001:SUBROUTINESGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, 002: $ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, 003: $ BERR, N_ERR_BNDS, ERR_BNDS_NORM, 004: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, 005: $ INFO ) 006:*007:* -- LAPACK driver routine (version 3.2) --008:* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --009:* -- Jason Riedy of Univ. of California Berkeley. --010:* -- November 2008 --011:*012:* -- LAPACK is a software package provided by Univ. of Tennessee, --013:* -- Univ. of California Berkeley and NAG Ltd. --014:*015:IMPLICITNONE 016:* ..017:* .. Scalar Arguments ..018: CHARACTER EQUED, FACT, TRANS 019: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 020: $ N_ERR_BNDS 021: REAL RCOND, RPVGRW 022:* ..023:* .. Array Arguments ..024: INTEGERIPIV( * ),IWORK( * ) 025: REALA( LDA, * ),AF( LDAF, * ),B( LDB, * ), 026: $X( LDX , * ),WORK( * ) 027: REALR( * ),C( * ),PARAMS( * ),BERR( * ), 028: $ERR_BNDS_NORM( NRHS, * ), 029: $ERR_BNDS_COMP( NRHS, * ) 030:* ..031:*032:* Purpose033:* =======034:*035:* SGESVXX uses the LU factorization to compute the solution to a036:* real system of linear equations A * X = B, where A is an037:* N-by-N matrix and X and B are N-by-NRHS matrices.038:*039:* If requested, both normwise and maximum componentwise error bounds040:* are returned. SGESVXX will return a solution with a tiny041:* guaranteed error (O(eps) where eps is the working machine042:* precision) unless the matrix is very ill-conditioned, in which043:* case a warning is returned. Relevant condition numbers also are044:* calculated and returned.045:*046:* SGESVXX accepts user-provided factorizations and equilibration047:* factors; see the definitions of the FACT and EQUED options.048:* Solving with refinement and using a factorization from a previous049:* SGESVXX call will also produce a solution with either O(eps)050:* errors or warnings, but we cannot make that claim for general051:* user-provided factorizations and equilibration factors if they052:* differ from what SGESVXX would itself produce.053:*054:* Description055:* ===========056:*057:* The following steps are performed:058:*059:* 1. If FACT = 'E', real scaling factors are computed to equilibrate060:* the system:061:*062:* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B063:* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B064:* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B065:*066:* Whether or not the system will be equilibrated depends on the067:* scaling of the matrix A, but if equilibration is used, A is068:* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')069:* or diag(C)*B (if TRANS = 'T' or 'C').070:*071:* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor072:* the matrix A (after equilibration if FACT = 'E') as073:*074:* A = P * L * U,075:*076:* where P is a permutation matrix, L is a unit lower triangular077:* matrix, and U is upper triangular.078:*079:* 3. If some U(i,i)=0, so that U is exactly singular, then the080:* routine returns with INFO = i. Otherwise, the factored form of A081:* is used to estimate the condition number of the matrix A (see082:* argument RCOND). If the reciprocal of the condition number is less083:* than machine precision, the routine still goes on to solve for X084:* and compute error bounds as described below.085:*086:* 4. The system of equations is solved for X using the factored form087:* of A.088:*089:* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),090:* the routine will use iterative refinement to try to get a small091:* error and error bounds. Refinement calculates the residual to at092:* least twice the working precision.093:*094:* 6. If equilibration was used, the matrix X is premultiplied by095:* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so096:* that it solves the original system before equilibration.097:*098:* Arguments099:* =========100:*101:* Some optional parameters are bundled in the PARAMS array. These102:* settings determine how refinement is performed, but often