:
.  B - the AIJ matrix

   Level: advanced

   Notes: The reference count on the AIJ matrix is not increased so you should not destroy it.

.seealso: MatCreateKAIJ()
@*/
PetscErrorCode  MatKAIJGetAIJ(Mat A,Mat *B)
{
  PetscErrorCode ierr;
  PetscBool      ismpikaij,isseqkaij;

  PetscFunctionBegin;
  ierr = PetscObjectTypeCompare((PetscObject)A,MATMPIKAIJ,&ismpikaij);CHKERRQ(ierr);
  ierr = PetscObjectTypeCompare((PetscObject)A,MATSEQKAIJ,&isseqkaij);CHKERRQ(ierr);
  if (ismpikaij) {
    Mat_MPIKAIJ *b = (Mat_MPIKAIJ*)A->data;

    *B = b->A;
  } else if (isseqkaij) {
    Mat_SeqKAIJ *b = (Mat_SeqKAIJ*)A->data;

    *B = b->AIJ;
  } else {
    *B = A;
  }
  PetscFunctionReturn(0);
}

/*@C
   MatKAIJGetS - Get the S matrix describing the shift action of the KAIJ matrix

   Not Collective, the entire S is stored and returned independently on all processes

   Input Parameter:
.  A - the KAIJ matrix

   Output Parameter:
.  S - the S matrix, in form of a scalar array in column-major format

   Notes:
   The dimensions of the output array can be determined by calling MatGetBlockSizes() on the KAIJ matrix.
   The row block and column block sizes returned will correspond to the number of rows and columns, respectively, in S.

   Level: advanced

.seealso: MatCreateKAIJ(), MatGetBlockSizes()
@*/
PetscErrorCode  MatKAIJGetS(Mat A,const PetscScalar **S)
{
  Mat_SeqKAIJ *b = (Mat_SeqKAIJ*)A->data;
  PetscFunctionBegin;
  *S = b->S;
  PetscFunctionReturn(0);
}

/*@C
   MatKAIJGetT - Get the transformation matrix T associated with the KAIJ matrix

   Not Collective, the entire T is stored and returned independently on all processes

   Input Parameter:
.  A - the KAIJ matrix

   Output Parameter:
.  T - the T matrix, in form of a scalar array in column-major format

   Notes:
   The dimensions of the output array can be determined by calling MatGetBlockSizes() on the KAIJ matrix.
   The row block and column block sizes returned will correspond to the number of rows and columns, respectively, in T.

   Level: advanced

.seealso: MatCreateKAIJ(), MatGetBlockSizes()
@*/
PetscErrorCode  MatKAIJGetT(Mat A,const PetscScalar **T)
{
  Mat_SeqKAIJ *b = (Mat_SeqKAIJ*)A->data;
  PetscFunctionBegin;
  *T = b->T;
  PetscFunctionReturn(0);
}

PetscErrorCode MatDestroy_SeqKAIJ(Mat A)
{
  PetscErrorCode ierr;
  Mat_SeqKAIJ    *b = (Mat_SeqKAIJ*)A->data;

  PetscFunctionBegin;
  ierr = MatDestroy(&b->AIJ);CHKERRQ(ierr);
  ierr = PetscFree3(b->S,b->T,b->ibdiag);CHKERRQ(ierr);
  ierr = PetscFree5(b->sor.w,b->sor.y,b->sor.work,b->sor.t,b->sor.arr);CHKERRQ(ierr);
  ierr = PetscFree(A->data);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatSetUp_KAIJ(Mat A)
{
  PetscFunctionBegin;
  SETERRQ(PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"Must use MatCreateKAIJ() to create KAIJ matrices");
  PetscFunctionReturn(0);
}

PetscErrorCode MatView_SeqKAIJ(Mat A,PetscViewer viewer)
{
  PetscErrorCode ierr;
  Mat            B;

  PetscFunctionBegin;
  ierr = MatConvert(A,MATSEQAIJ,MAT_INITIAL_MATRIX,&B);CHKERRQ(ierr);
  ierr = MatView(B,viewer);CHKERRQ(ierr);
  ierr = MatDestroy(&B);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatView_MPIKAIJ(Mat A,PetscViewer viewer)
{
  PetscErrorCode ierr;
  Mat            B;

  PetscFunctionBegin;
  ierr = MatConvert(A,MATMPIAIJ,MAT_INITIAL_MATRIX,&B);CHKERRQ(ierr);
  ierr = MatView(B,viewer);CHKERRQ(ierr);
  ierr = MatDestroy(&B);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatDestroy_MPIKAIJ(Mat A)
{
  PetscErrorCode ierr;
  Mat_MPIKAIJ    *b = (Mat_MPIKAIJ*)A->data;

  PetscFunctionBegin;
  ierr = MatDestroy(&b->AIJ);CHKERRQ(ierr);
  ierr = MatDestroy(&b->OAIJ);CHKERRQ(ierr);
  ierr = MatDestroy(&b->A);CHKERRQ(ierr);
  ierr = VecScatterDestroy(&b->ctx);CHKERRQ(ierr);
  ierr = VecDestroy(&b->w);CHKERRQ(ierr);
  ierr = PetscFree3(b->S,b->T,b->ibdiag);CHKERRQ(ierr);
  ierr = PetscFree(A->data);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*MC
  MATKAIJ - MATKAIJ = "kaij" - A matrix type to be used to evaluate matrices of the following form:

    [I \otimes S + A \otimes T]

  where
    S is a dense (p \times q) matrix
    T is a dense (p \times q) matrix
    A is an AIJ  (n \times n) matrix
    I is the identity matrix
  The resulting matrix is (np \times nq)

  The matrix type is based on MATSEQAIJ for a sequential matrix A, and MATMPIAIJ for a distributed matrix A.
  S and T are always stored independently on all processes as a PetscScalar array in column-major format.

