2: /*
3: This file implements the conjugate gradient method in PETSc as part of
4: KSP. You can use this as a starting point for implementing your own
5: Krylov method that is not provided with PETSc.
7: The following basic routines are required for each Krylov method.
8: KSPCreate_XXX() - Creates the Krylov context
9: KSPSetFromOptions_XXX() - Sets runtime options
10: KSPSolve_XXX() - Runs the Krylov method
11: KSPDestroy_XXX() - Destroys the Krylov context, freeing all
12: memory it needed
13: Here the "_XXX" denotes a particular implementation, in this case
14: we use _CG (e.g. KSPCreate_CG, KSPDestroy_CG). These routines are
15: are actually called via the common user interface routines
16: KSPSetType(), KSPSetFromOptions(), KSPSolve(), and KSPDestroy() so the
17: application code interface remains identical for all preconditioners.
19: Other basic routines for the KSP objects include
20: KSPSetUp_XXX()
21: KSPView_XXX() - Prints details of solver being used.
23: Detailed notes:
24: By default, this code implements the CG (Conjugate Gradient) method,
25: which is valid for real symmetric (and complex Hermitian) positive
26: definite matrices. Note that for the complex Hermitian case, the
27: VecDot() arguments within the code MUST remain in the order given
28: for correct computation of inner products.
30: Reference: Hestenes and Steifel, 1952.
32: By switching to the indefinite vector inner product, VecTDot(), the
33: same code is used for the complex symmetric case as well. The user
34: must call KSPCGSetType(ksp,KSP_CG_SYMMETRIC) or use the option
35: -ksp_cg_type symmetric to invoke this variant for the complex case.
36: Note, however, that the complex symmetric code is NOT valid for
37: all such matrices ... and thus we don't recommend using this method.
38: */
39: /*
40: cgimpl.h defines the simple data structured used to store information
41: related to the type of matrix (e.g. complex symmetric) being solved and
42: data used during the optional Lanczo process used to compute eigenvalues
43: */
44: #include <../src/ksp/ksp/impls/cg/cgimpl.h> 45: extern PetscErrorCode KSPComputeExtremeSingularValues_CG(KSP,PetscReal*,PetscReal*);
46: extern PetscErrorCode KSPComputeEigenvalues_CG(KSP,PetscInt,PetscReal*,PetscReal*,PetscInt*);
48: /*
49: KSPSetUp_CG - Sets up the workspace needed by the CG method.
51: This is called once, usually automatically by KSPSolve() or KSPSetUp()
52: but can be called directly by KSPSetUp()
53: */
54: static PetscErrorCode KSPSetUp_CG(KSP ksp) 55: {
56: KSP_CG *cgP = (KSP_CG*)ksp->data;
58: PetscInt maxit = ksp->max_it,nwork = 3;
61: /* get work vectors needed by CG */
62: if (cgP->singlereduction) nwork += 2;
63: KSPSetWorkVecs(ksp,nwork);
65: /*
66: If user requested computations of eigenvalues then allocate work
67: work space needed
68: */
69: if (ksp->calc_sings) {
70: /* get space to store tridiagonal matrix for Lanczos */
71: PetscMalloc4(maxit+1,&cgP->e,maxit+1,&cgP->d,maxit+1,&cgP->ee,maxit+1,&cgP->dd);
72: PetscLogObjectMemory((PetscObject)ksp,2*(maxit+1)*(sizeof(PetscScalar)+sizeof(PetscReal)));
74: ksp->ops->computeextremesingularvalues = KSPComputeExtremeSingularValues_CG;
75: ksp->ops->computeeigenvalues = KSPComputeEigenvalues_CG;
76: }
77: return(0);
78: }
80: /*
81: A macro used in the following KSPSolve_CG and KSPSolve_CG_SingleReduction routines
82: */
83: #define VecXDot(x,y,a) (((cg->type) == (KSP_CG_HERMITIAN)) ? VecDot(x,y,a) : VecTDot(x,y,a)) 85: /*
86: KSPSolve_CG - This routine actually applies the conjugate gradient method
88: Note : this routine can be replaced with another one (see below) which implements
89: another variant of CG.