the103:* defaults are acceptable. If the defaults are acceptable, users104:* can pass NPARAMS = 0 which prevents the source code from accessing105:* the PARAMS argument.106:*107:* FACT (input) CHARACTER*1108:* Specifies whether or not the factored form of the matrix A is109:* supplied on entry, and if not, whether the matrix A should be110:* equilibrated before it is factored.111:* = 'F': On entry, AF and IPIV contain the factored form of A.112:* If EQUED is not 'N', the matrix A has been113:* equilibrated with scaling factors given by R and C.114:* A, AF, and IPIV are not modified.115:* = 'N': The matrix A will be copied to AF and factored.116:* = 'E': The matrix A will be equilibrated if necessary, then117:* copied to AF and factored.118:*119:* TRANS (input) CHARACTER*1120:* Specifies the form of the system of equations:121:* = 'N': A * X = B (No transpose)122:* = 'T': A**T * X = B (Transpose)123:* = 'C': A**H * X = B (Conjugate Transpose = Transpose)124:*125:* N (input) INTEGER126:* The number of linear equations, i.e., the order of the127:* matrix A. N >= 0.128:*129:* NRHS (input) INTEGER130:* The number of right hand sides, i.e., the number of columns131:* of the matrices B and X. NRHS >= 0.132:*133:* A (input/output) REAL array, dimension (LDA,N)134:* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is135:* not 'N', then A must have been equilibrated by the scaling136:* factors in R and/or C. A is not modified if FACT = 'F' or137:* 'N', or if FACT = 'E' and EQUED = 'N' on exit.138:*139:* On exit, if EQUED .ne. 'N', A is scaled as follows:140:* EQUED = 'R': A := diag(R) * A141:* EQUED = 'C': A := A * diag(C)142:* EQUED = 'B': A := diag(R) * A * diag(C).143:*144:* LDA (input) INTEGER145:* The leading dimension of the array A. LDA >= max(1,N).146:*147:* AF (input or output) REAL array, dimension (LDAF,N)148:* If FACT = 'F', then AF is an input argument and on entry149:* contains the factors L and U from the factorization150:* A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then151:* AF is the factored form of the equilibrated matrix A.152:*153:* If FACT = 'N', then AF is an output argument and on exit154:* returns the factors L and U from the factorization A = P*L*U155:* of the original matrix A.156:*157:* If FACT = 'E', then AF is an output argument and on exit158:* returns the factors L and U from the factorization A = P*L*U159:* of the equilibrated matrix A (see the description of A for160:* the form of the equilibrated matrix).161:*162:* LDAF (input) INTEGER163:* The leading dimension of the array AF. LDAF >= max(1,N).164:*165:* IPIV (input or output) INTEGER array, dimension (N)166:* If FACT = 'F', then IPIV is an input argument and on entry167:* contains the pivot indices from the factorization A = P*L*U168:* as computed by SGETRF; row i of the matrix was interchanged169:* with row IPIV(i).170:*171:* If FACT = 'N', then IPIV is an output argument and on exit172:* contains the pivot indices from the factorization A = P*L*U173:* of the original matrix A.174:*175:* If FACT = 'E', then IPIV is an output argument and on exit176:* contains the pivot indices from the factorization A = P*L*U177:* of the equilibrated matrix A.178:*179:* EQUED (input or output) CHARACTER*1180:* Specifies the form of equilibration that was done.181:* = 'N': No equilibration (always true if FACT = 'N').182:* = 'R': Row equilibration, i.e., A has been premultiplied by183:* diag(R).184:* = 'C': Column equilibration, i.e., A has been postmultiplied185:* by diag(C).186:* = 'B': Both row and column equilibration, i.e., A has been187:* replaced by diag(R) * A * diag(C).188:* EQUED is an input argument if FACT = 'F'; otherwise, it is an189:* output argument.190:*191:* R (input or output) REAL array, dimension (N)192:* The row scale factors for A. If EQUED = 'R' or 'B', A is193:* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R194:* is not accessed. R is an input argument if FACT = 'F';195:* otherwise, R is an output argument. If FACT = 'F' and196:* EQUED = 'R' or 'B', each element of