Actual source code: fnlog.c

      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain

      6:    This file is part of SLEPc.
      7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
      8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      9: */
     10: /*
     11:    Logarithm function  log(x)
     12: */

     14: #include <slepc/private/fnimpl.h>
     15: #include <slepcblaslapack.h>

     17: static PetscErrorCode FNEvaluateFunction_Log(FN fn,PetscScalar x,PetscScalar *y)
     18: {
     19:   PetscFunctionBegin;
     20: #if !defined(PETSC_USE_COMPLEX)
     21:   PetscCheck(x>=0.0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Function not defined in the requested value");
     22: #endif
     23:   *y = PetscLogScalar(x);
     24:   PetscFunctionReturn(PETSC_SUCCESS);
     25: }

     27: static PetscErrorCode FNEvaluateDerivative_Log(FN fn,PetscScalar x,PetscScalar *y)
     28: {
     29:   PetscFunctionBegin;
     30:   PetscCheck(x!=0.0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Derivative not defined in the requested value");
     31: #if !defined(PETSC_USE_COMPLEX)
     32:   PetscCheck(x>0.0,PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Derivative not defined in the requested value");
     33: #endif
     34:   *y = 1.0/x;
     35:   PetscFunctionReturn(PETSC_SUCCESS);
     36: }

     38: /*
     39:    Block structure of a quasitriangular matrix T. Returns a list of n-1 numbers, where
     40:    structure(j) encodes the block type of the j:j+1,j:j+1 diagonal block as one of:
     41:       0 - not the start of a block
     42:       1 - start of a 2x2 triangular block
     43:       2 - start of a 2x2 quasi-triangular (full) block
     44: */
     45: static PetscErrorCode qtri_struct(PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscInt *structure)
     46: {
     47:   PetscBLASInt j;

     49:   PetscFunctionBegin;
     50: #if defined(PETSC_USE_COMPLEX)
     51:   for (j=0;j<n-1;j++) structure[j] = 1;
     52: #else
     53:   if (n==1) PetscFunctionReturn(PETSC_SUCCESS);
     54:   else if (n==2) {
     55:     structure[0] = (T[1]==0.0)? 1: 2;
     56:     PetscFunctionReturn(PETSC_SUCCESS);
     57:   }
     58:   j = 0;
     59:   while (j<n-2) {
     60:     if (T[j+1+j*ld] != 0.0) { /* Start of a 2x2 full block */
     61:       structure[j++] = 2;
     62:       structure[j++] = 0;
     63:     } else if (T[j+1+j*ld] == 0.0 && T[j+2+(j+1)*ld] == 0.0) { /* Start of a 2x2 triangular block */
     64:       structure[j++] = 1;
     65:     } else { /* Next block must start a 2x2 full block */
     66:       structure[j++] = 0;
     67:     }
     68:   }
     69:   if (T[n-1+(n-2)*ld] != 0.0) { /* 2x2 full block at the end */
     70:     structure[n-2] = 2;
     71:   } else if (structure[n-3] == 0 || structure[n-3] == 1) {
     72:     structure[n-2] = 1;
     73:   }
     74: #endif
     75:   PetscFunctionReturn(PETSC_SUCCESS);
     76: }

     78: /*
     79:    Compute scaling parameter (s) and order of Pade approximant (m).
     80:    wr,wi is overwritten. Required workspace is 3*n*n.
     81:    On output, Troot contains the sth square root of T.
     82: */
     83: static PetscErrorCode FNlogm_params(FN fn,PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscScalar *wr,PetscScalar *wi,PetscInt maxroots,PetscInt *s,PetscInt *m,PetscScalar *Troot,PetscScalar *work)
     84: {
     85:   PetscInt        i,j,k,p,s0;
     86:   PetscReal       inrm,eta,a2,a3,a4,d2,d3,d4,d5;
     87:   PetscScalar     *TrootmI=work+2*n*n;
     88:   PetscBool       foundm=PETSC_FALSE,more;
     89:   PetscRandom     rand;
     90:   const PetscReal xvals[] = { 1.586970738772063e-005, 2.313807884242979e-003, 1.938179313533253e-002,
     91:        6.209171588994762e-002, 1.276404810806775e-001, 2.060962623452836e-001, 2.879093714241194e-001 };
     92:   const PetscInt  mmax=PETSC_STATIC_ARRAY_LENGTH(xvals);