91: Input Parameter:
92: . ksp - the Krylov space object that was set to use conjugate gradient, by, for
93: example, KSPCreate(MPI_Comm,KSP *ksp); KSPSetType(ksp,KSPCG);
94: */
95: static PetscErrorCode KSPSolve_CG(KSP ksp) 96: {
98: PetscInt i,stored_max_it,eigs;
99: PetscScalar dpi = 0.0,a = 1.0,beta,betaold = 1.0,b = 0,*e = 0,*d = 0,dpiold;
100: PetscReal dp = 0.0;
101: Vec X,B,Z,R,P,W;
102: KSP_CG *cg;
103: Mat Amat,Pmat;
104: PetscBool diagonalscale;
107: PCGetDiagonalScale(ksp->pc,&diagonalscale);
108: if (diagonalscale) SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Krylov method %s does not support diagonal scaling",((PetscObject)ksp)->type_name);
110: cg = (KSP_CG*)ksp->data;
111: eigs = ksp->calc_sings;
112: stored_max_it = ksp->max_it;
113: X = ksp->vec_sol;
114: B = ksp->vec_rhs;
115: R = ksp->work[0];
116: Z = ksp->work[1];
117: P = ksp->work[2];
118: W = Z;
120: if (eigs) {e = cg->e; d = cg->d; e[0] = 0.0; }
121: PCGetOperators(ksp->pc,&Amat,&Pmat);
123: ksp->its = 0;
124: if (!ksp->guess_zero) {
125: KSP_MatMult(ksp,Amat,X,R); /* r <- b - Ax */
126: VecAYPX(R,-1.0,B);
127: } else {
128: VecCopy(B,R); /* r <- b (x is 0) */
129: }
131: switch (ksp->normtype) {
132: case KSP_NORM_PRECONDITIONED:
133: KSP_PCApply(ksp,R,Z); /* z <- Br */
134: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z = e'*A'*B'*B*A'*e' */
135: break;
136: case KSP_NORM_UNPRECONDITIONED:
137: VecNorm(R,NORM_2,&dp); /* dp <- r'*r = e'*A'*A*e */
138: break;
139: case KSP_NORM_NATURAL:
140: KSP_PCApply(ksp,R,Z); /* z <- Br */
141: VecXDot(Z,R,&beta); /* beta <- z'*r */
142: KSPCheckDot(ksp,beta);
143: dp = PetscSqrtReal(PetscAbsScalar(beta)); /* dp <- r'*z = r'*B*r = e'*A'*B*A*e */
144: break;
145: case KSP_NORM_NONE:
146: dp = 0.0;
147: break;
148: default:SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"%s",KSPNormTypes[ksp->normtype]);
149: }
150: KSPLogResidualHistory(ksp,dp);
151: KSPMonitor(ksp,0,dp);
152: ksp->rnorm = dp;
154: (*ksp->converged)(ksp,0,dp,&ksp->reason,ksp->cnvP); /* test for convergence */
155: if (ksp->reason) return(0);
157: if (ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) {
158: KSP_PCApply(ksp,R,Z); /* z <- Br */
159: }
160: if (ksp->normtype != KSP_NORM_NATURAL) {
161: VecXDot(Z,R,&beta); /* beta <- z'*r */
162: KSPCheckDot(ksp,beta);
163: }
165: i = 0;
166: do {
167: ksp->its = i+1;
168: if (beta == 0.0) {
169: ksp->reason = KSP_CONVERGED_ATOL;
170: PetscInfo(ksp,"converged due to beta = 0\n");
171: break;
172: #if !defined(PETSC_USE_COMPLEX)
