Date: 2023-09-05T10:17:41+00:00

Actual source code: epsimpl.h
slepc-3.19.2 2023-09-05
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      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: */

     11: #if !defined(SLEPCEPSIMPL_H)
     12: #define SLEPCEPSIMPL_H

     14: #include <slepceps.h>
     15: #include <slepc/private/slepcimpl.h>

     17: /* SUBMANSEC = EPS */

     19: SLEPC_EXTERN PetscBool EPSRegisterAllCalled;
     20: SLEPC_EXTERN PetscBool EPSMonitorRegisterAllCalled;
     21: SLEPC_EXTERN PetscErrorCode EPSRegisterAll(void);
     22: SLEPC_EXTERN PetscErrorCode EPSMonitorRegisterAll(void);
     23: SLEPC_EXTERN PetscLogEvent EPS_SetUp,EPS_Solve,EPS_CISS_SVD;

     25: typedef struct _EPSOps *EPSOps;

     27: struct _EPSOps {
     28:   PetscErrorCode (*solve)(EPS);
     29:   PetscErrorCode (*setup)(EPS);
     30:   PetscErrorCode (*setupsort)(EPS);
     31:   PetscErrorCode (*setfromoptions)(EPS,PetscOptionItems*);
     32:   PetscErrorCode (*publishoptions)(EPS);
     33:   PetscErrorCode (*destroy)(EPS);
     34:   PetscErrorCode (*reset)(EPS);
     35:   PetscErrorCode (*view)(EPS,PetscViewer);
     36:   PetscErrorCode (*backtransform)(EPS);
     37:   PetscErrorCode (*computevectors)(EPS);
     38:   PetscErrorCode (*setdefaultst)(EPS);
     39:   PetscErrorCode (*setdstype)(EPS);
     40: };

     42: /*
     43:    Maximum number of monitors you can run with a single EPS
     44: */
     45: #define MAXEPSMONITORS 5

     47: /*
     48:    The solution process goes through several states
     49: */
     50: typedef enum { EPS_STATE_INITIAL,
     51:                EPS_STATE_SETUP,
     52:                EPS_STATE_SOLVED,
     53:                EPS_STATE_EIGENVECTORS } EPSStateType;

     55: /*
     56:    To classify the different solvers into categories
     57: */
     58: typedef enum { EPS_CATEGORY_KRYLOV,      /* Krylov solver: relies on STApply and STBackTransform (same as OTHER) */
     59:                EPS_CATEGORY_PRECOND,     /* Preconditioned solver: uses ST only to manage preconditioner */
     60:                EPS_CATEGORY_CONTOUR,     /* Contour integral: ST used to solve linear systems at integration points */
     61:                EPS_CATEGORY_OTHER } EPSSolverType;

     63: /*
     64:    To check for unsupported features at EPSSetUp_XXX()
     65: */
     66: typedef enum { EPS_FEATURE_BALANCE=1,       /* balancing */
     67:                EPS_FEATURE_ARBITRARY=2,     /* arbitrary selection of eigepairs */
     68:                EPS_FEATURE_REGION=4,        /* nontrivial region for filtering */
     69:                EPS_FEATURE_EXTRACTION=8,    /* extraction technique different from Ritz */
     70:                EPS_FEATURE_CONVERGENCE=16,  /* convergence test selected by user */
     71:                EPS_FEATURE_STOPPING=32,     /* stopping test */
     72:                EPS_FEATURE_TWOSIDED=64      /* two-sided variant */
     73:              } EPSFeatureType;

     75: /*
     76:    Defines the EPS data structure
     77: */
     78: struct _p_EPS {
     79:   PETSCHEADER(struct _EPSOps);
     80:   /*------------------------- User parameters ---------------------------*/
     81:   PetscInt       max_it;           /* maximum number of iterations */
     82:   PetscInt       nev;              /* number of eigenvalues to compute */
     83:   PetscInt       ncv;              /* number of basis vectors */
     84:   PetscInt       mpd;              /* maximum dimension of projected problem */
     85:   PetscInt       nini,ninil;       /* number of initial vectors (negative means not copied yet) */
     86:   PetscInt       nds;              /* number of basis vectors of deflation space */
     87:   PetscScalar    target;           /* target value */
     88:   PetscReal      tol;              /* tolerance */
     89:   EPSConv        conv;             /* convergence test */
     90:   EPSStop        stop;             /* stopping test */
     91:   EPSWhich       which;            /* which part of the spectrum to be sought */
     92:   PetscReal      inta,intb;        /* interval [a,b] for spectrum slicing */
     93:   EPSProblemType problem_type;     /* which kind of problem to be solved */
     94:   EPSExtraction  extraction;       /* which kind of extraction to be applied */
     95:   EPSBalance     balance;          /* the balancing method */
     96:   PetscInt       balance_its;      /* number of iterations of the balancing method */
     97:   PetscReal      balance_cutoff;   /* cutoff value for balancing */
     98:   PetscBool      trueres;          /* whether the true residual norm must be computed */
     99:   PetscBool      trackall;         /* whether all the residuals must be computed */
    100:   PetscBool      purify;           /* whether eigenvectors need to be purified */
    101:   PetscBool      twosided;         /* whether to compute left eigenvectors (two-sided solver) */

    103:   /*-------------- User-provided functions and contexts -----------------*/
    104:   PetscErrorCode (*converged)(EPS,PetscScalar,PetscScalar,PetscReal,PetscReal*,void*);
    105:   PetscErrorCode (*convergeduser)(EPS,PetscScalar,PetscScalar,PetscReal,PetscReal*,void*);
    106:   PetscErrorCode (*convergeddestroy)(void*);
    107:   PetscErrorCode (*stopping)(EPS,PetscInt,PetscInt,PetscInt,PetscInt,EPSConvergedReason*,void*);
    108:   PetscErrorCode (*stoppinguser)(EPS,PetscInt,PetscInt,PetscInt,PetscInt,EPSConvergedReason*,void*);
    109:   PetscErrorCode (*stoppingdestroy)(void*);
    110:   PetscErrorCode (*arbitrary)(PetscScalar,PetscScalar,Vec,Vec,PetscScalar*,PetscScalar*,void*);
    111:   void           *convergedctx;
    112:   void           *stoppingctx;
    113:   void           *arbitraryctx;
    114:   PetscErrorCode (*monitor[MAXEPSMONITORS])(EPS,PetscInt,PetscInt,PetscScalar*,PetscScalar*,PetscReal*,PetscInt,void*);
    115:   PetscErrorCode (*monitordestroy[MAXEPSMONITORS])(void**);
    116:   void           *monitorcontext[MAXEPSMONITORS];
    117:   PetscInt       numbermonitors;

    119:   /*----------------- Child objects and working data -------------------*/
    120:   ST             st;               /* spectral transformation object */
    121:   DS             ds;               /* direct solver object */
    122:   BV             V;                /* set of basis vectors and computed eigenvectors */
    123:   BV             W;                /* left basis vectors (if left eigenvectors requested) */
    124:   RG             rg;               /* optional region for filtering */
    125:   SlepcSC        sc;               /* sorting criterion data */
    126:   Vec            D;                /* diagonal matrix for balancing */
    127:   Vec            *IS,*ISL;         /* references to user-provided initial spaces */
    128:   Vec            *defl;            /* references to user-provided deflation space */
    129:   PetscScalar    *eigr,*eigi;      /* real and imaginary parts of eigenvalues */
    130:   PetscReal      *errest;          /* error estimates */
    131:   PetscScalar    *rr,*ri;          /* values computed by user's arbitrary selection function */
    132:   PetscInt       *perm;            /* permutation for eigenvalue ordering */
    133:   PetscInt       nwork;            /* number of work vectors */
    134:   Vec            *work;            /* work vectors */
    135:   void           *data;            /* placeholder for solver-specific stuff */

