001:SUBROUTINECPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 002: $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 003: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 004: $ NPARAMS, PARAMS, WORK, RWORK, INFO ) 005:*006:* -- LAPACK driver routine (version 3.2.1) --007:* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --008:* -- Jason Riedy of Univ. of California Berkeley. --009:* -- April 2009 --010:*011:* -- LAPACK is a software package provided by Univ. of Tennessee, --012:* -- Univ. of California Berkeley and NAG Ltd. --013:*014:IMPLICITNONE 015:* ..016:* .. Scalar Arguments ..017: CHARACTER EQUED, FACT, UPLO 018: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 019: $ N_ERR_BNDS 020: REAL RCOND, RPVGRW 021:* ..022:* .. Array Arguments ..023: COMPLEXA( LDA, * ),AF( LDAF, * ),B( LDB, * ), 024: $WORK( * ),X( LDX, * ) 025: REALS( * ),PARAMS( * ),BERR( * ),RWORK( * ), 026: $ERR_BNDS_NORM( NRHS, * ), 027: $ERR_BNDS_COMP( NRHS, * ) 028:* ..029:*030:* Purpose031:* =======032:*033:* CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T034:* to compute the solution to a complex system of linear equations035:* A * X = B, where A is an N-by-N symmetric positive definite matrix036:* and X and B are N-by-NRHS matrices.037:*038:* If requested, both normwise and maximum componentwise error bounds039:* are returned. CPOSVXX will return a solution with a tiny040:* guaranteed error (O(eps) where eps is the working machine041:* precision) unless the matrix is very ill-conditioned, in which042:* case a warning is returned. Relevant condition numbers also are043:* calculated and returned.044:*045:* CPOSVXX accepts user-provided factorizations and equilibration046:* factors; see the definitions of the FACT and EQUED options.047:* Solving with refinement and using a factorization from a previous048:* CPOSVXX call will also produce a solution with either O(eps)049:* errors or warnings, but we cannot make that claim for general050:* user-provided factorizations and equilibration factors if they051:* differ from what CPOSVXX would itself produce.052:*053:* Description054:* ===========055:*056:* The following steps are performed:057:*058:* 1. If FACT = 'E', real scaling factors are computed to equilibrate059:* the system:060:*061:* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B062:*063:* Whether or not the system will be equilibrated depends on the064:* scaling of the matrix A, but if equilibration is used, A is065:* overwritten by diag(S)*A*diag(S) and B by diag(S)*B.066:*067:* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to068:* factor the matrix A (after equilibration if FACT = 'E') as069:* A = U**T* U, if UPLO = 'U', or070:* A = L * L**T, if UPLO = 'L',071:* where U is an upper triangular matrix and L is a lower triangular072:* matrix.073:*074:* 3. If the leading i-by-i principal minor is not positive definite,075:* then the routine returns with INFO = i. Otherwise, the factored076:* form of A is used to estimate the condition number of the matrix077:* A (see argument RCOND). If the reciprocal of the condition number078:* is less than machine precision, the routine still goes on to solve079:* for X and compute error bounds as described below.080:*081:* 4. The system of equations is solved for X using the factored form082:* of A.083:*084:* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),085:* the routine will use iterative refinement to try to get a small086:* error and error bounds. Refinement calculates the residual to at087:* least twice the working precision.088:*089:* 6. If equilibration was used, the matrix X is premultiplied by090:* diag(S) so that it solves the original system before091:* equilibration.092:*093:* Arguments094:* =========095:*096:* Some optional parameters are bundled in the PARAMS array. These097:* settings determine how refinement is performed, but often the098:* defaults are acceptable. If the defaults are acceptable, users099:* can pass NPARAMS = 0 which prevents the source code from accessing100:* the PARAMS argument.101:*102:* FACT (input) CHARACTER*1103:* Specifies whether or not the factored form of the matrix A is104:* supplied on entry, and if not, whether the matrix A should be105:* equilibrated before it is factored.106:* = 'F': On