173: } else if ((i > 0) && (beta*betaold < 0.0)) {
174: ksp->reason = KSP_DIVERGED_INDEFINITE_PC;
175: PetscInfo(ksp,"diverging due to indefinite preconditioner\n");
176: break;
177: #endif
178: }
179: if (!i) {
180: VecCopy(Z,P); /* p <- z */
181: b = 0.0;
182: } else {
183: b = beta/betaold;
184: if (eigs) {
185: if (ksp->max_it != stored_max_it) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Can not change maxit AND calculate eigenvalues");
186: e[i] = PetscSqrtReal(PetscAbsScalar(b))/a;
187: }
188: VecAYPX(P,b,Z); /* p <- z + b* p */
189: }
190: dpiold = dpi;
191: KSP_MatMult(ksp,Amat,P,W); /* w <- Ap */
192: VecXDot(P,W,&dpi); /* dpi <- p'w */
193: KSPCheckDot(ksp,dpi);
194: betaold = beta;
196: if ((dpi == 0.0) || ((i > 0) && (PetscRealPart(dpi*dpiold) <= 0.0))) {
197: ksp->reason = KSP_DIVERGED_INDEFINITE_MAT;
198: PetscInfo(ksp,"diverging due to indefinite or negative definite matrix\n");
199: break;
200: }
201: a = beta/dpi; /* a = beta/p'w */
202: if (eigs) d[i] = PetscSqrtReal(PetscAbsScalar(b))*e[i] + 1.0/a;
203: VecAXPY(X,a,P); /* x <- x + ap */
204: VecAXPY(R,-a,W); /* r <- r - aw */
205: if (ksp->normtype == KSP_NORM_PRECONDITIONED && ksp->chknorm < i+2) {
206: KSP_PCApply(ksp,R,Z); /* z <- Br */
207: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z */
208: } else if (ksp->normtype == KSP_NORM_UNPRECONDITIONED && ksp->chknorm < i+2) {
209: VecNorm(R,NORM_2,&dp); /* dp <- r'*r */
210: } else if (ksp->normtype == KSP_NORM_NATURAL) {
211: KSP_PCApply(ksp,R,Z); /* z <- Br */
212: VecXDot(Z,R,&beta); /* beta <- r'*z */
213: KSPCheckDot(ksp,beta);
214: dp = PetscSqrtReal(PetscAbsScalar(beta));
215: } else {
216: dp = 0.0;
217: }
218: ksp->rnorm = dp;
219: KSPLogResidualHistory(ksp,dp);
220: if (eigs) cg->ned = ksp->its;
221: KSPMonitor(ksp,i+1,dp);
222: (*ksp->converged)(ksp,i+1,dp,&ksp->reason,ksp->cnvP);
223: if (ksp->reason) break;
225: if ((ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) || (ksp->chknorm >= i+2)) {
226: KSP_PCApply(ksp,R,Z); /* z <- Br */
227: }
228: if ((ksp->normtype != KSP_NORM_NATURAL) || (ksp->chknorm >= i+2)) {
229: VecXDot(Z,R,&beta); /* beta <- z'*r */
230: KSPCheckDot(ksp,beta);
231: }
233: i++;
234: } while (i<ksp->max_it);
235: if (i >= ksp->max_it) ksp->reason = KSP_DIVERGED_ITS;
236: return(0);
237: }
239: /*
240: KSPSolve_CG_SingleReduction
242: This variant of CG is identical in exact arithmetic to the standard algorithm,
243: but is rearranged to use only a single reduction stage per iteration, using additional
244: intermediate vectors.