    137:   /* ----------------------- Status variables --------------------------*/
    138:   EPSStateType   state;            /* initial -> setup -> solved -> eigenvectors */
    139:   EPSSolverType  categ;            /* solver category */
    140:   PetscInt       nconv;            /* number of converged eigenvalues */
    141:   PetscInt       its;              /* number of iterations so far computed */
    142:   PetscInt       n,nloc;           /* problem dimensions (global, local) */
    143:   PetscReal      nrma,nrmb;        /* computed matrix norms */
    144:   PetscBool      useds;            /* whether the solver uses the DS object or not */
    145:   PetscBool      isgeneralized;
    146:   PetscBool      ispositive;
    147:   PetscBool      ishermitian;
    148:   EPSConvergedReason reason;
    149: };

    151: /*
    152:     Macros to test valid EPS arguments
    153: */
    154: #if !defined(PETSC_USE_DEBUG)

    156: #define EPSCheckSolved(h,arg) do {(void)(h);} while (0)

    158: #else

    160: #define EPSCheckSolved(h,arg) \
    161:   do { \
    162:     PetscCheck((h)->state>=EPS_STATE_SOLVED,PetscObjectComm((PetscObject)(h)),PETSC_ERR_ARG_WRONGSTATE,"Must call EPSSolve() first: Parameter #%d",arg); \
    163:   } while (0)

    165: #endif

    167: /*
    168:     Macros to check settings at EPSSetUp()
    169: */

    171: /* EPSCheckHermitianDefinite: the problem is HEP or GHEP */
    172: #define EPSCheckHermitianDefiniteCondition(eps,condition,msg) \
    173:   do { \
    174:     if (condition) { \
    175:       PetscCheck((eps)->ishermitian,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s cannot be used for non-%s problems",((PetscObject)(eps))->type_name,(msg),SLEPC_STRING_HERMITIAN); \
    176:       PetscCheck(!(eps)->isgeneralized || (eps)->ispositive,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s requires that the problem is %s-definite",((PetscObject)(eps))->type_name,(msg),SLEPC_STRING_HERMITIAN); \
    177:     } \
    178:   } while (0)
    179: #define EPSCheckHermitianDefinite(eps) EPSCheckHermitianDefiniteCondition(eps,PETSC_TRUE,"")

    181: /* EPSCheckHermitian: the problem is HEP, GHEP, or GHIEP */
    182: #define EPSCheckHermitianCondition(eps,condition,msg) \
    183:   do { \
    184:     if (condition) { \
    185:       PetscCheck((eps)->ishermitian,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s cannot be used for non-%s problems",((PetscObject)(eps))->type_name,(msg),SLEPC_STRING_HERMITIAN); \
    186:     } \
    187:   } while (0)
    188: #define EPSCheckHermitian(eps) EPSCheckHermitianCondition(eps,PETSC_TRUE,"")

    190: /* EPSCheckDefinite: the problem is not GHIEP */
    191: #define EPSCheckDefiniteCondition(eps,condition,msg) \
    192:   do { \
    193:     if (condition) { \
    194:       PetscCheck(!(eps)->isgeneralized || !(eps)->ishermitian || (eps)->ispositive,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s cannot be used for %s-indefinite problems",((PetscObject)(eps))->type_name,(msg),SLEPC_STRING_HERMITIAN); \
    195:     } \
    196:   } while (0)
    197: #define EPSCheckDefinite(eps) EPSCheckDefiniteCondition(eps,PETSC_TRUE,"")

    199: /* EPSCheckStandard: the problem is HEP or NHEP */
    200: #define EPSCheckStandardCondition(eps,condition,msg) \
    201:   do { \
    202:     if (condition) { \
    203:       PetscCheck(!(eps)->isgeneralized,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s cannot be used for generalized problems",((PetscObject)(eps))->type_name,(msg)); \
    204:     } \
    205:   } while (0)
    206: #define EPSCheckStandard(eps) EPSCheckStandardCondition(eps,PETSC_TRUE,"")

    208: /* EPSCheckSinvert: shift-and-invert ST */
    209: #define EPSCheckSinvertCondition(eps,condition,msg) \
    210:   do { \
    211:     if (condition) { \
    212:       PetscBool __flg; \
    213:       PetscCall(PetscObjectTypeCompare((PetscObject)(eps)->st,STSINVERT,&__flg)); \
    214:       PetscCheck(__flg,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s requires a shift-and-invert spectral transform",((PetscObject)(eps))->type_name,(msg)); \
    215:     } \
    216:   } while (0)
    217: #define EPSCheckSinvert(eps) EPSCheckSinvertCondition(eps,PETSC_TRUE,"")

    219: /* EPSCheckSinvertCayley: shift-and-invert or Cayley ST */
    220: #define EPSCheckSinvertCayleyCondition(eps,condition,msg) \
    221:   do { \
    222:     if (condition) { \
    223:       PetscBool __flg; \
    224:       PetscCall(PetscObjectTypeCompareAny((PetscObject)(eps)->st,&__flg,STSINVERT,STCAYLEY,"")); \
    225:       PetscCheck(__flg,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s requires shift-and-invert or Cayley transform",((PetscObject)(eps))->type_name,(msg)); \
    226:     } \
    227:   } while (0)
    228: #define EPSCheckSinvertCayley(eps) EPSCheckSinvertCayleyCondition(eps,PETSC_TRUE,"")

    230: /* Check for unsupported features */
    231: #define EPSCheckUnsupportedCondition(eps,mask,condition,msg) \
    232:   do { \
    233:     if (condition) { \
    234:       PetscCheck(!((mask) & EPS_FEATURE_BALANCE) || (eps)->balance==EPS_BALANCE_NONE,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s does not support balancing",((PetscObject)(eps))->type_name,(msg)); \
    235:       PetscCheck(!((mask) & EPS_FEATURE_ARBITRARY) || !(eps)->arbitrary,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s does not support arbitrary selection of eigenpairs",((PetscObject)(eps))->type_name,(msg)); \
    236:       if ((mask) & EPS_FEATURE_REGION) { \
    237:         PetscBool      __istrivial; \
    238:         PetscCall(RGIsTrivial((eps)->rg,&__istrivial)); \
    239:         PetscCheck(__istrivial,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s does not support region filtering",((PetscObject)(eps))->type_name,(msg)); \
    240:       } \
    241:       PetscCheck(!((mask) & EPS_FEATURE_EXTRACTION) || (eps)->extraction==EPS_RITZ,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s only supports Ritz extraction",((PetscObject)(eps))->type_name,(msg)); \
    242:       PetscCheck(!((mask) & EPS_FEATURE_CONVERGENCE) || (eps)->converged==EPSConvergedRelative,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s only supports the default convergence test",((PetscObject)(eps))->type_name,(msg)); \
    243:       PetscCheck(!((mask) & EPS_FEATURE_STOPPING) || (eps)->stopping==EPSStoppingBasic,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s only supports the default stopping test",((PetscObject)(eps))->type_name,(msg)); \
    244:       PetscCheck(!((mask) & EPS_FEATURE_TWOSIDED) || !(eps)->twosided,PetscObjectComm((PetscObject)(eps)),PETSC_ERR_SUP,"The solver '%s'%s cannot compute left eigenvectors (no two-sided variant)",((PetscObject)(eps))->type_name,(msg)); \
    245:     } \
    246:   } while (0)
    247: #define EPSCheckUnsupported(eps,mask) EPSCheckUnsupportedCondition(eps,mask,PETSC_TRUE,"")

    249: /* Check for ignored features */
    250: #define EPSCheckIgnoredCondition(eps,mask,condition,msg) \
    251:   do { \
    252:     if (condition) { \
    253:       if (((mask) & EPS_FEATURE_BALANCE) && (eps)->balance!=EPS_BALANCE_NONE) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the balancing settings\n",((PetscObject)(eps))->type_name,(msg))); \
    254:       if (((mask) & EPS_FEATURE_ARBITRARY) && (eps)->arbitrary) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the settings for arbitrary selection of eigenpairs\n",((PetscObject)(eps))->type_name,(msg))); \
    255:       if ((mask) & EPS_FEATURE_REGION) { \
    256:         PetscBool __istrivial; \
    257:         PetscCall(RGIsTrivial((eps)->rg,&__istrivial)); \
    258:         if (!__istrivial) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the specified region\n",((PetscObject)(eps))->type_name,(msg))); \
    259:       } \
    260:       if (((mask) & EPS_FEATURE_EXTRACTION) && (eps)->extraction!=EPS_RITZ) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the extraction settings\n",((PetscObject)(eps))->type_name,(msg))); \
    261:       if (((mask) & EPS_FEATURE_CONVERGENCE) && (eps)->converged!=EPSConvergedRelative) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the convergence test settings\n",((PetscObject)(eps))->type_name,(msg))); \
    262:       if (((mask) & EPS_FEATURE_STOPPING) && (eps)->stopping!=EPSStoppingBasic) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the stopping test settings\n",((PetscObject)(eps))->type_name,(msg))); \
    263:       if (((mask) & EPS_FEATURE_TWOSIDED) && (eps)->twosided) PetscCall(PetscInfo((eps),"The solver '%s'%s ignores the two-sided flag\n",((PetscObject)(eps))->type_name,(msg))); \
    264:     } \
    265:   } while (0)
    266: #define EPSCheckIgnored(eps,mask) EPSCheckIgnoredCondition(eps,mask,PETSC_TRUE,"")

    268: /*
    269:   EPS_SetInnerProduct - set B matrix for inner product if appropriate.
    270: */
    271: static inline PetscErrorCode EPS_SetInnerProduct(EPS eps)
    272: {
    273:   Mat            B;

    275:   PetscFunctionBegin;
    276:   if (!eps->V) PetscCall(EPSGetBV(eps,&eps->V));
    277:   if (eps->ispositive || (eps->isgeneralized && eps->ishermitian)) {
    278:     PetscCall(STGetBilinearForm(eps->st,&B));
    279:     PetscCall(BVSetMatrix(eps->V,B,PetscNot(eps->ispositive)));
    280:     PetscCall(MatDestroy(&B));
    281:   } else PetscCall(BVSetMatrix(eps->V,NULL,PETSC_FALSE));
    282:   PetscFunctionReturn(PETSC_SUCCESS);
    283: }

    285: /*
    286:   EPS_Purify - purify the first k vectors in the V basis
    287: */
    288: static inline PetscErrorCode EPS_Purify(EPS eps,PetscInt k)
    289: {
    290:   PetscInt       i;
    291:   Vec            v,z;

    293:   PetscFunctionBegin;
    294:   PetscCall(BVCreateVec(eps->V,&v));
    295:   for (i=0;i<k;i++) {
    296:     PetscCall(BVCopyVec(eps->V,i,v));
    297:     PetscCall(BVGetColumn(eps->V,i,&z));
    298:     PetscCall(STApply(eps->st,v,z));
    299:     PetscCall(BVRestoreColumn(eps->V,i,&z));
    300:   }
    301:   PetscCall(VecDestroy(&v));
    302:   PetscFunctionReturn(PETSC_SUCCESS);
    303: }

    305: /*
    306:   EPS_KSPSetOperators - Sets the KSP matrices, see also ST_KSPSetOperators()
    307: */
    308: static inline PetscErrorCode EPS_KSPSetOperators(KSP ksp,Mat A,Mat B)
    309: {
    310:   const char     *prefix;

    312:   PetscFunctionBegin;
    313:   PetscCall(KSPSetOperators(ksp,A,B));
    314:   PetscCall(MatGetOptionsPrefix(B,&prefix));
    315:   if (!prefix) {
    316:     /* set Mat prefix to be the same as KSP to enable setting command-line options (e.g. MUMPS)
    317:        only applies if the Mat has no user-defined prefix */
    318:     PetscCall(KSPGetOptionsPrefix(ksp,&prefix));
    319:     PetscCall(MatSetOptionsPrefix(B,prefix));
    320:   }
    321:   PetscFunctionReturn(PETSC_SUCCESS);
    322: }

    324: SLEPC_INTERN PetscErrorCode EPSSetWhichEigenpairs_Default(EPS);
    325: SLEPC_INTERN PetscErrorCode EPSSetDimensions_Default(EPS,PetscInt,PetscInt*,PetscInt*);
    326: SLEPC_INTERN PetscErrorCode EPSBackTransform_Default(EPS);
    327: SLEPC_INTERN PetscErrorCode EPSComputeVectors(EPS);
    328: SLEPC_INTERN PetscErrorCode EPSComputeVectors_Hermitian(EPS);
    329: SLEPC_INTERN PetscErrorCode EPSComputeVectors_Schur(EPS);
    330: SLEPC_INTERN PetscErrorCode EPSComputeVectors_Indefinite(EPS);
    331: SLEPC_INTERN PetscErrorCode EPSComputeVectors_Twosided(EPS);
    332: SLEPC_INTERN PetscErrorCode EPSComputeVectors_Slice(EPS);
    333: SLEPC_INTERN PetscErrorCode EPSComputeResidualNorm_Private(EPS,PetscBool,PetscScalar,PetscScalar,Vec,Vec,Vec*,PetscReal*);
    334: SLEPC_INTERN PetscErrorCode EPSComputeRitzVector(EPS,PetscScalar*,PetscScalar*,BV,Vec,Vec);
    335: SLEPC_INTERN PetscErrorCode EPSGetStartVector(EPS,PetscInt,PetscBool*);
    336: SLEPC_INTERN PetscErrorCode EPSGetLeftStartVector(EPS,PetscInt,PetscBool*);
    337: SLEPC_INTERN PetscErrorCode MatEstimateSpectralRange_EPS(Mat,PetscReal*,PetscReal*);

    339: /* Private functions of the solver implementations */

    341: SLEPC_INTERN PetscErrorCode EPSDelayedArnoldi(EPS,PetscScalar*,PetscInt,PetscInt,PetscInt*,PetscReal*,PetscBool*);
    342: SLEPC_INTERN PetscErrorCode EPSDelayedArnoldi1(EPS,PetscScalar*,PetscInt,PetscInt,PetscInt*,PetscReal*,PetscBool*);
    343: SLEPC_INTERN PetscErrorCode EPSKrylovConvergence(EPS,PetscBool,PetscInt,PetscInt,PetscReal,PetscReal,PetscReal,PetscInt*);
    344: SLEPC_INTERN PetscErrorCode EPSPseudoLanczos(EPS,PetscReal*,PetscReal*,PetscReal*,PetscInt,PetscInt*,PetscBool*,PetscBool*,PetscReal*,Vec);
    345: SLEPC_INTERN PetscErrorCode EPSBuildBalance_Krylov(EPS);
    346: SLEPC_INTERN PetscErrorCode EPSSetDefaultST(EPS);
    347: SLEPC_INTERN PetscErrorCode EPSSetDefaultST_Precond(EPS);
    348: SLEPC_INTERN PetscErrorCode EPSSetDefaultST_GMRES(EPS);
    349: SLEPC_INTERN PetscErrorCode EPSSetDefaultST_NoFactor(EPS);
    350: SLEPC_INTERN PetscErrorCode EPSSetUpSort_Basic(EPS);
    351: SLEPC_INTERN PetscErrorCode EPSSetUpSort_Default(EPS);

    353: #endif