entry, AF contains the factored form of A.107:* If EQUED is not 'N', the matrix A has been108:* equilibrated with scaling factors given by S.109:* A and AF are not modified.110:* = 'N': The matrix A will be copied to AF and factored.111:* = 'E': The matrix A will be equilibrated if necessary, then112:* copied to AF and factored.113:*114:* UPLO (input) CHARACTER*1115:* = 'U': Upper triangle of A is stored;116:* = 'L': Lower triangle of A is stored.117:*118:* N (input) INTEGER119:* The number of linear equations, i.e., the order of the120:* matrix A. N >= 0.121:*122:* NRHS (input) INTEGER123:* The number of right hand sides, i.e., the number of columns124:* of the matrices B and X. NRHS >= 0.125:*126:* A (input/output) COMPLEX array, dimension (LDA,N)127:* On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =128:* 'Y', then A must contain the equilibrated matrix129:* diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper130:* triangular part of A contains the upper triangular part of the131:* matrix A, and the strictly lower triangular part of A is not132:* referenced. If UPLO = 'L', the leading N-by-N lower triangular133:* part of A contains the lower triangular part of the matrix A, and134:* the strictly upper triangular part of A is not referenced. A is135:* not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =136:* 'N' on exit.137:*138:* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by139:* diag(S)*A*diag(S).140:*141:* LDA (input) INTEGER142:* The leading dimension of the array A. LDA >= max(1,N).143:*144:* AF (input or output) COMPLEX array, dimension (LDAF,N)145:* If FACT = 'F', then AF is an input argument and on entry146:* contains the triangular factor U or L from the Cholesky147:* factorization A = U**T*U or A = L*L**T, in the same storage148:* format as A. If EQUED .ne. 'N', then AF is the factored149:* form of the equilibrated matrix diag(S)*A*diag(S).150:*151:* If FACT = 'N', then AF is an output argument and on exit152:* returns the triangular factor U or L from the Cholesky153:* factorization A = U**T*U or A = L*L**T of the original154:* matrix A.155:*156:* If FACT = 'E', then AF is an output argument and on exit157:* returns the triangular factor U or L from the Cholesky158:* factorization A = U**T*U or A = L*L**T of the equilibrated159:* matrix A (see the description of A for the form of the160:* equilibrated matrix).161:*162:* LDAF (input) INTEGER163:* The leading dimension of the array AF. LDAF >= max(1,N).164:*165:* EQUED (input or output) CHARACTER*1166:* Specifies the form of equilibration that was done.167:* = 'N': No equilibration (always true if FACT = 'N').168:* = 'Y': Both row and column equilibration, i.e., A has been169:* replaced by diag(S) * A * diag(S).170:* EQUED is an input argument if FACT = 'F'; otherwise, it is an171:* output argument.172:*173:* S (input or output) REAL array, dimension (N)174:* The row scale factors for A. If EQUED = 'Y', A is multiplied on175:* the left and right by diag(S). S is an input argument if FACT =176:* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED177:* = 'Y', each element of S must be positive. If S is output, each178:* element of S is a power of the radix. If S is input, each element179:* of S should be a power of the radix to ensure a reliable solution180:* and error estimates. Scaling by powers of the radix does not cause181:* rounding errors unless the result underflows or overflows.182:* Rounding errors during scaling lead to refining with a matrix that183:* is not equivalent to the input matrix, producing error estimates184:* that may not be reliable.185:*186:* B (input/output) COMPLEX array, dimension (LDB,NRHS)187:* On entry, the N-by-NRHS right hand side matrix B.188:* On exit,189:* if EQUED = 'N', B is not modified;190:* if EQUED = 'Y', B is overwritten by diag(S)*B;191:*192:* LDB (input) INTEGER193:* The leading dimension of the array B. LDB >= max(1,N).194:*195:* X (output) COMPLEX array, dimension (LDX,NRHS)196:* If INFO = 0, the N-by-NRHS solution matrix X to the original197:* system of equations. Note that A and B are modified on exit if198:* EQUED .ne. 'N', and the solution to the equilibrated system is199:* inv(diag(S))*X.200:*201:* LDX (input) INTEGER202:* The leading dimension of the array X. LDX >= max(1,N).203:*204:* RCOND (output) REAL205:* Reciprocal scaled condition number. This is an estimate of the206:* reciprocal Skeel condition number of the matrix A after207:* equilibration (if done). If this is less than the machine208:* precision (in particular, if it is zero), the matrix is singular209:* to working precision. Note that the error may still be small even210:* if this number is very small and the matrix appears ill-211:* conditioned.212:*213:* RPVGRW (output) REAL214:* Reciprocal pivot growth. On exit, this contains the reciprocal215:* pivot growth factor norm(A)/norm(U). The "max absolute element"216:* norm is used. If this is much less than 1, then the stability of217:* the LU factorization of the (equilibrated) matrix A could be poor.218:* This also means that the solution X, estimated condition numbers,219:* and error bounds could be unreliable. If factorization fails with220:* 0<INFO<=N, then this contains the reciprocal pivot growth factor221:* for the leading INFO columns of A.222:*223:* BERR (output) REAL array, dimension (NRHS)224:* Componentwise relative backward error. This is the225:* componentwise relative backward error of each solution vector X(j)226:* (i.e., the smallest relative change in any element of A or B that227:* makes X(j) an exact solution).228:*229:* N_ERR_BNDS (input) INTEGER230:* Number of error bounds to return for each right hand side231:* and each type (normwise or componentwise). See ERR_BNDS_NORM and232:* ERR_BNDS_COMP below.233:*234:* ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)235:* For each right-hand side, this array contains information about236:* various error bounds and condition numbers corresponding to the237:* normwise relative error, which is defined as follows:238:*239:* Normwise relative error in the ith solution vector:240:* max_j (abs(XTRUE(j,i) - X(j,i)))241:* ------------------------------242:* max_j abs(X(j,i))243:*244:* The array is indexed by the type of error information as described245:* below. There currently are up to three pieces of information246:* returned.247:*248:* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith249:* right-hand side.250:*251:* The second index in ERR_BNDS_NORM(:,err) contains the following252:* three fields:253:* err = 1 "Trust/don't trust" boolean. Trust the answer if the254:* reciprocal condition number is less than the threshold255:* sqrt(n) * slamch('Epsilon').256:*257:* err = 2 "Guaranteed" error bound: The estimated forward error,258:* almost certainly within a factor of 10 of the true error259:* so long as the next entry is greater than the threshold260:* sqrt(n) * slamch('Epsilon'). This error bound should only261:* be trusted if the previous boolean is true.262:*263:* err = 3 Reciprocal condition number: Estimated normwise264:* reciprocal condition number. Compared with the threshold265:* sqrt(n) * slamch('Epsilon') to determine if the error266:* estimate is "guaranteed". These reciprocal condition267:* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some268:* appropriately scaled matrix Z.269:* Let Z = S*A, where S scales each row by a power of the270:* radix so all absolute row sums of Z are approximately 1.271:*272:* See Lapack Working Note 165 for further details and extra273:* cautions.274:*275:* ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)276:* For each right-hand side, this array contains information about277:* various error bounds and condition numbers corresponding to the278:* componentwise relative error, which is defined as follows:279:*280:* Componentwise relative error in the ith solution vector:281:* abs(XTRUE(j,i) - X(j,i))282:* max_j ----------------------283:* abs(X(j,i))284:*285:* The array is indexed by the right-hand side i (on which the286:* componentwise relative error depends), and the type of error287:* information as described below. There currently are up to three288:* pieces of information returned for each right-hand side. If289:* componentwise accuracy is not requested (PARAMS(3) = 0.0), then290:* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most291:* the first (:,N_ERR_BNDS) entries are returned.292:*293:* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith294:* right-hand side.295:*296:* The second index in