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: if (ksp->errorifnotconverged) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_NOT_CONVERGED,"Diverged due to indefinite preconditioner");
175: ksp->reason = KSP_DIVERGED_INDEFINITE_PC;
176: PetscInfo(ksp,"diverging due to indefinite preconditioner\n");
177: break;
178: #endif
179: }
180: if (!i) {
181: VecCopy(Z,P); /* p <- z */
182: b = 0.0;
183: } else {
184: b = beta/betaold;
185: if (eigs) {
186: if (ksp->max_it != stored_max_it) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Can not change maxit AND calculate eigenvalues");
187: e[i] = PetscSqrtReal(PetscAbsScalar(b))/a;
188: }
189: VecAYPX(P,b,Z); /* p <- z + b* p */
190: }
191: dpiold = dpi;
192: KSP_MatMult(ksp,Amat,P,W); /* w <- Ap */
193: VecXDot(P,W,&dpi); /* dpi <- p'w */
194: KSPCheckDot(ksp,dpi);
195: betaold = beta;
197: if ((dpi == 0.0) || ((i > 0) && (PetscRealPart(dpi*dpiold) <= 0.0))) {
198: if (ksp->errorifnotconverged) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_NOT_CONVERGED,"Diverged due to indefinite matrix");
199: ksp->reason = KSP_DIVERGED_INDEFINITE_MAT;
200: PetscInfo(ksp,"diverging due to indefinite or negative definite matrix\n");
201: break;
202: }
203: a = beta/dpi; /* a = beta/p'w */
204: if (eigs) d[i] = PetscSqrtReal(PetscAbsScalar(b))*e[i] + 1.0/a;
205: VecAXPY(X,a,P); /* x <- x + ap */
206: VecAXPY(R,-a,W); /* r <- r - aw */
207: if (ksp->normtype == KSP_NORM_PRECONDITIONED && ksp->chknorm < i+2) {
208: KSP_PCApply(ksp,R,Z); /* z <- Br */
209: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z */
210: } else if (ksp->normtype == KSP_NORM_UNPRECONDITIONED && ksp->chknorm < i+2) {
211: VecNorm(R,NORM_2,&dp); /* dp <- r'*r */
212: } else if (ksp->normtype == KSP_NORM_NATURAL) {
213: KSP_PCApply(ksp,R,Z); /* z <- Br */
214: VecXDot(Z,R,&beta); /* beta <- r'*z */
215: KSPCheckDot(ksp,beta);
216: dp = PetscSqrtReal(PetscAbsScalar(beta));
217: } else {
218: dp = 0.0;
219: }
220: ksp->rnorm = dp;
221: KSPLogResidualHistory(ksp,dp);
222: if (eigs) cg->ned = ksp->its;
223: KSPMonitor(ksp,i+1,dp);
224: (*ksp->converged)(ksp,i+1,dp,&ksp->reason,ksp->cnvP);
225: if (ksp->reason) break;
227: if ((ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) || (ksp->chknorm >= i+2)) {
228: KSP_PCApply(ksp,R,Z); /* z <- Br */
229: }
230: if ((ksp->normtype != KSP_NORM_NATURAL) || (ksp->chknorm >= i+2)) {
231: VecXDot(Z,R,&beta); /* beta <- z'*r */
232: KSPCheckDot(ksp,beta);
233: }
235: i++;
236: } while (i<ksp->max_it);
237: if (i >= ksp->max_it) ksp->reason = KSP_DIVERGED_ITS;
238: return(0);
239: }
241: /*
242: KSPSolve_CG_SingleReduction
244: This variant of CG is identical in exact arithmetic to the standard algorithm,
245: but is rearranged to use only a single reduction stage per iteration, using additional
246: intermediate vectors.
248: See KSPCGUseSingleReduction_CG()
250: */
251: static PetscErrorCode KSPSolve_CG_SingleReduction(KSP ksp)252: {
254: PetscInt i,stored_max_it,eigs;
255: PetscScalar dpi = 0.0,a = 1.0,beta,betaold = 1.0,b = 0,*e = 0,*d = 0,delta,dpiold,tmp[2];
256: PetscReal dp = 0.0;
257: Vec X,B,Z,R,P,S,W,tmpvecs[2];
258: KSP_CG *cg;
259: Mat Amat,Pmat;
260: PetscBool diagonalscale;
263: PCGetDiagonalScale(ksp->pc,&diagonalscale);
264: if (diagonalscale) SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Krylov method %s does not support diagonal scaling",((PetscObject)ksp)->type_name);
266: cg = (KSP_CG*)ksp->data;
267: eigs = ksp->calc_sings;
268: stored_max_it = ksp->max_it;
269: X = ksp->vec_sol;
270: B = ksp->vec_rhs;
271: R = ksp->work[0];
272: Z = ksp->work[1];
273: P = ksp->work[2];
274: S = ksp->work[3];
275: W = ksp->work[4];
277: if (eigs) {e = cg->e; d = cg->d; e[0] = 0.0; }
278: PCGetOperators(ksp->pc,&Amat,&Pmat);
280: ksp->its = 0;
281: if (!ksp->guess_zero) {
282: KSP_MatMult(ksp,Amat,X,R); /* r <- b - Ax */
283: VecAYPX(R,-1.0,B);
284: } else {
285: VecCopy(B,R); /* r <- b (x is 0) */
286: }
288: switch (ksp->normtype) {
289: case KSP_NORM_PRECONDITIONED:
290: KSP_PCApply(ksp,R,Z); /* z <- Br */
291: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z = e'*A'*B'*B*A'*e' */
292: break;
293: case KSP_NORM_UNPRECONDITIONED:
294: VecNorm(R,NORM_2,&dp); /* dp <- r'*r = e'*A'*A*e */
295: break;
296: case KSP_NORM_NATURAL:
297: KSP_PCApply(ksp,R,Z); /* z <- Br */
298: KSP_MatMult(ksp,Amat,Z,S);
299: VecXDot(Z,S,&delta); /* delta <- z'*A*z = r'*B*A*B*r */
300: VecXDot(Z,R,&beta); /* beta <- z'*r */
301: KSPCheckDot(ksp,beta);
302: dp = PetscSqrtReal(PetscAbsScalar(beta)); /* dp <- r'*z = r'*B*r = e'*A'*B*A*e */
303: break;
304: case KSP_NORM_NONE:
305: dp = 0.0;
306: break;
307: default:SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"%s",KSPNormTypes[ksp->normtype]);
308: }
309: KSPLogResidualHistory(ksp,dp);
310: KSPMonitor(ksp,0,dp);
311: ksp->rnorm = dp;
313: (*ksp->converged)(ksp,0,dp,&ksp->reason,ksp->cnvP); /* test for convergence */
314: if (ksp->reason) return(0);
316: if (ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) {
317: KSP_PCApply(ksp,R,Z); /* z <- Br */
318: }
319: if (ksp->normtype != KSP_NORM_NATURAL) {
320: KSP_MatMult(ksp,Amat,Z,S);
321: VecXDot(Z,S,&delta); /* delta <- z'*A*z = r'*B*A*B*r */
322: VecXDot(Z,R,&beta); /* beta <- z'*r */
323: KSPCheckDot(ksp,beta);
324: }
326: i = 0;
327: do {
328: ksp->its = i+1;
329: if (beta == 0.0) {
330: ksp->reason = KSP_CONVERGED_ATOL;
331: PetscInfo(ksp,"converged due to beta = 0\n");
332: break;
333: #if !defined(PETSC_USE_COMPLEX)
334: } else if ((i > 0) && (beta*betaold < 0.0)) {
335: if (ksp->errorifnotconverged) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_NOT_CONVERGED,"Diverged due to indefinite preconditioner");
336: ksp->reason = KSP_DIVERGED_INDEFINITE_PC;
337: PetscInfo(ksp,"diverging due to indefinite preconditioner\n");
338: break;
339: #endif
340: }
341: if (!i) {
342: VecCopy(Z,P); /* p <- z */
343: b = 0.0;
344: } else {
345: b = beta/betaold;
346: if (eigs) {
347: if (ksp->max_it != stored_max_it) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_SUP,"Can not change maxit AND calculate eigenvalues");
348: e[i] = PetscSqrtReal(PetscAbsScalar(b))/a;
349: }
350: VecAYPX(P,b,Z); /* p <- z + b* p */
351: }
352: dpiold = dpi;
353: if (!i) {
354: KSP_MatMult(ksp,Amat,P,W); /* w <- Ap */
355: VecXDot(P,W,&dpi); /* dpi <- p'w */
356: } else {
357: VecAYPX(W,beta/betaold,S); /* w <- Ap */
358: dpi = delta - beta*beta*dpiold/(betaold*betaold); /* dpi <- p'w */
359: }
360: betaold = beta;
361: KSPCheckDot(ksp,beta);
363: if ((dpi == 0.0) || ((i > 0) && (PetscRealPart(dpi*dpiold) <= 0.0))) {
364: if (ksp->errorifnotconverged) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_NOT_CONVERGED,"Diverged due to indefinite matrix");
365: ksp->reason = KSP_DIVERGED_INDEFINITE_MAT;
366: PetscInfo(ksp,"diverging due to indefinite or negative definite matrix\n");
367: break;
368: }
369: a = beta/dpi; /* a = beta/p'w */
370: if (eigs) d[i] = PetscSqrtReal(PetscAbsScalar(b))*e[i] + 1.0/a;
371: VecAXPY(X,a,P); /* x <- x + ap */
372: VecAXPY(R,-a,W); /* r <- r - aw */
373: if (ksp->normtype == KSP_NORM_PRECONDITIONED && ksp->chknorm < i+2) {
374: KSP_PCApply(ksp,R,Z); /* z <- Br */
375: KSP_MatMult(ksp,Amat,Z,S);
376: VecNorm(Z,NORM_2,&dp); /* dp <- z'*z */
377: } else if (ksp->normtype == KSP_NORM_UNPRECONDITIONED && ksp->chknorm < i+2) {
378: VecNorm(R,NORM_2,&dp); /* dp <- r'*r */
379: } else if (ksp->normtype == KSP_NORM_NATURAL) {
380: KSP_PCApply(ksp,R,Z); /* z <- Br */
381: tmpvecs[0] = S; tmpvecs[1] = R;
382: KSP_MatMult(ksp,Amat,Z,S);
383: VecMDot(Z,2,tmpvecs,tmp); /* delta <- z'*A*z = r'*B*A*B*r */
384: delta = tmp[0]; beta = tmp[1]; /* beta <- z'*r */
385: KSPCheckDot(ksp,beta);
386: dp = PetscSqrtReal(PetscAbsScalar(beta)); /* dp <- r'*z = r'*B*r = e'*A'*B*A*e */
387: } else {
388: dp = 0.0;
389: }
390: ksp->rnorm = dp;
391: KSPLogResidualHistory(ksp,dp);
392: if (eigs) cg->ned = ksp->its;
393: KSPMonitor(ksp,i+1,dp);
394: (*ksp->converged)(ksp,i+1,dp,&ksp->reason,ksp->cnvP);
395: if (ksp->reason) break;
397: if ((ksp->normtype != KSP_NORM_PRECONDITIONED && (ksp->normtype != KSP_NORM_NATURAL)) || (ksp->chknorm >= i+2)) {
398: KSP_PCApply(ksp,R,Z); /* z <- Br */
399: KSP_MatMult(ksp,Amat,Z,S);
400: }
401: if ((ksp->normtype != KSP_NORM_NATURAL) || (ksp->chknorm >= i+2)) {
402: tmpvecs[0] = S; tmpvecs[1] = R;
403: VecMDot(Z,2,tmpvecs,tmp);
404: delta = tmp[0]; beta = tmp[1]; /* delta <- z'*A*z = r'*B'*A*B*r */
405: KSPCheckDot(ksp,beta); /* beta <- z'*r */
406: }
408: i++;
409: } while (i<ksp->max_it);
410: if (i >= ksp->max_it) ksp->reason = KSP_DIVERGED_ITS;
411: return(0);
412: }
414: /*
415: KSPDestroy_CG - Frees resources allocated in KSPSetup_CG and clears function
416: compositions from KSPCreate_CG. If adding your own KSP implementation,
417: you must be sure to free all allocated resources here to prevent
418: leaks.
419: */
420: PetscErrorCode KSPDestroy_CG(KSP ksp)421: {
422: KSP_CG *cg = (KSP_CG*)ksp->data;
426: /* free space used for singular value calculations */
427: if (ksp->calc_sings) {
428: PetscFree4(cg->e,cg->d,cg->ee,cg->dd);
429: }
430: KSPDestroyDefault(ksp);
431: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGSetType_C",NULL);
432: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGUseSingleReduction_C",NULL);
433: return(0);
434: }
436: /*
437: KSPView_CG - Prints information about the current Krylov method being used.
438: If your Krylov method has special options or flags that information
439: should be printed here.
440: */
441: PetscErrorCode KSPView_CG(KSP ksp,PetscViewer viewer)442: {
443: KSP_CG *cg = (KSP_CG*)ksp->data;
445: PetscBool iascii;
448: PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
449: if (iascii) {
450: #if defined(PETSC_USE_COMPLEX)
451: PetscViewerASCIIPrintf(viewer," variant %s\n",KSPCGTypes[cg->type]);
452: #endif
453: if (cg->singlereduction) {
454: PetscViewerASCIIPrintf(viewer," using single-reduction variant\n");
455: }
456: }
457: return(0);
458: }
460: /*
461: KSPSetFromOptions_CG - Checks the options database for options related to the
462: conjugate gradient method.
463: */
464: PetscErrorCode KSPSetFromOptions_CG(PetscOptionItems *PetscOptionsObject,KSP ksp)465: {
467: KSP_CG *cg = (KSP_CG*)ksp->data;
470: PetscOptionsHead(PetscOptionsObject,"KSP CG and CGNE options");
471: #if defined(PETSC_USE_COMPLEX)
472: PetscOptionsEnum("-ksp_cg_type","Matrix is Hermitian or complex symmetric","KSPCGSetType",KSPCGTypes,(PetscEnum)cg->type,
473: (PetscEnum*)&cg->type,NULL);
474: #endif
475: PetscOptionsBool("-ksp_cg_single_reduction","Merge inner products into single MPIU_Allreduce()","KSPCGUseSingleReduction",cg->singlereduction,&cg->singlereduction,NULL);
476: PetscOptionsTail();
477: return(0);
478: }
480: /*
481: KSPCGSetType_CG - This is an option that is SPECIFIC to this particular Krylov method.
482: This routine is registered below in KSPCreate_CG() and called from the
483: routine KSPCGSetType() (see the file cgtype.c).
484: */
485: PetscErrorCode KSPCGSetType_CG(KSP ksp,KSPCGType type)486: {
487: KSP_CG *cg = (KSP_CG*)ksp->data;
490: cg->type = type;
491: return(0);
492: }
494: /*
495: KSPCGUseSingleReduction_CG
497: This routine sets a flag to use a variant of CG. Note that (in somewhat
498: atypical fashion) it also swaps out the routine called when KSPSolve()
499: is invoked.
500: */
501: static PetscErrorCode KSPCGUseSingleReduction_CG(KSP ksp,PetscBool flg)502: {
503: KSP_CG *cg = (KSP_CG*)ksp->data;
506: cg->singlereduction = flg;
507: if (cg->singlereduction) {
508: ksp->ops->solve = KSPSolve_CG_SingleReduction;
509: } else {
510: ksp->ops->solve = KSPSolve_CG;
511: }
512: return(0);
513: }
515: /*
516: KSPCreate_CG - Creates the data structure for the Krylov method CG and sets the
517: function pointers for all the routines it needs to call (KSPSolve_CG() etc)
519: It must be labeled as PETSC_EXTERN to be dynamically linkable in C++
520: */
521: /*MC
522: KSPCG - The Preconditioned Conjugate Gradient (PCG) iterative method
524: Options Database Keys:
525: + -ksp_cg_type Hermitian - (for complex matrices only) indicates the matrix is Hermitian, see KSPCGSetType()
526: . -ksp_cg_type symmetric - (for complex matrices only) indicates the matrix is symmetric
527: - -ksp_cg_single_reduction - performs both inner products needed in the algorithm with a single MPIU_Allreduce() call, see KSPCGUseSingleReduction()
529: Level: beginner
531: Notes:
532: The PCG method requires both the matrix and preconditioner to be symmetric positive (or negative) (semi) definite.
533: 534: Only left preconditioning is supported; there are several ways to motivate preconditioned CG, but they all produce the same algorithm.
535: One can interpret preconditioning A with B to mean any of the following\:
536: .n (1) Solve a left-preconditioned system BAx = Bb, using inv(B) to define an inner product in the algorithm.
537: .n (2) Solve a right-preconditioned system ABy = b, x = By, using B to define an inner product in the algorithm.
538: .n (3) Solve a symmetrically-preconditioned system, E^TAEy = E^Tb, x = Ey, where B = EE^T.
539: .n (4) Solve Ax=b with CG, but use the inner product defined by B to define the method [2].
540: .n In all cases, the resulting algorithm only requires application of B to vectors.
542: For complex numbers there are two different CG methods, one for Hermitian symmetric matrices and one for non-Hermitian symmetric matrices. Use
543: KSPCGSetType() to indicate which type you are using.
545: Developer Notes:
546: KSPSolve_CG() should actually query the matrix to determine if it is Hermitian symmetric or not and NOT require the user to
547: indicate it to the KSP object.
549: References:
550: . 1. - Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems,
551: Journal of Research of the National Bureau of Standards Vol. 49, No. 6, December 1952 Research Paper 2379
552: . 2. - Josef Malek and Zdenek Strakos, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs,
553: SIAM, 2014.
555: .seealso: KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP,
556: KSPCGSetType(), KSPCGUseSingleReduction(), KSPPIPECG, KSPGROPPCG558: M*/
559: PETSC_EXTERN PetscErrorCode KSPCreate_CG(KSP ksp)560: {
562: KSP_CG *cg;
565: PetscNewLog(ksp,&cg);
566: #if !defined(PETSC_USE_COMPLEX)
567: cg->type = KSP_CG_SYMMETRIC;
568: #else
569: cg->type = KSP_CG_HERMITIAN;
570: #endif
571: ksp->data = (void*)cg;
573: KSPSetSupportedNorm(ksp,KSP_NORM_PRECONDITIONED,PC_LEFT,3);
574: KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_LEFT,2);
575: KSPSetSupportedNorm(ksp,KSP_NORM_NATURAL,PC_LEFT,2);
576: KSPSetSupportedNorm(ksp,KSP_NORM_NONE,PC_LEFT,1);
578: /*
579: Sets the functions that are associated with this data structure
580: (in C++ this is the same as defining virtual functions)
581: */
582: ksp->ops->setup = KSPSetUp_CG;
583: ksp->ops->solve = KSPSolve_CG;
584: ksp->ops->destroy = KSPDestroy_CG;
585: ksp->ops->view = KSPView_CG;
586: ksp->ops->setfromoptions = KSPSetFromOptions_CG;
587: ksp->ops->buildsolution = KSPBuildSolutionDefault;
588: ksp->ops->buildresidual = KSPBuildResidualDefault;
590: /*
591: Attach the function KSPCGSetType_CG() to this object. The routine
592: KSPCGSetType() checks for this attached function and calls it if it finds
593: it. (Sort of like a dynamic member function that can be added at run time
594: */
595: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGSetType_C",KSPCGSetType_CG);
596: PetscObjectComposeFunction((PetscObject)ksp,"KSPCGUseSingleReduction_C",KSPCGUseSingleReduction_CG);
597: return(0);
598: }