  Operations provided:
. MatMult
. MatMultAdd
. MatInvertBlockDiagonal

  Level: advanced

.seealso: MatKAIJGetAIJ(), MatKAIJGetS(), MatKAIJGetT(), MatCreateKAIJ()
M*/

PETSC_EXTERN PetscErrorCode MatCreate_KAIJ(Mat A)
{
  PetscErrorCode ierr;
  Mat_MPIKAIJ    *b;
  PetscMPIInt    size;

  PetscFunctionBegin;
  ierr     = PetscNewLog(A,&b);CHKERRQ(ierr);
  A->data  = (void*)b;

  ierr = PetscMemzero(A->ops,sizeof(struct _MatOps));CHKERRQ(ierr);

  A->ops->setup = MatSetUp_KAIJ;

  b->w    = 0;
  ierr    = MPI_Comm_size(PetscObjectComm((PetscObject)A),&size);CHKERRQ(ierr);
  if (size == 1) {
    ierr = PetscObjectChangeTypeName((PetscObject)A,MATSEQKAIJ);CHKERRQ(ierr);
  } else {
    ierr = PetscObjectChangeTypeName((PetscObject)A,MATMPIKAIJ);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/* --------------------------------------------------------------------------------------*/

/* zz = yy + Axx */
PetscErrorCode KAIJMultAdd_Seq(Mat A,Vec xx,Vec yy,Vec zz)
{
  Mat_SeqKAIJ       *b = (Mat_SeqKAIJ*)A->data;
  Mat_SeqAIJ        *a = (Mat_SeqAIJ*)b->AIJ->data;
  const PetscScalar *s = b->S, *t = b->T;
  const PetscScalar *x,*v,*bx;
  PetscScalar       *y,*sums;
  PetscErrorCode    ierr;
  const PetscInt    m = b->AIJ->rmap->n,*idx,*ii;
  PetscInt          n,i,jrow,j,l,p=b->p,q=b->q,k;

  PetscFunctionBegin;
  if (!yy) {
    ierr = VecSet(zz,0.0);CHKERRQ(ierr);
  } else {
    ierr = VecCopy(yy,zz);CHKERRQ(ierr);
  }
  if ((!s) && (!t) && (!b->isTI)) PetscFunctionReturn(0);

  ierr = VecGetArrayRead(xx,&x);CHKERRQ(ierr);
  ierr = VecGetArray(zz,&y);CHKERRQ(ierr);
  idx  = a->j;
  v    = a->a;
  ii   = a->i;

  if (b->isTI) {
    for (i=0; i<m; i++) {
      jrow = ii[i];
      n    = ii[i+1] - jrow;
      sums = y + p*i;
      for (j=0; j<n; j++) {
        for (k=0; k<p; k++) {
          sums[k] += v[jrow+j]*x[q*idx[jrow+j]+k];
        }
      }
    }
  } else if (t) {
    for (i=0; i<m; i++) {
      jrow = ii[i];
      n    = ii[i+1] - jrow;
      sums = y + p*i;
      bx   = x + q*i;
      for (j=0; j<n; j++) {
        for (k=0; k<p; k++) {
          for (l=0; l<q; l++) {
            sums[k] += v[jrow+j]*t[k+l*p]*x[q*idx[jrow+j]+l];
          }
        }
      }
    }
  }
  if (s) {
    for (i=0; i<m; i++) {
      jrow = ii[i];
      n    = ii[i+1] - jrow;
      sums = y + p*i;
      bx   = x + q*i;
      if (i < b->AIJ->cmap->n) {
        for (j=0; j<q; j++) {
          for (k=0; k<p; k++) {
            sums[k] += s[k+j*p]*bx[j];
          }
        }
      }
    }
  }

  ierr = PetscLogFlops((2.0*p*q-p)*m+2*p*a->nz);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(xx,&x);CHKERRQ(ierr);
  ierr = VecRestoreArray(zz,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatMult_SeqKAIJ_N(Mat A,Vec xx,Vec yy)
{
  PetscErrorCode ierr;
  PetscFunctionBegin;
  ierr = KAIJMultAdd_Seq(A,xx,PETSC_NULL,yy);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatMultAdd_SeqKAIJ_N(Mat A,Vec xx,Vec yy,Vec zz)
{
  PetscErrorCode ierr;
  PetscFunctionBegin;
  ierr = KAIJMultAdd_Seq(A,xx,yy,zz);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#include <petsc/private/kernels/blockinvert.h>

PetscErrorCode MatInvertBlockDiagonal_SeqKAIJ_N(Mat A,const PetscScalar **values)
{
  Mat_SeqKAIJ       *b  = (Mat_SeqKAIJ*)A->data;
  Mat_SeqAIJ        *a  = (Mat_SeqAIJ*)b->AIJ->data;
  const PetscScalar *S  = b->S;
  const PetscScalar *T  = b->T;
  const PetscScalar *v  = a->a;
  const PetscInt     p  = b->p, q = b->q, m = b->AIJ->rmap->n, *idx = a->j, *ii = a->i;
  PetscErrorCode    ierr;
  PetscInt          i,j,*v_pivots,dof,dof2;
  PetscScalar       *diag,aval,*v_work;

  PetscFunctionBegin;
  if (p != q) SETERRQ(PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"MATKAIJ: Block size must be square to calculate inverse.");
  if ((!S) && (!T) && (!b->isTI)) SETERRQ(PetscObjectComm((PetscObject)A),PETSC_ERR_SUP,"MATKAIJ: Cannot invert a zero matrix.");

  dof  = p;
  dof2 = dof*dof;

  if (b->ibdiagvalid) {
    if (values) *values = b->ibdiag;
    PetscFunctionReturn(0);
  }
  if (!b->ibdiag) {
    ierr = PetscMalloc(dof2*m*sizeof(PetscScalar),&b->ibdiag);CHKERRQ(ierr);
    ierr = PetscLogObjectMemory((PetscObject)A,dof2*m*sizeof(PetscScalar));CHKERRQ(ierr);
  }
  if (values) *values = b->ibdiag;
  diag = b->ibdiag;

  ierr = PetscMalloc2(dof,&v_work,dof,&v_pivots);CHKERRQ(ierr);
  for (i=0; i<m; i++) {
    if (S) {
      ierr = PetscMemcpy(diag,S,dof2*sizeof(PetscScalar));CHKERRQ(ierr);
    } else {
      ierr = PetscMemzero(diag,dof2*sizeof(PetscScalar));CHKERRQ(ierr);
    }
    if (b->isTI) {
      aval = 0;
      for (j=ii[i]; j<ii[i+1]; j++) if (idx[j] == i) aval = v[j];
      for (j=0; j<dof; j++) diag[j+dof*j] += aval;
    } else if (T) {
      aval = 0;
      for (j=ii[i]; j<ii[i+1]; j++) if (idx[j] == i) aval = v[j];
      for (j=0; j<dof2; j++) diag[j] += aval*T[j];
    }
    ierr = PetscKernel_A_gets_inverse_A(dof,diag,v_pivots,v_work,PETSC_FALSE,NULL);CHKERRQ(ierr);
    diag += dof2;
  }
  ierr = PetscFree2(v_work,v_pivots);CHKERRQ(ierr);

  b->ibdiagvalid = PETSC_TRUE;
  PetscFunctionReturn(0);
}

static PetscErrorCode MatGetDiagonalBlock_MPIKAIJ(Mat A,Mat *B)
{
  Mat_MPIKAIJ *kaij = (Mat_MPIKAIJ*) A->data;

  PetscFunctionBegin;
  *B = kaij->AIJ;
  PetscFunctionReturn(0);
}

PetscErrorCode MatSOR_SeqKAIJ(Mat A,Vec bb,PetscReal omega,MatSORType flag,PetscReal fshift,PetscInt its,PetscInt lits,Vec xx)
{
  PetscErrorCode    ierr;
  Mat_SeqKAIJ       *kaij = (Mat_SeqKAIJ*) A->data;
  Mat_SeqAIJ        *a = (Mat_SeqAIJ*)kaij->AIJ->data;
  const PetscScalar *aa = a->a, *T = kaij->T, *v;
  const PetscInt    m  = kaij->AIJ->rmap->n, *ai=a->i, *aj=a->j, p = kaij->p, q = kaij->q, *diag, *vi;
  const PetscScalar *b, *xb, *idiag;
  PetscScalar       *x, *work, *workt, *w, *y, *arr, *t, *arrt;
  PetscInt          i, j, k, i2, bs, bs2, nz;


  PetscFunctionBegin;
  its = its*lits;
  if (flag & SOR_EISENSTAT) SETERRQ (PETSC_COMM_SELF,PETSC_ERR_SUP,"No support yet for Eisenstat");
  if (its <= 0)             SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Relaxation requires global its %D and local its %D both positive",its,lits);
  if (fshift)               SETERRQ (PETSC_COMM_SELF,PETSC_ERR_SUP,"Sorry, no support for diagonal shift");
  if ((flag & SOR_APPLY_UPPER) || (flag & SOR_APPLY_LOWER)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Sorry, no support for applying upper or lower triangular parts");
  if (p != q) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"MatSOR for KAIJ: Sorry, no support for non-square dense blocks");
  else        {bs = p; bs2 = bs*bs; }

  if (!m) PetscFunctionReturn(0);

  if (!kaij->ibdiagvalid) { ierr = MatInvertBlockDiagonal_SeqKAIJ_N(A,NULL);CHKERRQ(ierr); }
  idiag = kaij->ibdiag;
  diag  = a->diag;

  if (!kaij->sor.setup) {
    ierr = PetscMalloc5(bs,&kaij->sor.w,bs,&kaij->sor.y,m*bs,&kaij->sor.work,m*bs,&kaij->sor.t,m*bs2,&kaij->sor.arr);CHKERRQ(ierr);
    kaij->sor.setup = PETSC_TRUE;
  }
  y     = kaij->sor.y;
  w     = kaij->sor.w;
  work  = kaij->sor.work;
  t     = kaij->sor.t;
  arr   = kaij->sor.arr;

  ierr = VecGetArray(xx,&x);    CHKERRQ(ierr);
  ierr = VecGetArrayRead(bb,&b);CHKERRQ(ierr);

  if (flag & SOR_ZERO_INITIAL_GUESS) {
    if (flag & SOR_FORWARD_SWEEP || flag & SOR_LOCAL_FORWARD_SWEEP) {
      PetscKernel_w_gets_Ar_times_v(bs,bs,b,idiag,x);                            /* x[0:bs] <- D^{-1} b[0:bs] */
      ierr   =  PetscMemcpy(t,b,bs*sizeof(PetscScalar));CHKERRQ(ierr);
      i2     =  bs;
      idiag  += bs2;
      for (i=1; i<m; i++) {
        v  = aa + ai[i];
        vi = aj + ai[i];
        nz = diag[i] - ai[i];

        if (T) {                /* b - T (Arow * x) */
          for (k=0; k<bs; k++) w[k] = 0;
          for (j=0; j<nz; j++) {
            for (k=0; k<bs; k++) w[k] -= v[j] * x[vi[j]*bs+k];
          }
          PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs,w,T,&t[i2]);
          for (k=0; k<bs; k++) t[i2+k] += b[i2+k];
        } else if (kaij->isTI) {
          for (k=0; k<bs; k++) t[i2+k] = b[i2+k];
          for (j=0; j<nz; j++) {
            for (k=0; k<bs; k++) t[i2+k] -= v[j] * x[vi[j]*bs+k];
          }
        } else {
          for (k=0; k<bs; k++) t[i2+k] = b[i2+k];
        }

        PetscKernel_w_gets_Ar_times_v(bs,bs,t+i2,idiag,y);
        for (j=0; j<bs; j++) x[i2+j] = omega * y[j];

        idiag += bs2;
        i2    += bs;
      }
      /* for logging purposes assume number of nonzero in lower half is 1/2 of total */
      ierr = PetscLogFlops(1.0*bs2*a->nz);CHKERRQ(ierr);
      xb = t;
    } else xb = b;
    if (flag & SOR_BACKWARD_SWEEP || flag & SOR_LOCAL_BACKWARD_SWEEP) {
      idiag = kaij->ibdiag+bs2*(m-1);
      i2    = bs * (m-1);
      ierr  = PetscMemcpy(w,xb+i2,bs*sizeof(PetscScalar));CHKERRQ(ierr);
      PetscKernel_w_gets_Ar_times_v(bs,bs,w,idiag,x+i2);
      i2    -= bs;
      idiag -= bs2;
      for (i=m-2; i>=0; i--) {
        v  = aa + diag[i] + 1 ;
        vi = aj + diag[i] + 1;
        nz = ai[i+1] - diag[i] - 1;

        if (T) {                /* FIXME: This branch untested */
          ierr = PetscMemcpy(w,xb+i2,bs*sizeof(PetscScalar));CHKERRQ(ierr);
          /* copy all rows of x that are needed into contiguous space */
          workt = work;
          for (j=0; j<nz; j++) {
            ierr   = PetscMemcpy(workt,x + bs*(*vi++),bs*sizeof(PetscScalar));CHKERRQ(ierr);
            workt += bs;
          }
          arrt = arr;
          for (j=0; j<nz; j++) {
            ierr  = PetscMemcpy(arrt,T,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
            for (k=0; k<bs2; k++) arrt[k] *= v[j];
            arrt += bs2;
          }
          PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
        } else if (kaij->isTI) {
          for (k=0; k<bs; k++) w[k] = t[i2+k];
          for (j=0; j<nz; j++) {
            for (k=0; k<bs; k++) w[k] -= v[j] * x[vi[j]*bs+k];
          }
        }

        PetscKernel_w_gets_Ar_times_v(bs,bs,w,idiag,y); /* RHS incorrect for omega != 1.0 */
        for (j=0; j<bs; j++) x[i2+j] = (1.0-omega) * x[i2+j] + omega * y[j];

        idiag -= bs2;
        i2    -= bs;
      }
      ierr = PetscLogFlops(1.0*bs2*(a->nz));CHKERRQ(ierr);
    }
    its--;
  }
  while (its--) {               /* FIXME: This branch not updated */
    if (flag & SOR_FORWARD_SWEEP || flag & SOR_LOCAL_FORWARD_SWEEP) {
      i2     =  0;
      idiag  = kaij->ibdiag;
      for (i=0; i<m; i++) {
        ierr = PetscMemcpy(w,b+i2,bs*sizeof(PetscScalar));CHKERRQ(ierr);

        v  = aa + ai[i];
        vi = aj + ai[i];
        nz = diag[i] - ai[i];
        workt = work;
        for (j=0; j<nz; j++) {
          ierr   = PetscMemcpy(workt,x + bs*(*vi++),bs*sizeof(PetscScalar));CHKERRQ(ierr);
          workt += bs;
        }
        arrt = arr;
        if (T) {
          for (j=0; j<nz; j++) {
            ierr  = PetscMemcpy(arrt,T,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
            for (k=0; k<bs2; k++) arrt[k] *= v[j];
            arrt += bs2;
          }
          PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
        } else if (kaij->isTI) {
          for (j=0; j<nz; j++) {
            ierr = PetscMemzero(arrt,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
            for (k=0; k<bs; k++) arrt[k+bs*k] = v[j];
            arrt += bs2;
          }
          PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
        }
        ierr = PetscMemcpy(t+i2,w,bs*sizeof(PetscScalar));CHKERRQ(ierr);

        v  = aa + diag[i] + 1;
        vi = aj + diag[i] + 1;
        nz = ai[i+1] - diag[i] - 1;
        workt = work;
        for (j=0; j<nz; j++) {
          ierr   = PetscMemcpy(workt,x + bs*(*vi++),bs*sizeof(PetscScalar));CHKERRQ(ierr);
          workt += bs;
        }
        arrt = arr;
        if (T) {
          for (j=0; j<nz; j++) {
            ierr  = PetscMemcpy(arrt,T,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
            for (k=0; k<bs2; k++) arrt[k] *= v[j];
            arrt += bs2;
          }
          PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
        } else if (kaij->isTI) {
          for (j=0; j<nz; j++) {
            ierr = PetscMemzero(arrt,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
            for (k=0; k<bs; k++) arrt[k+bs*k] = v[j];
            arrt += bs2;
          }
          PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
        }

        PetscKernel_w_gets_Ar_times_v(bs,bs,w,idiag,y);
        for (j=0; j<bs; j++) *(x+i2+j) = (1.0-omega) * *(x+i2+j) + omega * *(y+j);

        idiag += bs2;
        i2    += bs;
      }
      xb = t;
    }
    else xb = b;
    if (flag & SOR_BACKWARD_SWEEP || flag & SOR_LOCAL_BACKWARD_SWEEP) {
      idiag = kaij->ibdiag+bs2*(m-1);
      i2    = bs * (m-1);
      if (xb == b) {
        for (i=m-1; i>=0; i--) {
          ierr = PetscMemcpy(w,b+i2,bs*sizeof(PetscScalar));CHKERRQ(ierr);

          v  = aa + ai[i];
          vi = aj + ai[i];
          nz = diag[i] - ai[i];
          workt = work;
          for (j=0; j<nz; j++) {
            ierr   = PetscMemcpy(workt,x + bs*(*vi++),bs*sizeof(PetscScalar));CHKERRQ(ierr);
            workt += bs;
          }
          arrt = arr;
          if (T) {
            for (j=0; j<nz; j++) {
              ierr  = PetscMemcpy(arrt,T,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
              for (k=0; k<bs2; k++) arrt[k] *= v[j];
              arrt += bs2;
            }
            PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
          } else if (kaij->isTI) {
            for (j=0; j<nz; j++) {
              ierr = PetscMemzero(arrt,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
              for (k=0; k<bs; k++) arrt[k+bs*k] = v[j];
              arrt += bs2;
            }
            PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
          }

          v  = aa + diag[i] + 1;
          vi = aj + diag[i] + 1;
          nz = ai[i+1] - diag[i] - 1;
          workt = work;
          for (j=0; j<nz; j++) {
            ierr   = PetscMemcpy(workt,x + bs*(*vi++),bs*sizeof(PetscScalar));CHKERRQ(ierr);
            workt += bs;
          }
          arrt = arr;
          if (T) {
            for (j=0; j<nz; j++) {
              ierr  = PetscMemcpy(arrt,T,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
              for (k=0; k<bs2; k++) arrt[k] *= v[j];
              arrt += bs2;
            }
            PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
          } else if (kaij->isTI) {
            for (j=0; j<nz; j++) {
              ierr = PetscMemzero(arrt,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
              for (k=0; k<bs; k++) arrt[k+bs*k] = v[j];
              arrt += bs2;
            }
            PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
          }

          PetscKernel_w_gets_Ar_times_v(bs,bs,w,idiag,y);
          for (j=0; j<bs; j++) *(x+i2+j) = (1.0-omega) * *(x+i2+j) + omega * *(y+j);
        }
      } else {
        for (i=m-1; i>=0; i--) {
          ierr = PetscMemcpy(w,xb+i2,bs*sizeof(PetscScalar));CHKERRQ(ierr);
          v  = aa + diag[i] + 1;
          vi = aj + diag[i] + 1;
          nz = ai[i+1] - diag[i] - 1;
          workt = work;
          for (j=0; j<nz; j++) {
            ierr   = PetscMemcpy(workt,x + bs*(*vi++),bs*sizeof(PetscScalar));CHKERRQ(ierr);
            workt += bs;
          }
          arrt = arr;
          if (T) {
            for (j=0; j<nz; j++) {
              ierr  = PetscMemcpy(arrt,T,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
              for (k=0; k<bs2; k++) arrt[k] *= v[j];
              arrt += bs2;
            }
            PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
          } else if (kaij->isTI) {
            for (j=0; j<nz; j++) {
              ierr = PetscMemzero(arrt,bs2*sizeof(PetscScalar));CHKERRQ(ierr);
              for (k=0; k<bs; k++) arrt[k+bs*k] = v[j];
              arrt += bs2;
            }
            PetscKernel_w_gets_w_minus_Ar_times_v(bs,bs*nz,w,arr,work);
          }
          PetscKernel_w_gets_Ar_times_v(bs,bs,w,idiag,y);
          for (j=0; j<bs; j++) *(x+i2+j) = (1.0-omega) * *(x+i2+j) + omega * *(y+j);
        }
        idiag -= bs2;
        i2    -= bs;
      }
      ierr = PetscLogFlops(1.0*bs2*(a->nz));CHKERRQ(ierr);
    }
  }

  ierr = VecRestoreArray(xx,&x);    CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(bb,&b);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*===================================================================================*/

PetscErrorCode KAIJMultAdd_MPI(Mat A,Vec xx,Vec yy,Vec zz)
{
  Mat_MPIKAIJ    *b = (Mat_MPIKAIJ*)A->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!yy) {
    ierr = VecSet(zz,0.0);CHKERRQ(ierr);
  } else {
    ierr = VecCopy(yy,zz);CHKERRQ(ierr);
  }
  /* start the scatter */
  ierr = VecScatterBegin(b->ctx,xx,b->w,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
  ierr = (*b->AIJ->ops->multadd)(b->AIJ,xx,zz,zz);CHKERRQ(ierr);
  ierr = VecScatterEnd(b->ctx,xx,b->w,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
  ierr = (*b->OAIJ->ops->multadd)(b->OAIJ,b->w,zz,zz);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatMult_MPIKAIJ_dof(Mat A,Vec xx,Vec yy)
{
  PetscErrorCode ierr;
  PetscFunctionBegin;
  ierr = KAIJMultAdd_MPI(A,xx,PETSC_NULL,yy);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatMultAdd_MPIKAIJ_dof(Mat A,Vec xx,Vec yy, Vec zz)
{
  PetscErrorCode ierr;
  PetscFunctionBegin;
  ierr = KAIJMultAdd_MPI(A,xx,yy,zz);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode MatInvertBlockDiagonal_MPIKAIJ_dof(Mat A,const PetscScalar **values)
{
  Mat_MPIKAIJ     *b = (Mat_MPIKAIJ*)A->data;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = (*b->AIJ->ops->invertblockdiagonal)(b->AIJ,values);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* ----------------------------------------------------------------*/

PetscErrorCode MatGetRow_SeqKAIJ(Mat A,PetscInt row,PetscInt *ncols,PetscInt **cols,PetscScalar **values)
{
  Mat_SeqKAIJ     *b    = (Mat_SeqKAIJ*) A->data;
  PetscErrorCode  ierr,diag;
  PetscInt        nzaij,nz,*colsaij,*idx,i,j,p=b->p,q=b->q,r=row/p,s=row%p,c;
  PetscScalar     *vaij,*v,*S=b->S,*T=b->T;

  PetscFunctionBegin;
  if (b->getrowactive) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Already active");
  b->getrowactive = PETSC_TRUE;
  if (row < 0 || row >= A->rmap->n) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Row %D out of range",row);

  if ((!S) && (!T) && (!b->isTI)) {
    if (ncols)    *ncols  = 0;
    if (cols)     *cols   = NULL;
    if (values)   *values = NULL;
    PetscFunctionReturn(0);
  }

  if (T || b->isTI) {
    ierr  = MatGetRow_SeqAIJ(b->AIJ,r,&nzaij,&colsaij,&vaij);CHKERRQ(ierr);
    diag  = PETSC_FALSE;
    c     = nzaij;
    for (i=0; i<nzaij; i++) {
      /* check if this row contains a diagonal entry */
      if (colsaij[i] == r) {
        diag = PETSC_TRUE;
        c = i;
      }
    }
  } else nzaij = c = 0;

  /* calculate size of row */
  nz = 0;
  if (S)            nz += q;
  if (T || b->isTI) nz += (diag && S ? (nzaij-1)*q : nzaij*q);

  if (cols || values) {
    ierr = PetscMalloc2(nz,&idx,nz,&v);CHKERRQ(ierr);
    if (b->isTI) {
      for (i=0; i<nzaij; i++) {
        for (j=0; j<q; j++) {
          idx[i*q+j] = colsaij[i]*q+j;
          v[i*q+j]   = (j==s ? vaij[i] : 0);
        }
      }
    } else if (T) {
      for (i=0; i<nzaij; i++) {
        for (j=0; j<q; j++) {
          idx[i*q+j] = colsaij[i]*q+j;
          v[i*q+j]   = vaij[i]*T[s+j*p];
        }
      }
    }
    if (S) {
      for (j=0; j<q; j++) {
        idx[c*q+j] = r*q+j;
        v[c*q+j]  += S[s+j*p];
      }
    }
  }

  if (ncols)    *ncols  = nz;
  if (cols)     *cols   = idx;
  if (values)   *values = v;

  PetscFunctionReturn(0);
}

PetscErrorCode MatRestoreRow_SeqKAIJ(Mat A,PetscInt row,PetscInt *nz,PetscInt **idx,PetscScalar **v)
{
  PetscErrorCode ierr;
  PetscFunctionBegin;
  ierr = PetscFree2(*idx,*v);CHKERRQ(ierr);
  ((Mat_SeqKAIJ*)A->data)->getrowactive = PETSC_FALSE;
  PetscFunctionReturn(0);
}

PetscErrorCode MatGetRow_MPIKAIJ(Mat A,PetscInt row,PetscInt *ncols,PetscInt **cols,PetscScalar **values)
{
  Mat_MPIKAIJ     *b      = (Mat_MPIKAIJ*) A->data;
  Mat             MatAIJ  = ((Mat_SeqKAIJ*)b->AIJ->data)->AIJ;
  Mat             MatOAIJ = ((Mat_SeqKAIJ*)b->OAIJ->data)->AIJ;
  Mat             AIJ     = b->A;
  PetscErrorCode  ierr;
  const PetscInt  rstart=A->rmap->rstart,rend=A->rmap->rend,p=b->p,q=b->q,*garray;
  PetscInt        nz,*idx,ncolsaij,ncolsoaij,*colsaij,*colsoaij,r,s,c,i,j,lrow;
  PetscScalar     *v,*vals,*ovals,*S=b->S,*T=b->T;
  PetscBool       diag;

  PetscFunctionBegin;
  if (b->getrowactive) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Already active");
  b->getrowactive = PETSC_TRUE;
  if (row < rstart || row >= rend) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Only local rows");
  lrow = row - rstart;

  if ((!S) && (!T) && (!b->isTI)) {
    if (ncols)    *ncols  = 0;
    if (cols)     *cols   = NULL;
    if (values)   *values = NULL;
    PetscFunctionReturn(0);
  }

  r = lrow/p;
  s = lrow%p;

  if (T || b->isTI) {
    ierr = MatMPIAIJGetSeqAIJ(AIJ,NULL,NULL,&garray);
    ierr = MatGetRow_SeqAIJ(MatAIJ,lrow/p,&ncolsaij,&colsaij,&vals);CHKERRQ(ierr);
    ierr = MatGetRow_SeqAIJ(MatOAIJ,lrow/p,&ncolsoaij,&colsoaij,&ovals);CHKERRQ(ierr);

    diag  = PETSC_FALSE;
    c     = ncolsaij + ncolsoaij;
    for (i=0; i<ncolsaij; i++) {
      /* check if this row contains a diagonal entry */
      if (colsaij[i] == r) {
        diag = PETSC_TRUE;
        c = i;
      }
    }
  } else c = 0;

  /* calculate size of row */
  nz = 0;
  if (S)            nz += q;
  if (T || b->isTI) nz += (diag && S ? (ncolsaij+ncolsoaij-1)*q : (ncolsaij+ncolsoaij)*q);

  if (cols || values) {
    ierr = PetscMalloc2(nz,&idx,nz,&v);CHKERRQ(ierr);
    if (b->isTI) {
      for (i=0; i<ncolsaij; i++) {
        for (j=0; j<q; j++) {
          idx[i*q+j] = (colsaij[i]+rstart/p)*q+j;
          v[i*q+j]   = (j==s ? vals[i] : 0.0);
        }
      }
      for (i=0; i<ncolsoaij; i++) {
        for (j=0; j<q; j++) {
          idx[(i+ncolsaij)*q+j] = garray[colsoaij[i]]*q+j;
          v[(i+ncolsaij)*q+j]   = (j==s ? ovals[i]: 0.0);
        }
      }
    } else if (T) {
      for (i=0; i<ncolsaij; i++) {
        for (j=0; j<q; j++) {
          idx[i*q+j] = (colsaij[i]+rstart/p)*q+j;
          v[i*q+j]   = vals[i]*T[s+j*p];
        }
      }
      for (i=0; i<ncolsoaij; i++) {
        for (j=0; j<q; j++) {
          idx[(i+ncolsaij)*q+j] = garray[colsoaij[i]]*q+j;
          v[(i+ncolsaij)*q+j]   = ovals[i]*T[s+j*p];
        }
      }
    }
    if (S) {
      for (j=0; j<q; j++) {
        idx[c*q+j] = (r+rstart/p)*q+j;
        v[c*q+j]  += S[s+j*p];
      }
    }
  }

  if (ncols)  *ncols  = nz;
  if (cols)   *cols   = idx;
  if (values) *values = v;
  PetscFunctionReturn(0);
}

PetscErrorCode MatRestoreRow_MPIKAIJ(Mat A,PetscInt row,PetscInt *nz,PetscInt **idx,PetscScalar **v)
{
  PetscErrorCode ierr;
  PetscFunctionBegin;
  ierr = PetscFree2(*idx,*v);CHKERRQ(ierr);
  ((Mat_SeqKAIJ*)A->data)->getrowactive = PETSC_FALSE;
  PetscFunctionReturn(0);
}

PetscErrorCode  MatCreateSubMatrix_KAIJ(Mat mat,IS isrow,IS iscol,MatReuse cll,Mat *newmat)
{
  PetscErrorCode ierr;
  Mat            A;

  PetscFunctionBegin;
  ierr = MatConvert(mat,MATAIJ,MAT_INITIAL_MATRIX,&A);CHKERRQ(ierr);
  ierr = MatCreateSubMatrix(A,isrow,iscol,cll,newmat);CHKERRQ(ierr);
  ierr = MatDestroy(&A);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* ---------------------------------------------------------------------------------- */
/*@C
  MatCreateKAIJ - Creates a matrix type to be used for matrices of the following form:

    [I \otimes S + A \otimes T]

  where
    S is a dense (p \times q) matrix
    T is a dense (p \times q) matrix
    A is an AIJ  (n \times n) matrix
    I is the identity matrix
  The resulting matrix is (np \times nq)

  The matrix type is based on MATSEQAIJ for a sequential matrix A, and MATMPIAIJ for a distributed matrix A.
  S is always stored independently on all processes as a PetscScalar array in column-major format.

  Collective

  Input Parameters:
+ A - the AIJ matrix
. S - the S matrix, stored as a PetscScalar array (column-major)
. T - the T matrix, stored as a PetscScalar array (column-major)
. p - number of rows in S and T
- q - number of columns in S and T

  Output Parameter:
. kaij - the new KAIJ matrix

  Operations provided:
+ MatMult
. MatMultAdd
. MatInvertBlockDiagonal
- MatView

  Level: advanced

.seealso: MatKAIJGetAIJ(), MATKAIJ
@*/
PetscErrorCode  MatCreateKAIJ(Mat A,PetscInt p,PetscInt q,const PetscScalar S[],const PetscScalar T[],Mat *kaij)
{
  PetscErrorCode ierr;
  PetscMPIInt    size;
  PetscInt       n,i,j;
  Mat            B;
  PetscBool      isTI = PETSC_FALSE;

  PetscFunctionBegin;

  /* check if T is an identity matrix */
  if (T && (p == q)) {
    isTI = PETSC_TRUE;
    for (i=0; i<p; i++) {
      for (j=0; j<q; j++) {
        if (i == j) {
          /* diagonal term must be 1 */
          if (T[i+j*p] != 1.0) isTI = PETSC_FALSE;
        } else {
          /* off-diagonal term must be 0 */
          if (T[i+j*p] != 0.0) isTI = PETSC_FALSE;
        }
      }
    }
  }

  ierr = PetscObjectReference((PetscObject)A);CHKERRQ(ierr);

  ierr = MatCreate(PetscObjectComm((PetscObject)A),&B);CHKERRQ(ierr);
  ierr = MatSetSizes(B,p*A->rmap->n,q*A->cmap->n,p*A->rmap->N,q*A->cmap->N);CHKERRQ(ierr);
  ierr = PetscLayoutSetBlockSize(B->rmap,p);CHKERRQ(ierr);
  ierr = PetscLayoutSetBlockSize(B->cmap,q);CHKERRQ(ierr);
  ierr = PetscLayoutSetUp(B->rmap);CHKERRQ(ierr);
  ierr = PetscLayoutSetUp(B->cmap);CHKERRQ(ierr);

  B->assembled = PETSC_TRUE;
  ierr = MPI_Comm_size(PetscObjectComm((PetscObject)A),&size);CHKERRQ(ierr);

  if (size == 1) {
    Mat_SeqKAIJ *b;

    ierr = MatSetType(B,MATSEQKAIJ);CHKERRQ(ierr);

    B->ops->setup   = NULL;
    B->ops->destroy = MatDestroy_SeqKAIJ;
    B->ops->view    = MatView_SeqKAIJ;
    b               = (Mat_SeqKAIJ*)B->data;
    b->p            = p;
    b->q            = q;
    b->AIJ          = A;
    b->isTI         = isTI;
    if (S) {
      ierr = PetscMalloc(p*q*sizeof(PetscScalar),&b->S);CHKERRQ(ierr);
      ierr = PetscMemcpy(b->S,S,p*q*sizeof(PetscScalar));CHKERRQ(ierr);
    } else  b->S = NULL;
    if (T && (!isTI)) {
      ierr = PetscMalloc(p*q*sizeof(PetscScalar),&b->T);CHKERRQ(ierr);
      ierr = PetscMemcpy(b->T,T,p*q*sizeof(PetscScalar));CHKERRQ(ierr);
    } else b->T = NULL;

    B->ops->mult                = MatMult_SeqKAIJ_N;
    B->ops->multadd             = MatMultAdd_SeqKAIJ_N;
    B->ops->invertblockdiagonal = MatInvertBlockDiagonal_SeqKAIJ_N;
    B->ops->getrow              = MatGetRow_SeqKAIJ;
    B->ops->restorerow          = MatRestoreRow_SeqKAIJ;
    B->ops->sor                 = MatSOR_SeqKAIJ;
  } else {
    Mat_MPIAIJ  *mpiaij = (Mat_MPIAIJ*)A->data;
    Mat_MPIKAIJ *b;
    IS          from,to;
    Vec         gvec;

    ierr = MatSetType(B,MATMPIKAIJ);CHKERRQ(ierr);

    B->ops->setup   = NULL;
    B->ops->destroy = MatDestroy_MPIKAIJ;
    B->ops->view    = MatView_MPIKAIJ;

    b       = (Mat_MPIKAIJ*)B->data;
    b->p    = p;
    b->q    = q;
    b->A    = A;
    b->isTI = isTI;
    if (S) {
      ierr = PetscMalloc(p*q*sizeof(PetscScalar),&b->S);CHKERRQ(ierr);
      ierr = PetscMemcpy(b->S,S,p*q*sizeof(PetscScalar));CHKERRQ(ierr);
    } else  b->S = NULL;
    if (T &&(!isTI)) {
      ierr = PetscMalloc(p*q*sizeof(PetscScalar),&b->T);CHKERRQ(ierr);
      ierr = PetscMemcpy(b->T,T,p*q*sizeof(PetscScalar));CHKERRQ(ierr);
    } else b->T = NULL;

    ierr = MatCreateKAIJ(mpiaij->A,p,q,S   ,T,&b->AIJ);CHKERRQ(ierr);
    ierr = MatCreateKAIJ(mpiaij->B,p,q,NULL,T,&b->OAIJ);CHKERRQ(ierr);

    ierr = VecGetSize(mpiaij->lvec,&n);CHKERRQ(ierr);
    ierr = VecCreate(PETSC_COMM_SELF,&b->w);CHKERRQ(ierr);
    ierr = VecSetSizes(b->w,n*q,n*q);CHKERRQ(ierr);
    ierr = VecSetBlockSize(b->w,q);CHKERRQ(ierr);
    ierr = VecSetType(b->w,VECSEQ);CHKERRQ(ierr);

    /* create two temporary Index sets for build scatter gather */
    ierr = ISCreateBlock(PetscObjectComm((PetscObject)A),q,n,mpiaij->garray,PETSC_COPY_VALUES,&from);CHKERRQ(ierr);
    ierr = ISCreateStride(PETSC_COMM_SELF,n*q,0,1,&to);CHKERRQ(ierr);

    /* create temporary global vector to generate scatter context */
    ierr = VecCreateMPIWithArray(PetscObjectComm((PetscObject)A),q,q*A->cmap->n,q*A->cmap->N,NULL,&gvec);CHKERRQ(ierr);

    /* generate the scatter context */
    ierr = VecScatterCreateWithData(gvec,from,b->w,to,&b->ctx);CHKERRQ(ierr);

    ierr = ISDestroy(&from);CHKERRQ(ierr);
    ierr = ISDestroy(&to);CHKERRQ(ierr);
    ierr = VecDestroy(&gvec);CHKERRQ(ierr);

    B->ops->mult                = MatMult_MPIKAIJ_dof;
    B->ops->multadd             = MatMultAdd_MPIKAIJ_dof;
    B->ops->invertblockdiagonal = MatInvertBlockDiagonal_MPIKAIJ_dof;
    B->ops->getrow              = MatGetRow_MPIKAIJ;
    B->ops->restorerow          = MatRestoreRow_MPIKAIJ;
    ierr = PetscObjectComposeFunction((PetscObject)B,"MatGetDiagonalBlock_C",MatGetDiagonalBlock_MPIKAIJ);CHKERRQ(ierr);
  }
  B->ops->createsubmatrix = MatCreateSubMatrix_KAIJ;
  B->assembled = PETSC_TRUE;
  ierr  = MatSetUp(B);CHKERRQ(ierr);
  *kaij = B;
  ierr  = MatViewFromOptions(B,NULL,"-mat_view");CHKERRQ(ierr);

  PetscFunctionReturn(0);
}