ERR_BNDS_COMP(:,err) contains the following297:* three fields:298:* err = 1 "Trust/don't trust" boolean. Trust the answer if the299:* reciprocal condition number is less than the threshold300:* sqrt(n) * slamch('Epsilon').301:*302:* err = 2 "Guaranteed" error bound: The estimated forward error,303:* almost certainly within a factor of 10 of the true error304:* so long as the next entry is greater than the threshold305:* sqrt(n) * slamch('Epsilon'). This error bound should only306:* be trusted if the previous boolean is true.307:*308:* err = 3 Reciprocal condition number: Estimated componentwise309:* reciprocal condition number. Compared with the threshold310:* sqrt(n) * slamch('Epsilon') to determine if the error311:* estimate is "guaranteed". These reciprocal condition312:* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some313:* appropriately scaled matrix Z.314:* Let Z = S*(A*diag(x)), where x is the solution for the315:* current right-hand side and S scales each row of316:* A*diag(x) by a power of the radix so all absolute row317:* sums of Z are approximately 1.318:*319:* See Lapack Working Note 165 for further details and extra320:* cautions.321:*322:* NPARAMS (input) INTEGER323:* Specifies the number of parameters set in PARAMS. If .LE. 0, the324:* PARAMS array is never referenced and default values are used.325:*326:* PARAMS (input / output) REAL array, dimension NPARAMS327:* Specifies algorithm parameters. If an entry is .LT. 0.0, then328:* that entry will be filled with default value used for that329:* parameter. Only positions up to NPARAMS are accessed; defaults330:* are used for higher-numbered parameters.331:*332:* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative333:* refinement or not.334:* Default: 1.0335:* = 0.0 : No refinement is performed, and no error bounds are336:* computed.337:* = 1.0 : Use the double-precision refinement algorithm,338:* possibly with doubled-single computations if the339:* compilation environment does not support DOUBLE340:* PRECISION.341:* (other values are reserved for future use)342:*343:* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual344:* computations allowed for refinement.345:* Default: 10346:* Aggressive: Set to 100 to permit convergence using approximate347:* factorizations or factorizations other than LU. If348:* the factorization uses a technique other than349:* Gaussian elimination, the guarantees in350:* err_bnds_norm and err_bnds_comp may no longer be351:* trustworthy.352:*353:* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code354:* will attempt to find a solution with small componentwise355:* relative error in the double-precision algorithm. Positive356:* is true, 0.0 is false.357:* Default: 1.0 (attempt componentwise convergence)358:*359:* WORK (workspace) COMPLEX array, dimension (2*N)360:*361:* RWORK (workspace) REAL array, dimension (2*N)362:*363:* INFO (output) INTEGER364:* = 0: Successful exit. The solution to every right-hand side is365:* guaranteed.366:* < 0: If INFO = -i, the i-th argument had an illegal value367:* > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization368:* has been completed, but the factor U is exactly singular, so369:* the solution and error bounds could not be computed. RCOND = 0370:* is returned.371:* = N+J: The solution corresponding to the Jth right-hand side is372:* not guaranteed. The solutions corresponding to other right-373:* hand sides K with K > J may not be guaranteed as well, but374:* only the first such right-hand side is reported. If a small375:* componentwise error is not requested (PARAMS(3) = 0.0) then376:* the Jth right-hand side is the first with a normwise error377:* bound that is not guaranteed (the smallest J such378:* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)379:* the Jth right-hand side is the first with either a normwise or380:* componentwise error bound that is not guaranteed (the smallest381:* J such that either ERR_BNDS_NORM(J,1) = 0.0 or382:* ERR_BNDS_COMP(J,1) = 0.0). See the definition of383:* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information384:* about all of the right-hand sides check ERR_BNDS_NORM or385:* ERR_BNDS_COMP.386:*387:* ==================================================================388:*389:* .. Parameters ..390: REAL