Date: 2022-03-31T04:49:06+00:00

Actual source code: gqt.c

      1: #include <petscsys.h>
      2: #include <petscblaslapack.h>

      4: static PetscErrorCode estsv(PetscInt n, PetscReal *r, PetscInt ldr, PetscReal *svmin, PetscReal *z)
      5: {
      6:   PetscBLASInt   blas1=1, blasn, blasnmi, blasj, blasldr;
      7:   PetscInt       i,j;
      8:   PetscReal      e,temp,w,wm,ynorm,znorm,s,sm;

     10:   PetscBLASIntCast(n,&blasn);
     11:   PetscBLASIntCast(ldr,&blasldr);
     12:   for (i=0;i<n;i++) {
     13:     z[i]=0.0;
     14:   }
     15:   e = PetscAbs(r[0]);
     16:   if (e == 0.0) {
     17:     *svmin = 0.0;
     18:     z[0] = 1.0;
     19:   } else {
     20:     /* Solve R'*y = e */
     21:     for (i=0;i<n;i++) {
     22:       /* Scale y. The scaling factor (0.01) reduces the number of scalings */
     23:       if (z[i] >= 0.0) e =-PetscAbs(e);
     24:       else             e = PetscAbs(e);

     26:       if (PetscAbs(e - z[i]) > PetscAbs(r[i + ldr*i])) {
     27:         temp = PetscMin(0.01,PetscAbs(r[i + ldr*i]))/PetscAbs(e-z[i]);
     28:         PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, z, &blas1));
     29:         e = temp*e;
     30:       }

     32:       /* Determine the two possible choices of y[i] */
     33:       if (r[i + ldr*i] == 0.0) {
     34:         w = wm = 1.0;
     35:       } else {
     36:         w  = (e - z[i]) / r[i + ldr*i];
     37:         wm = - (e + z[i]) / r[i + ldr*i];
     38:       }

     40:       /*  Chose y[i] based on the predicted value of y[j] for j>i */
     41:       s  = PetscAbs(e - z[i]);
     42:       sm = PetscAbs(e + z[i]);
     43:       for (j=i+1;j<n;j++) {
     44:         sm += PetscAbs(z[j] + wm * r[i + ldr*j]);
     45:       }
     46:       if (i < n-1) {
     47:         PetscBLASIntCast(n-i-1,&blasnmi);
     48:         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasnmi, &w, &r[i + ldr*(i+1)], &blasldr, &z[i+1], &blas1));
     49:         PetscStackCallBLAS("BLASasum",s += BLASasum_(&blasnmi, &z[i+1], &blas1));
     50:       }
     51:       if (s < sm) {
     52:         temp = wm - w;
     53:         w = wm;
     54:         if (i < n-1) {
     55:           PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasnmi, &temp, &r[i + ldr*(i+1)], &blasldr, &z[i+1], &blas1));
     56:         }
     57:       }
     58:       z[i] = w;
     59:     }

     61:     PetscStackCallBLAS("BLASnrm2",ynorm = BLASnrm2_(&blasn, z, &blas1));

     63:     /* Solve R*z = y */
     64:     for (j=n-1; j>=0; j--) {
     65:       /* Scale z */
     66:       if (PetscAbs(z[j]) > PetscAbs(r[j + ldr*j])) {
     67:         temp = PetscMin(0.01, PetscAbs(r[j + ldr*j] / z[j]));
     68:         PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, z, &blas1));
     69:         ynorm *=temp;
     70:       }
     71:       if (r[j + ldr*j] == 0) {
     72:         z[j] = 1.0;
     73:       } else {
     74:         z[j] = z[j] / r[j + ldr*j];
     75:       }
     76:       temp = -z[j];
     77:       PetscBLASIntCast(j,&blasj);
     78:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasj,&temp,&r[0+ldr*j],&blas1,z,&blas1));
     79:     }

     81:     /* Compute svmin and normalize z */
     82:     PetscStackCallBLAS("BLASnrm2",znorm = 1.0 / BLASnrm2_(&blasn, z, &blas1));
     83:     *svmin = ynorm*znorm;
     84:     PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &znorm, z, &blas1));
     85:   }
     86:   return 0;
     87: }

     89: /*
     90: c     ***********
     91: c
     92: c     Subroutine gqt
     93: c
     94: c     Given an n by n symmetric matrix A, an n-vector b, and a
     95: c     positive number delta, this subroutine determines a vector
     96: c     x which approximately minimizes the quadratic function
     97: c
     98: c           f(x) = (1/2)*x'*A*x + b'*x
     99: c
    100: c     subject to the Euclidean norm constraint
    101: c
    102: c           norm(x) <= delta.
    103: c
    104: c     This subroutine computes an approximation x and a Lagrange
    105: c     multiplier par such that either par is zero and
    106: c
    107: c            norm(x) <= (1+rtol)*delta,
    108: c
    109: c     or par is positive and
    110: c
    111: c            abs(norm(x) - delta) <= rtol*delta.
    112: c
    113: c     If xsol is the solution to the problem, the approximation x
    114: c     satisfies
    115: c
    116: c            f(x) <= ((1 - rtol)**2)*f(xsol)
    117: c
    118: c     The subroutine statement is
    119: c
    120: c       subroutine gqt(n,a,lda,b,delta,rtol,atol,itmax,
    121: c                        par,f,x,info,z,wa1,wa2)
    122: c
    123: c     where
    124: c
    125: c       n is an integer variable.
    126: c         On entry n is the order of A.
    127: c         On exit n is unchanged.
    128: c
    129: c       a is a double precision array of dimension (lda,n).
    130: c         On entry the full upper triangle of a must contain the
    131: c            full upper triangle of the symmetric matrix A.
    132: c         On exit the array contains the matrix A.
    133: c
    134: c       lda is an integer variable.
    135: c         On entry lda is the leading dimension of the array a.
    136: c         On exit lda is unchanged.
    137: c
    138: c       b is an double precision array of dimension n.
    139: c         On entry b specifies the linear term in the quadratic.
    140: c         On exit b is unchanged.
    141: c
    142: c       delta is a double precision variable.
    143: c         On entry delta is a bound on the Euclidean norm of x.
    144: c         On exit delta is unchanged.
    145: c
    146: c       rtol is a double precision variable.
    147: c         On entry rtol is the relative accuracy desired in the
    148: c            solution. Convergence occurs if
    149: c
    150: c              f(x) <= ((1 - rtol)**2)*f(xsol)
    151: c
    152: c         On exit rtol is unchanged.
    153: c
    154: c       atol is a double precision variable.
    155: c         On entry atol is the absolute accuracy desired in the
    156: c            solution. Convergence occurs when
    157: c
    158: c              norm(x) <= (1 + rtol)*delta
    159: c
    160: c              max(-f(x),-f(xsol)) <= atol
    161: c
    162: c         On exit atol is unchanged.
    163: c
    164: c       itmax is an integer variable.
    165: c         On entry itmax specifies the maximum number of iterations.
    166: c         On exit itmax is unchanged.
    167: c
    168: c       par is a double precision variable.
    169: c         On entry par is an initial estimate of the Lagrange
    170: c            multiplier for the constraint norm(x) <= delta.
    171: c         On exit par contains the final estimate of the multiplier.
    172: c
    173: c       f is a double precision variable.
    174: c         On entry f need not be specified.
    175: c         On exit f is set to f(x) at the output x.
    176: c
    177: c       x is a double precision array of dimension n.
    178: c         On entry x need not be specified.
    179: c         On exit x is set to the final estimate of the solution.
    180: c
    181: c       info is an integer variable.
    182: c         On entry info need not be specified.
    183: c         On exit info is set as follows:
    184: c
    185: c            info = 1  The function value f(x) has the relative
    186: c                      accuracy specified by rtol.
    187: c
    188: c            info = 2  The function value f(x) has the absolute
    189: c                      accuracy specified by atol.
    190: c
    191: c            info = 3  Rounding errors prevent further progress.
    192: c                      On exit x is the best available approximation.
    193: c
    194: c            info = 4  Failure to converge after itmax iterations.
    195: c                      On exit x is the best available approximation.
    196: c
    197: c       z is a double precision work array of dimension n.
    198: c
    199: c       wa1 is a double precision work array of dimension n.
    200: c
    201: c       wa2 is a double precision work array of dimension n.
    202: c
    203: c     Subprograms called
    204: c
    205: c       MINPACK-2  ......  destsv
    206: c
    207: c       LAPACK  .........  dpotrf
    208: c
    209: c       Level 1 BLAS  ...  daxpy, dcopy, ddot, dnrm2, dscal
    210: c
    211: c       Level 2 BLAS  ...  dtrmv, dtrsv
    212: c
    213: c     MINPACK-2 Project. October 1993.
    214: c     Argonne National Laboratory and University of Minnesota.
    215: c     Brett M. Averick, Richard Carter, and Jorge J. More'
    216: c
    217: c     ***********
    218: */
    219: PetscErrorCode gqt(PetscInt n, PetscReal *a, PetscInt lda, PetscReal *b,
    220:                    PetscReal delta, PetscReal rtol, PetscReal atol,
    221:                    PetscInt itmax, PetscReal *retpar, PetscReal *retf,
    222:                    PetscReal *x, PetscInt *retinfo, PetscInt *retits,
    223:                    PetscReal *z, PetscReal *wa1, PetscReal *wa2)
    224: {
    225:   PetscReal      f=0.0,p001=0.001,p5=0.5,minusone=-1,delta2=delta*delta;
    226:   PetscInt       iter, j, rednc,info;
    227:   PetscBLASInt   indef;
    228:   PetscBLASInt   blas1=1, blasn, iblas, blaslda, blasldap1, blasinfo;
    229:   PetscReal      alpha, anorm, bnorm, parc, parf, parl, pars, par=*retpar,paru, prod, rxnorm, rznorm=0.0, temp, xnorm;

