Date: 2022-11-02T11:47:26+00:00

Actual source code: ex40.c
slepc-3.18.1 2022-11-02
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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: static char help[] = "Checking the definite property in quadratic symmetric eigenproblem.\n\n"
     12:   "The command line options are:\n"
     13:   "  -n <n> ... dimension of the matrices.\n"
     14:   "  -transform... whether to transform to a hyperbolic problem or not.\n"
     15:   "  -nonhyperbolic... to test with a modified (definite) problem that is not hyperbolic.\n\n";

     17: #include <slepcpep.h>

     19: /*
     20:   This example is based on spring.c, for fixed values mu=1,tau=10,kappa=5

     22:   The transformations are based on the method proposed in [Niendorf and Voss, LAA 2010].
     23: */

     25: PetscErrorCode QEPDefiniteTransformGetMatrices(PEP,PetscBool,PetscReal,PetscReal,Mat[3]);
     26: PetscErrorCode QEPDefiniteTransformMap(PetscBool,PetscReal,PetscReal,PetscInt,PetscScalar*,PetscBool);
     27: PetscErrorCode QEPDefiniteCheckError(Mat*,PEP,PetscBool,PetscReal,PetscReal);
     28: PetscErrorCode TransformMatricesMoebius(Mat[3],MatStructure,PetscReal,PetscReal,PetscReal,PetscReal,Mat[3]);

     30: int main(int argc,char **argv)
     31: {
     32:   Mat            M,C,K,*Op,A[3],At[3],B[3]; /* problem matrices */
     33:   PEP            pep;        /* polynomial eigenproblem solver context */
     34:   ST             st;         /* spectral transformation context */
     35:   KSP            ksp;
     36:   PC             pc;
     37:   PEPProblemType type;
     38:   PetscBool      terse,transform=PETSC_FALSE,nohyp=PETSC_FALSE;
     39:   PetscInt       n=100,Istart,Iend,i,def=0,hyp;
     40:   PetscReal      muu=1,tau=10,kappa=5,inta,intb;
     41:   PetscReal      alpha,beta,xi,mu,at[2]={0.0,0.0},c=.857,s;
     42:   PetscScalar    target,targett,ats[2];

     45:   SlepcInitialize(&argc,&argv,(char*)0,help);

     47:   PetscOptionsGetInt(NULL,NULL,"-n",&n,NULL);
     48:   PetscPrintf(PETSC_COMM_WORLD,"\nPEP example that checks definite property, n=%" PetscInt_FMT "\n\n",n);

     50:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     51:      Compute the matrices that define the eigensystem, (k^2*M+k*C+K)x=0
     52:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     54:   /* K is a tridiagonal */
     55:   MatCreate(PETSC_COMM_WORLD,&K);
     56:   MatSetSizes(K,PETSC_DECIDE,PETSC_DECIDE,n,n);
     57:   MatSetFromOptions(K);
     58:   MatSetUp(K);

     60:   MatGetOwnershipRange(K,&Istart,&Iend);
     61:   for (i=Istart;i<Iend;i++) {
     62:     if (i>0) MatSetValue(K,i,i-1,-kappa,INSERT_VALUES);
     63:     MatSetValue(K,i,i,kappa*3.0,INSERT_VALUES);
     64:     if (i<n-1) MatSetValue(K,i,i+1,-kappa,INSERT_VALUES);
     65:   }

     67:   MatAssemblyBegin(K,MAT_FINAL_ASSEMBLY);
     68:   MatAssemblyEnd(K,MAT_FINAL_ASSEMBLY);

     70:   /* C is a tridiagonal */
     71:   MatCreate(PETSC_COMM_WORLD,&C);
     72:   MatSetSizes(C,PETSC_DECIDE,PETSC_DECIDE,n,n);
     73:   MatSetFromOptions(C);
     74:   MatSetUp(C);