246: See KSPCGUseSingleReduction_CG()
248: */
249: static PetscErrorCode KSPSolve_CG_SingleReduction(KSP ksp)250: {
252: PetscInt i,stored_max_it,eigs;
253: PetscScalar dpi = 0.0,a = 1.0,beta,betaold = 1.0,b = 0,*e = 0,*d = 0,delta,dpiold,tmp[2];
254: PetscReal dp = 0.0;
255: Vec X,B,Z,R,P,S,W,tmpvecs[2];
256: KSP_CG *cg;
257: Mat Amat,Pmat;
258: PetscBool diagonalscale;
261: PCGetDiagonalScale(ksp->pc,&diagonalscale);
262: if (diagonalscale) SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Krylov method %s does not support diagonal scaling",((PetscObject)ksp)->type_name);
264: cg = (KSP_CG*)ksp->data;
265: eigs = ksp->calc_sings;
266: stored_max_it = ksp->max_it;
267: X = ksp->vec_sol;
268: B = ksp->vec_rhs;
269: R = ksp->work[0];
270: Z = ksp->work[1];
271: P = ksp->work[2];
272: S = ksp->work[3];
273: W = ksp->work[4];
275: if (eigs) {e = cg->e; d = cg->d; e[0] = 0.0; }
276: PCGetOperators(ksp->pc,&Amat,&Pmat);
278: ksp->its = 0;
279: if (!ksp->guess_zero) {
280: KSP_MatMult(ksp,Amat,X,R); /* r <- b - Ax */
281: VecAYPX(R,-1.0,B);
282: } else {
283: VecCopy(B,R); /* r <- b (x is 0) */
284: }
286: switch (ksp->normtype) {
287: case KSP_NORM_PRECONDITIONED:
288: KSP_PCApply(ksp,R,Z); /* z <- Br */
289: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z = e'*A'*B'*B*A'*e' */
290: break;
291: case KSP_NORM_UNPRECONDITIONED:
292: VecNorm(R,NORM_2,&dp); /* dp <- r'*r = e'*A'*A*e */
293: break;
294: case KSP_NORM_NATURAL:
295: KSP_PCApply(ksp,R,Z); /* z <- Br */
296: KSP_MatMult(ksp,Amat,Z,S);
297: VecXDot(Z,S,&delta); /* delta <- z'*A*z = r'*B*A*B*r */
298: VecXDot(Z,R,&beta); /* beta <- z'*r */
299: KSPCheckDot(ksp,beta);
300: dp = PetscSqrtReal(PetscAbsScalar(beta)); /* dp <- r'*z = r'*B*r = e'*A'*B*A*e */
301: break;
302: case KSP_NORM_NONE:
303: dp = 0.0;
304: break;
305: default:SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"%s",KSPNormTypes[ksp->normtype]);
306: }
307: KSPLogResidualHistory(ksp,dp);
308: KSPMonitor(ksp,0,dp);
309: ksp->rnorm = dp;
311: (*ksp->converged)(ksp,0,dp,&ksp->reason,ksp->cnvP); /* test for convergence */
312: if (ksp->reason) return(0);
314: if (ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) {
315: KSP_PCApply(ksp,R,Z); /* z <- Br */
316: }
317: if (ksp->normtype != KSP_NORM_NATURAL) {
318: KSP_MatMult(ksp,Amat,Z,S);
319: VecXDot(Z,S,&delta); /* delta <- z'*A*z = r'*B*A*B*r */
320: VecXDot(Z,R,&beta); /* beta <- z'*r */
321: KSPCheckDot(ksp,beta);
322: }
324: i = 0;
325: do {
326: ksp->its = i+1;
327: if (beta == 0.0) {
328: ksp->reason = KSP_CONVERGED_ATOL;
329: PetscInfo(ksp,"converged due to beta = 0\n");
330: break;
331: #if !defined(PETSC_USE_COMPLEX)
332: } else if ((i > 0) && (beta*betaold < 0.0)) {
333: ksp->reason = KSP_DIVERGED_INDEFINITE_PC;
334: PetscInfo(ksp,"diverging due to indefinite preconditioner\n");
335: break;
336: #endif
337: }
338: if (!i) {
339: VecCopy(Z,P); /* p <- z */
340: b = 0.0;
341: } else {
342: b = beta/betaold;
343: if (eigs) {
344: if (ksp->max_it != stored_max_it) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Can not change maxit AND calculate eigenvalues");
345: e[i] = PetscSqrtReal(PetscAbsScalar(b))/a;
346: }
347: VecAYPX(P,b,Z); /* p <- z + b* p */
348: }
349: dpiold = dpi;
350: if (!i) {
351: KSP_MatMult(ksp,Amat,P,W); /* w <- Ap */
352: VecXDot(P,W,&dpi); /* dpi <- p'w */
353: } else {
354: VecAYPX(W,beta/betaold,S); /* w <- Ap */
355: dpi = delta - beta*beta*dpiold/(betaold*betaold); /* dpi <- p'w */
356: }
357: betaold = beta;
358: KSPCheckDot(ksp,beta);
360: if ((dpi == 0.0) || ((i > 0) && (PetscRealPart(dpi*dpiold) <= 0.0))) {
361: ksp->reason = KSP_DIVERGED_INDEFINITE_MAT;
362: PetscInfo(ksp,"diverging due to indefinite or negative definite matrix\n");
363: break;
364: }
365: a = beta/dpi; /* a = beta/p'w */
366: if (eigs) d[i] = PetscSqrtReal(PetscAbsScalar(b))*e[i] + 1.0/a;
367: VecAXPY(X,a,P); /* x <- x + ap */
368: VecAXPY(R,-a,W); /* r <- r - aw */
369: if (ksp->normtype == KSP_NORM_PRECONDITIONED && ksp->chknorm < i+2) {
370: KSP_PCApply(ksp,R,Z); /* z <- Br */
371: KSP_MatMult(ksp,Amat,Z,S);
372: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z */
373: } else if (ksp->normtype == KSP_NORM_UNPRECONDITIONED && ksp->chknorm < i+2) {
374: VecNorm(R,NORM_2,&dp); /* dp <- r'*r */
375: } else if (ksp->normtype == KSP_NORM_NATURAL) {
376: KSP_PCApply(ksp,R,Z); /* z <- Br */
377: tmpvecs[0] = S; tmpvecs[1] = R;
378: KSP_MatMult(ksp,Amat,Z,S);
379: VecMDot(Z,2,tmpvecs,tmp); /* delta <- z'*A*z = r'*B*A*B*r */
380: delta = tmp[0]; beta = tmp[1]; /* beta <- z'*r */
381: KSPCheckDot(ksp,beta);
382: dp = PetscSqrtReal(PetscAbsScalar(beta)); /* dp <- r'*z = r'*B*r = e'*A'*B*A*e */
383: } else {
384: dp = 0.0;
385: }
386: ksp->rnorm = dp;
387: KSPLogResidualHistory(ksp,dp);
388: if (eigs) cg->ned = ksp->its;
389: KSPMonitor(ksp,i+1,dp);
390: (*ksp->converged)(ksp,i+1,dp,&ksp->reason,ksp->cnvP);
391: if (ksp->reason) break;
393: if ((ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) || (ksp->chknorm >= i+2)) {
394: KSP_PCApply(ksp,R,Z); /* z <- Br */
395: KSP_MatMult(ksp,Amat,Z,S);
396: }
397: if ((ksp->normtype != KSP_NORM_NATURAL) || (ksp->chknorm >= i+2)) {
398: tmpvecs[0] = S; tmpvecs[1] = R;
399: VecMDot(Z,2,tmpvecs,tmp);
400: delta = tmp[0]; beta = tmp[1]; /* delta <- z'*A*z = r'*B'*A*B*r */
401: KSPCheckDot(ksp,beta); /* beta <- z'*r */
402: }
404: i++;
405: } while (i<ksp->max_it);
406: if (i >= ksp->max_it) ksp->reason = KSP_DIVERGED_ITS;
407: return(0);
408: }
410: /*
411: KSPDestroy_CG - Frees resources allocated in KSPSetup_CG and clears function
412: compositions from KSPCreate_CG. If adding your own KSP implementation,
413: you must be sure to free all allocated resources here to prevent
414: leaks.
415: */
416: PetscErrorCode KSPDestroy_CG(KSP ksp)417: {
418: KSP_CG *cg = (KSP_CG*)ksp->data;
422: /* free space used for singular value calculations */
423: if (ksp->calc_sings) {
424: PetscFree4(cg->e,cg->d,cg->ee,cg->dd);
425: }
426: KSPDestroyDefault(ksp);
427: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGSetType_C",NULL);
428: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGUseSingleReduction_C",NULL);
429: return(0);
430: }
432: /*
433: KSPView_CG - Prints information about the current Krylov method being used.
434: If your Krylov method has special options or flags that information
435: should be printed here.
436: */
437: PetscErrorCode KSPView_CG(KSP ksp,PetscViewer viewer)438: {
439: KSP_CG *cg = (KSP_CG*)ksp->data;
441: PetscBool iascii;
444: PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
445: if (iascii) {
446: #if defined(PETSC_USE_COMPLEX)
447: PetscViewerASCIIPrintf(viewer," variant %s\n",KSPCGTypes[cg->type]);
448: #endif
449: if (cg->singlereduction) {
450: PetscViewerASCIIPrintf(viewer," using single-reduction variant\n");
451: }
452: }
453: return(0);
454: }
456: /*
457: KSPSetFromOptions_CG - Checks the options database for options related to the
458: conjugate gradient method.
459: */
460: PetscErrorCode KSPSetFromOptions_CG(PetscOptionItems *PetscOptionsObject,KSP ksp)461: {
463: KSP_CG *cg = (KSP_CG*)ksp->data;
466: PetscOptionsHead(PetscOptionsObject,"KSP CG and CGNE options");
467: #if defined(PETSC_USE_COMPLEX)
468: PetscOptionsEnum("-ksp_cg_type","Matrix is Hermitian or complex symmetric","KSPCGSetType",KSPCGTypes,(PetscEnum)cg->type,
469: (PetscEnum*)&cg->type,NULL);
470: #endif
471: PetscOptionsBool("-ksp_cg_single_reduction","Merge inner products into single MPIU_Allreduce()","KSPCGUseSingleReduction",cg->singlereduction,&cg->singlereduction,NULL);
472: PetscOptionsTail();
473: return(0);
474: }
476: /*
477: KSPCGSetType_CG - This is an option that is SPECIFIC to this particular Krylov method.
478: This routine is registered below in KSPCreate_CG() and called from the
479: routine KSPCGSetType() (see the file cgtype.c).
480: */
481: PetscErrorCode KSPCGSetType_CG(KSP ksp,KSPCGType type)482: {
483: KSP_CG *cg = (KSP_CG*)ksp->data;
486: cg->type = type;
487: return(0);
488: }
490: /*
491: KSPCGUseSingleReduction_CG
493: This routine sets a flag to use a variant of CG. Note that (in somewhat
494: atypical fashion) it also swaps out the routine called when KSPSolve()
495: is invoked.
496: */
497: static PetscErrorCode KSPCGUseSingleReduction_CG(KSP ksp,PetscBool flg)498: {
499: KSP_CG *cg = (KSP_CG*)ksp->data;
502: cg->singlereduction = flg;
503: if (cg->singlereduction) {
504: ksp->ops->solve = KSPSolve_CG_SingleReduction;
505: } else {
506: ksp->ops->solve = KSPSolve_CG;
507: }
508: return(0);
509: }
511: /*
512: KSPCreate_CG - Creates the data structure for the Krylov method CG and sets the
513: function pointers for all the routines it needs to call (KSPSolve_CG() etc)
515: It must be labeled as PETSC_EXTERN to be dynamically linkable in C++
516: */
517: /*MC
518: KSPCG - The Preconditioned Conjugate Gradient (PCG) iterative method
520: Options Database Keys:
521: + -ksp_cg_type Hermitian - (for complex matrices only) indicates the matrix is Hermitian, see KSPCGSetType()
522: . -ksp_cg_type symmetric - (for complex matrices only) indicates the matrix is symmetric
523: - -ksp_cg_single_reduction - performs both inner products needed in the algorithm with a single MPIU_Allreduce() call, see KSPCGUseSingleReduction()
525: Level: beginner
527: Notes: The PCG method requires both the matrix and preconditioner to be symmetric positive (or negative) (semi) definite.