     94:   PetscFunctionBegin;
     95:   PetscCall(PetscRandomCreate(PETSC_COMM_SELF,&rand));
     96:   /* get initial s0 so that T^(1/2^s0) < xvals(mmax) */
     97:   *s = 0;
     98:   do {
     99:     inrm = SlepcAbsEigenvalue(wr[0]-PetscRealConstant(1.0),wi[0]);
    100:     for (i=1;i<n;i++) inrm = PetscMax(inrm,SlepcAbsEigenvalue(wr[i]-PetscRealConstant(1.0),wi[i]));
    101:     if (inrm < xvals[mmax-1]) break;
    102:     for (i=0;i<n;i++) {
    103: #if defined(PETSC_USE_COMPLEX)
    104:       wr[i] = PetscSqrtScalar(wr[i]);
    105: #else
    106: #if defined(PETSC_HAVE_COMPLEX)
    107:       PetscComplex z = PetscSqrtComplex(PetscCMPLX(wr[i],wi[i]));
    108:       wr[i] = PetscRealPartComplex(z);
    109:       wi[i] = PetscImaginaryPartComplex(z);
    110: #else
    111:       SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This operation requires a compiler with C99 or C++ complex support");
    112: #endif
    113: #endif
    114:     }
    115:     *s = *s + 1;
    116:   } while (*s<maxroots);
    117:   s0 = *s;
    118:   if (*s == maxroots) PetscCall(PetscInfo(fn,"Too many matrix square roots\n"));

    120:   /* Troot = T */
    121:   for (j=0;j<n;j++) PetscCall(PetscArraycpy(Troot+j*ld,T+j*ld,PetscMin(j+2,n)));
    122:   for (k=1;k<=PetscMin(*s,maxroots);k++) PetscCall(FNSqrtmSchur(fn,n,Troot,ld,PETSC_FALSE));
    123:   /* Compute value of s and m needed */
    124:   /* TrootmI = Troot - I */
    125:   for (j=0;j<n;j++) {
    126:     PetscCall(PetscArraycpy(TrootmI+j*ld,Troot+j*ld,PetscMin(j+2,n)));
    127:     TrootmI[j+j*ld] -= 1.0;
    128:   }
    129:   PetscCall(SlepcNormAm(n,TrootmI,2,work,rand,&d2));
    130:   d2 = PetscPowReal(d2,1.0/2.0);
    131:   PetscCall(SlepcNormAm(n,TrootmI,3,work,rand,&d3));
    132:   d3 = PetscPowReal(d3,1.0/3.0);
    133:   a2 = PetscMax(d2,d3);
    134:   if (a2 <= xvals[1]) {
    135:     *m = (a2 <= xvals[0])? 1: 2;
    136:     foundm = PETSC_TRUE;
    137:   }
    138:   p = 0;
    139:   while (!foundm) {
    140:     more = PETSC_FALSE;  /* More norm checks needed */
    141:     if (*s > s0) {
    142:       PetscCall(SlepcNormAm(n,TrootmI,3,work,rand,&d3));
    143:       d3 = PetscPowReal(d3,1.0/3.0);
    144:     }
    145:     PetscCall(SlepcNormAm(n,TrootmI,4,work,rand,&d4));
    146:     d4 = PetscPowReal(d4,1.0/4.0);
    147:     a3 = PetscMax(d3,d4);
    148:     if (a3 <= xvals[mmax-1]) {
    149:       for (j=2;j<mmax;j++) if (a3 <= xvals[j]) break;
    150:       if (j <= 5) {
    151:         *m = j+1;
    152:         break;
    153:       } else if (a3/2.0 <= xvals[4] && p < 2) {
    154:         more = PETSC_TRUE;
    155:         p = p + 1;
    156:       }
    157:     }
    158:     if (!more) {
    159:       PetscCall(SlepcNormAm(n,TrootmI,5,work,rand,&d5));
    160:       d5 = PetscPowReal(d5,1.0/5.0);
    161:       a4 = PetscMax(d4,d5);
    162:       eta = PetscMin(a3,a4);
    163:       if (eta <= xvals[mmax-1]) {
    164:         for (j=5;j<mmax;j++) if (eta <= xvals[j]) break;
    165:         *m = j + 1;
    166:         break;
    167:       }
    168:     }
    169:     if (*s == maxroots) {
    170:       PetscCall(PetscInfo(fn,"Too many matrix square roots\n"));
    171:       *m = mmax;  /* No good value found so take largest */
    172:       break;
    173:     }
    174:     PetscCall(FNSqrtmSchur(fn,n,Troot,ld,PETSC_FALSE));
    175:     /* TrootmI = Troot - I */
    176:     for (j=0;j<n;j++) {
    177:       PetscCall(PetscArraycpy(TrootmI+j*ld,Troot+j*ld,PetscMin(j+2,n)));
    178:       TrootmI[j+j*ld] -= 1.0;
    179:     }
    180:     *s = *s + 1;
    181:   }
    182:   PetscCall(PetscRandomDestroy(&rand));
    183:   PetscFunctionReturn(PETSC_SUCCESS);
    184: }

    186: #if !defined(PETSC_USE_COMPLEX)
    187: /*
    188:    Computes a^(1/2^s) - 1 accurately, avoiding subtractive cancellation
    189: */
    190: static PetscScalar sqrt_obo(PetscScalar a,PetscInt s)
    191: {
    192:   PetscScalar val,z0,r;
    193:   PetscReal   angle;
    194:   PetscInt    i,n0;

    196:   PetscFunctionBegin;
    197:   if (s == 0) val = a-1.0;
    198:   else {
    199:     n0 = s;
    200:     angle = PetscAtan2Real(PetscImaginaryPart(a),PetscRealPart(a));
    201:     if (angle >= PETSC_PI/2.0) {
    202:       a = PetscSqrtScalar(a);
    203:       n0 = s - 1;
    204:     }
    205:     z0 = a - 1.0;
    206:     a = PetscSqrtScalar(a);
    207:     r = 1.0 + a;
    208:     for (i=0;i<n0-1;i++) {
    209:       a = PetscSqrtScalar(a);
    210:       r = r*(1.0+a);
    211:     }
    212:     val = z0/r;
    213:   }
    214:   PetscFunctionReturn(val);
    215: }
    216: #endif

    218: /*
    219:    Square root of 2x2 matrix T from block diagonal of Schur form. Overwrites T.
    220: */
    221: static PetscErrorCode sqrtm_tbt(PetscScalar *T)
    222: {
    223:   PetscScalar t11,t12,t21,t22,r11,r22;
    224: #if !defined(PETSC_USE_COMPLEX)
    225:   PetscScalar mu;
    226: #endif

    228:   PetscFunctionBegin;
    229:   t11 = T[0]; t21 = T[1]; t12 = T[2]; t22 = T[3];
    230:   if (t21 != 0.0) {
    231:     /* Compute square root of 2x2 quasitriangular block */
    232:     /* The algorithm assumes the special structure of real Schur form */
    233: #if defined(PETSC_USE_COMPLEX)
    234:     SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Should not reach this line in complex scalars");
    235: #else
    236:     mu = PetscSqrtReal(-t21*t12);
    237:     if (t11 > 0.0) r11 = PetscSqrtReal((t11+SlepcAbsEigenvalue(t11,mu))/2.0);
    238:     else r11 = mu / PetscSqrtReal(2.0*(-t11+SlepcAbsEigenvalue(t11,mu)));
    239:     T[0] = r11;
    240:     T[1] = t21/(2.0*r11);
    241:     T[2] = t12/(2.0*r11);
    242:     T[3] = r11;
    243: #endif
    244:   } else {
    245:     /* Compute square root of 2x2 upper triangular block */
    246:     r11 = PetscSqrtScalar(t11);
    247:     r22 = PetscSqrtScalar(t22);
    248:     T[0] = r11;
    249:     T[2] = t12/(r11+r22);
    250:     T[3] = r22;
    251:   }
    252:   PetscFunctionReturn(PETSC_SUCCESS);
    253: }

    255: #if defined(PETSC_USE_COMPLEX)
    256: /*
    257:    Unwinding number of z
    258: */
    259: static inline PetscReal unwinding(PetscScalar z)
    260: {
    261:   PetscReal u;

    263:   PetscFunctionBegin;
    264:   u = PetscCeilReal((PetscImaginaryPart(z)-PETSC_PI)/(2.0*PETSC_PI));
    265:   PetscFunctionReturn(u);
    266: }
    267: #endif

    269: /*
    270:    Power of 2-by-2 upper triangular matrix A.
    271:    Returns the (1,2) element of the pth power of A, where p is an arbitrary real number
    272: */
    273: static PetscScalar powerm2by2(PetscScalar A11,PetscScalar A22,PetscScalar A12,PetscReal p)
    274: {
    275:   PetscScalar a1,a2,a1p,a2p,loga1,loga2,w,dd,x12;

    277:   PetscFunctionBegin;
    278:   a1 = A11;
    279:   a2 = A22;
    280:   if (a1 == a2) {
    281:     x12 = p*A12*PetscPowScalarReal(a1,p-1);
    282:   } else if (PetscAbsScalar(a1) < 0.5*PetscAbsScalar(a2) || PetscAbsScalar(a2) < 0.5*PetscAbsScalar(a1)) {
    283:     a1p = PetscPowScalarReal(a1,p);
    284:     a2p = PetscPowScalarReal(a2,p);
    285:     x12 = A12*(a2p-a1p)/(a2-a1);
    286:   } else {  /* Close eigenvalues */
    287:     loga1 = PetscLogScalar(a1);
    288:     loga2 = PetscLogScalar(a2);
    289:     w = PetscAtanhScalar((a2-a1)/(a2+a1));
    290: #if defined(PETSC_USE_COMPLEX)
    291:     w += PETSC_i*PETSC_PI*unwinding(loga2-loga1);
    292: #endif
    293:     dd = 2.0*PetscExpScalar((loga1+loga2)*p/2.0)*PetscSinhScalar(p*w)/(a2-a1);
    294:     x12 = A12*dd;
    295:   }
    296:   PetscFunctionReturn(x12);
    297: }

    299: /*
    300:    Recomputes diagonal blocks of T = X^(1/2^s) - 1 more accurately
    301: */
    302: static PetscErrorCode recompute_diag_blocks_sqrt(PetscBLASInt n,PetscScalar *Troot,PetscScalar *T,PetscBLASInt ld,PetscInt *blockStruct,PetscInt s)
    303: {
    304:   PetscScalar    A[4],P[4],M[4],Z0[4],det;
    305:   PetscInt       i,j;
    306: #if !defined(PETSC_USE_COMPLEX)
    307:   PetscInt       last_block=0;
    308:   PetscScalar    a;
    309: #endif

    311:   PetscFunctionBegin;
    312:   for (j=0;j<n-1;j++) {
    313: #if !defined(PETSC_USE_COMPLEX)
    314:     switch (blockStruct[j]) {
    315:       case 0: /* Not start of a block */
    316:         if (last_block != 0) {
    317:           last_block = 0;
    318:         } else { /* In a 1x1 block */
    319:           a = T[j+j*ld];
    320:           Troot[j+j*ld] = sqrt_obo(a,s);
    321:         }
    322:         break;
    323:       default: /* In a 2x2 block */
    324:         last_block = blockStruct[j];
    325: #endif
    326:         if (s == 0) {
    327:           Troot[j+j*ld]       = T[j+j*ld]-1.0;
    328:           Troot[j+1+j*ld]     = T[j+1+j*ld];
    329:           Troot[j+(j+1)*ld]   = T[j+(j+1)*ld];
    330:           Troot[j+1+(j+1)*ld] = T[j+1+(j+1)*ld]-1.0;
    331:           continue;
    332:         }
    333:         A[0] = T[j+j*ld]; A[1] = T[j+1+j*ld]; A[2] = T[j+(j+1)*ld]; A[3] = T[j+1+(j+1)*ld];
    334:         PetscCall(sqrtm_tbt(A));
    335:         /* Z0 = A - I */
    336:         Z0[0] = A[0]-1.0; Z0[1] = A[1]; Z0[2] = A[2]; Z0[3] = A[3]-1.0;
    337:         if (s == 1) {
    338:           Troot[j+j*ld]       = Z0[0];
    339:           Troot[j+1+j*ld]     = Z0[1];
    340:           Troot[j+(j+1)*ld]   = Z0[2];
    341:           Troot[j+1+(j+1)*ld] = Z0[3];
    342:           continue;
    343:         }
    344:         PetscCall(sqrtm_tbt(A));
    345:         /* P = A + I */
    346:         P[0] = A[0]+1.0; P[1] = A[1]; P[2] = A[2]; P[3] = A[3]+1.0;
    347:         for (i=0;i<s-2;i++) {
    348:           PetscCall(sqrtm_tbt(A));
    349:           /* P = P*(I + A) */
    350:           M[0] = P[0]*(A[0]+1.0)+P[2]*A[1];
    351:           M[1] = P[1]*(A[0]+1.0)+P[3]*A[1];
    352:           M[2] = P[0]*A[2]+P[2]*(A[3]+1.0);
    353:           M[3] = P[1]*A[2]+P[3]*(A[3]+1.0);
    354:           P[0] = M[0]; P[1] = M[1]; P[2] = M[2]; P[3] = M[3];
    355:         }
    356:         /* Troot(j:j+1,j:j+1) = Z0 / P (via Cramer) */
    357:         det = P[0]*P[3]-P[1]*P[2];
    358:         Troot[j+j*ld]       = (Z0[0]*P[3]-P[1]*Z0[2])/det;
    359:         Troot[j+(j+1)*ld]   = (P[0]*Z0[2]-Z0[0]*P[2])/det;
    360:         Troot[j+1+j*ld]     = (Z0[1]*P[3]-P[1]*Z0[3])/det;
    361:         Troot[j+1+(j+1)*ld] = (P[0]*Z0[3]-Z0[1]*P[2])/det;
    362:         /* If block is upper triangular recompute the (1,2) element.
    363:            Skip when T(j,j) or T(j+1,j+1) < 0 since the implementation of atanh is nonstandard */
    364:         if (T[j+1+j*ld]==0.0 && PetscRealPart(T[j+j*ld])>=0.0 && PetscRealPart(T[j+1+(j+1)*ld])>=0.0) {
    365:           Troot[j+(j+1)*ld] = powerm2by2(T[j+j*ld],T[j+1+(j+1)*ld],T[j+(j+1)*ld],1.0/PetscPowInt(2,s));
    366:         }
    367: #if !defined(PETSC_USE_COMPLEX)
    368:     }
    369: #endif
    370:   }
    371: #if !defined(PETSC_USE_COMPLEX)
    372:   /* If last diagonal entry is not in a block it will have been missed */
    373:   if (blockStruct[n-2] == 0) {
    374:     a = T[n-1+(n-1)*ld];
    375:     Troot[n-1+(n-1)*ld] = sqrt_obo(a,s);
    376:   }
    377: #endif
    378:   PetscFunctionReturn(PETSC_SUCCESS);
    379: }

    381: /*
    382:    Nodes x and weights w for n-point Gauss-Legendre quadrature (Q is n*n workspace)

    384:    G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature
    385:       rules, Math. Comp., 23(106):221-230, 1969.
    386: */
    387: static PetscErrorCode gauss_legendre(PetscBLASInt n,PetscScalar *x,PetscScalar *w,PetscScalar *Q)
    388: {
    389:   PetscScalar    v,a,*work;
    390:   PetscReal      *eig,dummy;
    391:   PetscBLASInt   k,ld=n,lwork,info;
    392: #if defined(PETSC_USE_COMPLEX)
    393:   PetscReal      *rwork,rdummy;
    394: #endif

    396:   PetscFunctionBegin;
    397:   PetscCall(PetscArrayzero(Q,n*n));
    398:   for (k=1;k<n;k++) {
    399:     v = k/PetscSqrtReal(4.0*k*k-1.0);
    400:     Q[k+(k-1)*n] = v;
    401:     Q[(k-1)+k*n] = v;
    402:   }

    404:   /* workspace query and memory allocation */
    405:   lwork = -1;
    406: #if defined(PETSC_USE_COMPLEX)
    407:   PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V","L",&n,Q,&ld,&dummy,&a,&lwork,&rdummy,&info));
    408:   PetscCall(PetscBLASIntCast((PetscInt)PetscRealPart(a),&lwork));
    409:   PetscCall(PetscMalloc3(n,&eig,lwork,&work,PetscMax(1,3*n-2),&rwork));
    410: #else
    411:   PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V","L",&n,Q,&ld,&dummy,&a,&lwork,&info));
    412:   PetscCall(PetscBLASIntCast((PetscInt)a,&lwork));
    413:   PetscCall(PetscMalloc2(n,&eig,lwork,&work));
    414: #endif

    416:   /* compute eigendecomposition */
    417: #if defined(PETSC_USE_COMPLEX)
    418:   PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V","L",&n,Q,&ld,eig,work,&lwork,rwork,&info));
    419: #else
    420:   PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V","L",&n,Q,&ld,eig,work,&lwork,&info));
    421: #endif
    422:   SlepcCheckLapackInfo("syev",info);

    424:   for (k=0;k<n;k++) {
    425:     x[k] = eig[k];
    426:     w[k] = 2.0*Q[k*n]*Q[k*n];
    427:   }
    428: #if defined(PETSC_USE_COMPLEX)
    429:   PetscCall(PetscFree3(eig,work,rwork));
    430: #else
    431:   PetscCall(PetscFree2(eig,work));
    432: #endif
    433:   PetscCall(PetscLogFlops(9.0*n*n*n+2.0*n*n*n));
    434:   PetscFunctionReturn(PETSC_SUCCESS);
    435: }

    437: /*
    438:    Pade approximation to log(1 + T) via partial fractions
    439: */
    440: static PetscErrorCode pade_approx(PetscBLASInt n,PetscScalar *T,PetscScalar *L,PetscBLASInt ld,PetscInt m,PetscScalar *work)
    441: {
    442:   PetscScalar    *K,*W,*nodes,*wts;
    443:   PetscBLASInt   *ipiv,info,mm;
    444:   PetscInt       i,j,k;

    446:   PetscFunctionBegin;
    447:   K     = work;
    448:   W     = work+n*n;
    449:   nodes = work+2*n*n;
    450:   wts   = work+2*n*n+m;
    451:   ipiv  = (PetscBLASInt*)(work+2*n*n+2*m);
    452:   PetscCall(PetscBLASIntCast(m,&mm));
    453:   PetscCall(gauss_legendre(mm,nodes,wts,L));
    454:   /* Convert from [-1,1] to [0,1] */
    455:   for (i=0;i<m;i++) {
    456:     nodes[i] = (nodes[i]+1.0)/2.0;
    457:     wts[i] = wts[i]/2.0;
    458:   }
    459:   PetscCall(PetscArrayzero(L,n*n));
    460:   for (k=0;k<m;k++) {
    461:     for (i=0;i<n;i++) for (j=0;j<n;j++) K[i+j*ld] = nodes[k]*T[i+j*ld];
    462:     for (i=0;i<n;i++) K[i+i*ld] += 1.0;
    463:     for (i=0;i<n;i++) for (j=0;j<n;j++) W[i+j*ld] = T[i+j*ld];
    464:     PetscCallBLAS("LAPACKgesv",LAPACKgesv_(&n,&n,K,&n,ipiv,W,&n,&info));
    465:     for (i=0;i<n;i++) for (j=0;j<n;j++) L[i+j*ld] += wts[k]*W[i+j*ld];
    466:   }
    467:   PetscFunctionReturn(PETSC_SUCCESS);
    468: }

    470: /*
    471:    Recomputes diagonal blocks of T = X^(1/2^s) - 1 more accurately
    472: */
    473: static PetscErrorCode recompute_diag_blocks_log(PetscBLASInt n,PetscScalar *L,PetscScalar *T,PetscBLASInt ld,PetscInt *blockStruct)
    474: {
    475:   PetscScalar a1,a2,a12,loga1,loga2,z,dd;
    476:   PetscInt    j;
    477: #if !defined(PETSC_USE_COMPLEX)
    478:   PetscInt    last_block=0;
    479:   PetscScalar f,t;
    480: #endif

    482:   PetscFunctionBegin;
    483:   for (j=0;j<n-1;j++) {
    484: #if !defined(PETSC_USE_COMPLEX)
    485:     switch (blockStruct[j]) {
    486:       case 0: /* Not start of a block */
    487:         if (last_block != 0) {
    488:           last_block = 0;
    489:         } else { /* In a 1x1 block */
    490:           L[j+j*ld] = PetscLogScalar(T[j+j*ld]);
    491:         }
    492:         break;
    493:       case 1: /* Start of upper-tri block */
    494:         last_block = 1;
    495: #endif
    496:         a1 = T[j+j*ld];
    497:         a2 = T[j+1+(j+1)*ld];
    498:         loga1 = PetscLogScalar(a1);
    499:         loga2 = PetscLogScalar(a2);
    500:         L[j+j*ld] = loga1;
    501:         L[j+1+(j+1)*ld] = loga2;
    502:         if ((PetscRealPart(a1)<0.0 && PetscImaginaryPart(a1)==0.0) || (PetscRealPart(a2)<0.0 && PetscImaginaryPart(a1)==0.0)) {
    503:          /* Problems with 2 x 2 formula for (1,2) block
    504:             since atanh is nonstandard, just redo diagonal part */
    505:           continue;
    506:         }
    507:         if (a1 == a2) {
    508:           a12 = T[j+(j+1)*ld]/a1;
    509:         } else if (PetscAbsScalar(a1)<0.5*PetscAbsScalar(a2) || PetscAbsScalar(a2)<0.5*PetscAbsScalar(a1)) {
    510:           a12 = T[j+(j+1)*ld]*(loga2-loga1)/(a2-a1);
    511:         } else {  /* Close eigenvalues */
    512:           z = (a2-a1)/(a2+a1);
    513:           dd = 2.0*PetscAtanhScalar(z);
    514: #if defined(PETSC_USE_COMPLEX)
    515:           dd += 2.0*PETSC_i*PETSC_PI*unwinding(loga2-loga1);
    516: #endif
    517:           dd /= (a2-a1);
    518:           a12 = T[j+(j+1)*ld]*dd;
    519:         }
    520:         L[j+(j+1)*ld] = a12;
    521: #if !defined(PETSC_USE_COMPLEX)
    522:         break;
    523:       case 2: /* Start of quasi-tri block */
    524:         last_block = 2;
    525:         f = 0.5*PetscLogScalar(T[j+j*ld]*T[j+j*ld]-T[j+(j+1)*ld]*T[j+1+j*ld]);
    526:         t = PetscAtan2Real(PetscSqrtScalar(-T[j+(j+1)*ld]*T[j+1+j*ld]),T[j+j*ld])/PetscSqrtScalar(-T[j+(j+1)*ld]*T[j+1+j*ld]);
    527:         L[j+j*ld]       = f;
    528:         L[j+1+j*ld]     = t*T[j+1+j*ld];
    529:         L[j+(j+1)*ld]   = t*T[j+(j+1)*ld];
    530:         L[j+1+(j+1)*ld] = f;
    531:     }
    532: #endif
    533:   }
    534:   PetscFunctionReturn(PETSC_SUCCESS);
    535: }
    536: /*
    537:  * Matrix logarithm implementation based on algorithm and matlab code by N. Higham and co-authors
    538:  *
    539:  *     H. Al-Mohy and N. J. Higham, "Improved inverse scaling and squaring
    540:  *     algorithms for the matrix logarithm", SIAM J. Sci. Comput. 34(4):C153-C169, 2012.
    541:  */
    542: static PetscErrorCode FNLogmPade(FN fn,PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscBool firstonly)
    543: {
    544:   PetscBLASInt   k,sdim,lwork,info;
    545:   PetscScalar    *wr,*wi=NULL,*W,*Q,*Troot,*L,*work,one=1.0,zero=0.0,alpha;
    546:   PetscInt       i,j,s=0,m=0,*blockformat;
    547: #if defined(PETSC_USE_COMPLEX)
    548:   PetscReal      *rwork;
    549: #endif

    551:   PetscFunctionBegin;
    552:   lwork = 3*n*n; /* gees needs only 5*n, but work is also passed to FNlogm_params */
    553:   k     = firstonly? 1: n;

    555:   /* compute Schur decomposition A*Q = Q*T */
    556:   PetscCall(PetscCalloc7(n,&wr,n*k,&W,n*n,&Q,n*n,&Troot,n*n,&L,lwork,&work,n-1,&blockformat));
    557: #if !defined(PETSC_USE_COMPLEX)
    558:   PetscCall(PetscMalloc1(n,&wi));
    559:   PetscCallBLAS("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,wi,Q,&ld,work,&lwork,NULL,&info));
    560: #else
    561:   PetscCall(PetscMalloc1(n,&rwork));
    562:   PetscCallBLAS("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,Q,&ld,work,&lwork,rwork,NULL,&info));
    563: #endif
    564:   SlepcCheckLapackInfo("gees",info);

    566: #if !defined(PETSC_USE_COMPLEX)
    567:   /* check for negative real eigenvalues */
    568:   for (i=0;i<n;i++) {
    569:     PetscCheck(wr[i]>=0.0 || wi[i]!=0.0,PETSC_COMM_SELF,PETSC_ERR_SUP,"Matrix has negative real eigenvalue; rerun with complex scalars");
    570:   }
    571: #endif

    573:   /* get block structure of Schur factor */
    574:   PetscCall(qtri_struct(n,T,ld,blockformat));

    576:   /* get parameters */
    577:   PetscCall(FNlogm_params(fn,n,T,ld,wr,wi,100,&s,&m,Troot,work));

    579:   /* compute Troot - I = T(1/2^s) - I more accurately */
    580:   PetscCall(recompute_diag_blocks_sqrt(n,Troot,T,ld,blockformat,s));

    582:   /* compute Pade approximant */
    583:   PetscCall(pade_approx(n,Troot,L,ld,m,work));

    585:   /* scale back up, L = 2^s * L */
    586:   alpha = PetscPowInt(2,s);
    587:   for (i=0;i<n;i++) for (j=0;j<n;j++) L[i+j*ld] *= alpha;

    589:   /* recompute diagonal blocks */
    590:   PetscCall(recompute_diag_blocks_log(n,L,T,ld,blockformat));

    592:   /* backtransform B = Q*L*Q' */
    593:   PetscCallBLAS("BLASgemm",BLASgemm_("N","C",&n,&k,&n,&one,L,&ld,Q,&ld,&zero,W,&ld));
    594:   PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&k,&n,&one,Q,&ld,W,&ld,&zero,T,&ld));

    596:   /* flop count: Schur decomposition, and backtransform */
    597:   PetscCall(PetscLogFlops(25.0*n*n*n+4.0*n*n*k));

    599:   PetscCall(PetscFree7(wr,W,Q,Troot,L,work,blockformat));
    600: #if !defined(PETSC_USE_COMPLEX)
    601:   PetscCall(PetscFree(wi));
    602: #else
    603:   PetscCall(PetscFree(rwork));
    604: #endif
    605:   PetscFunctionReturn(PETSC_SUCCESS);
    606: }

    608: static PetscErrorCode FNEvaluateFunctionMat_Log_Higham(FN fn,Mat A,Mat B)
    609: {
    610:   PetscBLASInt   n = 0;
    611:   PetscScalar    *T;
    612:   PetscInt       m;

    614:   PetscFunctionBegin;
    615:   if (A!=B) PetscCall(MatCopy(A,B,SAME_NONZERO_PATTERN));
    616:   PetscCall(MatDenseGetArray(B,&T));
    617:   PetscCall(MatGetSize(A,&m,NULL));
    618:   PetscCall(PetscBLASIntCast(m,&n));
    619:   PetscCall(FNLogmPade(fn,n,T,n,PETSC_FALSE));
    620:   PetscCall(MatDenseRestoreArray(B,&T));
    621:   PetscFunctionReturn(PETSC_SUCCESS);
    622: }

    624: static PetscErrorCode FNEvaluateFunctionMatVec_Log_Higham(FN fn,Mat A,Vec v)
    625: {
    626:   PetscBLASInt   n = 0;
    627:   PetscScalar    *T;
    628:   PetscInt       m;
    629:   Mat            B;

    631:   PetscFunctionBegin;
    632:   PetscCall(FN_AllocateWorkMat(fn,A,&B));
    633:   PetscCall(MatDenseGetArray(B,&T));
    634:   PetscCall(MatGetSize(A,&m,NULL));
    635:   PetscCall(PetscBLASIntCast(m,&n));
    636:   PetscCall(FNLogmPade(fn,n,T,n,PETSC_TRUE));
    637:   PetscCall(MatDenseRestoreArray(B,&T));
    638:   PetscCall(MatGetColumnVector(B,v,0));
    639:   PetscCall(FN_FreeWorkMat(fn,&B));
    640:   PetscFunctionReturn(PETSC_SUCCESS);
    641: }

    643: static PetscErrorCode FNView_Log(FN fn,PetscViewer viewer)
    644: {
    645:   PetscBool      isascii;
    646:   char           str[50];
    647:   const char     *methodname[] = {
    648:                   "scaling & squaring, [m/m] Pade approximant (Higham)"
    649:   };
    650:   const int      nmeth=PETSC_STATIC_ARRAY_LENGTH(methodname);

    652:   PetscFunctionBegin;
    653:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii));
    654:   if (isascii) {
    655:     if (fn->beta==(PetscScalar)1.0) {
    656:       if (fn->alpha==(PetscScalar)1.0) PetscCall(PetscViewerASCIIPrintf(viewer,"  logarithm: log(x)\n"));
    657:       else {
    658:         PetscCall(SlepcSNPrintfScalar(str,sizeof(str),fn->alpha,PETSC_TRUE));
    659:         PetscCall(PetscViewerASCIIPrintf(viewer,"  logarithm: log(%s*x)\n",str));
    660:       }
    661:     } else {
    662:       PetscCall(SlepcSNPrintfScalar(str,sizeof(str),fn->beta,PETSC_TRUE));
    663:       if (fn->alpha==(PetscScalar)1.0) PetscCall(PetscViewerASCIIPrintf(viewer,"  logarithm: %s*log(x)\n",str));
    664:       else {
    665:         PetscCall(PetscViewerASCIIPrintf(viewer,"  logarithm: %s",str));
    666:         PetscCall(PetscViewerASCIIUseTabs(viewer,PETSC_FALSE));
    667:         PetscCall(SlepcSNPrintfScalar(str,sizeof(str),fn->alpha,PETSC_TRUE));
    668:         PetscCall(PetscViewerASCIIPrintf(viewer,"*log(%s*x)\n",str));
    669:         PetscCall(PetscViewerASCIIUseTabs(viewer,PETSC_TRUE));
    670:       }
    671:     }
    672:     if (fn->method<nmeth) PetscCall(PetscViewerASCIIPrintf(viewer,"  computing matrix functions with: %s\n",methodname[fn->method]));
    673:   }
    674:   PetscFunctionReturn(PETSC_SUCCESS);
    675: }

    677: /*MC
    678:    FNLOG - FNLOG = "log" - The logarithm function $f(x)=\log x$.

    680:    Level: beginner

    682: .seealso: [](sec:fn), `FN`, `FNType`, `FNSetType()`
    683: M*/

    685: SLEPC_EXTERN PetscErrorCode FNCreate_Log(FN fn)
    686: {
    687:   PetscFunctionBegin;
    688:   fn->ops->evaluatefunction          = FNEvaluateFunction_Log;
    689:   fn->ops->evaluatederivative        = FNEvaluateDerivative_Log;
    690:   fn->ops->evaluatefunctionmat[0]    = FNEvaluateFunctionMat_Log_Higham;
    691:   fn->ops->evaluatefunctionmatvec[0] = FNEvaluateFunctionMatVec_Log_Higham;
    692:   fn->ops->view                      = FNView_Log;
    693:   PetscFunctionReturn(PETSC_SUCCESS);
    694: }