     76:   MatGetOwnershipRange(C,&Istart,&Iend);
     77:   for (i=Istart;i<Iend;i++) {
     78:     if (i>0) MatSetValue(C,i,i-1,-tau,INSERT_VALUES);
     79:     MatSetValue(C,i,i,tau*3.0,INSERT_VALUES);
     80:     if (i<n-1) MatSetValue(C,i,i+1,-tau,INSERT_VALUES);
     81:   }

     83:   MatAssemblyBegin(C,MAT_FINAL_ASSEMBLY);
     84:   MatAssemblyEnd(C,MAT_FINAL_ASSEMBLY);

     86:   /* M is a diagonal matrix */
     87:   MatCreate(PETSC_COMM_WORLD,&M);
     88:   MatSetSizes(M,PETSC_DECIDE,PETSC_DECIDE,n,n);
     89:   MatSetFromOptions(M);
     90:   MatSetUp(M);
     91:   MatGetOwnershipRange(M,&Istart,&Iend);
     92:   for (i=Istart;i<Iend;i++) MatSetValue(M,i,i,muu,INSERT_VALUES);
     93:   MatAssemblyBegin(M,MAT_FINAL_ASSEMBLY);
     94:   MatAssemblyEnd(M,MAT_FINAL_ASSEMBLY);

     96:   PetscOptionsGetBool(NULL,NULL,"-nonhyperbolic",&nohyp,NULL);
     97:   A[0] = K; A[1] = C; A[2] = M;
     98:   if (nohyp) {
     99:     s = c*.6;
    100:     TransformMatricesMoebius(A,UNKNOWN_NONZERO_PATTERN,c,s,-s,c,At);
    101:     for (i=0;i<3;i++) MatDestroy(&A[i]);
    102:     Op = At;
    103:   } else Op = A;

    105:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    106:                 Create the eigensolver and solve the problem
    107:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    109:   /*
    110:      Create eigensolver context
    111:   */
    112:   PEPCreate(PETSC_COMM_WORLD,&pep);
    113:   PEPSetProblemType(pep,PEP_HERMITIAN);
    114:   PEPSetType(pep,PEPSTOAR);
    115:   /*
    116:      Set operators and set problem type
    117:   */
    118:   PEPSetOperators(pep,3,Op);

    120:   /*
    121:      Set shift-and-invert with Cholesky; select MUMPS if available
    122:   */
    123:   PEPGetST(pep,&st);
    124:   STGetKSP(st,&ksp);
    125:   KSPSetType(ksp,KSPPREONLY);
    126:   KSPGetPC(ksp,&pc);
    127:   PCSetType(pc,PCCHOLESKY);

    129:   /*
    130:      Use MUMPS if available.
    131:      Note that in complex scalars we cannot use MUMPS for spectrum slicing,
    132:      because MatGetInertia() is not available in that case.
    133:   */
    134: #if defined(PETSC_HAVE_MUMPS) && !defined(PETSC_USE_COMPLEX)
    135:   PCFactorSetMatSolverType(pc,MATSOLVERMUMPS);
    136:   /*
    137:      Add several MUMPS options (see ex43.c for a better way of setting them in program):
    138:      '-st_mat_mumps_icntl_13 1': turn off ScaLAPACK for matrix inertia
    139:   */
    140:   PetscOptionsInsertString(NULL,"-st_mat_mumps_icntl_13 1 -st_mat_mumps_icntl_24 1 -st_mat_mumps_cntl_3 1e-12");
    141: #endif

    143:   /*
    144:      Set solver parameters at runtime
    145:   */
    146:   PEPSetFromOptions(pep);

    148:   PetscOptionsGetBool(NULL,NULL,"-transform",&transform,NULL);
    149:   if (transform) {
    150:     /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    151:                     Check if the problem is definite
    152:        - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    153:     PEPCheckDefiniteQEP(pep,&xi,&mu,&def,&hyp);
    154:     switch (def) {
    155:       case 1:
    156:         if (hyp==1) PetscPrintf(PETSC_COMM_WORLD,"Hyperbolic Problem xi=%g\n",(double)xi);
    157:         else PetscPrintf(PETSC_COMM_WORLD,"Definite Problem xi=%g mu=%g\n",(double)xi,(double)mu);
    158:         break;
    159:       case -1:
    160:         PetscPrintf(PETSC_COMM_WORLD,"Not Definite Problem\n");
    161:         break;
    162:       default:
    163:         PetscPrintf(PETSC_COMM_WORLD,"Cannot determine definiteness\n");
    164:         break;
    165:     }

    167:     /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    168:       Transform the QEP to have a definite inner product in the linearization
    169:        - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    170:     if (def==1) {
    171:       QEPDefiniteTransformGetMatrices(pep,hyp==1?PETSC_TRUE:PETSC_FALSE,xi,mu,B);
    172:       PEPSetOperators(pep,3,B);
    173:       PEPGetTarget(pep,&target);
    174:       targett = target;
    175:       QEPDefiniteTransformMap(hyp==1?PETSC_TRUE:PETSC_FALSE,xi,mu,1,&targett,PETSC_FALSE);
    176:       PEPSetTarget(pep,targett);
    177:       PEPGetProblemType(pep,&type);
    178:       PEPSetProblemType(pep,PEP_HYPERBOLIC);
    179:       PEPSTOARGetLinearization(pep,&alpha,&beta);
    180:       PEPSTOARSetLinearization(pep,1.0,0.0);
    181:       PEPGetInterval(pep,&inta,&intb);
    182:       if (inta!=intb) {
    183:         ats[0] = inta; ats[1] = intb;
    184:         QEPDefiniteTransformMap(hyp==1?PETSC_TRUE:PETSC_FALSE,xi,mu,2,ats,PETSC_FALSE);
    185:         at[0] = PetscRealPart(ats[0]); at[1] = PetscRealPart(ats[1]);
    186:         if (at[0]<at[1]) PEPSetInterval(pep,at[0],at[1]);
    187:         else PEPSetInterval(pep,PETSC_MIN_REAL,at[1]);
    188:       }
    189:     }
    190:   }

    192:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    193:                       Solve the eigensystem
    194:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    195:   PEPSolve(pep);

    197:   /* show detailed info unless -terse option is given by user */
    198:   if (def!=1) {
    199:     PetscOptionsHasName(NULL,NULL,"-terse",&terse);
    200:     if (terse) PEPErrorView(pep,PEP_ERROR_BACKWARD,NULL);
    201:     else {
    202:       PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO_DETAIL);
    203:       PEPConvergedReasonView(pep,PETSC_VIEWER_STDOUT_WORLD);
    204:       PEPErrorView(pep,PEP_ERROR_RELATIVE,PETSC_VIEWER_STDOUT_WORLD);
    205:       PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD);
    206:     }
    207:   } else {
    208:     /* Map the solution */
    209:     PEPConvergedReasonView(pep,PETSC_VIEWER_STDOUT_WORLD);
    210:     QEPDefiniteCheckError(Op,pep,hyp==1?PETSC_TRUE:PETSC_FALSE,xi,mu);
    211:     for (i=0;i<3;i++) MatDestroy(B+i);
    212:   }
    213:   if (at[0]>at[1]) {
    214:     PEPSetInterval(pep,at[0],PETSC_MAX_REAL);
    215:     PEPSolve(pep);
    216:     PEPConvergedReasonView(pep,PETSC_VIEWER_STDOUT_WORLD);
    217:     /* Map the solution */
    218:     QEPDefiniteCheckError(Op,pep,hyp==1?PETSC_TRUE:PETSC_FALSE,xi,mu);
    219:   }
    220:   if (def==1) {
    221:     PEPSetTarget(pep,target);
    222:     PEPSetProblemType(pep,type);
    223:     PEPSTOARSetLinearization(pep,alpha,beta);
    224:     if (inta!=intb) PEPSetInterval(pep,inta,intb);
    225:   }

    227:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    228:                     Clean up
    229:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    230:   PEPDestroy(&pep);
    231:   for (i=0;i<3;i++) MatDestroy(Op+i);
    232:   SlepcFinalize();
    233:   return 0;
    234: }

    236: /* ------------------------------------------------------------------- */
    237: /*
    238:   QEPDefiniteTransformMap_Initial - map a scalar value with a certain Moebius transform

    240:                    a theta + b
    241:          lambda = --------------
    242:                    c theta + d

    244:   Input:
    245:     xi,mu: real values such that Q(xi)<0 and Q(mu)>0
    246:     hyperbolic: if true the problem is assumed hyperbolic (mu is not used)
    247:   Input/Output:
    248:     val (array of length n)
    249:     if backtransform=true returns lambda from theta, else returns theta from lambda
    250: */
    251: static PetscErrorCode QEPDefiniteTransformMap_Initial(PetscBool hyperbolic,PetscReal xi,PetscReal mu,PetscInt n,PetscScalar *val,PetscBool backtransform)
    252: {
    253:   PetscInt  i;
    254:   PetscReal a,b,c,d,s;

    256:   if (hyperbolic) { a = 1.0; b = xi; c =0.0; d = 1.0; }
    257:   else { a = mu; b = mu*xi-1; c = 1.0; d = xi+mu; }
    258:   if (!backtransform) { s = a; a = -d; d = -s; }
    259:   for (i=0;i<n;i++) {
    260:     if (PetscRealPart(val[i]) >= PETSC_MAX_REAL || PetscRealPart(val[i]) <= PETSC_MIN_REAL) val[i] = a/c;
    261:     else if (val[i] == -d/c) val[i] = PETSC_MAX_REAL;
    262:     else val[i] = (a*val[i]+b)/(c*val[i]+d);
    263:   }
    264:   return 0;
    265: }

    267: /* ------------------------------------------------------------------- */
    268: /*
    269:   QEPDefiniteTransformMap - perform the mapping if the problem is hyperbolic, otherwise
    270:   modify the value of xi in advance
    271: */
    272: PetscErrorCode QEPDefiniteTransformMap(PetscBool hyperbolic,PetscReal xi,PetscReal mu,PetscInt n,PetscScalar *val,PetscBool backtransform)
    273: {
    274:   PetscReal      xit;
    275:   PetscScalar    alpha;

    277:   xit = xi;
    278:   if (!hyperbolic) {
    279:     alpha = xi;
    280:     QEPDefiniteTransformMap_Initial(PETSC_FALSE,0.0,mu,1,&alpha,PETSC_FALSE);
    281:     xit = PetscRealPart(alpha);
    282:   }
    283:   QEPDefiniteTransformMap_Initial(hyperbolic,xit,mu,n,val,backtransform);
    284:   return 0;
    285: }

    287: /* ------------------------------------------------------------------- */
    288: /*
    289:   TransformMatricesMoebius - transform the coefficient matrices of a QEP

    291:   Input:
    292:     A: coefficient matrices of the original QEP
    293:     a,b,c,d: parameters of the Moebius transform
    294:     str: structure flag for MatAXPY operations
    295:   Output:
    296:     B: transformed matrices
    297: */
    298: PetscErrorCode TransformMatricesMoebius(Mat A[3],MatStructure str,PetscReal a,PetscReal b,PetscReal c,PetscReal d,Mat B[3])
    299: {
    300:   PetscInt       i,k;
    301:   PetscReal      cf[9];

    303:   for (i=0;i<3;i++) MatDuplicate(A[2],MAT_COPY_VALUES,&B[i]);
    304:   /* Ct = b*b*A+b*d*B+d*d*C */
    305:   cf[0] = d*d; cf[1] = b*d; cf[2] = b*b;
    306:   /* Bt = 2*a*b*A+(b*c+a*d)*B+2*c*d*C*/
    307:   cf[3] = 2*c*d; cf[4] = b*c+a*d; cf[5] = 2*a*b;
    308:   /* At = a*a*A+a*c*B+c*c*C */
    309:   cf[6] = c*c; cf[7] = a*c; cf[8] = a*a;
    310:   for (k=0;k<3;k++) {
    311:     MatScale(B[k],cf[k*3+2]);
    312:     for (i=0;i<2;i++) MatAXPY(B[k],cf[3*k+i],A[i],str);
    313:   }
    314:   return 0;
    315: }

    317: /* ------------------------------------------------------------------- */
    318: /*
    319:   QEPDefiniteTransformGetMatrices - given a PEP of degree 2, transform the three
    320:   matrices with TransformMatricesMoebius

    322:   Input:
    323:     pep: polynomial eigenproblem to be transformed, with Q(.) being the quadratic polynomial
    324:     xi,mu: real values such that Q(xi)<0 and Q(mu)>0
    325:     hyperbolic: if true the problem is assumed hyperbolic (mu is not used)
    326:   Output:
    327:     T: coefficient matrices of the transformed polynomial
    328: */
    329: PetscErrorCode QEPDefiniteTransformGetMatrices(PEP pep,PetscBool hyperbolic,PetscReal xi,PetscReal mu,Mat T[3])
    330: {
    331:   MatStructure   str;
    332:   ST             st;
    333:   PetscInt       i;
    334:   PetscReal      a,b,c,d;
    335:   PetscScalar    xit;
    336:   Mat            A[3];

    338:   for (i=2;i>=0;i--) PEPGetOperators(pep,i,&A[i]);
    339:   if (hyperbolic) { a = 1.0; b = xi; c =0.0; d = 1.0; }
    340:   else {
    341:     xit = xi;
    342:     QEPDefiniteTransformMap_Initial(PETSC_FALSE,0.0,mu,1,&xit,PETSC_FALSE);
    343:     a = mu; b = mu*PetscRealPart(xit)-1.0; c = 1.0; d = PetscRealPart(xit)+mu;
    344:   }
    345:   PEPGetST(pep,&st);
    346:   STGetMatStructure(st,&str);
    347:   TransformMatricesMoebius(A,str,a,b,c,d,T);
    348:   return 0;
    349: }

    351: /* ------------------------------------------------------------------- */
    352: /*
    353:   Auxiliary function to compute the residual norm of an eigenpair of a QEP defined
    354:   by coefficient matrices A
    355: */
    356: static PetscErrorCode PEPResidualNorm(Mat *A,PetscScalar kr,PetscScalar ki,Vec xr,Vec xi,Vec *z,PetscReal *norm)
    357: {
    358:   PetscInt       i,nmat=3;
    359:   PetscScalar    vals[3];
    360:   Vec            u,w;
    361: #if !defined(PETSC_USE_COMPLEX)
    362:   Vec            ui,wi;
    363:   PetscReal      ni;
    364:   PetscBool      imag;
    365:   PetscScalar    ivals[3];
    366: #endif

    368:   u = z[0]; w = z[1];
    369:   VecSet(u,0.0);
    370: #if !defined(PETSC_USE_COMPLEX)
    371:   ui = z[2]; wi = z[3];
    372: #endif
    373:   vals[0] = 1.0;
    374:   vals[1] = kr;
    375:   vals[2] = kr*kr-ki*ki;
    376: #if !defined(PETSC_USE_COMPLEX)
    377:   ivals[0] = 0.0;
    378:   ivals[1] = ki;
    379:   ivals[2] = 2.0*kr*ki;
    380:   if (ki == 0 || PetscAbsScalar(ki) < PetscAbsScalar(kr*PETSC_MACHINE_EPSILON))
    381:     imag = PETSC_FALSE;
    382:   else {
    383:     imag = PETSC_TRUE;
    384:     VecSet(ui,0.0);
    385:   }
    386: #endif
    387:   for (i=0;i<nmat;i++) {
    388:     if (vals[i]!=0.0) {
    389:       MatMult(A[i],xr,w);
    390:       VecAXPY(u,vals[i],w);
    391:     }
    392: #if !defined(PETSC_USE_COMPLEX)
    393:     if (imag) {
    394:       if (ivals[i]!=0 || vals[i]!=0) {
    395:         MatMult(A[i],xi,wi);
    396:         if (vals[i]==0) MatMult(A[i],xr,w);
    397:       }
    398:       if (ivals[i]!=0) {
    399:         VecAXPY(u,-ivals[i],wi);
    400:         VecAXPY(ui,ivals[i],w);
    401:       }
    402:       if (vals[i]!=0) VecAXPY(ui,vals[i],wi);
    403:     }
    404: #endif
    405:   }
    406:   VecNorm(u,NORM_2,norm);
    407: #if !defined(PETSC_USE_COMPLEX)
    408:   if (imag) {
    409:     VecNorm(ui,NORM_2,&ni);
    410:     *norm = SlepcAbsEigenvalue(*norm,ni);
    411:   }
    412: #endif
    413:   return 0;
    414: }

    416: /* ------------------------------------------------------------------- */
    417: /*
    418:   QEPDefiniteCheckError - check and print the residual norm of a transformed PEP

    420:   Input:
    421:     A: coefficient matrices of the original problem
    422:     pep: solver containing the computed solution of the transformed problem
    423:     xi,mu,hyperbolic: parameters used in transformation
    424: */
    425: PetscErrorCode QEPDefiniteCheckError(Mat *A,PEP pep,PetscBool hyperbolic,PetscReal xi,PetscReal mu)
    426: {
    427:   PetscScalar    er,ei;
    428:   PetscReal      re,im,error;
    429:   Vec            vr,vi,w[4];
    430:   PetscInt       i,nconv;
    431:   BV             bv;
    432:   char           ex[30],sep[]=" ---------------------- --------------------\n";

    434:   PetscSNPrintf(ex,sizeof(ex),"||P(k)x||/||kx||");
    435:   PetscPrintf(PETSC_COMM_WORLD,"%s            k             %s\n%s",sep,ex,sep);
    436:   PEPGetConverged(pep,&nconv);
    437:   PEPGetBV(pep,&bv);
    438:   BVCreateVec(bv,w);
    439:   VecDuplicate(w[0],&vr);
    440:   VecDuplicate(w[0],&vi);
    441:   for (i=1;i<4;i++) VecDuplicate(w[0],w+i);
    442:   for (i=0;i<nconv;i++) {
    443:     PEPGetEigenpair(pep,i,&er,&ei,vr,vi);
    444:     QEPDefiniteTransformMap(hyperbolic,xi,mu,1,&er,PETSC_TRUE);
    445:     PEPResidualNorm(A,er,0.0,vr,vi,w,&error);
    446:     error /= SlepcAbsEigenvalue(er,0.0);
    447: #if defined(PETSC_USE_COMPLEX)
    448:     re = PetscRealPart(er);
    449:     im = PetscImaginaryPart(ei);
    450: #else
    451:     re = er;
    452:     im = ei;
    453: #endif
    454:     if (im!=0.0) PetscPrintf(PETSC_COMM_WORLD,"  % 9f%+9fi      %12g\n",(double)re,(double)im,(double)error);
    455:     else PetscPrintf(PETSC_COMM_WORLD,"    % 12f           %12g\n",(double)re,(double)error);
    456:   }
    457:   PetscPrintf(PETSC_COMM_WORLD,"%s",sep);
    458:   for (i=0;i<4;i++) VecDestroy(w+i);
    459:   VecDestroy(&vi);
    460:   VecDestroy(&vr);
    461:   return 0;
    462: }

    464: /*TEST

    466:    testset:
    467:       requires: !single
    468:       args: -pep_nev 3 -nonhyperbolic -pep_target 2
    469:       output_file: output/ex40_1.out
    470:       filter: grep -v "Definite" | sed -e "s/iterations [0-9]\([0-9]*\)/iterations xx/g" | sed -e "s/[0-9]\.[0-9]*e[+-]\([0-9]*\)/removed/g"
    471:       test:
    472:          suffix: 1
    473:          requires: !complex
    474:       test:
    475:          suffix: 1_complex
    476:          requires: complex !mumps
    477:       test:
    478:          suffix: 1_transform
    479:          requires: !complex
    480:          args: -transform
    481:       test:
    482:          suffix: 1_transform_complex
    483:          requires: complex !mumps
    484:          args: -transform

    486: TEST*/