ZERO, ONE 391:PARAMETER( ZERO = 0.0E+0, ONE = 1.0E+0 ) 392: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 393: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 394: INTEGER CMP_ERR_I, PIV_GROWTH_I 395:PARAMETER( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 396: $ BERR_I = 3 ) 397:PARAMETER( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 398:PARAMETER( CMP_RCOND_I = 7, CMP_ERR_I = 8, 399: $ PIV_GROWTH_I = 9 ) 400:* ..401:* .. Local Scalars ..402:LOGICALEQUIL, NOFACT, RCEQU 403: INTEGER INFEQU, J 404: REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 405:* ..406:* .. External Functions ..407:EXTERNALLSAME, SLAMCH, CLA_PORPVGRW 408:LOGICALLSAME 409: REAL SLAMCH, CLA_PORPVGRW 410:* ..411:* .. External Subroutines ..412:EXTERNALCPOCON, CPOEQUB, CPOTRF, CPOTRS, CLACPY, 413: $ CLAQHE, XERBLA, CLASCL2, CPORFSX 414:* ..415:* .. Intrinsic Functions ..416:INTRINSICMAX, MIN 417:* ..418:* .. Executable Statements ..419:*420: INFO = 0 421: NOFACT =LSAME( FACT, 'N' ) 422: EQUIL =LSAME( FACT, 'E' ) 423: SMLNUM =SLAMCH( 'Safe minimum' ) 424: BIGNUM = ONE / SMLNUM 425:IF( NOFACT .OR. EQUIL )THEN426: EQUED = 'N' 427: RCEQU = .FALSE. 428:ELSE429: RCEQU =LSAME( EQUED, 'Y' ) 430:ENDIF431:*432:* Default is failure. If an input parameter is wrong or433:* factorization fails, make everything look horrible. Only the434:* pivot growth is set here, the rest is initialized in CPORFSX.435:*436: RPVGRW = ZERO 437:*438:* Test the input parameters. PARAMS is not tested until CPORFSX.439:*440:IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 441: $LSAME( FACT, 'F' ) )THEN442: INFO = -1 443:ELSEIF( .NOT.LSAME( UPLO, 'U' ) .AND. 444: $ .NOT.LSAME( UPLO, 'L' ) )THEN445: INFO = -2 446:ELSEIF( N.LT.0 )THEN447: INFO = -3 448:ELSEIF( NRHS.LT.0 )THEN449: INFO = -4 450:ELSEIF( LDA.LT.MAX( 1, N ) )THEN451: INFO = -6 452:ELSEIF( LDAF.LT.MAX( 1, N ) )THEN453: INFO = -8 454:ELSEIF(LSAME( FACT, 'F' ) .AND. .NOT. 455: $ ( RCEQU .OR.LSAME( EQUED, 'N' ) ) )THEN456: INFO = -9 457:ELSE458:IF( RCEQU )THEN459: SMIN = BIGNUM 460: SMAX = ZERO 461:DO10 J = 1, N 462: SMIN =MIN( SMIN,S( J ) ) 463: SMAX =MAX( SMAX,S( J ) ) 464: 10CONTINUE465:IF( SMIN.LE.ZERO )THEN466: INFO = -10 467:ELSEIF( N.GT.0 )THEN468: SCOND =MAX( SMIN, SMLNUM ) /MIN( SMAX, BIGNUM ) 469:ELSE470: SCOND = ONE 471:ENDIF472:ENDIF473:IF( INFO.EQ.0 )THEN474:IF( LDB.LT.MAX( 1, N ) )THEN475: INFO = -12 476:ELSEIF( LDX.LT.MAX( 1, N ) )THEN477: INFO = -14 478:ENDIF479:ENDIF480:ENDIF481:*482:IF( INFO.NE.0 )THEN483:CALLXERBLA( 'CPOSVXX', -INFO ) 484:RETURN485:ENDIF486:*487:IF( EQUIL )THEN488:*489:* Compute row and column scalings to equilibrate the matrix A.490:*491:CALLCPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) 492:IF( INFEQU.EQ.0 )THEN493:*494:* Equilibrate the matrix.495:*496:CALLCLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 497: RCEQU =LSAME( EQUED, 'Y' ) 498:ENDIF499:ENDIF500:*501:* Scale the right-hand side.502:*503:IF( RCEQU )CALLCLASCL2( N, NRHS, S, B, LDB ) 504:*505:IF( NOFACT .OR. EQUIL )THEN506:*507:* Compute the LU factorization of A.508:*509:CALLCLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 510:CALLCPOTRF( UPLO, N, AF, LDAF, INFO ) 511:*512:* Return if INFO is non-zero.513:*514:IF( INFO.GT.0 )THEN515:*516:* Pivot in column INFO is exactly 0517:* Compute the reciprocal pivot growth factor of the518:* leading rank-deficient INFO columns of A.519:*520: RPVGRW =CLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) 521:RETURN522:ENDIF523:ENDIF524:*525:* Compute the reciprocal pivot growth factor RPVGRW.526:*527: RPVGRW =CLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) 528:*529:* Compute the solution matrix X.530:*531:CALLCLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 532:CALLCPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 533:*534:* Use iterative refinement to improve the computed solution and535:* compute error bounds and backward error estimates for it.536:*537:CALLCPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, 538: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 539: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) 540: 541:*542:* Scale solutions.543:*544:IF( RCEQU )THEN545:CALLCLASCL2( N, NRHS, S, X, LDX ) 546:ENDIF547:*548:RETURN549:*550:* End of CPOSVXX551:*552:END553: