Date: 2015-10-02T22:09:43+00:00

Actual source code: sbaijfact2.c
petsc-3.6.2 2015-10-02
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      2: /*
      3:     Factorization code for SBAIJ format.
      4: */

      6: #include <../src/mat/impls/sbaij/seq/sbaij.h>
      7: #include <../src/mat/impls/baij/seq/baij.h>
      8: #include <petsc/private/kernels/blockinvert.h>

     12: PetscErrorCode MatSolve_SeqSBAIJ_N_inplace(Mat A,Vec bb,Vec xx)
     13: {
     14:   Mat_SeqSBAIJ      *a   =(Mat_SeqSBAIJ*)A->data;
     15:   IS                isrow=a->row;
     16:   PetscInt          mbs  =a->mbs,*ai=a->i,*aj=a->j;
     17:   PetscErrorCode    ierr;
     18:   const PetscInt    *r;
     19:   PetscInt          nz,*vj,k,idx,k1;
     20:   PetscInt          bs =A->rmap->bs,bs2 = a->bs2;
     21:   const MatScalar   *aa=a->a,*v,*diag;
     22:   PetscScalar       *x,*xk,*xj,*xk_tmp,*t;
     23:   const PetscScalar *b;

     26:   VecGetArrayRead(bb,&b);
     27:   VecGetArray(xx,&x);
     28:   t    = a->solve_work;
     29:   ISGetIndices(isrow,&r);
     30:   PetscMalloc1(bs,&xk_tmp);

     32:   /* solve U^T * D * y = b by forward substitution */
     33:   xk = t;
     34:   for (k=0; k<mbs; k++) { /* t <- perm(b) */
     35:     idx = bs*r[k];
     36:     for (k1=0; k1<bs; k1++) *xk++ = b[idx+k1];
     37:   }
     38:   for (k=0; k<mbs; k++) {
     39:     v    = aa + bs2*ai[k];
     40:     xk   = t + k*bs;    /* Dk*xk = k-th block of x */
     41:     PetscMemcpy(xk_tmp,xk,bs*sizeof(PetscScalar)); /* xk_tmp <- xk */
     42:     nz   = ai[k+1] - ai[k];
     43:     vj   = aj + ai[k];
     44:     xj   = t + (*vj)*bs; /* *vj-th block of x, *vj>k */
     45:     while (nz--) {
     46:       /* x(:) += U(k,:)^T*(Dk*xk) */
     47:       PetscKernel_v_gets_v_plus_Atranspose_times_w(bs,xj,v,xk_tmp); /* xj <- xj + v^t * xk */
     48:       vj++; xj = t + (*vj)*bs;
     49:       v       += bs2;
     50:     }
     51:     /* xk = inv(Dk)*(Dk*xk) */
     52:     diag = aa+k*bs2;                            /* ptr to inv(Dk) */
     53:     PetscKernel_w_gets_A_times_v(bs,xk_tmp,diag,xk); /* xk <- diag * xk */
     54:   }

     56:   /* solve U*x = y by back substitution */
     57:   for (k=mbs-1; k>=0; k--) {
     58:     v  = aa + bs2*ai[k];
     59:     xk = t + k*bs;        /* xk */
     60:     nz = ai[k+1] - ai[k];
     61:     vj = aj + ai[k];
     62:     xj = t + (*vj)*bs;
     63:     while (nz--) {
     64:       /* xk += U(k,:)*x(:) */
     65:       PetscKernel_v_gets_v_plus_A_times_w(bs,xk,v,xj); /* xk <- xk + v*xj */
     66:       vj++;
     67:       v += bs2; xj = t + (*vj)*bs;
     68:     }
     69:     idx = bs*r[k];
     70:     for (k1=0; k1<bs; k1++) x[idx+k1] = *xk++;
     71:   }

     73:   PetscFree(xk_tmp);
     74:   ISRestoreIndices(isrow,&r);
     75:   VecRestoreArrayRead(bb,&b);
     76:   VecRestoreArray(xx,&x);
     77:   PetscLogFlops(4.0*bs2*a->nz -(bs+2.0*bs2)*mbs);
     78:   return(0);
     79: }

     83: PetscErrorCode MatForwardSolve_SeqSBAIJ_N_inplace(Mat A,Vec bb,Vec xx)
     84: {
     86:   SETERRQ(PETSC_COMM_SELF,1,"not implemented yet");
     87:   return(0);
     88: }

     92: PetscErrorCode MatBackwardSolve_SeqSBAIJ_N_inplace(Mat A,Vec bb,Vec xx)
     93: {
     95:   SETERRQ(PETSC_COMM_SELF,1,"not implemented yet");
     96:   return(0);
     97: }

    101: PetscErrorCode MatForwardSolve_SeqSBAIJ_N_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscInt bs,PetscScalar *x)
    102: {
    103:   PetscErrorCode  ierr;
    104:   PetscInt        nz,k;
    105:   const PetscInt  *vj,bs2 = bs*bs;
    106:   const MatScalar *v,*diag;
    107:   PetscScalar     *xk,*xj,*xk_tmp;

    110:   PetscMalloc1(bs,&xk_tmp);
    111:   for (k=0; k<mbs; k++) {
    112:     v    = aa + bs2*ai[k];
    113:     xk   = x + k*bs;    /* Dk*xk = k-th block of x */
    114:     PetscMemcpy(xk_tmp,xk,bs*sizeof(PetscScalar)); /* xk_tmp <- xk */
    115:     nz   = ai[k+1] - ai[k];
    116:     vj   = aj + ai[k];
    117:     xj   = x + (*vj)*bs; /* *vj-th block of x, *vj>k */
    118:     while (nz--) {
    119:       /* x(:) += U(k,:)^T*(Dk*xk) */
    120:       PetscKernel_v_gets_v_plus_Atranspose_times_w(bs,xj,v,xk_tmp); /* xj <- xj + v^t * xk */
    121:       vj++; xj = x + (*vj)*bs;
    122:       v       += bs2;
    123:     }
    124:     /* xk = inv(Dk)*(Dk*xk) */
    125:     diag = aa+k*bs2;                            /* ptr to inv(Dk) */
    126:     PetscKernel_w_gets_A_times_v(bs,xk_tmp,diag,xk); /* xk <- diag * xk */
    127:   }
    128:   PetscFree(xk_tmp);
    129:   return(0);
    130: }

    134: PetscErrorCode MatBackwardSolve_SeqSBAIJ_N_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscInt bs,PetscScalar *x)
    135: {
    136:   PetscInt        nz,k;
    137:   const PetscInt  *vj,bs2 = bs*bs;
    138:   const MatScalar *v;
    139:   PetscScalar     *xk,*xj;

    142:   for (k=mbs-1; k>=0; k--) {
    143:     v  = aa + bs2*ai[k];
    144:     xk = x + k*bs;        /* xk */
    145:     nz = ai[k+1] - ai[k];
    146:     vj = aj + ai[k];
    147:     xj = x + (*vj)*bs;
    148:     while (nz--) {
    149:       /* xk += U(k,:)*x(:) */
    150:       PetscKernel_v_gets_v_plus_A_times_w(bs,xk,v,xj); /* xk <- xk + v*xj */
    151:       vj++;
    152:       v += bs2; xj = x + (*vj)*bs;
    153:     }
    154:   }
    155:   return(0);
    156: }

    160: PetscErrorCode MatSolve_SeqSBAIJ_N_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    161: {
    162:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    163:   PetscErrorCode    ierr;
    164:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    165:   PetscInt          bs =A->rmap->bs;
    166:   const MatScalar   *aa=a->a;
    167:   PetscScalar       *x;
    168:   const PetscScalar *b;

    171:   VecGetArrayRead(bb,&b);
    172:   VecGetArray(xx,&x);

    174:   /* solve U^T * D * y = b by forward substitution */
    175:   PetscMemcpy(x,b,bs*mbs*sizeof(PetscScalar)); /* x <- b */
    176:   MatForwardSolve_SeqSBAIJ_N_NaturalOrdering(ai,aj,aa,mbs,bs,x);

    178:   /* solve U*x = y by back substitution */
    179:   MatBackwardSolve_SeqSBAIJ_N_NaturalOrdering(ai,aj,aa,mbs,bs,x);

    181:   VecRestoreArrayRead(bb,&b);
    182:   VecRestoreArray(xx,&x);
    183:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    184:   return(0);
    185: }

    189: PetscErrorCode MatForwardSolve_SeqSBAIJ_N_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    190: {
    191:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    192:   PetscErrorCode    ierr;
    193:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    194:   PetscInt          bs =A->rmap->bs;
    195:   const MatScalar   *aa=a->a;
    196:   const PetscScalar *b;
    197:   PetscScalar       *x;

    200:   VecGetArrayRead(bb,&b);
    201:   VecGetArray(xx,&x);
    202:   PetscMemcpy(x,b,bs*mbs*sizeof(PetscScalar)); /* x <- b */
    203:   MatForwardSolve_SeqSBAIJ_N_NaturalOrdering(ai,aj,aa,mbs,bs,x);
    204:   VecRestoreArrayRead(bb,&b);
    205:   VecRestoreArray(xx,&x);
    206:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    207:   return(0);
    208: }

    212: PetscErrorCode MatBackwardSolve_SeqSBAIJ_N_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    213: {
    214:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    215:   PetscErrorCode    ierr;
    216:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    217:   PetscInt          bs =A->rmap->bs;
    218:   const MatScalar   *aa=a->a;
    219:   const PetscScalar *b;
    220:   PetscScalar       *x;

    223:   VecGetArrayRead(bb,&b);
    224:   VecGetArray(xx,&x);
    225:   PetscMemcpy(x,b,bs*mbs*sizeof(PetscScalar));
    226:   MatBackwardSolve_SeqSBAIJ_N_NaturalOrdering(ai,aj,aa,mbs,bs,x);
    227:   VecRestoreArrayRead(bb,&b);
    228:   VecRestoreArray(xx,&x);
    229:   PetscLogFlops(2.0*bs2*(a->nz-mbs));
    230:   return(0);
    231: }

    235: PetscErrorCode MatSolve_SeqSBAIJ_7_inplace(Mat A,Vec bb,Vec xx)
    236: {
    237:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    238:   IS                isrow=a->row;
    239:   const PetscInt    mbs  =a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2,bs=A->rmap->bs,*r,*vj;
    240:   PetscErrorCode    ierr;
    241:   PetscInt          nz,k,idx;
    242:   const MatScalar   *aa=a->a,*v,*d;
    243:   PetscScalar       *x,x0,x1,x2,x3,x4,x5,x6,*t,*tp;
    244:   const PetscScalar *b;

    247:   VecGetArrayRead(bb,&b);
    248:   VecGetArray(xx,&x);
    249:   t    = a->solve_work;
    250:   ISGetIndices(isrow,&r);

    252:   /* solve U^T * D * y = b by forward substitution */
    253:   tp = t;
    254:   for (k=0; k<mbs; k++) { /* t <- perm(b) */
    255:     idx   = 7*r[k];
    256:     tp[0] = b[idx];
    257:     tp[1] = b[idx+1];
    258:     tp[2] = b[idx+2];
    259:     tp[3] = b[idx+3];
    260:     tp[4] = b[idx+4];
    261:     tp[5] = b[idx+5];
    262:     tp[6] = b[idx+6];
    263:     tp   += 7;
    264:   }

    266:   for (k=0; k<mbs; k++) {
    267:     v  = aa + 49*ai[k];
    268:     vj = aj + ai[k];
    269:     tp = t + k*7;
    270:     x0 =tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; x4=tp[4]; x5=tp[5]; x6=tp[6];
    271:     nz = ai[k+1] - ai[k];
    272:     tp = t + (*vj)*7;
    273:     while (nz--) {
    274:       tp[0]+=  v[0]*x0 +  v[1]*x1 +  v[2]*x2 + v[3]*x3 + v[4]*x4 + v[5]*x5 + v[6]*x6;
    275:       tp[1]+=  v[7]*x0 +  v[8]*x1 +  v[9]*x2+ v[10]*x3+ v[11]*x4+ v[12]*x5+ v[13]*x6;
    276:       tp[2]+= v[14]*x0 + v[15]*x1 + v[16]*x2+ v[17]*x3+ v[18]*x4+ v[19]*x5+ v[20]*x6;
    277:       tp[3]+= v[21]*x0 + v[22]*x1 + v[23]*x2+ v[24]*x3+ v[25]*x4+ v[26]*x5+ v[27]*x6;
    278:       tp[4]+= v[28]*x0 + v[29]*x1 + v[30]*x2+ v[31]*x3+ v[32]*x4+ v[33]*x5+ v[34]*x6;
    279:       tp[5]+= v[35]*x0 + v[36]*x1 + v[37]*x2+ v[38]*x3+ v[39]*x4+ v[40]*x5+ v[41]*x6;
    280:       tp[6]+= v[42]*x0 + v[43]*x1 + v[44]*x2+ v[45]*x3+ v[46]*x4+ v[47]*x5+ v[48]*x6;
    281:       vj++;
    282:       tp = t + (*vj)*7;
    283:       v += 49;
    284:     }

    286:     /* xk = inv(Dk)*(Dk*xk) */
    287:     d     = aa+k*49;       /* ptr to inv(Dk) */
    288:     tp    = t + k*7;
    289:     tp[0] = d[0]*x0 + d[7]*x1 + d[14]*x2 + d[21]*x3 + d[28]*x4 + d[35]*x5 + d[42]*x6;
    290:     tp[1] = d[1]*x0 + d[8]*x1 + d[15]*x2 + d[22]*x3 + d[29]*x4 + d[36]*x5 + d[43]*x6;
    291:     tp[2] = d[2]*x0 + d[9]*x1 + d[16]*x2 + d[23]*x3 + d[30]*x4 + d[37]*x5 + d[44]*x6;
    292:     tp[3] = d[3]*x0+ d[10]*x1 + d[17]*x2 + d[24]*x3 + d[31]*x4 + d[38]*x5 + d[45]*x6;
    293:     tp[4] = d[4]*x0+ d[11]*x1 + d[18]*x2 + d[25]*x3 + d[32]*x4 + d[39]*x5 + d[46]*x6;
    294:     tp[5] = d[5]*x0+ d[12]*x1 + d[19]*x2 + d[26]*x3 + d[33]*x4 + d[40]*x5 + d[47]*x6;
    295:     tp[6] = d[6]*x0+ d[13]*x1 + d[20]*x2 + d[27]*x3 + d[34]*x4 + d[41]*x5 + d[48]*x6;
    296:   }

    298:   /* solve U*x = y by back substitution */
    299:   for (k=mbs-1; k>=0; k--) {
    300:     v  = aa + 49*ai[k];
    301:     vj = aj + ai[k];
    302:     tp = t + k*7;
    303:     x0 = tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; x4=tp[4]; x5=tp[5];  x6=tp[6]; /* xk */
    304:     nz = ai[k+1] - ai[k];

    306:     tp = t + (*vj)*7;
    307:     while (nz--) {
    308:       /* xk += U(k,:)*x(:) */
    309:       x0 += v[0]*tp[0] + v[7]*tp[1] + v[14]*tp[2] + v[21]*tp[3] + v[28]*tp[4] + v[35]*tp[5] + v[42]*tp[6];
    310:       x1 += v[1]*tp[0] + v[8]*tp[1] + v[15]*tp[2] + v[22]*tp[3] + v[29]*tp[4] + v[36]*tp[5] + v[43]*tp[6];
    311:       x2 += v[2]*tp[0] + v[9]*tp[1] + v[16]*tp[2] + v[23]*tp[3] + v[30]*tp[4] + v[37]*tp[5] + v[44]*tp[6];
    312:       x3 += v[3]*tp[0]+ v[10]*tp[1] + v[17]*tp[2] + v[24]*tp[3] + v[31]*tp[4] + v[38]*tp[5] + v[45]*tp[6];
    313:       x4 += v[4]*tp[0]+ v[11]*tp[1] + v[18]*tp[2] + v[25]*tp[3] + v[32]*tp[4] + v[39]*tp[5] + v[46]*tp[6];
    314:       x5 += v[5]*tp[0]+ v[12]*tp[1] + v[19]*tp[2] + v[26]*tp[3] + v[33]*tp[4] + v[40]*tp[5] + v[47]*tp[6];
    315:       x6 += v[6]*tp[0]+ v[13]*tp[1] + v[20]*tp[2] + v[27]*tp[3] + v[34]*tp[4] + v[41]*tp[5] + v[48]*tp[6];
    316:       vj++;
    317:       tp = t + (*vj)*7;
    318:       v += 49;
    319:     }
    320:     tp       = t + k*7;
    321:     tp[0]    = x0; tp[1]=x1; tp[2]=x2; tp[3]=x3; tp[4]=x4; tp[5]=x5; tp[6]=x6;
    322:     idx      = 7*r[k];
    323:     x[idx]   = x0;
    324:     x[idx+1] = x1;
    325:     x[idx+2] = x2;
    326:     x[idx+3] = x3;
    327:     x[idx+4] = x4;
    328:     x[idx+5] = x5;
    329:     x[idx+6] = x6;
    330:   }

    332:   ISRestoreIndices(isrow,&r);
    333:   VecRestoreArrayRead(bb,&b);
    334:   VecRestoreArray(xx,&x);
    335:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    336:   return(0);
    337: }

    341: PetscErrorCode MatForwardSolve_SeqSBAIJ_7_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    342: {
    343:   const MatScalar *v,*d;
    344:   PetscScalar     *xp,x0,x1,x2,x3,x4,x5,x6;
    345:   PetscInt        nz,k;
    346:   const PetscInt  *vj;

    349:   for (k=0; k<mbs; k++) {
    350:     v  = aa + 49*ai[k];
    351:     xp = x + k*7;
    352:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; x4=xp[4]; x5=xp[5]; x6=xp[6]; /* Dk*xk = k-th block of x */
    353:     nz = ai[k+1] - ai[k];
    354:     vj = aj + ai[k];
    355:     xp = x + (*vj)*7;
    356:     PetscPrefetchBlock(vj+nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    357:     PetscPrefetchBlock(v+49*nz,49*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    358:     while (nz--) {
    359:       /* x(:) += U(k,:)^T*(Dk*xk) */
    360:       xp[0]+=  v[0]*x0 +  v[1]*x1 +  v[2]*x2 + v[3]*x3 + v[4]*x4 + v[5]*x5 + v[6]*x6;
    361:       xp[1]+=  v[7]*x0 +  v[8]*x1 +  v[9]*x2+ v[10]*x3+ v[11]*x4+ v[12]*x5+ v[13]*x6;
    362:       xp[2]+= v[14]*x0 + v[15]*x1 + v[16]*x2+ v[17]*x3+ v[18]*x4+ v[19]*x5+ v[20]*x6;
    363:       xp[3]+= v[21]*x0 + v[22]*x1 + v[23]*x2+ v[24]*x3+ v[25]*x4+ v[26]*x5+ v[27]*x6;
    364:       xp[4]+= v[28]*x0 + v[29]*x1 + v[30]*x2+ v[31]*x3+ v[32]*x4+ v[33]*x5+ v[34]*x6;
    365:       xp[5]+= v[35]*x0 + v[36]*x1 + v[37]*x2+ v[38]*x3+ v[39]*x4+ v[40]*x5+ v[41]*x6;
    366:       xp[6]+= v[42]*x0 + v[43]*x1 + v[44]*x2+ v[45]*x3+ v[46]*x4+ v[47]*x5+ v[48]*x6;
    367:       vj++;
    368:       xp = x + (*vj)*7;
    369:       v += 49;
    370:     }
    371:     /* xk = inv(Dk)*(Dk*xk) */
    372:     d     = aa+k*49;       /* ptr to inv(Dk) */
    373:     xp    = x + k*7;
    374:     xp[0] = d[0]*x0 + d[7]*x1 + d[14]*x2 + d[21]*x3 + d[28]*x4 + d[35]*x5 + d[42]*x6;
    375:     xp[1] = d[1]*x0 + d[8]*x1 + d[15]*x2 + d[22]*x3 + d[29]*x4 + d[36]*x5 + d[43]*x6;
    376:     xp[2] = d[2]*x0 + d[9]*x1 + d[16]*x2 + d[23]*x3 + d[30]*x4 + d[37]*x5 + d[44]*x6;
    377:     xp[3] = d[3]*x0+ d[10]*x1 + d[17]*x2 + d[24]*x3 + d[31]*x4 + d[38]*x5 + d[45]*x6;
    378:     xp[4] = d[4]*x0+ d[11]*x1 + d[18]*x2 + d[25]*x3 + d[32]*x4 + d[39]*x5 + d[46]*x6;
    379:     xp[5] = d[5]*x0+ d[12]*x1 + d[19]*x2 + d[26]*x3 + d[33]*x4 + d[40]*x5 + d[47]*x6;
    380:     xp[6] = d[6]*x0+ d[13]*x1 + d[20]*x2 + d[27]*x3 + d[34]*x4 + d[41]*x5 + d[48]*x6;
    381:   }
    382:   return(0);
    383: }

    387: PetscErrorCode MatBackwardSolve_SeqSBAIJ_7_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    388: {
    389:   const MatScalar *v;
    390:   PetscScalar     *xp,x0,x1,x2,x3,x4,x5,x6;
    391:   PetscInt        nz,k;
    392:   const PetscInt  *vj;

    395:   for (k=mbs-1; k>=0; k--) {
    396:     v  = aa + 49*ai[k];
    397:     xp = x + k*7;
    398:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; x4=xp[4]; x5=xp[5]; x6=xp[6]; /* xk */
    399:     nz = ai[k+1] - ai[k];
    400:     vj = aj + ai[k];
    401:     xp = x + (*vj)*7;
    402:     PetscPrefetchBlock(vj-nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    403:     PetscPrefetchBlock(v-49*nz,49*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    404:     while (nz--) {
    405:       /* xk += U(k,:)*x(:) */
    406:       x0 += v[0]*xp[0] + v[7]*xp[1] + v[14]*xp[2] + v[21]*xp[3] + v[28]*xp[4] + v[35]*xp[5] + v[42]*xp[6];
    407:       x1 += v[1]*xp[0] + v[8]*xp[1] + v[15]*xp[2] + v[22]*xp[3] + v[29]*xp[4] + v[36]*xp[5] + v[43]*xp[6];
    408:       x2 += v[2]*xp[0] + v[9]*xp[1] + v[16]*xp[2] + v[23]*xp[3] + v[30]*xp[4] + v[37]*xp[5] + v[44]*xp[6];
    409:       x3 += v[3]*xp[0]+ v[10]*xp[1] + v[17]*xp[2] + v[24]*xp[3] + v[31]*xp[4] + v[38]*xp[5] + v[45]*xp[6];
    410:       x4 += v[4]*xp[0]+ v[11]*xp[1] + v[18]*xp[2] + v[25]*xp[3] + v[32]*xp[4] + v[39]*xp[5] + v[46]*xp[6];
    411:       x5 += v[5]*xp[0]+ v[12]*xp[1] + v[19]*xp[2] + v[26]*xp[3] + v[33]*xp[4] + v[40]*xp[5] + v[47]*xp[6];
    412:       x6 += v[6]*xp[0]+ v[13]*xp[1] + v[20]*xp[2] + v[27]*xp[3] + v[34]*xp[4] + v[41]*xp[5] + v[48]*xp[6];
    413:       vj++;
    414:       v += 49; xp = x + (*vj)*7;
    415:     }
    416:     xp   = x + k*7;
    417:     xp[0]=x0; xp[1]=x1; xp[2]=x2; xp[3]=x3; xp[4]=x4; xp[5]=x5; xp[6]=x6;
    418:   }
    419:   return(0);
    420: }

    424: PetscErrorCode MatSolve_SeqSBAIJ_7_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    425: {
    426:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    427:   PetscErrorCode    ierr;
    428:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    429:   const MatScalar   *aa=a->a;
    430:   PetscScalar       *x;
    431:   const PetscScalar *b;

    434:   VecGetArrayRead(bb,&b);
    435:   VecGetArray(xx,&x);

    437:   /* solve U^T * D * y = b by forward substitution */
    438:   PetscMemcpy(x,b,7*mbs*sizeof(PetscScalar)); /* x <- b */
    439:   MatForwardSolve_SeqSBAIJ_7_NaturalOrdering(ai,aj,aa,mbs,x);

    441:   /* solve U*x = y by back substitution */
    442:   MatBackwardSolve_SeqSBAIJ_7_NaturalOrdering(ai,aj,aa,mbs,x);

    444:   VecRestoreArrayRead(bb,&b);
    445:   VecRestoreArray(xx,&x);
    446:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    447:   return(0);
    448: }

    452: PetscErrorCode MatForwardSolve_SeqSBAIJ_7_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    453: {
    454:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    455:   PetscErrorCode    ierr;
    456:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    457:   const MatScalar   *aa=a->a;
    458:   PetscScalar       *x;
    459:   const PetscScalar *b;

    462:   VecGetArrayRead(bb,&b);
    463:   VecGetArray(xx,&x);
    464:   PetscMemcpy(x,b,7*mbs*sizeof(PetscScalar));
    465:   MatForwardSolve_SeqSBAIJ_7_NaturalOrdering(ai,aj,aa,mbs,x);
    466:   VecRestoreArrayRead(bb,&b);
    467:   VecRestoreArray(xx,&x);
    468:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    469:   return(0);
    470: }

    474: PetscErrorCode MatBackwardSolve_SeqSBAIJ_7_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    475: {
    476:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    477:   PetscErrorCode    ierr;
    478:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    479:   const MatScalar   *aa=a->a;
    480:   PetscScalar       *x;
    481:   const PetscScalar *b;

    484:   VecGetArrayRead(bb,&b);
    485:   VecGetArray(xx,&x);
    486:   PetscMemcpy(x,b,7*mbs*sizeof(PetscScalar));
    487:   MatBackwardSolve_SeqSBAIJ_7_NaturalOrdering(ai,aj,aa,mbs,x);
    488:   VecRestoreArrayRead(bb,&b);
    489:   VecRestoreArray(xx,&x);
    490:   PetscLogFlops(2.0*bs2*(a->nz-mbs));
    491:   return(0);
    492: }

    496: PetscErrorCode MatSolve_SeqSBAIJ_6_inplace(Mat A,Vec bb,Vec xx)
    497: {
    498:   Mat_SeqSBAIJ      *a   =(Mat_SeqSBAIJ*)A->data;
    499:   IS                isrow=a->row;
    500:   const PetscInt    mbs  =a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2,*r,*vj;
    501:   PetscErrorCode    ierr;
    502:   PetscInt          nz,k,idx;
    503:   const MatScalar   *aa=a->a,*v,*d;
    504:   PetscScalar       *x,x0,x1,x2,x3,x4,x5,*t,*tp;
    505:   const PetscScalar *b;

    508:   VecGetArrayRead(bb,&b);
    509:   VecGetArray(xx,&x);
    510:   t    = a->solve_work;
    511:   ISGetIndices(isrow,&r);

    513:   /* solve U^T * D * y = b by forward substitution */
    514:   tp = t;
    515:   for (k=0; k<mbs; k++) { /* t <- perm(b) */
    516:     idx   = 6*r[k];
    517:     tp[0] = b[idx];
    518:     tp[1] = b[idx+1];
    519:     tp[2] = b[idx+2];
    520:     tp[3] = b[idx+3];
    521:     tp[4] = b[idx+4];
    522:     tp[5] = b[idx+5];
    523:     tp   += 6;
    524:   }

    526:   for (k=0; k<mbs; k++) {
    527:     v  = aa + 36*ai[k];
    528:     vj = aj + ai[k];
    529:     tp = t + k*6;
    530:     x0 =tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; x4=tp[4]; x5=tp[5];
    531:     nz = ai[k+1] - ai[k];
    532:     tp = t + (*vj)*6;
    533:     while (nz--) {
    534:       tp[0] +=  v[0]*x0 +  v[1]*x1 +  v[2]*x2 + v[3]*x3 + v[4]*x4 + v[5]*x5;
    535:       tp[1] +=  v[6]*x0 +  v[7]*x1 +  v[8]*x2 + v[9]*x3+ v[10]*x4+ v[11]*x5;
    536:       tp[2] += v[12]*x0 + v[13]*x1 + v[14]*x2+ v[15]*x3+ v[16]*x4+ v[17]*x5;
    537:       tp[3] += v[18]*x0 + v[19]*x1 + v[20]*x2+ v[21]*x3+ v[22]*x4+ v[23]*x5;
    538:       tp[4] += v[24]*x0 + v[25]*x1 + v[26]*x2+ v[27]*x3+ v[28]*x4+ v[29]*x5;
    539:       tp[5] += v[30]*x0 + v[31]*x1 + v[32]*x2+ v[33]*x3+ v[34]*x4+ v[35]*x5;
    540:       vj++;
    541:       tp = t + (*vj)*6;
    542:       v += 36;
    543:     }

    545:     /* xk = inv(Dk)*(Dk*xk) */
    546:     d     = aa+k*36;       /* ptr to inv(Dk) */
    547:     tp    = t + k*6;
    548:     tp[0] = d[0]*x0 + d[6]*x1 + d[12]*x2 + d[18]*x3 + d[24]*x4 + d[30]*x5;
    549:     tp[1] = d[1]*x0 + d[7]*x1 + d[13]*x2 + d[19]*x3 + d[25]*x4 + d[31]*x5;
    550:     tp[2] = d[2]*x0 + d[8]*x1 + d[14]*x2 + d[20]*x3 + d[26]*x4 + d[32]*x5;
    551:     tp[3] = d[3]*x0 + d[9]*x1 + d[15]*x2 + d[21]*x3 + d[27]*x4 + d[33]*x5;
    552:     tp[4] = d[4]*x0+ d[10]*x1 + d[16]*x2 + d[22]*x3 + d[28]*x4 + d[34]*x5;
    553:     tp[5] = d[5]*x0+ d[11]*x1 + d[17]*x2 + d[23]*x3 + d[29]*x4 + d[35]*x5;
    554:   }

    556:   /* solve U*x = y by back substitution */
    557:   for (k=mbs-1; k>=0; k--) {
    558:     v  = aa + 36*ai[k];
    559:     vj = aj + ai[k];
    560:     tp = t + k*6;
    561:     x0 = tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; x4=tp[4]; x5=tp[5]; /* xk */
    562:     nz = ai[k+1] - ai[k];

    564:     tp = t + (*vj)*6;
    565:     while (nz--) {
    566:       /* xk += U(k,:)*x(:) */
    567:       x0 += v[0]*tp[0] + v[6]*tp[1] + v[12]*tp[2] + v[18]*tp[3] + v[24]*tp[4] + v[30]*tp[5];
    568:       x1 += v[1]*tp[0] + v[7]*tp[1] + v[13]*tp[2] + v[19]*tp[3] + v[25]*tp[4] + v[31]*tp[5];
    569:       x2 += v[2]*tp[0] + v[8]*tp[1] + v[14]*tp[2] + v[20]*tp[3] + v[26]*tp[4] + v[32]*tp[5];
    570:       x3 += v[3]*tp[0] + v[9]*tp[1] + v[15]*tp[2] + v[21]*tp[3] + v[27]*tp[4] + v[33]*tp[5];
    571:       x4 += v[4]*tp[0]+ v[10]*tp[1] + v[16]*tp[2] + v[22]*tp[3] + v[28]*tp[4] + v[34]*tp[5];
    572:       x5 += v[5]*tp[0]+ v[11]*tp[1] + v[17]*tp[2] + v[23]*tp[3] + v[29]*tp[4] + v[35]*tp[5];
    573:       vj++;
    574:       tp = t + (*vj)*6;
    575:       v += 36;
    576:     }
    577:     tp       = t + k*6;
    578:     tp[0]    = x0; tp[1]=x1; tp[2]=x2; tp[3]=x3; tp[4]=x4; tp[5]=x5;
    579:     idx      = 6*r[k];
    580:     x[idx]   = x0;
    581:     x[idx+1] = x1;
    582:     x[idx+2] = x2;
    583:     x[idx+3] = x3;
    584:     x[idx+4] = x4;
    585:     x[idx+5] = x5;
    586:   }

    588:   ISRestoreIndices(isrow,&r);
    589:   VecRestoreArrayRead(bb,&b);
    590:   VecRestoreArray(xx,&x);
    591:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    592:   return(0);
    593: }

    597: PetscErrorCode MatForwardSolve_SeqSBAIJ_6_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    598: {
    599:   const MatScalar *v,*d;
    600:   PetscScalar     *xp,x0,x1,x2,x3,x4,x5;
    601:   PetscInt        nz,k;
    602:   const PetscInt  *vj;

    605:   for (k=0; k<mbs; k++) {
    606:     v  = aa + 36*ai[k];
    607:     xp = x + k*6;
    608:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; x4=xp[4]; x5=xp[5]; /* Dk*xk = k-th block of x */
    609:     nz = ai[k+1] - ai[k];
    610:     vj = aj + ai[k];
    611:     xp = x + (*vj)*6;
    612:     PetscPrefetchBlock(vj+nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    613:     PetscPrefetchBlock(v+36*nz,36*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    614:     while (nz--) {
    615:       /* x(:) += U(k,:)^T*(Dk*xk) */
    616:       xp[0] +=  v[0]*x0 +  v[1]*x1 +  v[2]*x2 + v[3]*x3 + v[4]*x4 + v[5]*x5;
    617:       xp[1] +=  v[6]*x0 +  v[7]*x1 +  v[8]*x2 + v[9]*x3+ v[10]*x4+ v[11]*x5;
    618:       xp[2] += v[12]*x0 + v[13]*x1 + v[14]*x2+ v[15]*x3+ v[16]*x4+ v[17]*x5;
    619:       xp[3] += v[18]*x0 + v[19]*x1 + v[20]*x2+ v[21]*x3+ v[22]*x4+ v[23]*x5;
    620:       xp[4] += v[24]*x0 + v[25]*x1 + v[26]*x2+ v[27]*x3+ v[28]*x4+ v[29]*x5;
    621:       xp[5] += v[30]*x0 + v[31]*x1 + v[32]*x2+ v[33]*x3+ v[34]*x4+ v[35]*x5;
    622:       vj++;
    623:       xp = x + (*vj)*6;
    624:       v += 36;
    625:     }
    626:     /* xk = inv(Dk)*(Dk*xk) */
    627:     d     = aa+k*36;       /* ptr to inv(Dk) */
    628:     xp    = x + k*6;
    629:     xp[0] = d[0]*x0 + d[6]*x1 + d[12]*x2 + d[18]*x3 + d[24]*x4 + d[30]*x5;
    630:     xp[1] = d[1]*x0 + d[7]*x1 + d[13]*x2 + d[19]*x3 + d[25]*x4 + d[31]*x5;
    631:     xp[2] = d[2]*x0 + d[8]*x1 + d[14]*x2 + d[20]*x3 + d[26]*x4 + d[32]*x5;
    632:     xp[3] = d[3]*x0 + d[9]*x1 + d[15]*x2 + d[21]*x3 + d[27]*x4 + d[33]*x5;
    633:     xp[4] = d[4]*x0+ d[10]*x1 + d[16]*x2 + d[22]*x3 + d[28]*x4 + d[34]*x5;
    634:     xp[5] = d[5]*x0+ d[11]*x1 + d[17]*x2 + d[23]*x3 + d[29]*x4 + d[35]*x5;
    635:   }
    636:   return(0);
    637: }
    640: PetscErrorCode MatBackwardSolve_SeqSBAIJ_6_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    641: {
    642:   const MatScalar   *v;
    643:   PetscScalar       *xp,x0,x1,x2,x3,x4,x5;
    644:   PetscInt          nz,k;
    645:   const PetscInt    *vj;

    648:   for (k=mbs-1; k>=0; k--) {
    649:     v  = aa + 36*ai[k];
    650:     xp = x + k*6;
    651:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; x4=xp[4]; x5=xp[5]; /* xk */
    652:     nz = ai[k+1] - ai[k];
    653:     vj = aj + ai[k];
    654:     xp = x + (*vj)*6;
    655:     PetscPrefetchBlock(vj-nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    656:     PetscPrefetchBlock(v-36*nz,36*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    657:     while (nz--) {
    658:       /* xk += U(k,:)*x(:) */
    659:       x0 += v[0]*xp[0] + v[6]*xp[1] + v[12]*xp[2] + v[18]*xp[3] + v[24]*xp[4] + v[30]*xp[5];
    660:       x1 += v[1]*xp[0] + v[7]*xp[1] + v[13]*xp[2] + v[19]*xp[3] + v[25]*xp[4] + v[31]*xp[5];
    661:       x2 += v[2]*xp[0] + v[8]*xp[1] + v[14]*xp[2] + v[20]*xp[3] + v[26]*xp[4] + v[32]*xp[5];
    662:       x3 += v[3]*xp[0] + v[9]*xp[1] + v[15]*xp[2] + v[21]*xp[3] + v[27]*xp[4] + v[33]*xp[5];
    663:       x4 += v[4]*xp[0]+ v[10]*xp[1] + v[16]*xp[2] + v[22]*xp[3] + v[28]*xp[4] + v[34]*xp[5];
    664:       x5 += v[5]*xp[0]+ v[11]*xp[1] + v[17]*xp[2] + v[23]*xp[3] + v[29]*xp[4] + v[35]*xp[5];
    665:       vj++;
    666:       v += 36; xp = x + (*vj)*6;
    667:     }
    668:     xp   = x + k*6;
    669:     xp[0]=x0; xp[1]=x1; xp[2]=x2; xp[3]=x3; xp[4]=x4; xp[5]=x5;
    670:   }
    671:   return(0);
    672: }

    677: PetscErrorCode MatSolve_SeqSBAIJ_6_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    678: {
    679:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    680:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    681:   const MatScalar   *aa=a->a;
    682:   PetscScalar       *x;
    683:   const PetscScalar *b;
    684:   PetscErrorCode    ierr;

    687:   VecGetArrayRead(bb,&b);
    688:   VecGetArray(xx,&x);

    690:   /* solve U^T * D * y = b by forward substitution */
    691:   PetscMemcpy(x,b,6*mbs*sizeof(PetscScalar)); /* x <- b */
    692:   MatForwardSolve_SeqSBAIJ_6_NaturalOrdering(ai,aj,aa,mbs,x);

    694:   /* solve U*x = y by back substitution */
    695:   MatBackwardSolve_SeqSBAIJ_6_NaturalOrdering(ai,aj,aa,mbs,x);

    697:   VecRestoreArrayRead(bb,&b);
    698:   VecRestoreArray(xx,&x);
    699:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    700:   return(0);
    701: }

    705: PetscErrorCode MatForwardSolve_SeqSBAIJ_6_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    706: {
    707:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    708:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    709:   const MatScalar   *aa=a->a;
    710:   PetscScalar       *x;
    711:   const PetscScalar *b;
    712:   PetscErrorCode    ierr;

    715:   VecGetArrayRead(bb,&b);
    716:   VecGetArray(xx,&x);
    717:   PetscMemcpy(x,b,6*mbs*sizeof(PetscScalar)); /* x <- b */
    718:   MatForwardSolve_SeqSBAIJ_6_NaturalOrdering(ai,aj,aa,mbs,x);
    719:   VecRestoreArrayRead(bb,&b);
    720:   VecRestoreArray(xx,&x);
    721:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    722:   return(0);
    723: }

    727: PetscErrorCode MatBackwardSolve_SeqSBAIJ_6_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    728: {
    729:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    730:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    731:   const MatScalar   *aa=a->a;
    732:   PetscScalar       *x;
    733:   const PetscScalar *b;
    734:   PetscErrorCode    ierr;

    737:   VecGetArrayRead(bb,&b);
    738:   VecGetArray(xx,&x);
    739:   PetscMemcpy(x,b,6*mbs*sizeof(PetscScalar)); /* x <- b */
    740:   MatBackwardSolve_SeqSBAIJ_6_NaturalOrdering(ai,aj,aa,mbs,x);
    741:   VecRestoreArrayRead(bb,&b);
    742:   VecRestoreArray(xx,&x);
    743:   PetscLogFlops(2.0*bs2*(a->nz - mbs));
    744:   return(0);
    745: }

    749: PetscErrorCode MatSolve_SeqSBAIJ_5_inplace(Mat A,Vec bb,Vec xx)
    750: {
    751:   Mat_SeqSBAIJ      *a=(Mat_SeqSBAIJ*)A->data;
    752:   IS                isrow=a->row;
    753:   const PetscInt    mbs  =a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2 = a->bs2;
    754:   PetscErrorCode    ierr;
    755:   const PetscInt    *r,*vj;
    756:   PetscInt          nz,k,idx;
    757:   const MatScalar   *aa=a->a,*v,*diag;
    758:   PetscScalar       *x,x0,x1,x2,x3,x4,*t,*tp;
    759:   const PetscScalar *b;

    762:   VecGetArrayRead(bb,&b);
    763:   VecGetArray(xx,&x);
    764:   t    = a->solve_work;
    765:   ISGetIndices(isrow,&r);

    767:   /* solve U^T * D * y = b by forward substitution */
    768:   tp = t;
    769:   for (k=0; k<mbs; k++) { /* t <- perm(b) */
    770:     idx   = 5*r[k];
    771:     tp[0] = b[idx];
    772:     tp[1] = b[idx+1];
    773:     tp[2] = b[idx+2];
    774:     tp[3] = b[idx+3];
    775:     tp[4] = b[idx+4];
    776:     tp   += 5;
    777:   }

    779:   for (k=0; k<mbs; k++) {
    780:     v  = aa + 25*ai[k];
    781:     vj = aj + ai[k];
    782:     tp = t + k*5;
    783:     x0 = tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; x4=tp[4];
    784:     nz = ai[k+1] - ai[k];

    786:     tp = t + (*vj)*5;
    787:     while (nz--) {
    788:       tp[0] +=  v[0]*x0 + v[1]*x1 + v[2]*x2 + v[3]*x3 + v[4]*x4;
    789:       tp[1] +=  v[5]*x0 + v[6]*x1 + v[7]*x2 + v[8]*x3 + v[9]*x4;
    790:       tp[2] += v[10]*x0+ v[11]*x1+ v[12]*x2+ v[13]*x3+ v[14]*x4;
    791:       tp[3] += v[15]*x0+ v[16]*x1+ v[17]*x2+ v[18]*x3+ v[19]*x4;
    792:       tp[4] += v[20]*x0+ v[21]*x1+ v[22]*x2+ v[23]*x3+ v[24]*x4;
    793:       vj++;
    794:       tp = t + (*vj)*5;
    795:       v += 25;
    796:     }

    798:     /* xk = inv(Dk)*(Dk*xk) */
    799:     diag  = aa+k*25;          /* ptr to inv(Dk) */
    800:     tp    = t + k*5;
    801:     tp[0] = diag[0]*x0 + diag[5]*x1 + diag[10]*x2 + diag[15]*x3 + diag[20]*x4;
    802:     tp[1] = diag[1]*x0 + diag[6]*x1 + diag[11]*x2 + diag[16]*x3 + diag[21]*x4;
    803:     tp[2] = diag[2]*x0 + diag[7]*x1 + diag[12]*x2 + diag[17]*x3 + diag[22]*x4;
    804:     tp[3] = diag[3]*x0 + diag[8]*x1 + diag[13]*x2 + diag[18]*x3 + diag[23]*x4;
    805:     tp[4] = diag[4]*x0 + diag[9]*x1 + diag[14]*x2 + diag[19]*x3 + diag[24]*x4;
    806:   }

    808:   /* solve U*x = y by back substitution */
    809:   for (k=mbs-1; k>=0; k--) {
    810:     v  = aa + 25*ai[k];
    811:     vj = aj + ai[k];
    812:     tp = t + k*5;
    813:     x0 = tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; x4=tp[4]; /* xk */
    814:     nz = ai[k+1] - ai[k];

    816:     tp = t + (*vj)*5;
    817:     while (nz--) {
    818:       /* xk += U(k,:)*x(:) */
    819:       x0 += v[0]*tp[0] + v[5]*tp[1] + v[10]*tp[2] + v[15]*tp[3] + v[20]*tp[4];
    820:       x1 += v[1]*tp[0] + v[6]*tp[1] + v[11]*tp[2] + v[16]*tp[3] + v[21]*tp[4];
    821:       x2 += v[2]*tp[0] + v[7]*tp[1] + v[12]*tp[2] + v[17]*tp[3] + v[22]*tp[4];
    822:       x3 += v[3]*tp[0] + v[8]*tp[1] + v[13]*tp[2] + v[18]*tp[3] + v[23]*tp[4];
    823:       x4 += v[4]*tp[0] + v[9]*tp[1] + v[14]*tp[2] + v[19]*tp[3] + v[24]*tp[4];
    824:       vj++;
    825:       tp = t + (*vj)*5;
    826:       v += 25;
    827:     }
    828:     tp       = t + k*5;
    829:     tp[0]    = x0; tp[1]=x1; tp[2]=x2; tp[3]=x3; tp[4]=x4;
    830:     idx      = 5*r[k];
    831:     x[idx]   = x0;
    832:     x[idx+1] = x1;
    833:     x[idx+2] = x2;
    834:     x[idx+3] = x3;
    835:     x[idx+4] = x4;
    836:   }

    838:   ISRestoreIndices(isrow,&r);
    839:   VecRestoreArrayRead(bb,&b);
    840:   VecRestoreArray(xx,&x);
    841:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    842:   return(0);
    843: }

    847: PetscErrorCode MatForwardSolve_SeqSBAIJ_5_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    848: {
    849:   const MatScalar *v,*diag;
    850:   PetscScalar     *xp,x0,x1,x2,x3,x4;
    851:   PetscInt        nz,k;
    852:   const PetscInt  *vj;

    855:   for (k=0; k<mbs; k++) {
    856:     v  = aa + 25*ai[k];
    857:     xp = x + k*5;
    858:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; x4=xp[4];/* Dk*xk = k-th block of x */
    859:     nz = ai[k+1] - ai[k];
    860:     vj = aj + ai[k];
    861:     xp = x + (*vj)*5;
    862:     PetscPrefetchBlock(vj+nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    863:     PetscPrefetchBlock(v+25*nz,25*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    864:     while (nz--) {
    865:       /* x(:) += U(k,:)^T*(Dk*xk) */
    866:       xp[0] +=  v[0]*x0 +  v[1]*x1 +  v[2]*x2 + v[3]*x3 + v[4]*x4;
    867:       xp[1] +=  v[5]*x0 +  v[6]*x1 +  v[7]*x2 + v[8]*x3 + v[9]*x4;
    868:       xp[2] += v[10]*x0 + v[11]*x1 + v[12]*x2+ v[13]*x3+ v[14]*x4;
    869:       xp[3] += v[15]*x0 + v[16]*x1 + v[17]*x2+ v[18]*x3+ v[19]*x4;
    870:       xp[4] += v[20]*x0 + v[21]*x1 + v[22]*x2+ v[23]*x3+ v[24]*x4;
    871:       vj++;
    872:       xp = x + (*vj)*5;
    873:       v += 25;
    874:     }
    875:     /* xk = inv(Dk)*(Dk*xk) */
    876:     diag  = aa+k*25;         /* ptr to inv(Dk) */
    877:     xp    = x + k*5;
    878:     xp[0] = diag[0]*x0 + diag[5]*x1 + diag[10]*x2 + diag[15]*x3 + diag[20]*x4;
    879:     xp[1] = diag[1]*x0 + diag[6]*x1 + diag[11]*x2 + diag[16]*x3 + diag[21]*x4;
    880:     xp[2] = diag[2]*x0 + diag[7]*x1 + diag[12]*x2 + diag[17]*x3 + diag[22]*x4;
    881:     xp[3] = diag[3]*x0 + diag[8]*x1 + diag[13]*x2 + diag[18]*x3 + diag[23]*x4;
    882:     xp[4] = diag[4]*x0 + diag[9]*x1 + diag[14]*x2 + diag[19]*x3 + diag[24]*x4;
    883:   }
    884:   return(0);
    885: }

    889: PetscErrorCode MatBackwardSolve_SeqSBAIJ_5_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    890: {
    891:   const MatScalar *v;
    892:   PetscScalar     *xp,x0,x1,x2,x3,x4;
    893:   PetscInt        nz,k;
    894:   const PetscInt  *vj;

    897:   for (k=mbs-1; k>=0; k--) {
    898:     v  = aa + 25*ai[k];
    899:     xp = x + k*5;
    900:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; x4=xp[4]; /* xk */
    901:     nz = ai[k+1] - ai[k];
    902:     vj = aj + ai[k];
    903:     xp = x + (*vj)*5;
    904:     PetscPrefetchBlock(vj-nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    905:     PetscPrefetchBlock(v-25*nz,25*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    906:     while (nz--) {
    907:       /* xk += U(k,:)*x(:) */
    908:       x0 += v[0]*xp[0] + v[5]*xp[1] + v[10]*xp[2] + v[15]*xp[3] + v[20]*xp[4];
    909:       x1 += v[1]*xp[0] + v[6]*xp[1] + v[11]*xp[2] + v[16]*xp[3] + v[21]*xp[4];
    910:       x2 += v[2]*xp[0] + v[7]*xp[1] + v[12]*xp[2] + v[17]*xp[3] + v[22]*xp[4];
    911:       x3 += v[3]*xp[0] + v[8]*xp[1] + v[13]*xp[2] + v[18]*xp[3] + v[23]*xp[4];
    912:       x4 += v[4]*xp[0] + v[9]*xp[1] + v[14]*xp[2] + v[19]*xp[3] + v[24]*xp[4];
    913:       vj++;
    914:       v += 25; xp = x + (*vj)*5;
    915:     }
    916:     xp   = x + k*5;
    917:     xp[0]=x0; xp[1]=x1; xp[2]=x2; xp[3]=x3; xp[4]=x4;
    918:   }
    919:   return(0);
    920: }

    924: PetscErrorCode MatSolve_SeqSBAIJ_5_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    925: {
    926:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    927:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2 = a->bs2;
    928:   const MatScalar   *aa=a->a;
    929:   PetscScalar       *x;
    930:   const PetscScalar *b;
    931:   PetscErrorCode    ierr;

    934:   VecGetArrayRead(bb,&b);
    935:   VecGetArray(xx,&x);

    937:   /* solve U^T * D * y = b by forward substitution */
    938:   PetscMemcpy(x,b,5*mbs*sizeof(PetscScalar)); /* x <- b */
    939:   MatForwardSolve_SeqSBAIJ_5_NaturalOrdering(ai,aj,aa,mbs,x);

    941:   /* solve U*x = y by back substitution */
    942:   MatBackwardSolve_SeqSBAIJ_5_NaturalOrdering(ai,aj,aa,mbs,x);

    944:   VecRestoreArrayRead(bb,&b);
    945:   VecRestoreArray(xx,&x);
    946:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    947:   return(0);
    948: }

    952: PetscErrorCode MatForwardSolve_SeqSBAIJ_5_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    953: {
    954:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    955:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    956:   const MatScalar   *aa=a->a;
    957:   PetscScalar       *x;
    958:   const PetscScalar *b;
    959:   PetscErrorCode    ierr;

    962:   VecGetArrayRead(bb,&b);
    963:   VecGetArray(xx,&x);
    964:   PetscMemcpy(x,b,5*mbs*sizeof(PetscScalar)); /* x <- b */
    965:   MatForwardSolve_SeqSBAIJ_5_NaturalOrdering(ai,aj,aa,mbs,x);
    966:   VecRestoreArrayRead(bb,&b);
    967:   VecRestoreArray(xx,&x);
    968:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    969:   return(0);
    970: }

    974: PetscErrorCode MatBackwardSolve_SeqSBAIJ_5_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    975: {
    976:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    977:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    978:   const MatScalar   *aa=a->a;
    979:   PetscScalar       *x;
    980:   const PetscScalar *b;
    981:   PetscErrorCode    ierr;

    984:   VecGetArrayRead(bb,&b);
    985:   VecGetArray(xx,&x);
    986:   PetscMemcpy(x,b,5*mbs*sizeof(PetscScalar));
    987:   MatBackwardSolve_SeqSBAIJ_5_NaturalOrdering(ai,aj,aa,mbs,x);
    988:   VecRestoreArrayRead(bb,&b);
    989:   VecRestoreArray(xx,&x);
    990:   PetscLogFlops(2.0*bs2*(a->nz-mbs));
    991:   return(0);
    992: }

    996: PetscErrorCode MatSolve_SeqSBAIJ_4_inplace(Mat A,Vec bb,Vec xx)
    997: {
    998:   Mat_SeqSBAIJ      *a   =(Mat_SeqSBAIJ*)A->data;
    999:   IS                isrow=a->row;
    1000:   const PetscInt    mbs  =a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1001:   PetscErrorCode    ierr;
    1002:   const PetscInt    *r,*vj;
    1003:   PetscInt          nz,k,idx;
    1004:   const MatScalar   *aa=a->a,*v,*diag;
    1005:   PetscScalar       *x,x0,x1,x2,x3,*t,*tp;
    1006:   const PetscScalar *b;

    1009:   VecGetArrayRead(bb,&b);
    1010:   VecGetArray(xx,&x);
    1011:   t    = a->solve_work;
    1012:   ISGetIndices(isrow,&r);

    1014:   /* solve U^T * D * y = b by forward substitution */
    1015:   tp = t;
    1016:   for (k=0; k<mbs; k++) { /* t <- perm(b) */
    1017:     idx   = 4*r[k];
    1018:     tp[0] = b[idx];
    1019:     tp[1] = b[idx+1];
    1020:     tp[2] = b[idx+2];
    1021:     tp[3] = b[idx+3];
    1022:     tp   += 4;
    1023:   }

    1025:   for (k=0; k<mbs; k++) {
    1026:     v  = aa + 16*ai[k];
    1027:     vj = aj + ai[k];
    1028:     tp = t + k*4;
    1029:     x0 = tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3];
    1030:     nz = ai[k+1] - ai[k];

    1032:     tp = t + (*vj)*4;
    1033:     while (nz--) {
    1034:       tp[0] += v[0]*x0 + v[1]*x1 + v[2]*x2 + v[3]*x3;
    1035:       tp[1] += v[4]*x0 + v[5]*x1 + v[6]*x2 + v[7]*x3;
    1036:       tp[2] += v[8]*x0 + v[9]*x1 + v[10]*x2+ v[11]*x3;
    1037:       tp[3] += v[12]*x0+ v[13]*x1+ v[14]*x2+ v[15]*x3;
    1038:       vj++;
    1039:       tp = t + (*vj)*4;
    1040:       v += 16;
    1041:     }

    1043:     /* xk = inv(Dk)*(Dk*xk) */
    1044:     diag  = aa+k*16;          /* ptr to inv(Dk) */
    1045:     tp    = t + k*4;
    1046:     tp[0] = diag[0]*x0 + diag[4]*x1 + diag[8]*x2 + diag[12]*x3;
    1047:     tp[1] = diag[1]*x0 + diag[5]*x1 + diag[9]*x2 + diag[13]*x3;
    1048:     tp[2] = diag[2]*x0 + diag[6]*x1 + diag[10]*x2+ diag[14]*x3;
    1049:     tp[3] = diag[3]*x0 + diag[7]*x1 + diag[11]*x2+ diag[15]*x3;
    1050:   }

    1052:   /* solve U*x = y by back substitution */
    1053:   for (k=mbs-1; k>=0; k--) {
    1054:     v  = aa + 16*ai[k];
    1055:     vj = aj + ai[k];
    1056:     tp = t + k*4;
    1057:     x0 = tp[0]; x1=tp[1]; x2=tp[2]; x3=tp[3]; /* xk */
    1058:     nz = ai[k+1] - ai[k];

    1060:     tp = t + (*vj)*4;
    1061:     while (nz--) {
    1062:       /* xk += U(k,:)*x(:) */
    1063:       x0 += v[0]*tp[0] + v[4]*tp[1] + v[8]*tp[2] + v[12]*tp[3];
    1064:       x1 += v[1]*tp[0] + v[5]*tp[1] + v[9]*tp[2] + v[13]*tp[3];
    1065:       x2 += v[2]*tp[0] + v[6]*tp[1]+ v[10]*tp[2] + v[14]*tp[3];
    1066:       x3 += v[3]*tp[0] + v[7]*tp[1]+ v[11]*tp[2] + v[15]*tp[3];
    1067:       vj++;
    1068:       tp = t + (*vj)*4;
    1069:       v += 16;
    1070:     }
    1071:     tp       = t + k*4;
    1072:     tp[0]    =x0; tp[1]=x1; tp[2]=x2; tp[3]=x3;
    1073:     idx      = 4*r[k];
    1074:     x[idx]   = x0;
    1075:     x[idx+1] = x1;
    1076:     x[idx+2] = x2;
    1077:     x[idx+3] = x3;
    1078:   }

    1080:   ISRestoreIndices(isrow,&r);
    1081:   VecRestoreArrayRead(bb,&b);
    1082:   VecRestoreArray(xx,&x);
    1083:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    1084:   return(0);
    1085: }

    1089: PetscErrorCode MatForwardSolve_SeqSBAIJ_4_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    1090: {
    1091:   const MatScalar *v,*diag;
    1092:   PetscScalar     *xp,x0,x1,x2,x3;
    1093:   PetscInt        nz,k;
    1094:   const PetscInt  *vj;

    1097:   for (k=0; k<mbs; k++) {
    1098:     v  = aa + 16*ai[k];
    1099:     xp = x + k*4;
    1100:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; /* Dk*xk = k-th block of x */
    1101:     nz = ai[k+1] - ai[k];
    1102:     vj = aj + ai[k];
    1103:     xp = x + (*vj)*4;
    1104:     PetscPrefetchBlock(vj+nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    1105:     PetscPrefetchBlock(v+16*nz,16*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    1106:     while (nz--) {
    1107:       /* x(:) += U(k,:)^T*(Dk*xk) */
    1108:       xp[0] += v[0]*x0 + v[1]*x1 + v[2]*x2 + v[3]*x3;
    1109:       xp[1] += v[4]*x0 + v[5]*x1 + v[6]*x2 + v[7]*x3;
    1110:       xp[2] += v[8]*x0 + v[9]*x1 + v[10]*x2+ v[11]*x3;
    1111:       xp[3] += v[12]*x0+ v[13]*x1+ v[14]*x2+ v[15]*x3;
    1112:       vj++;
    1113:       xp = x + (*vj)*4;
    1114:       v += 16;
    1115:     }
    1116:     /* xk = inv(Dk)*(Dk*xk) */
    1117:     diag  = aa+k*16;         /* ptr to inv(Dk) */
    1118:     xp    = x + k*4;
    1119:     xp[0] = diag[0]*x0 + diag[4]*x1 + diag[8]*x2 + diag[12]*x3;
    1120:     xp[1] = diag[1]*x0 + diag[5]*x1 + diag[9]*x2 + diag[13]*x3;
    1121:     xp[2] = diag[2]*x0 + diag[6]*x1 + diag[10]*x2+ diag[14]*x3;
    1122:     xp[3] = diag[3]*x0 + diag[7]*x1 + diag[11]*x2+ diag[15]*x3;
    1123:   }
    1124:   return(0);
    1125: }

    1129: PetscErrorCode MatBackwardSolve_SeqSBAIJ_4_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    1130: {
    1131:   const MatScalar *v;
    1132:   PetscScalar     *xp,x0,x1,x2,x3;
    1133:   PetscInt        nz,k;
    1134:   const PetscInt  *vj;

    1137:   for (k=mbs-1; k>=0; k--) {
    1138:     v  = aa + 16*ai[k];
    1139:     xp = x + k*4;
    1140:     x0 = xp[0]; x1=xp[1]; x2=xp[2]; x3=xp[3]; /* xk */
    1141:     nz = ai[k+1] - ai[k];
    1142:     vj = aj + ai[k];
    1143:     xp = x + (*vj)*4;
    1144:     PetscPrefetchBlock(vj-nz,nz,0,PETSC_PREFETCH_HINT_NTA);      /* Indices for the next row (assumes same size as this one) */
    1145:     PetscPrefetchBlock(v-16*nz,16*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    1146:     while (nz--) {
    1147:       /* xk += U(k,:)*x(:) */
    1148:       x0 += v[0]*xp[0] + v[4]*xp[1] + v[8]*xp[2] + v[12]*xp[3];
    1149:       x1 += v[1]*xp[0] + v[5]*xp[1] + v[9]*xp[2] + v[13]*xp[3];
    1150:       x2 += v[2]*xp[0] + v[6]*xp[1]+ v[10]*xp[2] + v[14]*xp[3];
    1151:       x3 += v[3]*xp[0] + v[7]*xp[1]+ v[11]*xp[2] + v[15]*xp[3];
    1152:       vj++;
    1153:       v += 16; xp = x + (*vj)*4;
    1154:     }
    1155:     xp    = x + k*4;
    1156:     xp[0] = x0; xp[1] = x1; xp[2] = x2; xp[3] = x3;
    1157:   }
    1158:   return(0);
    1159: }

    1163: PetscErrorCode MatSolve_SeqSBAIJ_4_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1164: {
    1165:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1166:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1167:   const MatScalar   *aa=a->a;
    1168:   PetscScalar       *x;
    1169:   const PetscScalar *b;
    1170:   PetscErrorCode    ierr;

    1173:   VecGetArrayRead(bb,&b);
    1174:   VecGetArray(xx,&x);

    1176:   /* solve U^T * D * y = b by forward substitution */
    1177:   PetscMemcpy(x,b,4*mbs*sizeof(PetscScalar)); /* x <- b */
    1178:   MatForwardSolve_SeqSBAIJ_4_NaturalOrdering(ai,aj,aa,mbs,x);

    1180:   /* solve U*x = y by back substitution */
    1181:   MatBackwardSolve_SeqSBAIJ_4_NaturalOrdering(ai,aj,aa,mbs,x);
    1182:   VecRestoreArrayRead(bb,&b);
    1183:   VecRestoreArray(xx,&x);
    1184:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    1185:   return(0);
    1186: }

    1190: PetscErrorCode MatForwardSolve_SeqSBAIJ_4_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1191: {
    1192:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1193:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1194:   const MatScalar   *aa=a->a;
    1195:   PetscScalar       *x;
    1196:   const PetscScalar *b;
    1197:   PetscErrorCode    ierr;

    1200:   VecGetArrayRead(bb,&b);
    1201:   VecGetArray(xx,&x);
    1202:   PetscMemcpy(x,b,4*mbs*sizeof(PetscScalar)); /* x <- b */
    1203:   MatForwardSolve_SeqSBAIJ_4_NaturalOrdering(ai,aj,aa,mbs,x);
    1204:   VecRestoreArrayRead(bb,&b);
    1205:   VecRestoreArray(xx,&x);
    1206:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    1207:   return(0);
    1208: }

    1212: PetscErrorCode MatBackwardSolve_SeqSBAIJ_4_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1213: {
    1214:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1215:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    1216:   const MatScalar   *aa=a->a;
    1217:   PetscScalar       *x;
    1218:   const PetscScalar *b;
    1219:   PetscErrorCode    ierr;

    1222:   VecGetArrayRead(bb,&b);
    1223:   VecGetArray(xx,&x);
    1224:   PetscMemcpy(x,b,4*mbs*sizeof(PetscScalar));
    1225:   MatBackwardSolve_SeqSBAIJ_4_NaturalOrdering(ai,aj,aa,mbs,x);
    1226:   VecRestoreArrayRead(bb,&b);
    1227:   VecRestoreArray(xx,&x);
    1228:   PetscLogFlops(2.0*bs2*(a->nz-mbs));
    1229:   return(0);
    1230: }

    1234: PetscErrorCode MatSolve_SeqSBAIJ_3_inplace(Mat A,Vec bb,Vec xx)
    1235: {
    1236:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1237:   IS                isrow=a->row;
    1238:   const PetscInt    mbs  =a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1239:   PetscErrorCode    ierr;
    1240:   const PetscInt    *r;
    1241:   PetscInt          nz,k,idx;
    1242:   const PetscInt    *vj;
    1243:   const MatScalar   *aa=a->a,*v,*diag;
    1244:   PetscScalar       *x,x0,x1,x2,*t,*tp;
    1245:   const PetscScalar *b;

    1248:   VecGetArrayRead(bb,&b);
    1249:   VecGetArray(xx,&x);
    1250:   t    = a->solve_work;
    1251:   ISGetIndices(isrow,&r);

    1253:   /* solve U^T * D * y = b by forward substitution */
    1254:   tp = t;
    1255:   for (k=0; k<mbs; k++) { /* t <- perm(b) */
    1256:     idx   = 3*r[k];
    1257:     tp[0] = b[idx];
    1258:     tp[1] = b[idx+1];
    1259:     tp[2] = b[idx+2];
    1260:     tp   += 3;
    1261:   }

    1263:   for (k=0; k<mbs; k++) {
    1264:     v  = aa + 9*ai[k];
    1265:     vj = aj + ai[k];
    1266:     tp = t + k*3;
    1267:     x0 = tp[0]; x1 = tp[1]; x2 = tp[2];
    1268:     nz = ai[k+1] - ai[k];

    1270:     tp = t + (*vj)*3;
    1271:     while (nz--) {
    1272:       tp[0] += v[0]*x0 + v[1]*x1 + v[2]*x2;
    1273:       tp[1] += v[3]*x0 + v[4]*x1 + v[5]*x2;
    1274:       tp[2] += v[6]*x0 + v[7]*x1 + v[8]*x2;
    1275:       vj++;
    1276:       tp = t + (*vj)*3;
    1277:       v += 9;
    1278:     }

    1280:     /* xk = inv(Dk)*(Dk*xk) */
    1281:     diag  = aa+k*9;          /* ptr to inv(Dk) */
    1282:     tp    = t + k*3;
    1283:     tp[0] = diag[0]*x0 + diag[3]*x1 + diag[6]*x2;
    1284:     tp[1] = diag[1]*x0 + diag[4]*x1 + diag[7]*x2;
    1285:     tp[2] = diag[2]*x0 + diag[5]*x1 + diag[8]*x2;
    1286:   }

    1288:   /* solve U*x = y by back substitution */
    1289:   for (k=mbs-1; k>=0; k--) {
    1290:     v  = aa + 9*ai[k];
    1291:     vj = aj + ai[k];
    1292:     tp = t + k*3;
    1293:     x0 = tp[0]; x1 = tp[1]; x2 = tp[2];  /* xk */
    1294:     nz = ai[k+1] - ai[k];

    1296:     tp = t + (*vj)*3;
    1297:     while (nz--) {
    1298:       /* xk += U(k,:)*x(:) */
    1299:       x0 += v[0]*tp[0] + v[3]*tp[1] + v[6]*tp[2];
    1300:       x1 += v[1]*tp[0] + v[4]*tp[1] + v[7]*tp[2];
    1301:       x2 += v[2]*tp[0] + v[5]*tp[1] + v[8]*tp[2];
    1302:       vj++;
    1303:       tp = t + (*vj)*3;
    1304:       v += 9;
    1305:     }
    1306:     tp       = t + k*3;
    1307:     tp[0]    = x0; tp[1] = x1; tp[2] = x2;
    1308:     idx      = 3*r[k];
    1309:     x[idx]   = x0;
    1310:     x[idx+1] = x1;
    1311:     x[idx+2] = x2;
    1312:   }

    1314:   ISRestoreIndices(isrow,&r);
    1315:   VecRestoreArrayRead(bb,&b);
    1316:   VecRestoreArray(xx,&x);
    1317:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    1318:   return(0);
    1319: }

    1323: PetscErrorCode MatForwardSolve_SeqSBAIJ_3_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    1324: {
    1325:   const MatScalar *v,*diag;
    1326:   PetscScalar     *xp,x0,x1,x2;
    1327:   PetscInt        nz,k;
    1328:   const PetscInt  *vj;

    1331:   for (k=0; k<mbs; k++) {
    1332:     v  = aa + 9*ai[k];
    1333:     xp = x + k*3;
    1334:     x0 = xp[0]; x1 = xp[1]; x2 = xp[2]; /* Dk*xk = k-th block of x */
    1335:     nz = ai[k+1] - ai[k];
    1336:     vj = aj + ai[k];
    1337:     xp = x + (*vj)*3;
    1338:     PetscPrefetchBlock(vj+nz,nz,0,PETSC_PREFETCH_HINT_NTA);    /* Indices for the next row (assumes same size as this one) */
    1339:     PetscPrefetchBlock(v+9*nz,9*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    1340:     while (nz--) {
    1341:       /* x(:) += U(k,:)^T*(Dk*xk) */
    1342:       xp[0] += v[0]*x0 + v[1]*x1 + v[2]*x2;
    1343:       xp[1] += v[3]*x0 + v[4]*x1 + v[5]*x2;
    1344:       xp[2] += v[6]*x0 + v[7]*x1 + v[8]*x2;
    1345:       vj++;
    1346:       xp = x + (*vj)*3;
    1347:       v += 9;
    1348:     }
    1349:     /* xk = inv(Dk)*(Dk*xk) */
    1350:     diag  = aa+k*9;         /* ptr to inv(Dk) */
    1351:     xp    = x + k*3;
    1352:     xp[0] = diag[0]*x0 + diag[3]*x1 + diag[6]*x2;
    1353:     xp[1] = diag[1]*x0 + diag[4]*x1 + diag[7]*x2;
    1354:     xp[2] = diag[2]*x0 + diag[5]*x1 + diag[8]*x2;
    1355:   }
    1356:   return(0);
    1357: }

    1361: PetscErrorCode MatBackwardSolve_SeqSBAIJ_3_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    1362: {
    1363:   const MatScalar *v;
    1364:   PetscScalar     *xp,x0,x1,x2;
    1365:   PetscInt        nz,k;
    1366:   const PetscInt  *vj;

    1369:   for (k=mbs-1; k>=0; k--) {
    1370:     v  = aa + 9*ai[k];
    1371:     xp = x + k*3;
    1372:     x0 = xp[0]; x1 = xp[1]; x2 = xp[2];  /* xk */
    1373:     nz = ai[k+1] - ai[k];
    1374:     vj = aj + ai[k];
    1375:     xp = x + (*vj)*3;
    1376:     PetscPrefetchBlock(vj-nz,nz,0,PETSC_PREFETCH_HINT_NTA);    /* Indices for the next row (assumes same size as this one) */
    1377:     PetscPrefetchBlock(v-9*nz,9*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    1378:     while (nz--) {
    1379:       /* xk += U(k,:)*x(:) */
    1380:       x0 += v[0]*xp[0] + v[3]*xp[1] + v[6]*xp[2];
    1381:       x1 += v[1]*xp[0] + v[4]*xp[1] + v[7]*xp[2];
    1382:       x2 += v[2]*xp[0] + v[5]*xp[1] + v[8]*xp[2];
    1383:       vj++;
    1384:       v += 9; xp = x + (*vj)*3;
    1385:     }
    1386:     xp    = x + k*3;
    1387:     xp[0] = x0; xp[1] = x1; xp[2] = x2;
    1388:   }
    1389:   return(0);
    1390: }

    1394: PetscErrorCode MatSolve_SeqSBAIJ_3_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1395: {
    1396:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1397:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1398:   const MatScalar   *aa=a->a;
    1399:   PetscScalar       *x;
    1400:   const PetscScalar *b;
    1401:   PetscErrorCode    ierr;

    1404:   VecGetArrayRead(bb,&b);
    1405:   VecGetArray(xx,&x);

    1407:   /* solve U^T * D * y = b by forward substitution */
    1408:   PetscMemcpy(x,b,3*mbs*sizeof(PetscScalar));
    1409:   MatForwardSolve_SeqSBAIJ_3_NaturalOrdering(ai,aj,aa,mbs,x);

    1411:   /* solve U*x = y by back substitution */
    1412:   MatBackwardSolve_SeqSBAIJ_3_NaturalOrdering(ai,aj,aa,mbs,x);

    1414:   VecRestoreArrayRead(bb,&b);
    1415:   VecRestoreArray(xx,&x);
    1416:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    1417:   return(0);
    1418: }

    1422: PetscErrorCode MatForwardSolve_SeqSBAIJ_3_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1423: {
    1424:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1425:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1426:   const MatScalar   *aa=a->a;
    1427:   PetscScalar       *x;
    1428:   const PetscScalar *b;
    1429:   PetscErrorCode    ierr;

    1432:   VecGetArrayRead(bb,&b);
    1433:   VecGetArray(xx,&x);
    1434:   PetscMemcpy(x,b,3*mbs*sizeof(PetscScalar));
    1435:   MatForwardSolve_SeqSBAIJ_3_NaturalOrdering(ai,aj,aa,mbs,x);
    1436:   VecRestoreArrayRead(bb,&b);
    1437:   VecRestoreArray(xx,&x);
    1438:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    1439:   return(0);
    1440: }

    1444: PetscErrorCode MatBackwardSolve_SeqSBAIJ_3_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1445: {
    1446:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1447:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    1448:   const MatScalar   *aa=a->a;
    1449:   PetscScalar       *x;
    1450:   const PetscScalar *b;
    1451:   PetscErrorCode    ierr;

    1454:   VecGetArrayRead(bb,&b);
    1455:   VecGetArray(xx,&x);
    1456:   PetscMemcpy(x,b,3*mbs*sizeof(PetscScalar));
    1457:   MatBackwardSolve_SeqSBAIJ_3_NaturalOrdering(ai,aj,aa,mbs,x);
    1458:   VecRestoreArrayRead(bb,&b);
    1459:   VecRestoreArray(xx,&x);
    1460:   PetscLogFlops(2.0*bs2*(a->nz-mbs));
    1461:   return(0);
    1462: }

    1466: PetscErrorCode MatSolve_SeqSBAIJ_2_inplace(Mat A,Vec bb,Vec xx)
    1467: {
    1468:   Mat_SeqSBAIJ      *a   =(Mat_SeqSBAIJ*)A->data;
    1469:   IS                isrow=a->row;
    1470:   const PetscInt    mbs  =a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1471:   PetscErrorCode    ierr;
    1472:   const PetscInt    *r,*vj;
    1473:   PetscInt          nz,k,k2,idx;
    1474:   const MatScalar   *aa=a->a,*v,*diag;
    1475:   PetscScalar       *x,x0,x1,*t;
    1476:   const PetscScalar *b;

    1479:   VecGetArrayRead(bb,&b);
    1480:   VecGetArray(xx,&x);
    1481:   t    = a->solve_work;
    1482:   ISGetIndices(isrow,&r);

    1484:   /* solve U^T * D * y = perm(b) by forward substitution */
    1485:   for (k=0; k<mbs; k++) {  /* t <- perm(b) */
    1486:     idx      = 2*r[k];
    1487:     t[k*2]   = b[idx];
    1488:     t[k*2+1] = b[idx+1];
    1489:   }
    1490:   for (k=0; k<mbs; k++) {
    1491:     v  = aa + 4*ai[k];
    1492:     vj = aj + ai[k];
    1493:     k2 = k*2;
    1494:     x0 = t[k2]; x1 = t[k2+1];
    1495:     nz = ai[k+1] - ai[k];
    1496:     while (nz--) {
    1497:       t[(*vj)*2]   += v[0]*x0 + v[1]*x1;
    1498:       t[(*vj)*2+1] += v[2]*x0 + v[3]*x1;
    1499:       vj++; v      += 4;
    1500:     }
    1501:     diag    = aa+k*4; /* ptr to inv(Dk) */
    1502:     t[k2]   = diag[0]*x0 + diag[2]*x1;
    1503:     t[k2+1] = diag[1]*x0 + diag[3]*x1;
    1504:   }

    1506:   /* solve U*x = y by back substitution */
    1507:   for (k=mbs-1; k>=0; k--) {
    1508:     v  = aa + 4*ai[k];
    1509:     vj = aj + ai[k];
    1510:     k2 = k*2;
    1511:     x0 = t[k2]; x1 = t[k2+1];
    1512:     nz = ai[k+1] - ai[k];
    1513:     while (nz--) {
    1514:       x0 += v[0]*t[(*vj)*2] + v[2]*t[(*vj)*2+1];
    1515:       x1 += v[1]*t[(*vj)*2] + v[3]*t[(*vj)*2+1];
    1516:       vj++;
    1517:       v += 4;
    1518:     }
    1519:     t[k2]    = x0;
    1520:     t[k2+1]  = x1;
    1521:     idx      = 2*r[k];
    1522:     x[idx]   = x0;
    1523:     x[idx+1] = x1;
    1524:   }

    1526:   ISRestoreIndices(isrow,&r);
    1527:   VecRestoreArrayRead(bb,&b);
    1528:   VecRestoreArray(xx,&x);
    1529:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    1530:   return(0);
    1531: }

    1535: PetscErrorCode MatForwardSolve_SeqSBAIJ_2_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    1536: {
    1537:   const MatScalar *v,*diag;
    1538:   PetscScalar     x0,x1;
    1539:   PetscInt        nz,k,k2;
    1540:   const PetscInt  *vj;
    1541:
    1543:   for (k=0; k<mbs; k++) {
    1544:     v  = aa + 4*ai[k];
    1545:     vj = aj + ai[k];
    1546:     k2 = k*2;
    1547:     x0 = x[k2]; x1 = x[k2+1];  /* Dk*xk = k-th block of x */
    1548:     nz = ai[k+1] - ai[k];
    1549:     PetscPrefetchBlock(vj+nz,nz,0,PETSC_PREFETCH_HINT_NTA);    /* Indices for the next row (assumes same size as this one) */
    1550:     PetscPrefetchBlock(v+4*nz,4*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    1551:     while (nz--) {
    1552:       /* x(:) += U(k,:)^T*(Dk*xk) */
    1553:       x[(*vj)*2]   += v[0]*x0 + v[1]*x1;
    1554:       x[(*vj)*2+1] += v[2]*x0 + v[3]*x1;
    1555:       vj++; v      += 4;
    1556:     }
    1557:     /* xk = inv(Dk)*(Dk*xk) */
    1558:     diag    = aa+k*4;       /* ptr to inv(Dk) */
    1559:     x[k2]   = diag[0]*x0 + diag[2]*x1;
    1560:     x[k2+1] = diag[1]*x0 + diag[3]*x1;
    1561:   }
    1562:   return(0);
    1563: }

    1567: PetscErrorCode MatBackwardSolve_SeqSBAIJ_2_NaturalOrdering(const PetscInt *ai,const PetscInt *aj,const MatScalar *aa,PetscInt mbs,PetscScalar *x)
    1568: {
    1569:   const MatScalar *v;
    1570:   PetscScalar     x0,x1;
    1571:   PetscInt        nz,k,k2;
    1572:   const PetscInt  *vj;

    1575:   for (k=mbs-1; k>=0; k--) {
    1576:     v  = aa + 4*ai[k];
    1577:     vj = aj + ai[k];
    1578:     k2 = k*2;
    1579:     x0 = x[k2]; x1 = x[k2+1];  /* xk */
    1580:     nz = ai[k+1] - ai[k];
    1581:     PetscPrefetchBlock(vj-nz,nz,0,PETSC_PREFETCH_HINT_NTA);    /* Indices for the next row (assumes same size as this one) */
    1582:     PetscPrefetchBlock(v-4*nz,4*nz,0,PETSC_PREFETCH_HINT_NTA); /* Entries for the next row */
    1583:     while (nz--) {
    1584:       /* xk += U(k,:)*x(:) */
    1585:       x0 += v[0]*x[(*vj)*2] + v[2]*x[(*vj)*2+1];
    1586:       x1 += v[1]*x[(*vj)*2] + v[3]*x[(*vj)*2+1];
    1587:       vj++;
    1588:       v += 4;
    1589:     }
    1590:     x[k2]   = x0;
    1591:     x[k2+1] = x1;
    1592:   }
    1593:   return(0);
    1594: }

    1598: PetscErrorCode MatSolve_SeqSBAIJ_2_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1599: {
    1600:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1601:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1602:   const MatScalar   *aa=a->a;
    1603:   PetscScalar       *x;
    1604:   const PetscScalar *b;
    1605:   PetscErrorCode    ierr;

    1608:   VecGetArrayRead(bb,&b);
    1609:   VecGetArray(xx,&x);

    1611:   /* solve U^T * D * y = b by forward substitution */
    1612:   PetscMemcpy(x,b,2*mbs*sizeof(PetscScalar));
    1613:   MatForwardSolve_SeqSBAIJ_2_NaturalOrdering(ai,aj,aa,mbs,x);

    1615:   /* solve U*x = y by back substitution */
    1616:   MatBackwardSolve_SeqSBAIJ_2_NaturalOrdering(ai,aj,aa,mbs,x);

    1618:   VecRestoreArrayRead(bb,&b);
    1619:   VecRestoreArray(xx,&x);
    1620:   PetscLogFlops(4.0*bs2*a->nz - (bs+2.0*bs2)*mbs);
    1621:   return(0);
    1622: }

    1626: PetscErrorCode MatForwardSolve_SeqSBAIJ_2_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1627: {
    1628:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1629:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs=A->rmap->bs,bs2=a->bs2;
    1630:   const MatScalar   *aa=a->a;
    1631:   PetscScalar       *x;
    1632:   const PetscScalar *b;
    1633:   PetscErrorCode    ierr;

    1636:   VecGetArrayRead(bb,&b);
    1637:   VecGetArray(xx,&x);
    1638:   PetscMemcpy(x,b,2*mbs*sizeof(PetscScalar));
    1639:   MatForwardSolve_SeqSBAIJ_2_NaturalOrdering(ai,aj,aa,mbs,x);
    1640:   VecRestoreArrayRead(bb,&b);
    1641:   VecRestoreArray(xx,&x);
    1642:   PetscLogFlops(2.0*bs2*a->nz - bs*mbs);
    1643:   return(0);
    1644: }

    1648: PetscErrorCode MatBackwardSolve_SeqSBAIJ_2_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    1649: {
    1650:   Mat_SeqSBAIJ      *a =(Mat_SeqSBAIJ*)A->data;
    1651:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,bs2=a->bs2;
    1652:   const MatScalar   *aa=a->a;
    1653:   PetscScalar       *x;
    1654:   const PetscScalar *b;
    1655:   PetscErrorCode    ierr;

    1658:   VecGetArrayRead(bb,&b);
    1659:   VecGetArray(xx,&x);
    1660:   PetscMemcpy(x,b,2*mbs*sizeof(PetscScalar));
    1661:   MatBackwardSolve_SeqSBAIJ_2_NaturalOrdering(ai,aj,aa,mbs,x);
    1662:   VecRestoreArrayRead(bb,&b);
    1663:   VecRestoreArray(xx,&x);
    1664:   PetscLogFlops(2.0*bs2*(a->nz - mbs));
    1665:   return(0);
    1666: }

    1670: PetscErrorCode MatSolve_SeqSBAIJ_1(Mat A,Vec bb,Vec xx)
    1671: {
    1672:   Mat_SeqSBAIJ      *a   = (Mat_SeqSBAIJ*)A->data;
    1673:   IS                isrow=a->row;
    1674:   PetscErrorCode    ierr;
    1675:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*rp,*vj,*adiag = a->diag;
    1676:   const MatScalar   *aa=a->a,*v;
    1677:   const PetscScalar *b;
    1678:   PetscScalar       *x,xk,*t;
    1679:   PetscInt          nz,k,j;

    1682:   VecGetArrayRead(bb,&b);
    1683:   VecGetArray(xx,&x);
    1684:   t    = a->solve_work;
    1685:   ISGetIndices(isrow,&rp);

    1687:   /* solve U^T*D*y = perm(b) by forward substitution */
    1688:   for (k=0; k<mbs; k++) t[k] = b[rp[k]];
    1689:   for (k=0; k<mbs; k++) {
    1690:     v  = aa + ai[k];
    1691:     vj = aj + ai[k];
    1692:     xk = t[k];
    1693:     nz = ai[k+1] - ai[k] - 1;
    1694:     for (j=0; j<nz; j++) t[vj[j]] += v[j]*xk;
    1695:     t[k] = xk*v[nz];   /* v[nz] = 1/D(k) */
    1696:   }

    1698:   /* solve U*perm(x) = y by back substitution */
    1699:   for (k=mbs-1; k>=0; k--) {
    1700:     v  = aa + adiag[k] - 1;
    1701:     vj = aj + adiag[k] - 1;
    1702:     nz = ai[k+1] - ai[k] - 1;
    1703:     for (j=0; j<nz; j++) t[k] += v[-j]*t[vj[-j]];
    1704:     x[rp[k]] = t[k];
    1705:   }

    1707:   ISRestoreIndices(isrow,&rp);
    1708:   VecRestoreArrayRead(bb,&b);
    1709:   VecRestoreArray(xx,&x);
    1710:   PetscLogFlops(4.0*a->nz - 3.0*mbs);
    1711:   return(0);
    1712: }

    1716: PetscErrorCode MatSolve_SeqSBAIJ_1_inplace(Mat A,Vec bb,Vec xx)
    1717: {
    1718:   Mat_SeqSBAIJ      *a   = (Mat_SeqSBAIJ*)A->data;
    1719:   IS                isrow=a->row;
    1720:   PetscErrorCode    ierr;
    1721:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*rp,*vj;
    1722:   const MatScalar   *aa=a->a,*v;
    1723:   PetscScalar       *x,xk,*t;
    1724:   const PetscScalar *b;
    1725:   PetscInt          nz,k;

    1728:   VecGetArrayRead(bb,&b);
    1729:   VecGetArray(xx,&x);
    1730:   t    = a->solve_work;
    1731:   ISGetIndices(isrow,&rp);

    1733:   /* solve U^T*D*y = perm(b) by forward substitution */
    1734:   for (k=0; k<mbs; k++) t[k] = b[rp[k]];
    1735:   for (k=0; k<mbs; k++) {
    1736:     v  = aa + ai[k] + 1;
    1737:     vj = aj + ai[k] + 1;
    1738:     xk = t[k];
    1739:     nz = ai[k+1] - ai[k] - 1;
    1740:     while (nz--) t[*vj++] += (*v++) * xk;
    1741:     t[k] = xk*aa[ai[k]];  /* aa[k] = 1/D(k) */
    1742:   }

    1744:   /* solve U*perm(x) = y by back substitution */
    1745:   for (k=mbs-1; k>=0; k--) {
    1746:     v  = aa + ai[k] + 1;
    1747:     vj = aj + ai[k] + 1;
    1748:     nz = ai[k+1] - ai[k] - 1;
    1749:     while (nz--) t[k] += (*v++) * t[*vj++];
    1750:     x[rp[k]] = t[k];
    1751:   }

    1753:   ISRestoreIndices(isrow,&rp);
    1754:   VecRestoreArrayRead(bb,&b);
    1755:   VecRestoreArray(xx,&x);
    1756:   PetscLogFlops(4.0*a->nz - 3*mbs);
    1757:   return(0);
    1758: }

    1762: PetscErrorCode MatForwardSolve_SeqSBAIJ_1(Mat A,Vec bb,Vec xx)
    1763: {
    1764:   Mat_SeqSBAIJ      *a   = (Mat_SeqSBAIJ*)A->data;
    1765:   IS                isrow=a->row;
    1766:   PetscErrorCode    ierr;
    1767:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*rp,*vj,*adiag = a->diag;
    1768:   const MatScalar   *aa=a->a,*v;
    1769:   PetscReal         diagk;
    1770:   PetscScalar       *x,xk;
    1771:   const PetscScalar *b;
    1772:   PetscInt          nz,k;

    1775:   /* solve U^T*D^(1/2)*x = perm(b) by forward substitution */
    1776:   VecGetArrayRead(bb,&b);
    1777:   VecGetArray(xx,&x);
    1778:   ISGetIndices(isrow,&rp);

    1780:   for (k=0; k<mbs; k++) x[k] = b[rp[k]];
    1781:   for (k=0; k<mbs; k++) {
    1782:     v  = aa + ai[k];
    1783:     vj = aj + ai[k];
    1784:     xk = x[k];
    1785:     nz = ai[k+1] - ai[k] - 1;
    1786:     while (nz--) x[*vj++] += (*v++) * xk;

    1788:     diagk = PetscRealPart(aa[adiag[k]]); /* note: aa[diag[k]] = 1/D(k) */
    1789:     if (PetscImaginaryPart(aa[adiag[k]]) || diagk < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal must be real and nonnegative");
    1790:     x[k] = xk*PetscSqrtReal(diagk);
    1791:   }
    1792:   ISRestoreIndices(isrow,&rp);
    1793:   VecRestoreArrayRead(bb,&b);
    1794:   VecRestoreArray(xx,&x);
    1795:   PetscLogFlops(2.0*a->nz - mbs);
    1796:   return(0);
    1797: }

    1801: PetscErrorCode MatForwardSolve_SeqSBAIJ_1_inplace(Mat A,Vec bb,Vec xx)
    1802: {
    1803:   Mat_SeqSBAIJ      *a   = (Mat_SeqSBAIJ*)A->data;
    1804:   IS                isrow=a->row;
    1805:   PetscErrorCode    ierr;
    1806:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*rp,*vj;
    1807:   const MatScalar   *aa=a->a,*v;
    1808:   PetscReal         diagk;
    1809:   PetscScalar       *x,xk;
    1810:   const PetscScalar *b;
    1811:   PetscInt          nz,k;

    1814:   /* solve U^T*D^(1/2)*x = perm(b) by forward substitution */
    1815:   VecGetArrayRead(bb,&b);
    1816:   VecGetArray(xx,&x);
    1817:   ISGetIndices(isrow,&rp);

    1819:   for (k=0; k<mbs; k++) x[k] = b[rp[k]];
    1820:   for (k=0; k<mbs; k++) {
    1821:     v  = aa + ai[k] + 1;
    1822:     vj = aj + ai[k] + 1;
    1823:     xk = x[k];
    1824:     nz = ai[k+1] - ai[k] - 1;
    1825:     while (nz--) x[*vj++] += (*v++) * xk;

    1827:     diagk = PetscRealPart(aa[ai[k]]); /* note: aa[diag[k]] = 1/D(k) */
    1828:     if (PetscImaginaryPart(aa[ai[k]]) || diagk < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal must be real and nonnegative");
    1829:     x[k] = xk*PetscSqrtReal(diagk);
    1830:   }
    1831:   ISRestoreIndices(isrow,&rp);
    1832:   VecRestoreArrayRead(bb,&b);
    1833:   VecRestoreArray(xx,&x);
    1834:   PetscLogFlops(2.0*a->nz - mbs);
    1835:   return(0);
    1836: }

    1840: PetscErrorCode MatBackwardSolve_SeqSBAIJ_1(Mat A,Vec bb,Vec xx)
    1841: {
    1842:   Mat_SeqSBAIJ      *a   = (Mat_SeqSBAIJ*)A->data;
    1843:   IS                isrow=a->row;
    1844:   PetscErrorCode    ierr;
    1845:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*rp,*vj,*adiag = a->diag;
    1846:   const MatScalar   *aa=a->a,*v;
    1847:   PetscReal         diagk;
    1848:   PetscScalar       *x,*t;
    1849:   const PetscScalar *b;
    1850:   PetscInt          nz,k;

    1853:   /* solve D^(1/2)*U*perm(x) = b by back substitution */
    1854:   VecGetArrayRead(bb,&b);
    1855:   VecGetArray(xx,&x);
    1856:   t    = a->solve_work;
    1857:   ISGetIndices(isrow,&rp);

    1859:   for (k=mbs-1; k>=0; k--) {
    1860:     v     = aa + ai[k];
    1861:     vj    = aj + ai[k];
    1862:     diagk = PetscRealPart(aa[adiag[k]]);
    1863:     if (PetscImaginaryPart(aa[adiag[k]]) || diagk < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal must be real and nonnegative");
    1864:     t[k] = b[k] * PetscSqrtReal(diagk);
    1865:     nz   = ai[k+1] - ai[k] - 1;
    1866:     while (nz--) t[k] += (*v++) * t[*vj++];
    1867:     x[rp[k]] = t[k];
    1868:   }
    1869:   ISRestoreIndices(isrow,&rp);
    1870:   VecRestoreArrayRead(bb,&b);
    1871:   VecRestoreArray(xx,&x);
    1872:   PetscLogFlops(2.0*a->nz - mbs);
    1873:   return(0);
    1874: }

    1878: PetscErrorCode MatBackwardSolve_SeqSBAIJ_1_inplace(Mat A,Vec bb,Vec xx)
    1879: {
    1880:   Mat_SeqSBAIJ      *a   = (Mat_SeqSBAIJ*)A->data;
    1881:   IS                isrow=a->row;
    1882:   PetscErrorCode    ierr;
    1883:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*rp,*vj;
    1884:   const MatScalar   *aa=a->a,*v;
    1885:   PetscReal         diagk;
    1886:   PetscScalar       *x,*t;
    1887:   const PetscScalar *b;
    1888:   PetscInt          nz,k;

    1891:   /* solve D^(1/2)*U*perm(x) = b by back substitution */
    1892:   VecGetArrayRead(bb,&b);
    1893:   VecGetArray(xx,&x);
    1894:   t    = a->solve_work;
    1895:   ISGetIndices(isrow,&rp);

    1897:   for (k=mbs-1; k>=0; k--) {
    1898:     v     = aa + ai[k] + 1;
    1899:     vj    = aj + ai[k] + 1;
    1900:     diagk = PetscRealPart(aa[ai[k]]);
    1901:     if (PetscImaginaryPart(aa[ai[k]]) || diagk < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal must be real and nonnegative");
    1902:     t[k] = b[k] * PetscSqrtReal(diagk);
    1903:     nz   = ai[k+1] - ai[k] - 1;
    1904:     while (nz--) t[k] += (*v++) * t[*vj++];
    1905:     x[rp[k]] = t[k];
    1906:   }
    1907:   ISRestoreIndices(isrow,&rp);
    1908:   VecRestoreArrayRead(bb,&b);
    1909:   VecRestoreArray(xx,&x);
    1910:   PetscLogFlops(2.0*a->nz - mbs);
    1911:   return(0);
    1912: }

    1916: PetscErrorCode MatSolves_SeqSBAIJ_1(Mat A,Vecs bb,Vecs xx)
    1917: {
    1918:   Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ*)A->data;

    1922:   if (A->rmap->bs == 1) {
    1923:     MatSolve_SeqSBAIJ_1(A,bb->v,xx->v);
    1924:   } else {
    1925:     IS                isrow=a->row;
    1926:     const PetscInt    *vj,mbs=a->mbs,*ai=a->i,*aj=a->j,*rp;
    1927:     const MatScalar   *aa=a->a,*v;
    1928:     PetscScalar       *x,*t;
    1929:     const PetscScalar *b;
    1930:     PetscInt          nz,k,n,i,j;

    1932:     if (bb->n > a->solves_work_n) {
    1933:       PetscFree(a->solves_work);
    1934:       PetscMalloc1(bb->n*A->rmap->N,&a->solves_work);
    1935:       a->solves_work_n = bb->n;
    1936:     }
    1937:     n    = bb->n;
    1938:     VecGetArrayRead(bb->v,&b);
    1939:     VecGetArray(xx->v,&x);
    1940:     t    = a->solves_work;

    1942:     ISGetIndices(isrow,&rp);

    1944:     /* solve U^T*D*y = perm(b) by forward substitution */
    1945:     for (k=0; k<mbs; k++) {
    1946:       for (i=0; i<n; i++) t[n*k+i] = b[rp[k]+i*mbs]; /* values are stored interlaced in t */
    1947:     }
    1948:     for (k=0; k<mbs; k++) {
    1949:       v  = aa + ai[k];
    1950:       vj = aj + ai[k];
    1951:       nz = ai[k+1] - ai[k] - 1;
    1952:       for (j=0; j<nz; j++) {
    1953:         for (i=0; i<n; i++) t[n*(*vj)+i] += (*v) * t[n*k+i];
    1954:         v++;vj++;
    1955:       }
    1956:       for (i=0; i<n; i++) t[n*k+i] *= aa[nz];  /* note: aa[nz] = 1/D(k) */
    1957:     }

    1959:     /* solve U*perm(x) = y by back substitution */
    1960:     for (k=mbs-1; k>=0; k--) {
    1961:       v  = aa + ai[k] - 1;
    1962:       vj = aj + ai[k] - 1;
    1963:       nz = ai[k+1] - ai[k] - 1;
    1964:       for (j=0; j<nz; j++) {
    1965:         for (i=0; i<n; i++) t[n*k+i] += (*v) * t[n*(*vj)+i];
    1966:         v++;vj++;
    1967:       }
    1968:       for (i=0; i<n; i++) x[rp[k]+i*mbs] = t[n*k+i];
    1969:     }

    1971:     ISRestoreIndices(isrow,&rp);
    1972:     VecRestoreArrayRead(bb->v,&b);
    1973:     VecRestoreArray(xx->v,&x);
    1974:     PetscLogFlops(bb->n*(4.0*a->nz - 3.0*mbs));
    1975:   }
    1976:   return(0);
    1977: }

    1981: PetscErrorCode MatSolves_SeqSBAIJ_1_inplace(Mat A,Vecs bb,Vecs xx)
    1982: {
    1983:   Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ*)A->data;

    1987:   if (A->rmap->bs == 1) {
    1988:     MatSolve_SeqSBAIJ_1_inplace(A,bb->v,xx->v);
    1989:   } else {
    1990:     IS                isrow=a->row;
    1991:     const PetscInt    *vj,mbs=a->mbs,*ai=a->i,*aj=a->j,*rp;
    1992:     const MatScalar   *aa=a->a,*v;
    1993:     PetscScalar       *x,*t;
    1994:     const PetscScalar *b;
    1995:     PetscInt          nz,k,n,i;

    1997:     if (bb->n > a->solves_work_n) {
    1998:       PetscFree(a->solves_work);
    1999:       PetscMalloc1(bb->n*A->rmap->N,&a->solves_work);
    2000:       a->solves_work_n = bb->n;
    2001:     }
    2002:     n    = bb->n;
    2003:     VecGetArrayRead(bb->v,&b);
    2004:     VecGetArray(xx->v,&x);
    2005:     t    = a->solves_work;

    2007:     ISGetIndices(isrow,&rp);

    2009:     /* solve U^T*D*y = perm(b) by forward substitution */
    2010:     for (k=0; k<mbs; k++) {
    2011:       for (i=0; i<n; i++) t[n*k+i] = b[rp[k]+i*mbs];  /* values are stored interlaced in t */
    2012:     }
    2013:     for (k=0; k<mbs; k++) {
    2014:       v  = aa + ai[k];
    2015:       vj = aj + ai[k];
    2016:       nz = ai[k+1] - ai[k];
    2017:       while (nz--) {
    2018:         for (i=0; i<n; i++) t[n*(*vj)+i] += (*v) * t[n*k+i];
    2019:         v++;vj++;
    2020:       }
    2021:       for (i=0; i<n; i++) t[n*k+i] *= aa[k];  /* note: aa[k] = 1/D(k) */
    2022:     }

    2024:     /* solve U*perm(x) = y by back substitution */
    2025:     for (k=mbs-1; k>=0; k--) {
    2026:       v  = aa + ai[k];
    2027:       vj = aj + ai[k];
    2028:       nz = ai[k+1] - ai[k];
    2029:       while (nz--) {
    2030:         for (i=0; i<n; i++) t[n*k+i] += (*v) * t[n*(*vj)+i];
    2031:         v++;vj++;
    2032:       }
    2033:       for (i=0; i<n; i++) x[rp[k]+i*mbs] = t[n*k+i];
    2034:     }

    2036:     ISRestoreIndices(isrow,&rp);
    2037:     VecRestoreArrayRead(bb->v,&b);
    2038:     VecRestoreArray(xx->v,&x);
    2039:     PetscLogFlops(bb->n*(4.0*a->nz - 3.0*mbs));
    2040:   }
    2041:   return(0);
    2042: }

    2046: PetscErrorCode MatSolve_SeqSBAIJ_1_NaturalOrdering(Mat A,Vec bb,Vec xx)
    2047: {
    2048:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    2049:   PetscErrorCode    ierr;
    2050:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*vj,*adiag = a->diag;
    2051:   const MatScalar   *aa=a->a,*v;
    2052:   const PetscScalar *b;
    2053:   PetscScalar       *x,xi;
    2054:   PetscInt          nz,i,j;

    2057:   VecGetArrayRead(bb,&b);
    2058:   VecGetArray(xx,&x);

    2060:   /* solve U^T*D*y = b by forward substitution */
    2061:   PetscMemcpy(x,b,mbs*sizeof(PetscScalar));
    2062:   for (i=0; i<mbs; i++) {
    2063:     v  = aa + ai[i];
    2064:     vj = aj + ai[i];
    2065:     xi = x[i];
    2066:     nz = ai[i+1] - ai[i] - 1; /* exclude diag[i] */
    2067:     for (j=0; j<nz; j++) x[vj[j]] += v[j]* xi;
    2068:     x[i] = xi*v[nz];  /* v[nz] = aa[diag[i]] = 1/D(i) */
    2069:   }

    2071:   /* solve U*x = y by backward substitution */
    2072:   for (i=mbs-2; i>=0; i--) {
    2073:     xi = x[i];
    2074:     v  = aa + adiag[i] - 1; /* end of row i, excluding diag */
    2075:     vj = aj + adiag[i] - 1;
    2076:     nz = ai[i+1] - ai[i] - 1;
    2077:     for (j=0; j<nz; j++) xi += v[-j]*x[vj[-j]];
    2078:     x[i] = xi;
    2079:   }

    2081:   VecRestoreArrayRead(bb,&b);
    2082:   VecRestoreArray(xx,&x);
    2083:   PetscLogFlops(4.0*a->nz - 3*mbs);
    2084:   return(0);
    2085: }

    2089: PetscErrorCode MatSolve_SeqSBAIJ_1_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    2090: {
    2091:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    2092:   PetscErrorCode    ierr;
    2093:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*vj;
    2094:   const MatScalar   *aa=a->a,*v;
    2095:   PetscScalar       *x,xk;
    2096:   const PetscScalar *b;
    2097:   PetscInt          nz,k;

    2100:   VecGetArrayRead(bb,&b);
    2101:   VecGetArray(xx,&x);

    2103:   /* solve U^T*D*y = b by forward substitution */
    2104:   PetscMemcpy(x,b,mbs*sizeof(PetscScalar));
    2105:   for (k=0; k<mbs; k++) {
    2106:     v  = aa + ai[k] + 1;
    2107:     vj = aj + ai[k] + 1;
    2108:     xk = x[k];
    2109:     nz = ai[k+1] - ai[k] - 1;     /* exclude diag[k] */
    2110:     while (nz--) x[*vj++] += (*v++) * xk;
    2111:     x[k] = xk*aa[ai[k]];  /* note: aa[diag[k]] = 1/D(k) */
    2112:   }

    2114:   /* solve U*x = y by back substitution */
    2115:   for (k=mbs-2; k>=0; k--) {
    2116:     v  = aa + ai[k] + 1;
    2117:     vj = aj + ai[k] + 1;
    2118:     xk = x[k];
    2119:     nz = ai[k+1] - ai[k] - 1;
    2120:     while (nz--) {
    2121:       xk += (*v++) * x[*vj++];
    2122:     }
    2123:     x[k] = xk;
    2124:   }

    2126:   VecRestoreArrayRead(bb,&b);
    2127:   VecRestoreArray(xx,&x);
    2128:   PetscLogFlops(4.0*a->nz - 3*mbs);
    2129:   return(0);
    2130: }

    2134: PetscErrorCode MatForwardSolve_SeqSBAIJ_1_NaturalOrdering(Mat A,Vec bb,Vec xx)
    2135: {
    2136:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    2137:   PetscErrorCode    ierr;
    2138:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*adiag = a->diag,*vj;
    2139:   const MatScalar   *aa=a->a,*v;
    2140:   PetscReal         diagk;
    2141:   PetscScalar       *x;
    2142:   const PetscScalar *b;
    2143:   PetscInt          nz,k;

    2146:   /* solve U^T*D^(1/2)*x = b by forward substitution */
    2147:   VecGetArrayRead(bb,&b);
    2148:   VecGetArray(xx,&x);
    2149:   PetscMemcpy(x,b,mbs*sizeof(PetscScalar));
    2150:   for (k=0; k<mbs; k++) {
    2151:     v  = aa + ai[k];
    2152:     vj = aj + ai[k];
    2153:     nz = ai[k+1] - ai[k] - 1;     /* exclude diag[k] */
    2154:     while (nz--) x[*vj++] += (*v++) * x[k];
    2155:     diagk = PetscRealPart(aa[adiag[k]]); /* note: aa[adiag[k]] = 1/D(k) */
    2156:     if (PetscImaginaryPart(aa[adiag[k]]) || diagk < 0) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal (%g,%g) must be real and nonnegative",PetscRealPart(aa[adiag[k]]),PetscImaginaryPart(aa[adiag[k]]));
    2157:     x[k] *= PetscSqrtReal(diagk);
    2158:   }
    2159:   VecRestoreArrayRead(bb,&b);
    2160:   VecRestoreArray(xx,&x);
    2161:   PetscLogFlops(2.0*a->nz - mbs);
    2162:   return(0);
    2163: }

    2167: PetscErrorCode MatForwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    2168: {
    2169:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    2170:   PetscErrorCode    ierr;
    2171:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*vj;
    2172:   const MatScalar   *aa=a->a,*v;
    2173:   PetscReal         diagk;
    2174:   PetscScalar       *x;
    2175:   const PetscScalar *b;
    2176:   PetscInt          nz,k;

    2179:   /* solve U^T*D^(1/2)*x = b by forward substitution */
    2180:   VecGetArrayRead(bb,&b);
    2181:   VecGetArray(xx,&x);
    2182:   PetscMemcpy(x,b,mbs*sizeof(PetscScalar));
    2183:   for (k=0; k<mbs; k++) {
    2184:     v  = aa + ai[k] + 1;
    2185:     vj = aj + ai[k] + 1;
    2186:     nz = ai[k+1] - ai[k] - 1;     /* exclude diag[k] */
    2187:     while (nz--) x[*vj++] += (*v++) * x[k];
    2188:     diagk = PetscRealPart(aa[ai[k]]); /* note: aa[diag[k]] = 1/D(k) */
    2189:     if (PetscImaginaryPart(aa[ai[k]]) || diagk < 0) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal (%g,%g) must be real and nonnegative",PetscRealPart(aa[ai[k]]),PetscImaginaryPart(aa[ai[k]]));
    2190:     x[k] *= PetscSqrtReal(diagk);
    2191:   }
    2192:   VecRestoreArrayRead(bb,&b);
    2193:   VecRestoreArray(xx,&x);
    2194:   PetscLogFlops(2.0*a->nz - mbs);
    2195:   return(0);
    2196: }

    2200: PetscErrorCode MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering(Mat A,Vec bb,Vec xx)
    2201: {
    2202:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    2203:   PetscErrorCode    ierr;
    2204:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*adiag = a->diag,*vj;
    2205:   const MatScalar   *aa=a->a,*v;
    2206:   PetscReal         diagk;
    2207:   PetscScalar       *x;
    2208:   const PetscScalar *b;
    2209:   PetscInt          nz,k;

    2212:   /* solve D^(1/2)*U*x = b by back substitution */
    2213:   VecGetArrayRead(bb,&b);
    2214:   VecGetArray(xx,&x);

    2216:   for (k=mbs-1; k>=0; k--) {
    2217:     v     = aa + ai[k];
    2218:     vj    = aj + ai[k];
    2219:     diagk = PetscRealPart(aa[adiag[k]]); /* note: aa[diag[k]] = 1/D(k) */
    2220:     if (PetscImaginaryPart(aa[adiag[k]]) || diagk < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal must be real and nonnegative");
    2221:     x[k] = PetscSqrtReal(diagk)*b[k];
    2222:     nz   = ai[k+1] - ai[k] - 1;
    2223:     while (nz--) x[k] += (*v++) * x[*vj++];
    2224:   }
    2225:   VecRestoreArrayRead(bb,&b);
    2226:   VecRestoreArray(xx,&x);
    2227:   PetscLogFlops(2.0*a->nz - mbs);
    2228:   return(0);
    2229: }

    2233: PetscErrorCode MatBackwardSolve_SeqSBAIJ_1_NaturalOrdering_inplace(Mat A,Vec bb,Vec xx)
    2234: {
    2235:   Mat_SeqSBAIJ      *a = (Mat_SeqSBAIJ*)A->data;
    2236:   PetscErrorCode    ierr;
    2237:   const PetscInt    mbs=a->mbs,*ai=a->i,*aj=a->j,*vj;
    2238:   const MatScalar   *aa=a->a,*v;
    2239:   PetscReal         diagk;
    2240:   PetscScalar       *x;
    2241:   const PetscScalar *b;
    2242:   PetscInt          nz,k;

    2245:   /* solve D^(1/2)*U*x = b by back substitution */
    2246:   VecGetArrayRead(bb,&b);
    2247:   VecGetArray(xx,&x);

    2249:   for (k=mbs-1; k>=0; k--) {
    2250:     v     = aa + ai[k] + 1;
    2251:     vj    = aj + ai[k] + 1;
    2252:     diagk = PetscRealPart(aa[ai[k]]); /* note: aa[diag[k]] = 1/D(k) */
    2253:     if (PetscImaginaryPart(aa[ai[k]]) || diagk < 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Diagonal must be real and nonnegative");
    2254:     x[k] = PetscSqrtReal(diagk)*b[k];
    2255:     nz   = ai[k+1] - ai[k] - 1;
    2256:     while (nz--) x[k] += (*v++) * x[*vj++];
    2257:   }
    2258:   VecRestoreArrayRead(bb,&b);
    2259:   VecRestoreArray(xx,&x);
    2260:   PetscLogFlops(2.0*a->nz - mbs);
    2261:   return(0);
    2262: }

    2264: /* Use Modified Sparse Row storage for u and ju, see Saad pp.85 */
    2267: PetscErrorCode MatICCFactorSymbolic_SeqSBAIJ_MSR(Mat B,Mat A,IS perm,const MatFactorInfo *info)
    2268: {
    2269:   Mat_SeqSBAIJ   *a = (Mat_SeqSBAIJ*)A->data,*b;
    2271:   const PetscInt *rip,mbs = a->mbs,*ai,*aj;
    2272:   PetscInt       *jutmp,bs = A->rmap->bs,bs2=a->bs2,i;
    2273:   PetscInt       m,reallocs = 0,*levtmp;
    2274:   PetscInt       *prowl,*q,jmin,jmax,juidx,nzk,qm,*iu,*ju,k,j,vj,umax,maxadd;
    2275:   PetscInt       incrlev,*lev,shift,prow,nz;
    2276:   PetscReal      f = info->fill,levels = info->levels;
    2277:   PetscBool      perm_identity;

    2280:   /* check whether perm is the identity mapping */
    2281:   ISIdentity(perm,&perm_identity);

    2283:   if (perm_identity) {
    2284:     a->permute = PETSC_FALSE;
    2285:     ai         = a->i; aj = a->j;
    2286:   } else { /*  non-trivial permutation */
    2287:     SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Matrix reordering is not supported for sbaij matrix. Use aij format");
    2288:     a->permute = PETSC_TRUE;
    2289:     MatReorderingSeqSBAIJ(A, perm);
    2290:     ai         = a->inew; aj = a->jnew;
    2291:   }

    2293:   /* initialization */
    2294:   ISGetIndices(perm,&rip);
    2295:   umax  = (PetscInt)(f*ai[mbs] + 1);
    2296:   PetscMalloc1(umax,&lev);
    2297:   umax += mbs + 1;
    2298:   shift = mbs + 1;
    2299:   PetscMalloc1(mbs+1,&iu);
    2300:   PetscMalloc1(umax,&ju);
    2301:   iu[0] = mbs + 1;
    2302:   juidx = mbs + 1;
    2303:   /* prowl: linked list for pivot row */
    2304:   PetscMalloc3(mbs,&prowl,mbs,&q,mbs,&levtmp);
    2305:   /* q: linked list for col index */

    2307:   for (i=0; i<mbs; i++) {
    2308:     prowl[i] = mbs;
    2309:     q[i]     = 0;
    2310:   }

    2312:   /* for each row k */
    2313:   for (k=0; k<mbs; k++) {
    2314:     nzk  = 0;
    2315:     q[k] = mbs;
    2316:     /* copy current row into linked list */
    2317:     nz = ai[rip[k]+1] - ai[rip[k]];
    2318:     j  = ai[rip[k]];
    2319:     while (nz--) {
    2320:       vj = rip[aj[j++]];
    2321:       if (vj > k) {
    2322:         qm = k;
    2323:         do {
    2324:           m = qm; qm = q[m];
    2325:         } while (qm < vj);
    2326:         if (qm == vj) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Duplicate entry in A\n");
    2327:         nzk++;
    2328:         q[m]       = vj;
    2329:         q[vj]      = qm;
    2330:         levtmp[vj] = 0;   /* initialize lev for nonzero element */
    2331:       }
    2332:     }

    2334:     /* modify nonzero structure of k-th row by computing fill-in
    2335:        for each row prow to be merged in */
    2336:     prow = k;
    2337:     prow = prowl[prow]; /* next pivot row (== 0 for symbolic factorization) */

    2339:     while (prow < k) {
    2340:       /* merge row prow into k-th row */
    2341:       jmin = iu[prow] + 1;
    2342:       jmax = iu[prow+1];
    2343:       qm   = k;
    2344:       for (j=jmin; j<jmax; j++) {
    2345:         incrlev = lev[j-shift] + 1;
    2346:         if (incrlev > levels) continue;
    2347:         vj = ju[j];
    2348:         do {
    2349:           m = qm; qm = q[m];
    2350:         } while (qm < vj);
    2351:         if (qm != vj) {      /* a fill */
    2352:           nzk++; q[m] = vj; q[vj] = qm; qm = vj;
    2353:           levtmp[vj] = incrlev;
    2354:         } else if (levtmp[vj] > incrlev) levtmp[vj] = incrlev;
    2355:       }
    2356:       prow = prowl[prow]; /* next pivot row */
    2357:     }

    2359:     /* add k to row list for first nonzero element in k-th row */
    2360:     if (nzk > 1) {
    2361:       i        = q[k]; /* col value of first nonzero element in k_th row of U */
    2362:       prowl[k] = prowl[i]; prowl[i] = k;
    2363:     }
    2364:     iu[k+1] = iu[k] + nzk;

    2366:     /* allocate more space to ju and lev if needed */
    2367:     if (iu[k+1] > umax) {
    2368:       /* estimate how much additional space we will need */
    2369:       /* use the strategy suggested by David Hysom <hysom@perch-t.icase.edu> */
    2370:       /* just double the memory each time */
    2371:       maxadd = umax;
    2372:       if (maxadd < nzk) maxadd = (mbs-k)*(nzk+1)/2;
    2373:       umax += maxadd;

    2375:       /* allocate a longer ju */
    2376:       PetscMalloc1(umax,&jutmp);
    2377:       PetscMemcpy(jutmp,ju,iu[k]*sizeof(PetscInt));
    2378:       PetscFree(ju);
    2379:       ju   = jutmp;

    2381:       PetscMalloc1(umax,&jutmp);
    2382:       PetscMemcpy(jutmp,lev,(iu[k]-shift)*sizeof(PetscInt));
    2383:       PetscFree(lev);
    2384:       lev       = jutmp;
    2385:       reallocs += 2; /* count how many times we realloc */
    2386:     }

    2388:     /* save nonzero structure of k-th row in ju */
    2389:     i=k;
    2390:     while (nzk--) {
    2391:       i                = q[i];
    2392:       ju[juidx]        = i;
    2393:       lev[juidx-shift] = levtmp[i];
    2394:       juidx++;
    2395:     }
    2396:   }

    2398: #if defined(PETSC_USE_INFO)
    2399:   if (ai[mbs] != 0) {
    2400:     PetscReal af = ((PetscReal)iu[mbs])/((PetscReal)ai[mbs]);
    2401:     PetscInfo3(A,"Reallocs %D Fill ratio:given %g needed %g\n",reallocs,(double)f,(double)af);
    2402:     PetscInfo1(A,"Run with -pc_factor_fill %g or use \n",(double)af);
    2403:     PetscInfo1(A,"PCFactorSetFill(pc,%g);\n",(double)af);
    2404:     PetscInfo(A,"for best performance.\n");
    2405:   } else {
    2406:     PetscInfo(A,"Empty matrix.\n");
    2407:   }
    2408: #endif

    2410:   ISRestoreIndices(perm,&rip);
    2411:   PetscFree3(prowl,q,levtmp);
    2412:   PetscFree(lev);

    2414:   /* put together the new matrix */
    2415:   MatSeqSBAIJSetPreallocation_SeqSBAIJ(B,bs,0,NULL);

    2417:   /* PetscLogObjectParent((PetscObject)B,(PetscObject)iperm); */
    2418:   b    = (Mat_SeqSBAIJ*)(B)->data;
    2419:   PetscFree2(b->imax,b->ilen);

    2421:   b->singlemalloc = PETSC_FALSE;
    2422:   b->free_a       = PETSC_TRUE;
    2423:   b->free_ij      = PETSC_TRUE;
    2424:   /* the next line frees the default space generated by the Create() */
    2425:   PetscFree3(b->a,b->j,b->i);
    2426:   PetscMalloc1((iu[mbs]+1)*bs2,&b->a);
    2427:   b->j    = ju;
    2428:   b->i    = iu;
    2429:   b->diag = 0;
    2430:   b->ilen = 0;
    2431:   b->imax = 0;

    2433:   ISDestroy(&b->row);
    2434:   ISDestroy(&b->icol);
    2435:   b->row  = perm;
    2436:   b->icol = perm;
    2437:   PetscObjectReference((PetscObject)perm);
    2438:   PetscObjectReference((PetscObject)perm);
    2439:   PetscMalloc1(bs*mbs+bs,&b->solve_work);
    2440:   /* In b structure:  Free imax, ilen, old a, old j.
    2441:      Allocate idnew, solve_work, new a, new j */
    2442:   PetscLogObjectMemory((PetscObject)B,(iu[mbs]-mbs)*(sizeof(PetscInt)+sizeof(MatScalar)));
    2443:   b->maxnz = b->nz = iu[mbs];

    2445:   (B)->info.factor_mallocs   = reallocs;
    2446:   (B)->info.fill_ratio_given = f;
    2447:   if (ai[mbs] != 0) {
    2448:     (B)->info.fill_ratio_needed = ((PetscReal)iu[mbs])/((PetscReal)ai[mbs]);
    2449:   } else {
    2450:     (B)->info.fill_ratio_needed = 0.0;
    2451:   }
    2452:   MatSeqSBAIJSetNumericFactorization_inplace(B,perm_identity);
    2453:   return(0);
    2454: }

    2456: /*
    2457:   See MatICCFactorSymbolic_SeqAIJ() for description of its data structure
    2458: */
    2459: #include <petscbt.h>
    2460: #include <../src/mat/utils/freespace.h>
    2463: PetscErrorCode MatICCFactorSymbolic_SeqSBAIJ(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
    2464: {
    2465:   Mat_SeqSBAIJ       *a = (Mat_SeqSBAIJ*)A->data,*b;
    2466:   PetscErrorCode     ierr;
    2467:   PetscBool          perm_identity,free_ij = PETSC_TRUE,missing;
    2468:   PetscInt           bs=A->rmap->bs,am=a->mbs,d,*ai=a->i,*aj= a->j;
    2469:   const PetscInt     *rip;
    2470:   PetscInt           reallocs=0,i,*ui,*udiag,*cols;
    2471:   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
    2472:   PetscInt           nlnk,*lnk,*lnk_lvl=NULL,ncols,*uj,**uj_ptr,**uj_lvl_ptr;
    2473:   PetscReal          fill          =info->fill,levels=info->levels;
    2474:   PetscFreeSpaceList free_space    =NULL,current_space=NULL;
    2475:   PetscFreeSpaceList free_space_lvl=NULL,current_space_lvl=NULL;
    2476:   PetscBT            lnkbt;

    2479:   if (A->rmap->n != A->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Must be square matrix, rows %D columns %D",A->rmap->n,A->cmap->n);
    2480:   MatMissingDiagonal(A,&missing,&d);
    2481:   if (missing) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Matrix is missing diagonal entry %D",d);
    2482:   if (bs > 1) {
    2483:     MatICCFactorSymbolic_SeqSBAIJ_inplace(fact,A,perm,info);
    2484:     return(0);
    2485:   }

    2487:   /* check whether perm is the identity mapping */
    2488:   ISIdentity(perm,&perm_identity);
    2489:   if (!perm_identity) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Matrix reordering is not supported for sbaij matrix. Use aij format");
    2490:   a->permute = PETSC_FALSE;

    2492:   PetscMalloc1(am+1,&ui);
    2493:   PetscMalloc1(am+1,&udiag);
    2494:   ui[0] = 0;

    2496:   /* ICC(0) without matrix ordering: simply rearrange column indices */
    2497:   if (!levels) {
    2498:     /* reuse the column pointers and row offsets for memory savings */
    2499:     for (i=0; i<am; i++) {
    2500:       ncols    = ai[i+1] - ai[i];
    2501:       ui[i+1]  = ui[i] + ncols;
    2502:       udiag[i] = ui[i+1] - 1; /* points to the last entry of U(i,:) */
    2503:     }
    2504:     PetscMalloc1(ui[am]+1,&uj);
    2505:     cols = uj;
    2506:     for (i=0; i<am; i++) {
    2507:       aj    = a->j + ai[i] + 1; /* 1st entry of U(i,:) without diagonal */
    2508:       ncols = ai[i+1] - ai[i] -1;
    2509:       for (j=0; j<ncols; j++) *cols++ = aj[j];
    2510:       *cols++ = i; /* diagoanl is located as the last entry of U(i,:) */
    2511:     }
    2512:   } else { /* case: levels>0 */
    2513:     ISGetIndices(perm,&rip);

    2515:     /* initialization */
    2516:     /* jl: linked list for storing indices of the pivot rows
    2517:        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
    2518:     PetscMalloc4(am,&uj_ptr,am,&uj_lvl_ptr,am,&il,am,&jl);
    2519:     for (i=0; i<am; i++) {
    2520:       jl[i] = am; il[i] = 0;
    2521:     }

    2523:     /* create and initialize a linked list for storing column indices of the active row k */
    2524:     nlnk = am + 1;
    2525:     PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);

    2527:     /* initial FreeSpace size is fill*(ai[am]+1) */
    2528:     PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);

    2530:     current_space = free_space;

    2532:     PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);

    2534:     current_space_lvl = free_space_lvl;

    2536:     for (k=0; k<am; k++) {  /* for each active row k */
    2537:       /* initialize lnk by the column indices of row k */
    2538:       nzk   = 0;
    2539:       ncols = ai[k+1] - ai[k];
    2540:       if (!ncols) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_MAT_CH_ZRPVT,"Empty row %D in matrix ",k);
    2541:       cols = aj+ai[k];
    2542:       PetscIncompleteLLInit(ncols,cols,am,rip,nlnk,lnk,lnk_lvl,lnkbt);
    2543:       nzk += nlnk;

    2545:       /* update lnk by computing fill-in for each pivot row to be merged in */
    2546:       prow = jl[k]; /* 1st pivot row */

    2548:       while (prow < k) {
    2549:         nextprow = jl[prow];

    2551:         /* merge prow into k-th row */
    2552:         jmin  = il[prow] + 1; /* index of the 2nd nzero entry in U(prow,k:am-1) */
    2553:         jmax  = ui[prow+1];
    2554:         ncols = jmax-jmin;
    2555:         i     = jmin - ui[prow];

    2557:         cols = uj_ptr[prow] + i;  /* points to the 2nd nzero entry in U(prow,k:am-1) */
    2558:         uj   = uj_lvl_ptr[prow] + i;  /* levels of cols */
    2559:         j    = *(uj - 1);
    2560:         PetscICCLLAddSorted(ncols,cols,levels,uj,am,nlnk,lnk,lnk_lvl,lnkbt,j);
    2561:         nzk += nlnk;

    2563:         /* update il and jl for prow */
    2564:         if (jmin < jmax) {
    2565:           il[prow] = jmin;

    2567:           j = *cols; jl[prow] = jl[j]; jl[j] = prow;
    2568:         }
    2569:         prow = nextprow;
    2570:       }

    2572:       /* if free space is not available, make more free space */
    2573:       if (current_space->local_remaining<nzk) {
    2574:         i    = am - k + 1; /* num of unfactored rows */
    2575:         i   *= PetscMin(nzk, i-1); /* i*nzk, i*(i-1): estimated and max additional space needed */
    2576:         PetscFreeSpaceGet(i,&current_space);
    2577:         PetscFreeSpaceGet(i,&current_space_lvl);
    2578:         reallocs++;
    2579:       }

    2581:       /* copy data into free_space and free_space_lvl, then initialize lnk */
    2582:       if (nzk == 0) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Empty row %D in ICC matrix factor",k);
    2583:       PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);

    2585:       /* add the k-th row into il and jl */
    2586:       if (nzk > 1) {
    2587:         i     = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
    2588:         jl[k] = jl[i]; jl[i] = k;
    2589:         il[k] = ui[k] + 1;
    2590:       }
    2591:       uj_ptr[k]     = current_space->array;
    2592:       uj_lvl_ptr[k] = current_space_lvl->array;

    2594:       current_space->array               += nzk;
    2595:       current_space->local_used          += nzk;
    2596:       current_space->local_remaining     -= nzk;
    2597:       current_space_lvl->array           += nzk;
    2598:       current_space_lvl->local_used      += nzk;
    2599:       current_space_lvl->local_remaining -= nzk;

    2601:       ui[k+1] = ui[k] + nzk;
    2602:     }

    2604:     ISRestoreIndices(perm,&rip);
    2605:     PetscFree4(uj_ptr,uj_lvl_ptr,il,jl);

    2607:     /* destroy list of free space and other temporary array(s) */
    2608:     PetscMalloc1(ui[am]+1,&uj);
    2609:     PetscFreeSpaceContiguous_Cholesky(&free_space,uj,am,ui,udiag); /* store matrix factor  */
    2610:     PetscIncompleteLLDestroy(lnk,lnkbt);
    2611:     PetscFreeSpaceDestroy(free_space_lvl);

    2613:   } /* end of case: levels>0 || (levels=0 && !perm_identity) */

    2615:   /* put together the new matrix in MATSEQSBAIJ format */
    2616:   MatSeqSBAIJSetPreallocation_SeqSBAIJ(fact,bs,MAT_SKIP_ALLOCATION,NULL);

    2618:   b    = (Mat_SeqSBAIJ*)(fact)->data;
    2619:   PetscFree2(b->imax,b->ilen);

    2621:   b->singlemalloc = PETSC_FALSE;
    2622:   b->free_a       = PETSC_TRUE;
    2623:   b->free_ij      = free_ij;

    2625:   PetscMalloc1(ui[am]+1,&b->a);

    2627:   b->j         = uj;
    2628:   b->i         = ui;
    2629:   b->diag      = udiag;
    2630:   b->free_diag = PETSC_TRUE;
    2631:   b->ilen      = 0;
    2632:   b->imax      = 0;
    2633:   b->row       = perm;
    2634:   b->col       = perm;

    2636:   PetscObjectReference((PetscObject)perm);
    2637:   PetscObjectReference((PetscObject)perm);

    2639:   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */

    2641:   PetscMalloc1(am+1,&b->solve_work);
    2642:   PetscLogObjectMemory((PetscObject)fact,ui[am]*(sizeof(PetscInt)+sizeof(MatScalar)));

    2644:   b->maxnz = b->nz = ui[am];

    2646:   fact->info.factor_mallocs   = reallocs;
    2647:   fact->info.fill_ratio_given = fill;
    2648:   if (ai[am] != 0) {
    2649:     fact->info.fill_ratio_needed = ((PetscReal)ui[am])/ai[am];
    2650:   } else {
    2651:     fact->info.fill_ratio_needed = 0.0;
    2652:   }
    2653: #if defined(PETSC_USE_INFO)
    2654:   if (ai[am] != 0) {
    2655:     PetscReal af = fact->info.fill_ratio_needed;
    2656:     PetscInfo3(A,"Reallocs %D Fill ratio:given %g needed %g\n",reallocs,(double)fill,(double)af);
    2657:     PetscInfo1(A,"Run with -pc_factor_fill %g or use \n",(double)af);
    2658:     PetscInfo1(A,"PCFactorSetFill(pc,%g) for best performance.\n",(double)af);
    2659:   } else {
    2660:     PetscInfo(A,"Empty matrix.\n");
    2661:   }
    2662: #endif
    2663:   fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering;
    2664:   return(0);
    2665: }

    2669: PetscErrorCode MatICCFactorSymbolic_SeqSBAIJ_inplace(Mat fact,Mat A,IS perm,const MatFactorInfo *info)
    2670: {
    2671:   Mat_SeqSBAIJ       *a = (Mat_SeqSBAIJ*)A->data;
    2672:   Mat_SeqSBAIJ       *b;
    2673:   PetscErrorCode     ierr;
    2674:   PetscBool          perm_identity,free_ij = PETSC_TRUE;
    2675:   PetscInt           bs=A->rmap->bs,am=a->mbs;
    2676:   const PetscInt     *cols,*rip,*ai=a->i,*aj=a->j;
    2677:   PetscInt           reallocs=0,i,*ui;
    2678:   PetscInt           jmin,jmax,nzk,k,j,*jl,prow,*il,nextprow;
    2679:   PetscInt           nlnk,*lnk,*lnk_lvl=NULL,ncols,*cols_lvl,*uj,**uj_ptr,**uj_lvl_ptr;
    2680:   PetscReal          fill          =info->fill,levels=info->levels,ratio_needed;
    2681:   PetscFreeSpaceList free_space    =NULL,current_space=NULL;
    2682:   PetscFreeSpaceList free_space_lvl=NULL,current_space_lvl=NULL;
    2683:   PetscBT            lnkbt;

    2686:   /*
    2687:    This code originally uses Modified Sparse Row (MSR) storage
    2688:    (see page 85, "Iterative Methods ..." by Saad) for the output matrix B - bad choice!
    2689:    Then it is rewritten so the factor B takes seqsbaij format. However the associated
    2690:    MatCholeskyFactorNumeric_() have not been modified for the cases of bs>1,
    2691:    thus the original code in MSR format is still used for these cases.
    2692:    The code below should replace MatICCFactorSymbolic_SeqSBAIJ_MSR() whenever
    2693:    MatCholeskyFactorNumeric_() is modified for using sbaij symbolic factor.
    2694:   */
    2695:   if (bs > 1) {
    2696:     MatICCFactorSymbolic_SeqSBAIJ_MSR(fact,A,perm,info);
    2697:     return(0);
    2698:   }

    2700:   /* check whether perm is the identity mapping */
    2701:   ISIdentity(perm,&perm_identity);
    2702:   if (!perm_identity) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Matrix reordering is not supported for sbaij matrix. Use aij format");
    2703:   a->permute = PETSC_FALSE;

    2705:   /* special case that simply copies fill pattern */
    2706:   if (!levels) {
    2707:     /* reuse the column pointers and row offsets for memory savings */
    2708:     ui           = a->i;
    2709:     uj           = a->j;
    2710:     free_ij      = PETSC_FALSE;
    2711:     ratio_needed = 1.0;
    2712:   } else { /* case: levels>0 */
    2713:     ISGetIndices(perm,&rip);

    2715:     /* initialization */
    2716:     PetscMalloc1(am+1,&ui);
    2717:     ui[0] = 0;

    2719:     /* jl: linked list for storing indices of the pivot rows
    2720:        il: il[i] points to the 1st nonzero entry of U(i,k:am-1) */
    2721:     PetscMalloc4(am,&uj_ptr,am,&uj_lvl_ptr,am,&il,am,&jl);
    2722:     for (i=0; i<am; i++) {
    2723:       jl[i] = am; il[i] = 0;
    2724:     }

    2726:     /* create and initialize a linked list for storing column indices of the active row k */
    2727:     nlnk = am + 1;
    2728:     PetscIncompleteLLCreate(am,am,nlnk,lnk,lnk_lvl,lnkbt);

    2730:     /* initial FreeSpace size is fill*(ai[am]+1) */
    2731:     PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space);

    2733:     current_space = free_space;

    2735:     PetscFreeSpaceGet((PetscInt)(fill*(ai[am]+1)),&free_space_lvl);

    2737:     current_space_lvl = free_space_lvl;

    2739:     for (k=0; k<am; k++) {  /* for each active row k */
    2740:       /* initialize lnk by the column indices of row rip[k] */
    2741:       nzk   = 0;
    2742:       ncols = ai[rip[k]+1] - ai[rip[k]];
    2743:       cols  = aj+ai[rip[k]];
    2744:       PetscIncompleteLLInit(ncols,cols,am,rip,nlnk,lnk,lnk_lvl,lnkbt);
    2745:       nzk  += nlnk;

    2747:       /* update lnk by computing fill-in for each pivot row to be merged in */
    2748:       prow = jl[k]; /* 1st pivot row */

    2750:       while (prow < k) {
    2751:         nextprow = jl[prow];

    2753:         /* merge prow into k-th row */
    2754:         jmin     = il[prow] + 1; /* index of the 2nd nzero entry in U(prow,k:am-1) */
    2755:         jmax     = ui[prow+1];
    2756:         ncols    = jmax-jmin;
    2757:         i        = jmin - ui[prow];
    2758:         cols     = uj_ptr[prow] + i; /* points to the 2nd nzero entry in U(prow,k:am-1) */
    2759:         j        = *(uj_lvl_ptr[prow] + i - 1);
    2760:         cols_lvl = uj_lvl_ptr[prow]+i;
    2761:         PetscICCLLAddSorted(ncols,cols,levels,cols_lvl,am,nlnk,lnk,lnk_lvl,lnkbt,j);
    2762:         nzk     += nlnk;

    2764:         /* update il and jl for prow */
    2765:         if (jmin < jmax) {
    2766:           il[prow] = jmin;
    2767:           j        = *cols;
    2768:           jl[prow] = jl[j];
    2769:           jl[j]    = prow;
    2770:         }
    2771:         prow = nextprow;
    2772:       }

    2774:       /* if free space is not available, make more free space */
    2775:       if (current_space->local_remaining<nzk) {
    2776:         i    = am - k + 1; /* num of unfactored rows */
    2777:         i    = PetscMin(i*nzk, i*(i-1)); /* i*nzk, i*(i-1): estimated and max additional space needed */
    2778:         PetscFreeSpaceGet(i,&current_space);
    2779:         PetscFreeSpaceGet(i,&current_space_lvl);
    2780:         reallocs++;
    2781:       }

    2783:       /* copy data into free_space and free_space_lvl, then initialize lnk */
    2784:       PetscIncompleteLLClean(am,am,nzk,lnk,lnk_lvl,current_space->array,current_space_lvl->array,lnkbt);

    2786:       /* add the k-th row into il and jl */
    2787:       if (nzk-1 > 0) {
    2788:         i     = current_space->array[1]; /* col value of the first nonzero element in U(k, k+1:am-1) */
    2789:         jl[k] = jl[i]; jl[i] = k;
    2790:         il[k] = ui[k] + 1;
    2791:       }
    2792:       uj_ptr[k]     = current_space->array;
    2793:       uj_lvl_ptr[k] = current_space_lvl->array;

    2795:       current_space->array               += nzk;
    2796:       current_space->local_used          += nzk;
    2797:       current_space->local_remaining     -= nzk;
    2798:       current_space_lvl->array           += nzk;
    2799:       current_space_lvl->local_used      += nzk;
    2800:       current_space_lvl->local_remaining -= nzk;

    2802:       ui[k+1] = ui[k] + nzk;
    2803:     }

    2805:     ISRestoreIndices(perm,&rip);
    2806:     PetscFree4(uj_ptr,uj_lvl_ptr,il,jl);

    2808:     /* destroy list of free space and other temporary array(s) */
    2809:     PetscMalloc1(ui[am]+1,&uj);
    2810:     PetscFreeSpaceContiguous(&free_space,uj);
    2811:     PetscIncompleteLLDestroy(lnk,lnkbt);
    2812:     PetscFreeSpaceDestroy(free_space_lvl);
    2813:     if (ai[am] != 0) {
    2814:       ratio_needed = ((PetscReal)ui[am])/((PetscReal)ai[am]);
    2815:     } else {
    2816:       ratio_needed = 0.0;
    2817:     }
    2818:   } /* end of case: levels>0 || (levels=0 && !perm_identity) */

    2820:   /* put together the new matrix in MATSEQSBAIJ format */
    2821:   MatSeqSBAIJSetPreallocation_SeqSBAIJ(fact,bs,MAT_SKIP_ALLOCATION,NULL);

    2823:   b = (Mat_SeqSBAIJ*)(fact)->data;

    2825:   PetscFree2(b->imax,b->ilen);

    2827:   b->singlemalloc = PETSC_FALSE;
    2828:   b->free_a       = PETSC_TRUE;
    2829:   b->free_ij      = free_ij;

    2831:   PetscMalloc1(ui[am]+1,&b->a);

    2833:   b->j             = uj;
    2834:   b->i             = ui;
    2835:   b->diag          = 0;
    2836:   b->ilen          = 0;
    2837:   b->imax          = 0;
    2838:   b->row           = perm;
    2839:   b->pivotinblocks = PETSC_FALSE; /* need to get from MatFactorInfo */

    2841:   PetscObjectReference((PetscObject)perm);
    2842:   b->icol = perm;
    2843:   PetscObjectReference((PetscObject)perm);
    2844:   PetscMalloc1(am+1,&b->solve_work);

    2846:   b->maxnz = b->nz = ui[am];

    2848:   fact->info.factor_mallocs    = reallocs;
    2849:   fact->info.fill_ratio_given  = fill;
    2850:   fact->info.fill_ratio_needed = ratio_needed;
    2851: #if defined(PETSC_USE_INFO)
    2852:   if (ai[am] != 0) {
    2853:     PetscReal af = fact->info.fill_ratio_needed;
    2854:     PetscInfo3(A,"Reallocs %D Fill ratio:given %g needed %g\n",reallocs,(double)fill,(double)af);
    2855:     PetscInfo1(A,"Run with -pc_factor_fill %g or use \n",(double)af);
    2856:     PetscInfo1(A,"PCFactorSetFill(pc,%g) for best performance.\n",(double)af);
    2857:   } else {
    2858:     PetscInfo(A,"Empty matrix.\n");
    2859:   }
    2860: #endif
    2861:   if (perm_identity) {
    2862:     fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqSBAIJ_1_NaturalOrdering_inplace;
    2863:   } else {
    2864:     fact->ops->choleskyfactornumeric = MatCholeskyFactorNumeric_SeqSBAIJ_1_inplace;
    2865:   }
    2866:   return(0);
    2867: }