    231:   PetscBLASIntCast(n,&blasn);
    232:   PetscBLASIntCast(lda,&blaslda);
    233:   PetscBLASIntCast(lda+1,&blasldap1);
    234:   parf   = 0.0;
    235:   xnorm  = 0.0;
    236:   rxnorm = 0.0;
    237:   rednc  = 0;
    238:   for (j=0; j<n; j++) {
    239:     x[j] = 0.0;
    240:     z[j] = 0.0;
    241:   }

    243:   /* Copy the diagonal and save A in its lower triangle */
    244:   PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,a,&blasldap1, wa1, &blas1));
    245:   for (j=0;j<n-1;j++) {
    246:     PetscBLASIntCast(n - j - 1,&iblas);
    247:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j + lda*(j+1)], &blaslda, &a[j+1 + lda*j], &blas1));
    248:   }

    250:   /* Calculate the l1-norm of A, the Gershgorin row sums, and the
    251:    l2-norm of b */
    252:   anorm = 0.0;
    253:   for (j=0;j<n;j++) {
    254:     PetscStackCallBLAS("BLASasum",wa2[j] = BLASasum_(&blasn, &a[0 + lda*j], &blas1));CHKMEMQ;
    255:     anorm = PetscMax(anorm,wa2[j]);
    256:   }
    257:   for (j=0;j<n;j++) {
    258:     wa2[j] = wa2[j] - PetscAbs(wa1[j]);
    259:   }
    260:   PetscStackCallBLAS("BLASnrm2",bnorm = BLASnrm2_(&blasn,b,&blas1));CHKMEMQ;
    261:   /* Calculate a lower bound, pars, for the domain of the problem.
    262:    Also calculate an upper bound, paru, and a lower bound, parl,
    263:    for the Lagrange multiplier. */
    264:   pars = parl = paru = -anorm;
    265:   for (j=0;j<n;j++) {
    266:     pars = PetscMax(pars, -wa1[j]);
    267:     parl = PetscMax(parl, wa1[j] + wa2[j]);
    268:     paru = PetscMax(paru, -wa1[j] + wa2[j]);
    269:   }
    270:   parl = PetscMax(bnorm/delta - parl,pars);
    271:   parl = PetscMax(0.0,parl);
    272:   paru = PetscMax(0.0, bnorm/delta + paru);

    274:   /* If the input par lies outside of the interval (parl, paru),
    275:    set par to the closer endpoint. */

    277:   par = PetscMax(par,parl);
    278:   par = PetscMin(par,paru);

    280:   /* Special case: parl == paru */
    281:   paru = PetscMax(paru, (1.0 + rtol)*parl);

    283:   /* Beginning of an iteration */

    285:   info = 0;
    286:   for (iter=1;iter<=itmax;iter++) {
    287:     /* Safeguard par */
    288:     if (par <= pars && paru > 0) {
    289:       par = PetscMax(p001, PetscSqrtScalar(parl/paru)) * paru;
    290:     }

    292:     /* Copy the lower triangle of A into its upper triangle and  compute A + par*I */

    294:     for (j=0;j<n-1;j++) {
    295:       PetscBLASIntCast(n - j - 1,&iblas);
    296:       PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j+1 + j*lda], &blas1,&a[j + (j+1)*lda], &blaslda));
    297:     }
    298:     for (j=0;j<n;j++) {
    299:       a[j + j*lda] = wa1[j] + par;
    300:     }

    302:     /* Attempt the Cholesky factorization of A without referencing the lower triangular part. */
    303:     PetscStackCallBLAS("LAPACKpotrf",LAPACKpotrf_("U",&blasn,a,&blaslda,&indef));

    305:     /* Case 1: A + par*I is pos. def. */
    306:     if (indef == 0) {

    308:       /* Compute an approximate solution x and save the last value of par with A + par*I pos. def. */

    310:       parf = par;
    311:       PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, b, &blas1, wa2, &blas1));
    312:       PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&blasn,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
    314:       PetscStackCallBLAS("BLASnrm2",rxnorm = BLASnrm2_(&blasn, wa2, &blas1));
    315:       PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","N","N",&blasn,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));

    318:       PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, wa2, &blas1, x, &blas1));
    319:       PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &minusone, x, &blas1));
    320:       PetscStackCallBLAS("BLASnrm2",xnorm = BLASnrm2_(&blasn, x, &blas1));CHKMEMQ;

    322:       /* Test for convergence */
    323:       if (PetscAbs(xnorm - delta) <= rtol*delta || (par == 0  && xnorm <= (1.0+rtol)*delta)) {
    324:         info = 1;
    325:       }

    327:       /* Compute a direction of negative curvature and use this information to improve pars. */
    328:       estsv(n,a,lda,&rznorm,z);CHKMEMQ;
    329:       pars = PetscMax(pars, par-rznorm*rznorm);

    331:       /* Compute a negative curvature solution of the form x + alpha*z,  where norm(x+alpha*z)==delta */

    333:       rednc = 0;
    334:       if (xnorm < delta) {
    335:         /* Compute alpha */
    336:         PetscStackCallBLAS("BLASdot",prod = BLASdot_(&blasn, z, &blas1, x, &blas1)/delta);
    337:         temp = (delta - xnorm)*((delta + xnorm)/delta);
    338:         alpha = temp/(PetscAbs(prod) + PetscSqrtScalar(prod*prod + temp/delta));
    339:         if (prod >= 0) alpha = PetscAbs(alpha);
    340:         else alpha =-PetscAbs(alpha);

    342:         /* Test to decide if the negative curvature step produces a larger reduction than with z=0 */
    343:         rznorm = PetscAbs(alpha) * rznorm;
    344:         if ((rznorm*rznorm + par*xnorm*xnorm)/(delta2) <= par) {
    345:           rednc = 1;
    346:         }
    347:         /* Test for convergence */
    348:         if (p5 * rznorm*rznorm / delta2 <= rtol*(1.0-p5*rtol)*(par + rxnorm*rxnorm/delta2)) {
    349:           info = 1;
    350:         } else if (info == 0 && (p5*(par + rxnorm*rxnorm/delta2) <= atol/delta2)) {
    351:           info = 2;
    352:         }
    353:       }

    355:       /* Compute the Newton correction parc to par. */
    356:       if (xnorm == 0) {
    357:         parc = -par;
    358:       } else {
    359:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, x, &blas1, wa2, &blas1));
    360:         temp = 1.0/xnorm;
    361:         PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, wa2, &blas1));
    362:         PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&blasn, &blas1, a, &blaslda, wa2, &blasn, &blasinfo));
    364:         PetscStackCallBLAS("BLASnrm2",temp = BLASnrm2_(&blasn, wa2, &blas1));
    365:         parc = (xnorm - delta)/(delta*temp*temp);
    366:       }

    368:       /* update parl or paru */
    369:       if (xnorm > delta) {
    370:         parl = PetscMax(parl, par);
    371:       } else if (xnorm < delta) {
    372:         paru = PetscMin(paru, par);
    373:       }
    374:     } else {
    375:       /* Case 2: A + par*I is not pos. def. */

    377:       /* Use the rank information from the Cholesky decomposition to update par. */

    379:       if (indef > 1) {
    380:         /* Restore column indef to A + par*I. */
    381:         iblas = indef - 1;
    382:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[indef-1 + 0*lda],&blaslda,&a[0 + (indef-1)*lda],&blas1));
    383:         a[indef-1 + (indef-1)*lda] = wa1[indef-1] + par;

    385:         /* compute parc. */
    386:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[0 + (indef-1)*lda], &blas1, wa2, &blas1));
    387:         PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&iblas,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
    389:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,wa2,&blas1,&a[0 + (indef-1)*lda],&blas1));
    390:         PetscStackCallBLAS("BLASnrm2",temp = BLASnrm2_(&iblas,&a[0 + (indef-1)*lda],&blas1));CHKMEMQ;
    391:         a[indef-1 + (indef-1)*lda] -= temp*temp;
    392:         PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","N","N",&iblas,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
    394:       }

    396:       wa2[indef-1] = -1.0;
    397:       iblas = indef;
    398:       PetscStackCallBLAS("BLASnrm2",temp = BLASnrm2_(&iblas,wa2,&blas1));
    399:       parc = - a[indef-1 + (indef-1)*lda]/(temp*temp);
    400:       pars = PetscMax(pars,par+parc);

    402:       /* If necessary, increase paru slightly.
    403:        This is needed because in some exceptional situations
    404:        paru is the optimal value of par. */

    406:       paru = PetscMax(paru, (1.0+rtol)*pars);
    407:     }

    409:     /* Use pars to update parl */
    410:     parl = PetscMax(parl,pars);

    412:     /* Test for converged. */
    413:     if (info == 0) {
    414:       if (iter == itmax) info=4;
    415:       if (paru <= (1.0+p5*rtol)*pars) info=3;
    416:       if (paru == 0.0) info = 2;
    417:     }

    419:     /* If exiting, store the best approximation and restore
    420:      the upper triangle of A. */

    422:     if (info != 0) {
    423:       /* Compute the best current estimates for x and f. */
    424:       par = parf;
    425:       f = -p5 * (rxnorm*rxnorm + par*xnorm*xnorm);
    426:       if (rednc) {
    427:         f = -p5 * (rxnorm*rxnorm + par*delta*delta - rznorm*rznorm);
    428:         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn, &alpha, z, &blas1, x, &blas1));
    429:       }
    430:       /* Restore the upper triangle of A */
    431:       for (j = 0; j<n; j++) {
    432:         PetscBLASIntCast(n - j - 1,&iblas);
    433:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j+1 + j*lda],&blas1, &a[j + (j+1)*lda],&blaslda));
    434:       }
    435:       PetscBLASIntCast(lda+1,&iblas);
    436:       PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,wa1,&blas1,a,&iblas));
    437:       break;
    438:     }
    439:     par = PetscMax(parl,par+parc);
    440:   }
    441:   *retpar  = par;
    442:   *retf    = f;
    443:   *retinfo = info;
    444:   *retits  = iter;
    445:   CHKMEMQ;
    446:   return 0;
    447: }