Actual source code: minres.c

      1: #include <petsc/private/kspimpl.h>
      2: #include <petscblaslapack.h>
      3: PETSC_INTERN PetscErrorCode KSPComputeExtremeSingularValues_MINRES(KSP, PetscReal *, PetscReal *);
      4: PETSC_INTERN PetscErrorCode KSPComputeEigenvalues_MINRES(KSP, PetscInt, PetscReal *, PetscReal *, PetscInt *);

      6: PetscBool  QLPcite       = PETSC_FALSE;
      7: const char QLPCitation[] = "@article{choi2011minres,\n"
      8:                            "  title={MINRES-QLP: A Krylov subspace method for indefinite or singular symmetric systems},\n"
      9:                            "  author={Choi, Sou-Cheng T and Paige, Christopher C and Saunders, Michael A},\n"
     10:                            "  journal={SIAM Journal on Scientific Computing},\n"
     11:                            "  volume={33},\n"
     12:                            "  number={4},\n"
     13:                            "  pages={1810--1836},\n"
     14:                            "  year={2011},\n}\n";

     16: typedef struct {
     17:   PetscReal         haptol;
     18:   PetscReal         nutol;
     19:   PetscBool         qlp;
     20:   PetscReal         maxxnorm;
     21:   PetscReal         TranCond;
     22:   PetscBool         monitor;
     23:   PetscViewer       viewer;
     24:   PetscViewerFormat viewer_fmt;
     25:   // The following arrays are of size ksp->maxit
     26:   PetscScalar *e, *d;
     27:   PetscReal   *ee, *dd; /* work space for Lanczos algorithm */
     28: } KSP_MINRES;

     30: static PetscErrorCode KSPSetUp_MINRES(KSP ksp)
     31: {
     32:   PetscFunctionBegin;
     33:   PetscCall(KSPSetWorkVecs(ksp, 9));
     34:   /*
     35:      If user requested computations of eigenvalues then allocate
     36:      work space needed
     37:   */
     38:   if (ksp->calc_sings) {
     39:     KSP_MINRES *minres = (KSP_MINRES *)ksp->data;
     40:     PetscInt    maxit  = ksp->max_it;
     41:     PetscCall(PetscFree4(minres->e, minres->d, minres->ee, minres->dd));
     42:     PetscCall(PetscMalloc4(maxit + 1, &minres->e, maxit, &minres->d, maxit, &minres->ee, maxit, &minres->dd));

     44:     ksp->ops->computeextremesingularvalues = KSPComputeExtremeSingularValues_MINRES;
     45:     ksp->ops->computeeigenvalues           = KSPComputeEigenvalues_MINRES;
     46:   }
     47:   PetscFunctionReturn(PETSC_SUCCESS);
     48: }

     50: /* Convenience functions */
     51: #define KSPMinresSwap3(V1, V2, V3) \
     52:   do { \
     53:     Vec T = V1; \
     54:     V1    = V2; \
     55:     V2    = V3; \
     56:     V3    = T; \
     57:   } while (0)

     59: static inline PetscReal Norm3(PetscReal a, PetscReal b, PetscReal c)
     60: {
     61:   return PetscSqrtReal(PetscSqr(a) + PetscSqr(b) + PetscSqr(c));
     62: }

     64: static inline void SymOrtho(PetscReal a, PetscReal b, PetscReal *c, PetscReal *s, PetscReal *r)
     65: {
     66:   if (b == 0.0) {
     67:     if (a == 0.0) *c = 1.0;
     68:     else *c = PetscCopysignReal(1.0, a);
     69:     *s = 0.0;
     70:     *r = PetscAbsReal(a);
     71:   } else if (a == 0.0) {
     72:     *c = 0.0;
     73:     *s = PetscCopysignReal(1.0, b);
     74:     *r = PetscAbsReal(b);
     75:   } else if (PetscAbsReal(b) > PetscAbsReal(a)) {
     76:     PetscReal t = a / b;

     78:     *s = PetscCopysignReal(1.0, b) / PetscSqrtReal(1.0 + t * t);
     79:     *c = (*s) * t;
     80:     *r = b / (*s); // computationally better than d = a / c since |c| <= |s|
     81:   } else {
     82:     PetscReal t = b / a;

     84:     *c = PetscCopysignReal(1.0, a) / PetscSqrtReal(1.0 + t * t);
     85:     *s = (*c) * t;
     86:     *r = a / (*c); // computationally better than d = b / s since |s| <= |c|
     87:   }
     88: }

     90: /*
     91:    Code adapted from https://stanford.edu/group/SOL/software/minresqlp/minresqlp-matlab/CPS11.zip
     92:       CSP11/Algorithms/MINRESQLP/minresQLP.m
     93: */
     94: static PetscErrorCode KSPSolve_MINRES(KSP ksp)
     95: {
     96:   KSP_MINRES  *minres = (KSP_MINRES *)ksp->data;
     97:   Mat          Amat;
     98:   Vec          X, B, R1, R2, R3, V, W, WL, WL2, XL2, RN;
     99:   PetscReal    alpha, beta, beta1, betan, betal;
    100:   PetscBool    diagonalscale;
    101:   PetscReal    zero = 0.0, dbar, dltan = 0.0, dlta, cs = -1.0, sn = 0.0, epln, eplnn = 0.0, gbar, dlta_QLP;
    102:   PetscReal    gamal3 = 0.0, gamal2 = 0.0, gamal = 0.0, gama = 0.0, gama_tmp;
    103:   PetscReal    taul2 = 0.0, taul = 0.0, tau = 0.0, phi, phi0, phir;
    104:   PetscReal    Axnorm, xnorm, xnorm_tmp, xl2norm = 0.0, pnorm, Anorm = 0.0, gmin = 0.0, gminl = 0.0, gminl2 = 0.0;
    105:   PetscReal    Acond = 1.0, Acondl = 0.0, rnorml, rnorm, rootl, relAresl, relres, relresl, Arnorml, Anorml = 0.0;
    106:   PetscReal    epsx, realmin = PETSC_REAL_MIN, eps = PETSC_MACHINE_EPSILON;
    107:   PetscReal    veplnl2 = 0.0, veplnl = 0.0, vepln = 0.0, etal2 = 0.0, etal = 0.0, eta = 0.0;
    108:   PetscReal    dlta_tmp, sr2 = 0.0, cr2 = -1.0, cr1 = -1.0, sr1 = 0.0;
    109:   PetscReal    ul4 = 0.0, ul3 = 0.0, ul2 = 0.0, ul = 0.0, u = 0.0, ul_QLP = 0.0, u_QLP = 0.0;
    110:   PetscReal    vepln_QLP = 0.0, gamal_QLP = 0.0, gama_QLP = 0.0, gamal_tmp, abs_gama;
    111:   PetscInt     flag = -2, flag0 = -2, QLPiter = 0;
    112:   PetscInt     stored_max_it, eigs;
    113:   PetscScalar *e = NULL, *d = NULL;

    115:   PetscFunctionBegin;
    116:   PetscCall(PetscCitationsRegister(QLPCitation, &QLPcite));
    117:   PetscCall(PCGetDiagonalScale(ksp->pc, &diagonalscale));
    118:   PetscCheck(!diagonalscale, PetscObjectComm((PetscObject)ksp), PETSC_ERR_SUP, "Krylov method %s does not support diagonal scaling", ((PetscObject)ksp)->type_name);

    120:   eigs          = ksp->calc_sings;
    121:   stored_max_it = ksp->max_it;
    122:   if (eigs) {
    123:     e = minres->e;
    124:     d = minres->d;
    125:   }

    127:   X   = ksp->vec_sol;
    128:   B   = ksp->vec_rhs;
    129:   R1  = ksp->work[0];
    130:   R2  = ksp->work[1];
    131:   R3  = ksp->work[2];
    132:   V   = ksp->work[3];
    133:   W   = ksp->work[4];
    134:   WL  = ksp->work[5];
    135:   WL2 = ksp->work[6];
    136:   XL2 = ksp->work[7];
    137:   RN  = ksp->work[8];
    138:   PetscCall(PCGetOperators(ksp->pc, &Amat, NULL));

    140:   ksp->its   = 0;
    141:   ksp->rnorm = 0.0;
    142:   if (!ksp->guess_zero) {
    143:     PetscCall(KSP_MatMult(ksp, Amat, X, R2));
    144:     PetscCall(VecNorm(R2, NORM_2, &Axnorm));
    145:     PetscCall(VecNorm(X, NORM_2, &xnorm));
    146:     PetscCall(VecAYPX(R2, -1.0, B));
    147:   } else {
    148:     PetscCall(VecCopy(B, R2));
    149:     Axnorm = 0.0;
    150:     xnorm  = 0.0;
    151:   }
    152:   PetscCall(KSP_PCApply(ksp, R2, R3));
    153:   if (ksp->converged_neg_curve) PetscCall(VecCopy(R3, RN));
    154:   PetscCall(VecDotRealPart(R3, R2, &beta1));
    155:   KSPCheckDot(ksp, beta1);
    156:   if (beta1 < 0.0) {
    157:     PetscCheck(!ksp->errorifnotconverged, PetscObjectComm((PetscObject)ksp), PETSC_ERR_CONV_FAILED, "Detected indefinite operator %g", (double)beta1);
    158:     ksp->reason = KSP_DIVERGED_INDEFINITE_PC;
    159:     PetscFunctionReturn(PETSC_SUCCESS);
    160:   }
    161:   beta1 = PetscSqrtReal(beta1);

    163:   rnorm = beta1;
    164:   if (ksp->normtype == KSP_NORM_PRECONDITIONED) ksp->rnorm = rnorm;
    165:   else if (ksp->normtype == KSP_NORM_UNPRECONDITIONED) PetscCall(VecNorm(R2, NORM_2, &ksp->rnorm));
    166:   PetscCall(KSPLogResidualHistory(ksp, ksp->rnorm));
    167:   PetscCall(KSPMonitor(ksp, 0, ksp->rnorm));
    168:   PetscCall((*ksp->converged)(ksp, 0, ksp->rnorm, &ksp->reason, ksp->cnvP)); /* test for convergence */
    169:   if (ksp->reason) PetscFunctionReturn(PETSC_SUCCESS);

    171:   relres = rnorm / beta1;
    172:   betan  = beta1;
    173:   phi0   = beta1;
    174:   phi    = beta1;
    175:   betan  = beta1;
    176:   beta   = 0.0;
    177:   do {
    178:     /* Lanczos */
    179:     ksp->its++;
    180:     betal = beta;
    181:     beta  = betan;
    182:     PetscCall(VecAXPBY(V, 1.0 / beta, 0.0, R3));
    183:     PetscCall(KSP_MatMult(ksp, Amat, V, R3));
    184:     if (ksp->its > 1) PetscCall(VecAXPY(R3, -beta / betal, R1));
    185:     PetscCall(VecDotRealPart(R3, V, &alpha));
    186:     PetscCall(VecAXPY(R3, -alpha / beta, R2));
    187:     KSPMinresSwap3(R1, R2, R3);
    188:     if (eigs) {
    189:       PetscCheck(ksp->max_it == stored_max_it, PetscObjectComm((PetscObject)ksp), PETSC_ERR_SUP, "Cannot change maxit AND calculate eigenvalues");
    190:       d[ksp->its - 1] = alpha;
    191:       e[ksp->its - 1] = beta;
    192:     }

    194:     PetscCall(KSP_PCApply(ksp, R2, R3));
    195:     PetscCall(VecDotRealPart(R3, R2, &betan));
    196:     KSPCheckDot(ksp, betan);
    197:     if (betan < 0.0) {
    198:       PetscCheck(!ksp->errorifnotconverged, PetscObjectComm((PetscObject)ksp), PETSC_ERR_CONV_FAILED, "Detected indefinite preconditioner %g", (double)betan);
    199:       ksp->reason = KSP_DIVERGED_INDEFINITE_PC;
    200:       PetscFunctionReturn(PETSC_SUCCESS);
    201:     }
    202:     betan = PetscSqrtReal(betan);

    204:     pnorm = Norm3(betal, alpha, betan);

    206:     // Apply previous left rotation Q_{k-1}
    207:     dbar     = dltan;
    208:     epln     = eplnn;
    209:     dlta     = cs * dbar + sn * alpha;
    210:     gbar     = sn * dbar - cs * alpha;
    211:     eplnn    = sn * betan;
    212:     dltan    = -cs * betan;
    213:     dlta_QLP = dlta;

    215:     // Stop if negative curvature is detected and return residual
    216:     // This is very experimental and maybe changed in the future
    217:     // based on https://arxiv.org/pdf/2208.07095.pdf
    218:     if (ksp->converged_neg_curve) {
    219:       if (cs * gbar >= 0.0) {
    220:         PetscCall(PetscInfo(ksp, "Detected negative curvature c_nm1 %g, gbar %g\n", (double)cs, (double)gbar));
    221:         ksp->reason = KSP_CONVERGED_NEG_CURVE;
    222:         PetscCall(VecCopy(RN, X));
    223:         break;
    224:       } else {
    225:         PetscCall(VecAXPBY(RN, -phi * cs, PetscSqr(sn), V));
    226:       }
    227:     }

    229:     // Compute the current left plane rotation Q_k
    230:     gamal3 = gamal2;
    231:     gamal2 = gamal;
    232:     gamal  = gama;
    233:     SymOrtho(gbar, betan, &cs, &sn, &gama);

    235:     // Inexactness condition from https://arxiv.org/pdf/2208.07095.pdf
    236:     rootl = Norm3(gbar, dltan, zero);
    237:     phir  = PetscSqr(phi0 / phi);
    238:     if (ksp->its > 2 && minres->nutol > 0.0) {
    239:       PetscReal tmp;

    241:       phir = PetscSqrtReal(phir - 1.0);
    242:       tmp  = rootl / phir;
    243:       PetscCall(PetscInfo(ksp, "it = %" PetscInt_FMT ": inexact check %g (%g / %g)\n", ksp->its - 2, (double)tmp, (double)rootl, (double)phir));
    244:       if (tmp < minres->nutol) {
    245:         ksp->its--;
    246:         ksp->reason = KSP_CONVERGED_RTOL;
    247:         break;
    248:       }
    249:     }

    251:     gama_tmp = gama;
    252:     taul2    = taul;
    253:     taul     = tau;
    254:     tau      = cs * phi;
    255:     Axnorm   = Norm3(Axnorm, tau, zero);
    256:     phi      = sn * phi;

    258:     //Apply the previous right plane rotation P{k-2,k}
    259:     if (ksp->its > 2) {
    260:       veplnl2  = veplnl;
    261:       etal2    = etal;
    262:       etal     = eta;
    263:       dlta_tmp = sr2 * vepln - cr2 * dlta;
    264:       veplnl   = cr2 * vepln + sr2 * dlta;
    265:       dlta     = dlta_tmp;
    266:       eta      = sr2 * gama;
    267:       gama     = -cr2 * gama;
    268:     }

    270:     // Compute the current right plane rotation P{k-1,k}, P_12, P_23,...
    271:     if (ksp->its > 1) {
    272:       SymOrtho(gamal, dlta, &cr1, &sr1, &gamal);
    273:       vepln = sr1 * gama;
    274:       gama  = -cr1 * gama;
    275:     }

    277:     // Update xnorm
    278:     ul4 = ul3;
    279:     ul3 = ul2;
    280:     if (ksp->its > 2) ul2 = (taul2 - etal2 * ul4 - veplnl2 * ul3) / gamal2;
    281:     if (ksp->its > 1) ul = (taul - etal * ul3 - veplnl * ul2) / gamal;
    282:     xnorm_tmp = Norm3(xl2norm, ul2, ul);
    283:     if (PetscAbsReal(gama) > realmin && xnorm_tmp < minres->maxxnorm) {
    284:       u = (tau - eta * ul2 - vepln * ul) / gama;
    285:       if (Norm3(xnorm_tmp, u, zero) > minres->maxxnorm) {
    286:         u    = 0;
    287:         flag = 6;
    288:       }
    289:     } else {
    290:       u    = 0;
    291:       flag = 9;
    292:     }
    293:     xl2norm = Norm3(xl2norm, ul2, zero);
    294:     xnorm   = Norm3(xl2norm, ul, u);

    296:     // Update w. Update x except if it will become too big
    297:     //if (Acond < minres->TranCond && flag != flag0 && QLPiter == 0) { // I believe they have a typo in the MATLAB code
    298:     if ((Acond < minres->TranCond || !minres->qlp) && flag == flag0 && QLPiter == 0) { // MINRES
    299:       KSPMinresSwap3(WL2, WL, W);
    300:       PetscCall(VecAXPBY(W, 1.0 / gama_tmp, 0.0, V));
    301:       if (ksp->its > 1) {
    302:         Vec         T[]      = {WL, WL2};
    303:         PetscScalar alphas[] = {-dlta_QLP / gama_tmp, -epln / gama_tmp};
    304:         PetscInt    nv       = (ksp->its == 2 ? 1 : 2);

    306:         PetscCall(VecMAXPY(W, nv, alphas, T));
    307:       }
    308:       if (xnorm < minres->maxxnorm) {
    309:         PetscCall(VecAXPY(X, tau, W));
    310:       } else {
    311:         flag = 6;
    312:       }
    313:     } else if (minres->qlp) { //MINRES-QLP updates
    314:       QLPiter = QLPiter + 1;
    315:       if (QLPiter == 1) {
    316:         // xl2 = x - wl*ul_QLP - w*u_QLP;
    317:         PetscScalar maxpys[] = {1.0, -ul_QLP, -u_QLP};
    318:         Vec         maxpyv[] = {X, WL, W};

    320:         PetscCall(VecSet(XL2, 0.0));
    321:         // construct w_{k-3}, w_{k-2}, w_{k-1}
    322:         if (ksp->its > 1) {
    323:           if (ksp->its > 3) { // w_{k-3}
    324:             //wl2 = gamal3*wl2 + veplnl2*wl + etal*w;
    325:             PetscCall(VecAXPBYPCZ(WL2, veplnl2, etal, gamal3, WL, W));
    326:           }
    327:           if (ksp->its > 2) { // w_{k-2}
    328:             //wl = gamal_QLP*wl + vepln_QLP*w;
    329:             PetscCall(VecAXPBY(WL, vepln_QLP, gamal_QLP, W));
    330:           }
    331:           // w = gama_QLP*w;
    332:           PetscCall(VecScale(W, gama_QLP));
    333:           // xl2 = x - wl*ul_QLP - w*u_QLP;
    334:           PetscCall(VecMAXPY(XL2, 3, maxpys, maxpyv));
    335:         }
    336:       }
    337:       if (ksp->its == 1) {
    338:         //wl2 = wl;      wl = v*sr1;     w  = -v*cr1;
    339:         PetscCall(VecCopy(WL, WL2));
    340:         PetscCall(VecAXPBY(WL, sr1, 0, V));
    341:         PetscCall(VecAXPBY(W, -cr1, 0, V));
    342:       } else if (ksp->its == 2) {
    343:         //wl2 = wl;
    344:         //wl  = w*cr1 + v*sr1;
    345:         //w   = w*sr1 - v*cr1;
    346:         PetscCall(VecCopy(WL, WL2));
    347:         PetscCall(VecAXPBYPCZ(WL, cr1, sr1, 0.0, W, V));
    348:         PetscCall(VecAXPBY(W, -cr1, sr1, V));
    349:       } else {
    350:         //wl2 = wl;      wl = w;         w  = wl2*sr2 - v*cr2;
    351:         //wl2 = wl2*cr2 + v*sr2;         v  = wl *cr1 + w*sr1;
    352:         //w   = wl *sr1 - w*cr1;         wl = v;
    353:         PetscCall(VecCopy(WL, WL2));
    354:         PetscCall(VecCopy(W, WL));
    355:         PetscCall(VecAXPBYPCZ(W, sr2, -cr2, 0.0, WL2, V));
    356:         PetscCall(VecAXPBY(WL2, sr2, cr2, V));
    357:         PetscCall(VecAXPBYPCZ(V, cr1, sr1, 0.0, WL, W));
    358:         PetscCall(VecAXPBY(W, sr1, -cr1, WL));
    359:         PetscCall(VecCopy(V, WL));
    360:       }

    362:       //xl2 = xl2 + wl2*ul2;
    363:       PetscCall(VecAXPY(XL2, ul2, WL2));
    364:       //x   = xl2 + wl *ul + w*u;
    365:       PetscCall(VecCopy(XL2, X));
    366:       PetscCall(VecAXPBYPCZ(X, ul, u, 1.0, WL, W));
    367:     }
    368:     // Compute the next right plane rotation P{k-1,k+1}
    369:     gamal_tmp = gamal;
    370:     SymOrtho(gamal, eplnn, &cr2, &sr2, &gamal);

    372:     //Store quantities for transferring from MINRES to MINRES-QLP
    373:     gamal_QLP = gamal_tmp;
    374:     vepln_QLP = vepln;
    375:     gama_QLP  = gama;
    376:     ul_QLP    = ul;
    377:     u_QLP     = u;

    379:     // Estimate various norms
    380:     abs_gama = PetscAbsReal(gama);
    381:     Anorml   = Anorm;
    382:     Anorm    = PetscMax(PetscMax(Anorm, pnorm), PetscMax(gamal, abs_gama));
    383:     if (ksp->its == 1) {
    384:       gmin  = gama;
    385:       gminl = gmin;
    386:     } else {
    387:       gminl2 = gminl;
    388:       gminl  = gmin;
    389:       gmin   = PetscMin(gminl2, PetscMin(gamal, abs_gama));
    390:     }
    391:     Acondl  = Acond;
    392:     Acond   = Anorm / gmin;
    393:     rnorml  = rnorm;
    394:     relresl = relres;
    395:     if (flag != 9) rnorm = phi;
    396:     relres   = rnorm / (Anorm * xnorm + beta1);
    397:     Arnorml  = rnorml * rootl;
    398:     relAresl = rootl / Anorm;

    400:     // See if any of the stopping criteria are satisfied.
    401:     epsx = Anorm * xnorm * eps;
    402:     if (flag == flag0 || flag == 9) {
    403:       //if (Acond >= Acondlim) flag = 7; // Huge Acond
    404:       if (epsx >= beta1) flag = 5; // x is an eigenvector
    405:       if (minres->qlp) {           /* We use these indicators only if the QLP variant has been selected */
    406:         PetscReal t1 = 1.0 + relres;
    407:         PetscReal t2 = 1.0 + relAresl;
    408:         if (xnorm >= minres->maxxnorm) flag = 6; // xnorm exceeded its limit
    409:         if (t2 <= 1) flag = 4;                   // Accurate LS solution
    410:         if (t1 <= 1) flag = 3;                   // Accurate Ax=b solution
    411:         if (relAresl <= ksp->rtol) flag = 2;     // Good enough LS solution
    412:         if (relres <= ksp->rtol) flag = 1;       // Good enough Ax=b solution
    413:       }
    414:     }

    416:     if (flag == 2 || flag == 4 || flag == 6 || flag == 7) {
    417:       Acond  = Acondl;
    418:       rnorm  = rnorml;
    419:       relres = relresl;
    420:     }

    422:     if (minres->monitor) { /* Mimics MATLAB code with extra flag */
    423:       PetscCall(PetscViewerPushFormat(minres->viewer, minres->viewer_fmt));
    424:       if (ksp->its == 1) PetscCall(PetscViewerASCIIPrintf(minres->viewer, "        flag      rnorm     Arnorm   Compatible         LS      Anorm      Acond      xnorm\n"));
    425:       PetscCall(PetscViewerASCIIPrintf(minres->viewer, "%s %5" PetscInt_FMT "   %2" PetscInt_FMT " %10.2e %10.2e   %10.2e %10.2e %10.2e %10.2e %10.2e\n", QLPiter == 1 ? "P" : " ", ksp->its - 1, flag, (double)rnorml, (double)Arnorml, (double)relresl, (double)relAresl, (double)Anorml, (double)Acondl, (double)xnorm));
    426:       PetscCall(PetscViewerPopFormat(minres->viewer));
    427:     }

    429:     if (ksp->normtype == KSP_NORM_PRECONDITIONED) ksp->rnorm = rnorm;
    430:     else if (ksp->normtype == KSP_NORM_UNPRECONDITIONED) {
    431:       PetscCall(KSP_MatMult(ksp, Amat, X, V));
    432:       PetscCall(VecAYPX(V, -1.0, B));
    433:       PetscCall(VecNorm(V, NORM_2, &ksp->rnorm));
    434:     }
    435:     PetscCall(KSPLogResidualHistory(ksp, ksp->rnorm));
    436:     PetscCall(KSPMonitor(ksp, ksp->its, ksp->rnorm));
    437:     PetscCall((*ksp->converged)(ksp, ksp->its, ksp->rnorm, &ksp->reason, ksp->cnvP));
    438:     if (!ksp->reason) {
    439:       switch (flag) {
    440:       case 1:
    441:       case 2:
    442:       case 5: /* XXX */
    443:         ksp->reason = KSP_CONVERGED_RTOL;
    444:         break;
    445:       case 3:
    446:       case 4:
    447:         ksp->reason = KSP_CONVERGED_HAPPY_BREAKDOWN;
    448:         break;
    449:       case 6:
    450:         ksp->reason = KSP_CONVERGED_STEP_LENGTH;
    451:         break;
    452:       default:
    453:         break;
    454:       }
    455:     }
    456:     if (ksp->reason) break;
    457:   } while (ksp->its < ksp->max_it);

    459:   if (minres->monitor && flag != 2 && flag != 4 && flag != 6 && flag != 7) {
    460:     PetscCall(VecNorm(X, NORM_2, &xnorm));
    461:     PetscCall(KSP_MatMult(ksp, Amat, X, R1));
    462:     PetscCall(VecAYPX(R1, -1.0, B));
    463:     PetscCall(VecNorm(R1, NORM_2, &rnorml));
    464:     PetscCall(KSP_MatMult(ksp, Amat, R1, R2));
    465:     PetscCall(VecNorm(R2, NORM_2, &Arnorml));
    466:     relresl  = rnorml / (Anorm * xnorm + beta1);
    467:     relAresl = rnorml > realmin ? Arnorml / (Anorm * rnorml) : 0.0;
    468:     PetscCall(PetscViewerPushFormat(minres->viewer, minres->viewer_fmt));
    469:     PetscCall(PetscViewerASCIIPrintf(minres->viewer, "%s %5" PetscInt_FMT "   %2" PetscInt_FMT " %10.2e %10.2e   %10.2e %10.2e %10.2e %10.2e %10.2e\n", QLPiter == 1 ? "P" : " ", ksp->its, flag, (double)rnorml, (double)Arnorml, (double)relresl, (double)relAresl, (double)Anorml, (double)Acondl, (double)xnorm));
    470:     PetscCall(PetscViewerPopFormat(minres->viewer));
    471:   }
    472:   if (!ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
    473:   PetscFunctionReturn(PETSC_SUCCESS);
    474: }

    476: /* This was the original implementation provided by R. Scheichl */
    477: static PetscErrorCode KSPSolve_MINRES_OLD(KSP ksp)
    478: {
    479:   PetscInt          i;
    480:   PetscScalar       alpha, beta, betaold, eta, c = 1.0, ceta, cold = 1.0, coold, s = 0.0, sold = 0.0, soold;
    481:   PetscScalar       rho0, rho1, rho2, rho3, dp = 0.0;
    482:   const PetscScalar none = -1.0;
    483:   PetscReal         np;
    484:   Vec               X, B, R, Z, U, V, W, UOLD, VOLD, WOLD, WOOLD;
    485:   Mat               Amat;
    486:   KSP_MINRES       *minres = (KSP_MINRES *)ksp->data;
    487:   PetscBool         diagonalscale;
    488:   PetscInt          stored_max_it, eigs;
    489:   PetscScalar      *e = NULL, *d = NULL;

    491:   PetscFunctionBegin;
    492:   PetscCall(PCGetDiagonalScale(ksp->pc, &diagonalscale));
    493:   PetscCheck(!diagonalscale, PetscObjectComm((PetscObject)ksp), PETSC_ERR_SUP, "Krylov method %s does not support diagonal scaling", ((PetscObject)ksp)->type_name);

    495:   X     = ksp->vec_sol;
    496:   B     = ksp->vec_rhs;
    497:   R     = ksp->work[0];
    498:   Z     = ksp->work[1];
    499:   U     = ksp->work[2];
    500:   V     = ksp->work[3];
    501:   W     = ksp->work[4];
    502:   UOLD  = ksp->work[5];
    503:   VOLD  = ksp->work[6];
    504:   WOLD  = ksp->work[7];
    505:   WOOLD = ksp->work[8];

    507:   PetscCall(PCGetOperators(ksp->pc, &Amat, NULL));

    509:   ksp->its      = 0;
    510:   eigs          = ksp->calc_sings;
    511:   stored_max_it = ksp->max_it;
    512:   if (eigs) {
    513:     e = minres->e;
    514:     d = minres->d;
    515:   }

    517:   if (!ksp->guess_zero) {
    518:     PetscCall(KSP_MatMult(ksp, Amat, X, R)); /*     r <- b - A*x    */
    519:     PetscCall(VecAYPX(R, -1.0, B));
    520:   } else {
    521:     PetscCall(VecCopy(B, R)); /*     r <- b (x is 0) */
    522:   }
    523:   PetscCall(KSP_PCApply(ksp, R, Z));  /*     z  <- B*r       */
    524:   PetscCall(VecNorm(Z, NORM_2, &np)); /*   np <- ||z||        */
    525:   KSPCheckNorm(ksp, np);
    526:   PetscCall(VecDot(R, Z, &dp));
    527:   KSPCheckDot(ksp, dp);

    529:   if (PetscRealPart(dp) < minres->haptol && np > minres->haptol) {
    530:     PetscCheck(!ksp->errorifnotconverged, PetscObjectComm((PetscObject)ksp), PETSC_ERR_CONV_FAILED, "Detected indefinite operator %g tolerance %g", (double)PetscRealPart(dp), (double)minres->haptol);
    531:     PetscCall(PetscInfo(ksp, "Detected indefinite operator %g tolerance %g\n", (double)PetscRealPart(dp), (double)minres->haptol));
    532:     ksp->reason = KSP_DIVERGED_INDEFINITE_MAT;
    533:     PetscFunctionReturn(PETSC_SUCCESS);
    534:   }

    536:   ksp->rnorm = 0.0;
    537:   if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = np;
    538:   PetscCall(KSPLogResidualHistory(ksp, ksp->rnorm));
    539:   PetscCall(KSPMonitor(ksp, 0, ksp->rnorm));
    540:   PetscCall((*ksp->converged)(ksp, 0, ksp->rnorm, &ksp->reason, ksp->cnvP)); /* test for convergence */
    541:   if (ksp->reason) PetscFunctionReturn(PETSC_SUCCESS);

    543:   dp   = PetscAbsScalar(dp);
    544:   dp   = PetscSqrtScalar(dp);
    545:   beta = dp; /*  beta <- sqrt(r'*z)  */
    546:   eta  = beta;
    547:   PetscCall(VecAXPBY(V, 1.0 / beta, 0, R)); /* v <- r / beta */
    548:   PetscCall(VecAXPBY(U, 1.0 / beta, 0, Z)); /* u <- z / beta */

    550:   i = 0;
    551:   do {
    552:     ksp->its = i + 1;

    554:     /*   Lanczos  */

    556:     PetscCall(KSP_MatMult(ksp, Amat, U, R)); /*      r <- A*u   */
    557:     PetscCall(VecDot(U, R, &alpha));         /*  alpha <- r'*u  */
    558:     PetscCall(KSP_PCApply(ksp, R, Z));       /*      z <- B*r   */
    559:     if (eigs) {
    560:       PetscCheck(ksp->max_it == stored_max_it, PetscObjectComm((PetscObject)ksp), PETSC_ERR_SUP, "Cannot change maxit AND calculate eigenvalues");
    561:       d[i] = alpha;
    562:       e[i] = beta;
    563:     }

    565:     if (ksp->its > 1) {
    566:       Vec         T[2];
    567:       PetscScalar alphas[] = {-alpha, -beta};
    568:       /*  r <- r - alpha v - beta v_old    */
    569:       T[0] = V;
    570:       T[1] = VOLD;
    571:       PetscCall(VecMAXPY(R, 2, alphas, T));
    572:       /*  z <- z - alpha u - beta u_old    */
    573:       T[0] = U;
    574:       T[1] = UOLD;
    575:       PetscCall(VecMAXPY(Z, 2, alphas, T));
    576:     } else {
    577:       PetscCall(VecAXPY(R, -alpha, V)); /*  r <- r - alpha v     */
    578:       PetscCall(VecAXPY(Z, -alpha, U)); /*  z <- z - alpha u     */
    579:     }

    581:     betaold = beta;

    583:     PetscCall(VecDot(R, Z, &dp));
    584:     KSPCheckDot(ksp, dp);
    585:     dp   = PetscAbsScalar(dp);
    586:     beta = PetscSqrtScalar(dp); /*  beta <- sqrt(r'*z)   */

    588:     /*    QR factorisation    */

    590:     coold = cold;
    591:     cold  = c;
    592:     soold = sold;
    593:     sold  = s;

    595:     rho0 = cold * alpha - coold * sold * betaold;
    596:     rho1 = PetscSqrtScalar(rho0 * rho0 + beta * beta);
    597:     rho2 = sold * alpha + coold * cold * betaold;
    598:     rho3 = soold * betaold;

    600:     /* Stop if negative curvature is detected */
    601:     if (ksp->converged_neg_curve && PetscRealPart(cold * rho0) <= 0.0) {
    602:       PetscCall(PetscInfo(ksp, "Detected negative curvature c_nm1=%g, gbar %g\n", (double)PetscRealPart(cold), -(double)PetscRealPart(rho0)));
    603:       ksp->reason = KSP_CONVERGED_NEG_CURVE;
    604:       break;
    605:     }

    607:     /*     Givens rotation    */

    609:     c = rho0 / rho1;
    610:     s = beta / rho1;

    612:     /* Update */
    613:     /*  w_oold <- w_old */
    614:     /*  w_old  <- w     */
    615:     KSPMinresSwap3(WOOLD, WOLD, W);

    617:     /* w <- (u - rho2 w_old - rho3 w_oold)/rho1 */
    618:     PetscCall(VecAXPBY(W, 1.0 / rho1, 0.0, U));
    619:     if (ksp->its > 1) {
    620:       Vec         T[]      = {WOLD, WOOLD};
    621:       PetscScalar alphas[] = {-rho2 / rho1, -rho3 / rho1};
    622:       PetscInt    nv       = (ksp->its == 2 ? 1 : 2);

    624:       PetscCall(VecMAXPY(W, nv, alphas, T));
    625:     }

    627:     ceta = c * eta;
    628:     PetscCall(VecAXPY(X, ceta, W)); /*  x <- x + c eta w     */

    630:     /*
    631:         when dp is really small we have either convergence or an indefinite operator so compute true
    632:         residual norm to check for convergence
    633:     */
    634:     if (PetscRealPart(dp) < minres->haptol) {
    635:       PetscCall(PetscInfo(ksp, "Possible indefinite operator %g tolerance %g\n", (double)PetscRealPart(dp), (double)minres->haptol));
    636:       PetscCall(KSP_MatMult(ksp, Amat, X, VOLD));
    637:       PetscCall(VecAXPY(VOLD, none, B));
    638:       PetscCall(VecNorm(VOLD, NORM_2, &np));
    639:       KSPCheckNorm(ksp, np);
    640:     } else {
    641:       /* otherwise compute new residual norm via recurrence relation */
    642:       np *= PetscAbsScalar(s);
    643:     }

    645:     if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = np;
    646:     PetscCall(KSPLogResidualHistory(ksp, ksp->rnorm));
    647:     PetscCall(KSPMonitor(ksp, i + 1, ksp->rnorm));
    648:     PetscCall((*ksp->converged)(ksp, i + 1, ksp->rnorm, &ksp->reason, ksp->cnvP)); /* test for convergence */
    649:     if (ksp->reason) break;

    651:     if (PetscRealPart(dp) < minres->haptol) {
    652:       PetscCheck(!ksp->errorifnotconverged, PetscObjectComm((PetscObject)ksp), PETSC_ERR_CONV_FAILED, "Detected indefinite operator %g tolerance %g", (double)PetscRealPart(dp), (double)minres->haptol);
    653:       PetscCall(PetscInfo(ksp, "Detected indefinite operator %g tolerance %g\n", (double)PetscRealPart(dp), (double)minres->haptol));
    654:       ksp->reason = KSP_DIVERGED_INDEFINITE_MAT;
    655:       break;
    656:     }

    658:     eta = -s * eta;
    659:     KSPMinresSwap3(VOLD, V, R);
    660:     KSPMinresSwap3(UOLD, U, Z);
    661:     PetscCall(VecScale(V, 1.0 / beta)); /* v <- r / beta */
    662:     PetscCall(VecScale(U, 1.0 / beta)); /* u <- z / beta */

    664:     i++;
    665:   } while (i < ksp->max_it);
    666:   if (i >= ksp->max_it) ksp->reason = KSP_DIVERGED_ITS;
    667:   PetscFunctionReturn(PETSC_SUCCESS);
    668: }

    670: static PetscErrorCode KSPDestroy_MINRES(KSP ksp)
    671: {
    672:   KSP_MINRES *minres = (KSP_MINRES *)ksp->data;

    674:   PetscFunctionBegin;
    675:   PetscCall(PetscFree4(minres->e, minres->d, minres->ee, minres->dd));
    676:   PetscCall(PetscViewerDestroy(&minres->viewer));
    677:   PetscCall(PetscFree(ksp->data));
    678:   PetscCall(PetscObjectComposeFunction((PetscObject)ksp, "KSPMINRESSetRadius_C", NULL));
    679:   PetscCall(PetscObjectComposeFunction((PetscObject)ksp, "KSPMINRESSetUseQLP_C", NULL));
    680:   PetscCall(PetscObjectComposeFunction((PetscObject)ksp, "KSPMINRESGetUseQLP_C", NULL));
    681:   PetscFunctionReturn(PETSC_SUCCESS);
    682: }

    684: static PetscErrorCode KSPMINRESSetUseQLP_MINRES(KSP ksp, PetscBool qlp)
    685: {
    686:   KSP_MINRES *minres = (KSP_MINRES *)ksp->data;

    688:   PetscFunctionBegin;
    689:   minres->qlp = qlp;
    690:   PetscFunctionReturn(PETSC_SUCCESS);
    691: }

    693: static PetscErrorCode KSPMINRESSetRadius_MINRES(KSP ksp, PetscReal radius)
    694: {
    695:   KSP_MINRES *minres = (KSP_MINRES *)ksp->data;

    697:   PetscFunctionBegin;
    698:   minres->maxxnorm = radius;
    699:   PetscFunctionReturn(PETSC_SUCCESS);
    700: }

    702: static PetscErrorCode KSPMINRESGetUseQLP_MINRES(KSP ksp, PetscBool *qlp)
    703: {
    704:   KSP_MINRES *minres = (KSP_MINRES *)ksp->data;

    706:   PetscFunctionBegin;
    707:   *qlp = minres->qlp;
    708:   PetscFunctionReturn(PETSC_SUCCESS);
    709: }

    711: static PetscErrorCode KSPSetFromOptions_MINRES(KSP ksp, PetscOptionItems PetscOptionsObject)
    712: {
    713:   KSP_MINRES *minres = (KSP_MINRES *)ksp->data;

    715:   PetscFunctionBegin;
    716:   PetscOptionsHeadBegin(PetscOptionsObject, "KSP MINRES options");
    717:   { /* Allow comparing with the old code (to be removed in a few releases) */
    718:     PetscBool flg = PETSC_FALSE;
    719:     PetscCall(PetscOptionsBool("-ksp_minres_old", "Use old implementation (to be removed)", "None", flg, &flg, NULL));
    720:     if (flg) ksp->ops->solve = KSPSolve_MINRES_OLD;
    721:     else ksp->ops->solve = KSPSolve_MINRES;
    722:   }
    723:   PetscCall(PetscOptionsBool("-ksp_minres_qlp", "Solve with QLP variant", "KSPMINRESSetUseQLP", minres->qlp, &minres->qlp, NULL));
    724:   PetscCall(PetscOptionsReal("-ksp_minres_radius", "Maximum allowed norm of solution", "KSPMINRESSetRadius", minres->maxxnorm, &minres->maxxnorm, NULL));
    725:   PetscCall(PetscOptionsReal("-ksp_minres_trancond", "Threshold on condition number to dynamically switch to QLP", "None", minres->TranCond, &minres->TranCond, NULL));
    726:   PetscCall(PetscOptionsCreateViewer(PetscObjectComm((PetscObject)ksp), PetscOptionsObject->options, PetscOptionsObject->prefix, "-ksp_minres_monitor", &minres->viewer, &minres->viewer_fmt, &minres->monitor));
    727:   PetscCall(PetscOptionsReal("-ksp_minres_nutol", "Inexactness tolerance", NULL, minres->nutol, &minres->nutol, NULL));
    728:   PetscOptionsHeadEnd();
    729:   PetscFunctionReturn(PETSC_SUCCESS);
    730: }

    732: /*@
    733:   KSPMINRESSetUseQLP - Use the QLP variant of `KSPMINRES`

    735:   Logically Collective

    737:   Input Parameters:
    738: + ksp - the iterative context
    739: - qlp - a Boolean indicating if the QLP variant should be used

    741:   Level: beginner

    743:   Note:
    744:   By default, the QLP variant is not used.

    746: .seealso: [](ch_ksp), `KSP`, `KSPMINRES`, `KSPMINRESGetUseQLP()`
    747: @*/
    748: PetscErrorCode KSPMINRESSetUseQLP(KSP ksp, PetscBool qlp)
    749: {
    750:   PetscFunctionBegin;
    753:   PetscTryMethod(ksp, "KSPMINRESSetUseQLP_C", (KSP, PetscBool), (ksp, qlp));
    754:   PetscFunctionReturn(PETSC_SUCCESS);
    755: }

    757: /*@
    758:   KSPMINRESSetRadius - Set the maximum solution norm allowed for use with trust region methods

    760:   Logically Collective

    762:   Input Parameters:
    763: + ksp    - the iterative context
    764: - radius - the value

    766:   Level: beginner

    768:   Options Database Key:
    769: . -ksp_minres_radius radius - maximum allowed solution norm

    771:   Developer Note:
    772:   Perhaps the KSPXXXSetRadius() should be unified

    774: .seealso: [](ch_ksp), `KSP`, `KSPMINRES`, `KSPMINRESSetUseQLP()`
    775: @*/
    776: PetscErrorCode KSPMINRESSetRadius(KSP ksp, PetscReal radius)
    777: {
    778:   PetscFunctionBegin;
    781:   PetscTryMethod(ksp, "KSPMINRESSetRadius_C", (KSP, PetscReal), (ksp, radius));
    782:   PetscFunctionReturn(PETSC_SUCCESS);
    783: }

    785: /*@
    786:   KSPMINRESGetUseQLP - Get the flag that indicates if the QLP variant is being used

    788:   Logically Collective

    790:   Input Parameter:
    791: . ksp - the iterative context

    793:   Output Parameter:
    794: . qlp - a Boolean indicating if the QLP variant is used

    796:   Level: beginner

    798: .seealso: [](ch_ksp), `KSP`, `KSPMINRES`, `KSPMINRESSetUseQLP()`
    799: @*/
    800: PetscErrorCode KSPMINRESGetUseQLP(KSP ksp, PetscBool *qlp)
    801: {
    802:   PetscFunctionBegin;
    804:   PetscAssertPointer(qlp, 2);
    805:   PetscUseMethod(ksp, "KSPMINRESGetUseQLP_C", (KSP, PetscBool *), (ksp, qlp));
    806:   PetscFunctionReturn(PETSC_SUCCESS);
    807: }

    809: /*MC
    810:    KSPMINRES - This code implements the MINRES (Minimum Residual) method and its QLP variant {cite}`paige.saunders:solution`, {cite}`choi2011minres`,
    811:    {cite}`liu2022newton` for solving linear systems using `KSP`.

    813:    Options Database Keys:
    814: +   -ksp_minres_qlp (true|false)       - activates QLP code
    815: .   -ksp_minres_radius maxnorm         - maximum allowed solution norm
    816: .   -ksp_minres_trancond condthreshold - threshold on condition number to dynamically switch to QLP iterations when QLP has been activated
    817: .   -ksp_minres_monitor                - monitors convergence quantities
    818: -   -ksp_minres_nutol tol              - inexactness tolerance (see https://arxiv.org/pdf/2208.07095.pdf)

    820:    Level: beginner

    822:    Notes:
    823:    The matrix (operator) and the preconditioner must be symmetric and the preconditioner must also be positive definite for this method.

    825:    `KSPMINRES` is often the best Krylov method for symmetric indefinite matrices.

    827:    Supports only left preconditioning.

    829:    Contributed by:
    830:    Original MINRES code - Robert Scheichl: maprs@maths.bath.ac.uk
    831:    QLP variant adapted from: https://stanford.edu/group/SOL/software/minresqlp/minresqlp-matlab/CPS11.zip

    833: .seealso: [](ch_ksp), `KSPCreate()`, `KSPSetType()`, `KSPType`, `KSP`, `KSPCG`, `KSPCR`, `KSPMINRESGetUseQLP()`, `KSPMINRESSetUseQLP()`, `KSPMINRESSetRadius()`
    834: M*/
    835: PETSC_EXTERN PetscErrorCode KSPCreate_MINRES(KSP ksp)
    836: {
    837:   KSP_MINRES *minres;

    839:   PetscFunctionBegin;
    840:   PetscCall(KSPSetSupportedNorm(ksp, KSP_NORM_PRECONDITIONED, PC_LEFT, 3));
    841:   PetscCall(KSPSetSupportedNorm(ksp, KSP_NORM_UNPRECONDITIONED, PC_LEFT, 2));
    842:   PetscCall(KSPSetSupportedNorm(ksp, KSP_NORM_NONE, PC_LEFT, 1));
    843:   PetscCall(PetscNew(&minres));

    845:   /* this parameter is arbitrary and belongs to the old implementation; but e-50 didn't work for __float128 in one example */
    846: #if defined(PETSC_USE_REAL___FLOAT128)
    847:   minres->haptol = 1.e-100;
    848: #elif defined(PETSC_USE_REAL_SINGLE)
    849:   minres->haptol = 1.e-25;
    850: #else
    851:   minres->haptol = 1.e-50;
    852: #endif
    853:   /* those are set as 1.e7 in the MATLAB code -> use 1.0/sqrt(eps) to support single precision */
    854:   minres->maxxnorm = 1.0 / PETSC_SQRT_MACHINE_EPSILON;
    855:   minres->TranCond = 1.0 / PETSC_SQRT_MACHINE_EPSILON;

    857:   ksp->data = (void *)minres;

    859:   ksp->ops->setup          = KSPSetUp_MINRES;
    860:   ksp->ops->solve          = KSPSolve_MINRES;
    861:   ksp->ops->destroy        = KSPDestroy_MINRES;
    862:   ksp->ops->setfromoptions = KSPSetFromOptions_MINRES;
    863:   ksp->ops->buildsolution  = KSPBuildSolutionDefault;
    864:   ksp->ops->buildresidual  = KSPBuildResidualDefault;

    866:   PetscCall(PetscObjectComposeFunction((PetscObject)ksp, "KSPMINRESSetRadius_C", KSPMINRESSetRadius_MINRES));
    867:   PetscCall(PetscObjectComposeFunction((PetscObject)ksp, "KSPMINRESSetUseQLP_C", KSPMINRESSetUseQLP_MINRES));
    868:   PetscCall(PetscObjectComposeFunction((PetscObject)ksp, "KSPMINRESGetUseQLP_C", KSPMINRESGetUseQLP_MINRES));
    869:   PetscFunctionReturn(PETSC_SUCCESS);
    870: }

    872: PetscErrorCode KSPComputeEigenvalues_MINRES(KSP ksp, PetscInt nmax, PetscReal *r, PetscReal *c, PetscInt *neig)
    873: {
    874:   KSP_MINRES  *minres = (KSP_MINRES *)ksp->data;
    875:   PetscScalar *d, *e;
    876:   PetscReal   *ee;
    877:   PetscInt     n = ksp->its;
    878:   PetscBLASInt bn, lierr = 0, ldz = 1;

    880:   PetscFunctionBegin;
    881:   PetscCheck(nmax >= n, PetscObjectComm((PetscObject)ksp), PETSC_ERR_ARG_SIZ, "Not enough room in work space r and c for eigenvalues");
    882:   *neig = n;

    884:   PetscCall(PetscArrayzero(c, nmax));
    885:   if (!n) PetscFunctionReturn(PETSC_SUCCESS);
    886:   d  = minres->d;
    887:   e  = minres->e;
    888:   ee = minres->ee;

    890:   /* copy tridiagonal matrix to work space */
    891:   for (PetscInt j = 0; j < n; j++) {
    892:     r[j]  = PetscRealPart(d[j]);
    893:     ee[j] = PetscRealPart(e[j + 1]);
    894:   }

    896:   PetscCall(PetscBLASIntCast(n, &bn));
    897:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
    898:   PetscCallBLAS("LAPACKREALstev", LAPACKREALstev_("N", &bn, r, ee, NULL, &ldz, NULL, &lierr));
    899:   PetscCheck(!lierr, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEV error");
    900:   PetscCall(PetscFPTrapPop());
    901:   PetscCall(PetscSortReal(n, r));
    902:   PetscFunctionReturn(PETSC_SUCCESS);
    903: }

    905: PetscErrorCode KSPComputeExtremeSingularValues_MINRES(KSP ksp, PetscReal *emax, PetscReal *emin)
    906: {
    907:   KSP_MINRES  *minres = (KSP_MINRES *)ksp->data;
    908:   PetscScalar *d, *e;
    909:   PetscReal   *dd, *ee;
    910:   PetscInt     n = ksp->its;
    911:   PetscBLASInt bn, lierr = 0, ldz = 1;

    913:   PetscFunctionBegin;
    914:   if (!n) {
    915:     *emax = *emin = 1.0;
    916:     PetscFunctionReturn(PETSC_SUCCESS);
    917:   }
    918:   d  = minres->d;
    919:   e  = minres->e;
    920:   dd = minres->dd;
    921:   ee = minres->ee;

    923:   /* copy tridiagonal matrix to work space */
    924:   for (PetscInt j = 0; j < n; j++) {
    925:     dd[j] = PetscRealPart(d[j]);
    926:     ee[j] = PetscRealPart(e[j + 1]);
    927:   }

    929:   PetscCall(PetscBLASIntCast(n, &bn));
    930:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
    931:   PetscCallBLAS("LAPACKREALstev", LAPACKREALstev_("N", &bn, dd, ee, NULL, &ldz, NULL, &lierr));
    932:   PetscCheck(!lierr, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEV error");
    933:   PetscCall(PetscFPTrapPop());
    934:   for (PetscInt j = 0; j < n; j++) dd[j] = PetscAbsReal(dd[j]);
    935:   PetscCall(PetscSortReal(n, dd));
    936:   *emin = dd[0];
    937:   *emax = dd[n - 1];
    938:   PetscFunctionReturn(PETSC_SUCCESS);
    939: }