528: 529: Only left preconditioning is supported; there are several ways to motivate preconditioned CG, but they all produce the same algorithm.
530: One can interpret preconditioning A with B to mean any of the following\:
531: .n (1) Solve a left-preconditioned system BAx = Bb, using inv(B) to define an inner product in the algorithm.
532: .n (2) Solve a right-preconditioned system ABy = b, x = By, using B to define an inner product in the algorithm.
533: .n (3) Solve a symmetrically-preconditioned system, E^TAEy = E^Tb, x = Ey, where B = EE^T.
534: .n (4) Solve Ax=b with CG, but use the inner product defined by B to define the method [2].
535: .n In all cases, the resulting algorithm only requires application of B to vectors.
537: For complex numbers there are two different CG methods, one for Hermitian symmetric matrices and one for non-Hermitian symmetric matrices. Use
538: KSPCGSetType() to indicate which type you are using.
540: Developer Notes: KSPSolve_CG() should actually query the matrix to determine if it is Hermitian symmetric or not and NOT require the user to
541: indicate it to the KSP object.
543: References:
544: . 1. - Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems,
545: Journal of Research of the National Bureau of Standards Vol. 49, No. 6, December 1952 Research Paper 2379
546: . 2. - Josef Malek and Zdenek Strakos, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs,
547: SIAM, 2014.
549: .seealso: KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP,
550: KSPCGSetType(), KSPCGUseSingleReduction(), KSPPIPECG, KSPGROPPCG552: M*/
553: PETSC_EXTERN PetscErrorCode KSPCreate_CG(KSP ksp)554: {
556: KSP_CG *cg;
559: PetscNewLog(ksp,&cg);
560: #if !defined(PETSC_USE_COMPLEX)
561: cg->type = KSP_CG_SYMMETRIC;
562: #else
563: cg->type = KSP_CG_HERMITIAN;
564: #endif
565: ksp->data = (void*)cg;
567: KSPSetSupportedNorm(ksp,KSP_NORM_PRECONDITIONED,PC_LEFT,3);
568: KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_LEFT,2);
569: KSPSetSupportedNorm(ksp,KSP_NORM_NATURAL,PC_LEFT,2);
570: KSPSetSupportedNorm(ksp,KSP_NORM_NONE,PC_LEFT,1);
572: /*
573: Sets the functions that are associated with this data structure
574: (in C++ this is the same as defining virtual functions)
575: */
576: ksp->ops->setup = KSPSetUp_CG;
577: ksp->ops->solve = KSPSolve_CG;
578: ksp->ops->destroy = KSPDestroy_CG;
579: ksp->ops->view = KSPView_CG;
580: ksp->ops->setfromoptions = KSPSetFromOptions_CG;
581: ksp->ops->buildsolution = KSPBuildSolutionDefault;
582: ksp->ops->buildresidual = KSPBuildResidualDefault;
584: /*
585: Attach the function KSPCGSetType_CG() to this object. The routine
586: KSPCGSetType() checks for this attached function and calls it if it finds
587: it. (Sort of like a dynamic member function that can be added at run time
588: */
589: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGSetType_C",KSPCGSetType_CG);
590: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGUseSingleReduction_C",KSPCGUseSingleReduction_CG);
591: return(0);
592: }