R must be positive.197:* If R is output, each element of R is a power of the radix.198:* If R is input, each element of R should be a power of the radix199:* to ensure a reliable solution and error estimates. Scaling by200:* powers of the radix does not cause rounding errors unless the201:* result underflows or overflows. Rounding errors during scaling202:* lead to refining with a matrix that is not equivalent to the203:* input matrix, producing error estimates that may not be204:* reliable.205:*206:* C (input or output) REAL array, dimension (N)207:* The column scale factors for A. If EQUED = 'C' or 'B', A is208:* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C209:* is not accessed. C is an input argument if FACT = 'F';210:* otherwise, C is an output argument. If FACT = 'F' and211:* EQUED = 'C' or 'B', each element of C must be positive.212:* If C is output, each element of C is a power of the radix.213:* If C is input, each element of C should be a power of the radix214:* to ensure a reliable solution and error estimates. Scaling by215:* powers of the radix does not cause rounding errors unless the216:* result underflows or overflows. Rounding errors during scaling217:* lead to refining with a matrix that is not equivalent to the218:* input matrix, producing error estimates that may not be219:* reliable.220:*221:* B (input/output) REAL array, dimension (LDB,NRHS)222:* On entry, the N-by-NRHS right hand side matrix B.223:* On exit,224:* if EQUED = 'N', B is not modified;225:* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by226:* diag(R)*B;227:* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is228:* overwritten by diag(C)*B.229:*230:* LDB (input) INTEGER231:* The leading dimension of the array B. LDB >= max(1,N).232:*233:* X (output) REAL array, dimension (LDX,NRHS)234:* If INFO = 0, the N-by-NRHS solution matrix X to the original235:* system of equations. Note that A and B are modified on exit236:* if EQUED .ne. 'N', and the solution to the equilibrated system is237:* inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or238:* inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.239:*240:* LDX (input) INTEGER241:* The leading dimension of the array X. LDX >= max(1,N).242:*243:* RCOND (output) REAL244:* Reciprocal scaled condition number. This is an estimate of the245:* reciprocal Skeel condition number of the matrix A after246:* equilibration (if done). If this is less than the machine247:* precision (in particular, if it is zero), the matrix is singular248:* to working precision. Note that the error may still be small even249:* if this number is very small and the matrix appears ill-250:* conditioned.251:*252:* RPVGRW (output) REAL253:* Reciprocal pivot growth. On exit, this contains the reciprocal254:* pivot growth factor norm(A)/norm(U). The "max absolute element"255:* norm is used. If this is much less than 1, then the stability of256:* the LU factorization of the (equilibrated) matrix A could be poor.257:* This also means that the solution X, estimated condition numbers,258:* and error bounds could be unreliable. If factorization fails with259:* 0<INFO<=N, then this contains the reciprocal pivot growth factor260:* for the leading INFO columns of A. In SGESVX, this quantity is261:* returned in WORK(1).262:*263:* BERR (output) REAL array, dimension (NRHS)264:* Componentwise relative backward error. This is the265:* componentwise relative backward error of each solution vector X(j)266:* (i.e., the smallest relative change in any element of A or B that267:* makes X(j) an exact solution).268:*269:* N_ERR_BNDS (input) INTEGER270:* Number of error bounds to return for each right hand side271:* and each type (normwise or componentwise). See ERR_BNDS_NORM and272:* ERR_BNDS_COMP below.273:*274:* ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)275:* For each right-hand side, this array contains information about276:* various error bounds and condition numbers corresponding to the277:* normwise relative error, which is defined as follows:278:*279:* Normwise relative error in the ith solution vector:280:* max_j (abs(XTRUE(j,i) - X(j,i)))281:* ------------------------------282:* max_j abs(X(j,i))283:*284:* The array is indexed by the type of error information as described285:* below. There currently are up to three pieces of information286:* returned.287:*288:* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith289:* right-hand side.290:*291:* The second index in ERR_BNDS_NORM(:,err) contains the following292:* three fields:293:* err = 1 "Trust/don't trust" boolean. Trust the answer if the294:* reciprocal condition number is less than the threshold295:* sqrt(n) * slamch('Epsilon').296:*297:* err = 2 "Guaranteed" error bound: The estimated forward error,298:* almost certainly within a factor of 10 of the true error299:* so long as the next entry is greater than the threshold300:* sqrt(n) * slamch('Epsilon'). This error bound should only301:* be trusted if the previous boolean is true.302:*303:* err = 3 Reciprocal condition number: Estimated normwise304:* reciprocal condition number. Compared with the threshold305:* sqrt(n) * slamch('Epsilon') to determine if the error306:* estimate is "guaranteed". These reciprocal condition307:* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some308:* appropriately scaled matrix Z.309:* Let Z = S*A, where S scales each row by a power of the310:* radix so all absolute row sums of Z are approximately 1.311:*312:* See Lapack Working Note 165 for further details and extra313:* cautions.314:*315:* ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)316:* For each right-hand side, this array contains information about317:* various error bounds and condition numbers corresponding to the318:* componentwise relative error, which is defined as follows:319:*320:* Componentwise relative error in the ith solution vector:321:* abs(XTRUE(j,i) - X(j,i))322:* max_j ----------------------323:* abs(X(j,i))324:*325:* The array is indexed by the right-hand side i (on which the326:* componentwise relative error depends), and the type of error327:* information as described below. There currently are up to three328:* pieces of information returned for each right-hand side. If329:* componentwise accuracy is not requested (PARAMS(3) = 0.0), then330:* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most331:* the first (:,N_ERR_BNDS) entries are returned.332:*333:* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith334:* right-hand side.335:*336:* The second index in ERR_BNDS_COMP(:,err) contains the following337:* three fields:338:* err = 1 "Trust/don't trust" boolean. Trust the answer if the339:* reciprocal condition number is less than the threshold340:* sqrt(n) * slamch('Epsilon').341:*342:* err = 2 "Guaranteed" error bound: The estimated forward error,343:* almost certainly within a factor of 10 of the true error344:* so long as the next entry is greater than the threshold345:* sqrt(n) * slamch('Epsilon'). This error bound should only346:* be trusted if the previous boolean is true.347:*348:* err = 3 Reciprocal condition number: Estimated componentwise349:* reciprocal condition number. Compared with the threshold350:* sqrt(n) * slamch('Epsilon') to determine if the error351:* estimate is "guaranteed". These reciprocal condition352:* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some353:* appropriately scaled matrix Z.354:* Let Z = S*(A*diag(x)), where x is the solution for the355:* current right-hand side and S scales each row of356:* A*diag(x) by a power of the radix so all absolute row357:* sums of Z are approximately 1.358:*359:* See Lapack Working Note 165 for further details and extra360:* cautions.361:*362:* NPARAMS (input) INTEGER363:* Specifies the number of parameters set in PARAMS. If .LE. 0, the364:* PARAMS array is never referenced and default values are used.365:*366:* PARAMS (input / output) REAL array, dimension NPARAMS367:* Specifies algorithm parameters. If an entry is .LT. 0.0, then368:* that entry will be filled with default value used for that369:* parameter. Only positions up to NPARAMS are accessed; defaults370:* are used for higher-numbered parameters.371:*372:* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative373:* refinement or not.374:* Default: 1.0375:* = 0.0 : No refinement is performed, and no error bounds are376:* computed.377:* = 1.0 : Use the double-precision refinement algorithm,378:* possibly with doubled-single computations if the379:* compilation environment does not support DOUBLE380:* PRECISION.381:* (other values are reserved for future use)382:*383:* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual384:* computations allowed for refinement.385:* Default: 10386:* Aggressive: Set to 100 to permit convergence using approximate387:* factorizations or factorizations other than LU. If388:* the factorization uses a technique other than389:* Gaussian elimination, the guarantees in390:* err_bnds_norm and err_bnds_comp may no longer be391:* trustworthy.392:*393:* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code394:* will attempt to find a solution with small componentwise395:* relative error in the double-precision algorithm. Positive396:* is true, 0.0 is false.397:* Default: 1.0 (attempt componentwise convergence)398:*399:* WORK (workspace) REAL array, dimension (4*N)400:*401:* IWORK (workspace) INTEGER array, dimension (N)402:*403:* INFO (output) INTEGER404:* = 0: Successful exit. The solution to every right-hand side is405:* guaranteed.406:* < 0: If INFO = -i, the i-th argument had an illegal value407:* > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization408:* has been completed, but the factor U is exactly singular, so409:* the solution and error bounds could not be computed. RCOND = 0410:* is returned.411:* = N+J: The solution corresponding to the Jth right-hand side is412:* not guaranteed. The solutions corresponding to other right-413:* hand sides K with K > J may not be guaranteed as well, but414:* only the first such right-hand side is reported. If a small415:* componentwise error is not requested (PARAMS(3) = 0.0) then416:* the Jth right-hand side is the first with a normwise error417:* bound that is not guaranteed (the smallest J such418:* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)419:* the Jth right-hand side is the first with either a normwise or420:* componentwise error bound that is not guaranteed (the smallest421:* J such that either ERR_BNDS_NORM(J,1) = 0.0 or422:* ERR_BNDS_COMP(J,1) = 0.0). See the definition of423:* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information424:* about all of the right-hand sides check ERR_BNDS_NORM or425:* ERR_BNDS_COMP.426:*427:* ==================================================================428:*429:* .. Parameters ..430: REAL ZERO, ONE 431:PARAMETER( ZERO = 0.0D+0, ONE = 1.0D+0 ) 432: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 433: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 434: INTEGER CMP_ERR_I, PIV_GROWTH_I 435:PARAMETER( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 436: $ BERR_I = 3 ) 437:PARAMETER( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 438:PARAMETER( CMP_RCOND_I = 7, CMP_ERR_I = 8, 439: $ PIV_GROWTH_I = 9 ) 440:* ..441:* .. Local Scalars ..442:LOGICALCOLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 443: INTEGER INFEQU, J 444: REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN, ROWCND, 445: $ SMLNUM 446:* ..447:* .. External Functions ..448:EXTERNALLSAME, SLAMCH, SLA_RPVGRW 449:LOGICALLSAME 450: REAL SLAMCH, SLA_RPVGRW 451:* ..452:* .. External Subroutines ..453:EXTERNALSGEEQUB, SGETRF, SGETRS, SLACPY, SLAQGE, 454: $ XERBLA, SLASCL2, SGERFSX 455:* ..456:* .. Intrinsic Functions ..457:INTRINSICMAX, MIN 458:* ..459:* .. Executable Statements ..460:*461: INFO = 0 462: NOFACT =LSAME( FACT, 'N' ) 463: EQUIL =LSAME( FACT, 'E' ) 464: NOTRAN =LSAME( TRANS, 'N' ) 465: SMLNUM =SLAMCH( 'Safe minimum' ) 466: BIGNUM = ONE / SMLNUM 467:IF( NOFACT .OR. EQUIL )THEN468: EQUED = 'N' 469: ROWEQU = .FALSE. 470: COLEQU = .FALSE. 471:ELSE472: ROWEQU =LSAME( EQUED, 'R' ) .OR.LSAME( EQUED, 'B' ) 473: COLEQU =LSAME( EQUED, 'C' ) .OR.LSAME( EQUED, 'B' ) 474:ENDIF475:*476:* Default is failure. If an input parameter is wrong or477:* factorization fails, make everything look horrible. Only the478:* pivot growth is set here, the rest is initialized in SGERFSX.479:*480: RPVGRW = ZERO 481:*482:* Test the input parameters. PARAMS is not tested until SGERFSX.483:*484:IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 485: $LSAME( FACT, 'F' ) )THEN486: INFO = -1 487:ELSEIF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 488: $LSAME( TRANS, 'C' ) )THEN489: INFO = -2 490:ELSEIF( N.LT.0 )THEN491: INFO = -3 492:ELSEIF( NRHS.LT.0 )THEN493: INFO = -4 494:ELSEIF( LDA.LT.MAX( 1, N ) )THEN495: INFO = -6 496:ELSEIF( LDAF.LT.MAX( 1, N ) )THEN497: INFO = -8 498:ELSEIF(LSAME( FACT, 'F' ) .AND. .NOT. 499: $ ( ROWEQU .OR. COLEQU .OR.LSAME( EQUED, 'N' ) ) )THEN500: INFO = -10 501:ELSE502:IF( ROWEQU )THEN503: RCMIN = BIGNUM 504: RCMAX = ZERO 505:DO10 J = 1, N 506: RCMIN =MIN( RCMIN,R( J ) ) 507: RCMAX =MAX( RCMAX,R( J ) ) 508: 10CONTINUE509:IF( RCMIN.LE.ZERO )THEN510: INFO = -11 511:ELSEIF( N.GT.0 )THEN512: ROWCND =MAX( RCMIN, SMLNUM ) /MIN( RCMAX, BIGNUM ) 513:ELSE514: ROWCND = ONE 515:ENDIF516:ENDIF517:IF( COLEQU .AND. INFO.EQ.0 )THEN518: RCMIN = BIGNUM 519: RCMAX = ZERO 520:DO20 J = 1, N 521: RCMIN =MIN( RCMIN,C( J ) ) 522: RCMAX =MAX( RCMAX,C( J ) ) 523: 20CONTINUE524:IF( RCMIN.LE.ZERO )THEN525: INFO = -12 526:ELSEIF( N.GT.0 )THEN527: COLCND =MAX( RCMIN, SMLNUM ) /MIN( RCMAX, BIGNUM ) 528:ELSE529: COLCND = ONE 530:ENDIF531:ENDIF532:IF( INFO.EQ.0 )THEN533:IF( LDB.LT.MAX( 1, N ) )THEN534: INFO = -14 535:ELSEIF( LDX.LT.MAX( 1, N ) )THEN536: INFO = -16 537:ENDIF538:ENDIF539:ENDIF540:*541:IF( INFO.NE.0 )THEN542:CALLXERBLA( 'SGESVXX', -INFO ) 543:RETURN544:ENDIF545:*546:IF( EQUIL )THEN547:*548:* Compute row and column scalings to equilibrate the matrix A.549:*550:CALLSGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, 551: $ INFEQU ) 552:IF( INFEQU.EQ.0 )THEN553:*554:* Equilibrate the matrix.555:*556:CALLSLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, 557: $ EQUED ) 558: ROWEQU =LSAME( EQUED, 'R' ) .OR.LSAME( EQUED, 'B' ) 559: COLEQU =LSAME( EQUED, 'C' ) .OR.LSAME( EQUED, 'B' ) 560:ENDIF561:*562:* If the scaling factors are not applied, set them to 1.0.563:*564:IF( .NOT.ROWEQU )THEN565:DOJ = 1, N 566:R( J ) = 1.0 567:ENDDO568:ENDIF569:IF( .NOT.COLEQU )THEN570:DOJ = 1, N 571:C( J ) = 1.0 572:ENDDO573:ENDIF574:ENDIF575:*576:* Scale the right-hand side.577:*578:IF( NOTRAN )THEN579:IF( ROWEQU )CALLSLASCL2( N, NRHS, R, B, LDB ) 580:ELSE581:IF( COLEQU )CALLSLASCL2( N, NRHS, C, B, LDB ) 582:ENDIF583:*584:IF( NOFACT .OR. EQUIL )THEN585:*586:* Compute the LU factorization of A.587:*588:CALLSLACPY( 'Full', N, N, A, LDA, AF, LDAF ) 589:CALLSGETRF( N, N, AF, LDAF, IPIV, INFO ) 590:*591:* Return if INFO is non-zero.592:*593:IF( INFO.GT.0 )THEN594:*595:* Pivot in column INFO is exactly 0596:* Compute the reciprocal pivot growth factor of the597:* leading rank-deficient INFO columns of A.598:*599: RPVGRW =SLA_RPVGRW( N, INFO, A, LDA, AF, LDAF ) 600:RETURN601:ENDIF602:ENDIF603:*604:* Compute the reciprocal pivot growth factor RPVGRW.605:*606: RPVGRW =SLA_RPVGRW( N, N, A, LDA, AF, LDAF ) 607:*608:* Compute the solution matrix X.609:*610:CALLSLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 611:CALLSGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 612:*613:* Use iterative refinement to improve the computed solution and614:* compute error bounds and backward error estimates for it.615:*616:CALLSGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, 617: $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, 618: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 619: $ WORK, IWORK, INFO ) 620:*621:* Scale solutions.622:*623:IF( COLEQU .AND. NOTRAN )THEN624:CALLSLASCL2( N, NRHS, C, X, LDX ) 625:ELSEIF( ROWEQU .AND. .NOT.NOTRAN )THEN626:CALLSLASCL2( N, NRHS, R, X, LDX ) 627:ENDIF628:*629:RETURN630:*631:* End of SGESVXX632: 633:END634: