1*> \brief <b> ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices</b> 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZPOSVXX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zposvxx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zposvxx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zposvxx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 22* S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 23* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 24* NPARAMS, PARAMS, WORK, RWORK, INFO ) 25* 26* .. Scalar Arguments .. 27* CHARACTER EQUED, FACT, UPLO 28* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 29* $ N_ERR_BNDS 30* DOUBLE PRECISION RCOND, RPVGRW 31* .. 32* .. Array Arguments .. 33* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 34* $ WORK( * ), X( LDX, * ) 35* DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ), 36* $ ERR_BNDS_NORM( NRHS, * ), 37* $ ERR_BNDS_COMP( NRHS, * ) 38* .. 39* 40* 41*> \par Purpose: 42* ============= 43*> 44*> \verbatim 45*> 46*> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T 47*> to compute the solution to a complex*16 system of linear equations 48*> A * X = B, where A is an N-by-N Hermitian positive definite matrix 49*> and X and B are N-by-NRHS matrices. 50*> 51*> If requested, both normwise and maximum componentwise error bounds 52*> are returned. ZPOSVXX will return a solution with a tiny 53*> guaranteed error (O(eps) where eps is the working machine 54*> precision) unless the matrix is very ill-conditioned, in which 55*> case a warning is returned. Relevant condition numbers also are 56*> calculated and returned. 57*> 58*> ZPOSVXX accepts user-provided factorizations and equilibration 59*> factors; see the definitions of the FACT and EQUED options. 60*> Solving with refinement and using a factorization from a previous 61*> ZPOSVXX call will also produce a solution with either O(eps) 62*> errors or warnings, but we cannot make that claim for general 63*> user-provided factorizations and equilibration factors if they 64*> differ from what ZPOSVXX would itself produce. 65*> \endverbatim 66* 67*> \par Description: 68* ================= 69*> 70*> \verbatim 71*> 72*> The following steps are performed: 73*> 74*> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate 75*> the system: 76*> 77*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B 78*> 79*> Whether or not the system will be equilibrated depends on the 80*> scaling of the matrix A, but if equilibration is used, A is 81*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 82*> 83*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 84*> factor the matrix A (after equilibration if FACT = 'E') as 85*> A = U**T* U, if UPLO = 'U', or 86*> A = L * L**T, if UPLO = 'L', 87*> where U is an upper triangular matrix and L is a lower triangular 88*> matrix. 89*> 90*> 3. If the leading i-by-i principal minor is not positive definite, 91*> then the routine returns with INFO = i. Otherwise, the factored 92*> form of A is used to estimate the condition number of the matrix 93*> A (see argument RCOND). If the reciprocal of the condition number 94*> is less than machine precision, the routine still goes on to solve 95*> for X and compute error bounds as described below. 96*> 97*> 4. The system of equations is solved for X using the factored form 98*> of A. 99*> 100*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 101*> the routine will use iterative refinement to try to get a small 102*> error and error bounds. Refinement calculates the residual to at 103*> least twice the working precision. 104*> 105*> 6. If equilibration was used, the matrix X is premultiplied by 106*> diag(S) so that it solves the original system before 107*> equilibration. 108*> \endverbatim 109* 110* Arguments: 111* ========== 112* 113*> \verbatim 114*> Some optional parameters are bundled in the PARAMS array. These 115*> settings determine how refinement is performed, but often the 116*> defaults are acceptable. If the defaults are acceptable, users 117*> can pass NPARAMS = 0 which prevents the source code from accessing 118*> the PARAMS argument. 119*> \endverbatim 120*> 121*> \param[in] FACT 122*> \verbatim 123*> FACT is CHARACTER*1 124*> Specifies whether or not the factored form of the matrix A is 125*> supplied on entry, and if not, whether the matrix A should be 126*> equilibrated before it is factored. 127*> = 'F': On entry, AF contains the factored form of A. 128*> If EQUED is not 'N', the matrix A has been 129*> equilibrated with scaling factors given by S. 130*> A and AF are not modified. 131*> = 'N': The matrix A will be copied to AF and factored. 132*> = 'E': The matrix A will be equilibrated if necessary, then 133*> copied to AF and factored. 134*> \endverbatim 135*> 136*> \param[in] UPLO 137*> \verbatim 138*> UPLO is CHARACTER*1 139*> = 'U': Upper triangle of A is stored; 140*> = 'L': Lower triangle of A is stored. 141*> \endverbatim 142*> 143*> \param[in] N 144*> \verbatim 145*> N is INTEGER 146*> The number of linear equations, i.e., the order of the 147*> matrix A. N >= 0. 148*> \endverbatim 149*> 150*> \param[in] NRHS 151*> \verbatim 152*> NRHS is INTEGER 153*> The number of right hand sides, i.e., the number of columns 154*> of the matrices B and X. NRHS >= 0. 155*> \endverbatim 156*> 157*> \param[in,out] A 158*> \verbatim 159*> A is COMPLEX*16 array, dimension (LDA,N) 160*> On entry, the Hermitian matrix A, except if FACT = 'F' and EQUED = 161*> 'Y', then A must contain the equilibrated matrix 162*> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper 163*> triangular part of A contains the upper triangular part of the 164*> matrix A, and the strictly lower triangular part of A is not 165*> referenced. If UPLO = 'L', the leading N-by-N lower triangular 166*> part of A contains the lower triangular part of the matrix A, and 167*> the strictly upper triangular part of A is not referenced. A is 168*> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 169*> 'N' on exit. 170*> 171*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 172*> diag(S)*A*diag(S). 173*> \endverbatim 174*> 175*> \param[in] LDA 176*> \verbatim 177*> LDA is INTEGER 178*> The leading dimension of the array A. LDA >= max(1,N). 179*> \endverbatim 180*> 181*> \param[in,out] AF 182*> \verbatim 183*> AF is COMPLEX*16 array, dimension (LDAF,N) 184*> If FACT = 'F', then AF is an input argument and on entry 185*> contains the triangular factor U or L from the Cholesky 186*> factorization A = U**T*U or A = L*L**T, in the same storage 187*> format as A. If EQUED .ne. 'N', then AF is the factored 188*> form of the equilibrated matrix diag(S)*A*diag(S). 189*> 190*> If FACT = 'N', then AF is an output argument and on exit 191*> returns the triangular factor U or L from the Cholesky 192*> factorization A = U**T*U or A = L*L**T of the original 193*> matrix A. 194*> 195*> If FACT = 'E', then AF is an output argument and on exit 196*> returns the triangular factor U or L from the Cholesky 197*> factorization A = U**T*U or A = L*L**T of the equilibrated 198*> matrix A (see the description of A for the form of the 199*> equilibrated matrix). 200*> \endverbatim 201*> 202*> \param[in] LDAF 203*> \verbatim 204*> LDAF is INTEGER 205*> The leading dimension of the array AF. LDAF >= max(1,N). 206*> \endverbatim 207*> 208*> \param[in,out] EQUED 209*> \verbatim 210*> EQUED is CHARACTER*1 211*> Specifies the form of equilibration that was done. 212*> = 'N': No equilibration (always true if FACT = 'N'). 213*> = 'Y': Both row and column equilibration, i.e., A has been 214*> replaced by diag(S) * A * diag(S). 215*> EQUED is an input argument if FACT = 'F'; otherwise, it is an 216*> output argument. 217*> \endverbatim 218*> 219*> \param[in,out] S 220*> \verbatim 221*> S is DOUBLE PRECISION array, dimension (N) 222*> The row scale factors for A. If EQUED = 'Y', A is multiplied on 223*> the left and right by diag(S). S is an input argument if FACT = 224*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 225*> = 'Y', each element of S must be positive. If S is output, each 226*> element of S is a power of the radix. If S is input, each element 227*> of S should be a power of the radix to ensure a reliable solution 228*> and error estimates. Scaling by powers of the radix does not cause 229*> rounding errors unless the result underflows or overflows. 230*> Rounding errors during scaling lead to refining with a matrix that 231*> is not equivalent to the input matrix, producing error estimates 232*> that may not be reliable. 233*> \endverbatim 234*> 235*> \param[in,out] B 236*> \verbatim 237*> B is COMPLEX*16 array, dimension (LDB,NRHS) 238*> On entry, the N-by-NRHS right hand side matrix B. 239*> On exit, 240*> if EQUED = 'N', B is not modified; 241*> if EQUED = 'Y', B is overwritten by diag(S)*B; 242*> \endverbatim 243*> 244*> \param[in] LDB 245*> \verbatim 246*> LDB is INTEGER 247*> The leading dimension of the array B. LDB >= max(1,N). 248*> \endverbatim 249*> 250*> \param[out] X 251*> \verbatim 252*> X is COMPLEX*16 array, dimension (LDX,NRHS) 253*> If INFO = 0, the N-by-NRHS solution matrix X to the original 254*> system of equations. Note that A and B are modified on exit if 255*> EQUED .ne. 'N', and the solution to the equilibrated system is 256*> inv(diag(S))*X. 257*> \endverbatim 258*> 259*> \param[in] LDX 260*> \verbatim 261*> LDX is INTEGER 262*> The leading dimension of the array X. LDX >= max(1,N). 263*> \endverbatim 264*> 265*> \param[out] RCOND 266*> \verbatim 267*> RCOND is DOUBLE PRECISION 268*> Reciprocal scaled condition number. This is an estimate of the 269*> reciprocal Skeel condition number of the matrix A after 270*> equilibration (if done). If this is less than the machine 271*> precision (in particular, if it is zero), the matrix is singular 272*> to working precision. Note that the error may still be small even 273*> if this number is very small and the matrix appears ill- 274*> conditioned. 275*> \endverbatim 276*> 277*> \param[out] RPVGRW 278*> \verbatim 279*> RPVGRW is DOUBLE PRECISION 280*> Reciprocal pivot growth. On exit, this contains the reciprocal 281*> pivot growth factor norm(A)/norm(U). The "max absolute element" 282*> norm is used. If this is much less than 1, then the stability of 283*> the LU factorization of the (equilibrated) matrix A could be poor. 284*> This also means that the solution X, estimated condition numbers, 285*> and error bounds could be unreliable. If factorization fails with 286*> 0<INFO<=N, then this contains the reciprocal pivot growth factor 287*> for the leading INFO columns of A. 288*> \endverbatim 289*> 290*> \param[out] BERR 291*> \verbatim 292*> BERR is DOUBLE PRECISION array, dimension (NRHS) 293*> Componentwise relative backward error. This is the 294*> componentwise relative backward error of each solution vector X(j) 295*> (i.e., the smallest relative change in any element of A or B that 296*> makes X(j) an exact solution). 297*> \endverbatim 298*> 299*> \param[in] N_ERR_BNDS 300*> \verbatim 301*> N_ERR_BNDS is INTEGER 302*> Number of error bounds to return for each right hand side 303*> and each type (normwise or componentwise). See ERR_BNDS_NORM and 304*> ERR_BNDS_COMP below. 305*> \endverbatim 306*> 307*> \param[out] ERR_BNDS_NORM 308*> \verbatim 309*> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 310*> For each right-hand side, this array contains information about 311*> various error bounds and condition numbers corresponding to the 312*> normwise relative error, which is defined as follows: 313*> 314*> Normwise relative error in the ith solution vector: 315*> max_j (abs(XTRUE(j,i) - X(j,i))) 316*> ------------------------------ 317*> max_j abs(X(j,i)) 318*> 319*> The array is indexed by the type of error information as described 320*> below. There currently are up to three pieces of information 321*> returned. 322*> 323*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 324*> right-hand side. 325*> 326*> The second index in ERR_BNDS_NORM(:,err) contains the following 327*> three fields: 328*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 329*> reciprocal condition number is less than the threshold 330*> sqrt(n) * dlamch('Epsilon'). 331*> 332*> err = 2 "Guaranteed" error bound: The estimated forward error, 333*> almost certainly within a factor of 10 of the true error 334*> so long as the next entry is greater than the threshold 335*> sqrt(n) * dlamch('Epsilon'). This error bound should only 336*> be trusted if the previous boolean is true. 337*> 338*> err = 3 Reciprocal condition number: Estimated normwise 339*> reciprocal condition number. Compared with the threshold 340*> sqrt(n) * dlamch('Epsilon') to determine if the error 341*> estimate is "guaranteed". These reciprocal condition 342*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 343*> appropriately scaled matrix Z. 344*> Let Z = S*A, where S scales each row by a power of the 345*> radix so all absolute row sums of Z are approximately 1. 346*> 347*> See Lapack Working Note 165 for further details and extra 348*> cautions. 349*> \endverbatim 350*> 351*> \param[out] ERR_BNDS_COMP 352*> \verbatim 353*> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 354*> For each right-hand side, this array contains information about 355*> various error bounds and condition numbers corresponding to the 356*> componentwise relative error, which is defined as follows: 357*> 358*> Componentwise relative error in the ith solution vector: 359*> abs(XTRUE(j,i) - X(j,i)) 360*> max_j ---------------------- 361*> abs(X(j,i)) 362*> 363*> The array is indexed by the right-hand side i (on which the 364*> componentwise relative error depends), and the type of error 365*> information as described below. There currently are up to three 366*> pieces of information returned for each right-hand side. If 367*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 368*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most 369*> the first (:,N_ERR_BNDS) entries are returned. 370*> 371*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 372*> right-hand side. 373*> 374*> The second index in ERR_BNDS_COMP(:,err) contains the following 375*> three fields: 376*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 377*> reciprocal condition number is less than the threshold 378*> sqrt(n) * dlamch('Epsilon'). 379*> 380*> err = 2 "Guaranteed" error bound: The estimated forward error, 381*> almost certainly within a factor of 10 of the true error 382*> so long as the next entry is greater than the threshold 383*> sqrt(n) * dlamch('Epsilon'). This error bound should only 384*> be trusted if the previous boolean is true. 385*> 386*> err = 3 Reciprocal condition number: Estimated componentwise 387*> reciprocal condition number. Compared with the threshold 388*> sqrt(n) * dlamch('Epsilon') to determine if the error 389*> estimate is "guaranteed". These reciprocal condition 390*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 391*> appropriately scaled matrix Z. 392*> Let Z = S*(A*diag(x)), where x is the solution for the 393*> current right-hand side and S scales each row of 394*> A*diag(x) by a power of the radix so all absolute row 395*> sums of Z are approximately 1. 396*> 397*> See Lapack Working Note 165 for further details and extra 398*> cautions. 399*> \endverbatim 400*> 401*> \param[in] NPARAMS 402*> \verbatim 403*> NPARAMS is INTEGER 404*> Specifies the number of parameters set in PARAMS. If <= 0, the 405*> PARAMS array is never referenced and default values are used. 406*> \endverbatim 407*> 408*> \param[in,out] PARAMS 409*> \verbatim 410*> PARAMS is DOUBLE PRECISION array, dimension NPARAMS 411*> Specifies algorithm parameters. If an entry is < 0.0, then 412*> that entry will be filled with default value used for that 413*> parameter. Only positions up to NPARAMS are accessed; defaults 414*> are used for higher-numbered parameters. 415*> 416*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 417*> refinement or not. 418*> Default: 1.0D+0 419*> = 0.0: No refinement is performed, and no error bounds are 420*> computed. 421*> = 1.0: Use the extra-precise refinement algorithm. 422*> (other values are reserved for future use) 423*> 424*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 425*> computations allowed for refinement. 426*> Default: 10 427*> Aggressive: Set to 100 to permit convergence using approximate 428*> factorizations or factorizations other than LU. If 429*> the factorization uses a technique other than 430*> Gaussian elimination, the guarantees in 431*> err_bnds_norm and err_bnds_comp may no longer be 432*> trustworthy. 433*> 434*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 435*> will attempt to find a solution with small componentwise 436*> relative error in the double-precision algorithm. Positive 437*> is true, 0.0 is false. 438*> Default: 1.0 (attempt componentwise convergence) 439*> \endverbatim 440*> 441*> \param[out] WORK 442*> \verbatim 443*> WORK is COMPLEX*16 array, dimension (2*N) 444*> \endverbatim 445*> 446*> \param[out] RWORK 447*> \verbatim 448*> RWORK is DOUBLE PRECISION array, dimension (2*N) 449*> \endverbatim 450*> 451*> \param[out] INFO 452*> \verbatim 453*> INFO is INTEGER 454*> = 0: Successful exit. The solution to every right-hand side is 455*> guaranteed. 456*> < 0: If INFO = -i, the i-th argument had an illegal value 457*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 458*> has been completed, but the factor U is exactly singular, so 459*> the solution and error bounds could not be computed. RCOND = 0 460*> is returned. 461*> = N+J: The solution corresponding to the Jth right-hand side is 462*> not guaranteed. The solutions corresponding to other right- 463*> hand sides K with K > J may not be guaranteed as well, but 464*> only the first such right-hand side is reported. If a small 465*> componentwise error is not requested (PARAMS(3) = 0.0) then 466*> the Jth right-hand side is the first with a normwise error 467*> bound that is not guaranteed (the smallest J such 468*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 469*> the Jth right-hand side is the first with either a normwise or 470*> componentwise error bound that is not guaranteed (the smallest 471*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 472*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 473*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 474*> about all of the right-hand sides check ERR_BNDS_NORM or 475*> ERR_BNDS_COMP. 476*> \endverbatim 477* 478* Authors: 479* ======== 480* 481*> \author Univ. of Tennessee 482*> \author Univ. of California Berkeley 483*> \author Univ. of Colorado Denver 484*> \author NAG Ltd. 485* 486*> \ingroup complex16POsolve 487* 488* ===================================================================== 489 SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 490 $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 491 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 492 $ NPARAMS, PARAMS, WORK, RWORK, INFO ) 493* 494* -- LAPACK driver routine -- 495* -- LAPACK is a software package provided by Univ. of Tennessee, -- 496* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 497* 498* .. Scalar Arguments .. 499 CHARACTER EQUED, FACT, UPLO 500 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 501 $ N_ERR_BNDS 502 DOUBLE PRECISION RCOND, RPVGRW 503* .. 504* .. Array Arguments .. 505 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 506 $ WORK( * ), X( LDX, * ) 507 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ), 508 $ ERR_BNDS_NORM( NRHS, * ), 509 $ ERR_BNDS_COMP( NRHS, * ) 510* .. 511* 512* ================================================================== 513* 514* .. Parameters .. 515 DOUBLE PRECISION ZERO, ONE 516 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 517 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 518 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 519 INTEGER CMP_ERR_I, PIV_GROWTH_I 520 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 521 $ BERR_I = 3 ) 522 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 523 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 524 $ PIV_GROWTH_I = 9 ) 525* .. 526* .. Local Scalars .. 527 LOGICAL EQUIL, NOFACT, RCEQU 528 INTEGER INFEQU, J 529 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 530* .. 531* .. External Functions .. 532 EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW 533 LOGICAL LSAME 534 DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW 535* .. 536* .. External Subroutines .. 537 EXTERNAL ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY, 538 $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX 539* .. 540* .. Intrinsic Functions .. 541 INTRINSIC MAX, MIN 542* .. 543* .. Executable Statements .. 544* 545 INFO = 0 546 NOFACT = LSAME( FACT, 'N' ) 547 EQUIL = LSAME( FACT, 'E' ) 548 SMLNUM = DLAMCH( 'Safe minimum' ) 549 BIGNUM = ONE / SMLNUM 550 IF( NOFACT .OR. EQUIL ) THEN 551 EQUED = 'N' 552 RCEQU = .FALSE. 553 ELSE 554 RCEQU = LSAME( EQUED, 'Y' ) 555 ENDIF 556* 557* Default is failure. If an input parameter is wrong or 558* factorization fails, make everything look horrible. Only the 559* pivot growth is set here, the rest is initialized in ZPORFSX. 560* 561 RPVGRW = ZERO 562* 563* Test the input parameters. PARAMS is not tested until ZPORFSX. 564* 565 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 566 $ LSAME( FACT, 'F' ) ) THEN 567 INFO = -1 568 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. 569 $ .NOT.LSAME( UPLO, 'L' ) ) THEN 570 INFO = -2 571 ELSE IF( N.LT.0 ) THEN 572 INFO = -3 573 ELSE IF( NRHS.LT.0 ) THEN 574 INFO = -4 575 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 576 INFO = -6 577 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 578 INFO = -8 579 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 580 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 581 INFO = -9 582 ELSE 583 IF ( RCEQU ) THEN 584 SMIN = BIGNUM 585 SMAX = ZERO 586 DO 10 J = 1, N 587 SMIN = MIN( SMIN, S( J ) ) 588 SMAX = MAX( SMAX, S( J ) ) 589 10 CONTINUE 590 IF( SMIN.LE.ZERO ) THEN 591 INFO = -10 592 ELSE IF( N.GT.0 ) THEN 593 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 594 ELSE 595 SCOND = ONE 596 END IF 597 END IF 598 IF( INFO.EQ.0 ) THEN 599 IF( LDB.LT.MAX( 1, N ) ) THEN 600 INFO = -12 601 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 602 INFO = -14 603 END IF 604 END IF 605 END IF 606* 607 IF( INFO.NE.0 ) THEN 608 CALL XERBLA( 'ZPOSVXX', -INFO ) 609 RETURN 610 END IF 611* 612 IF( EQUIL ) THEN 613* 614* Compute row and column scalings to equilibrate the matrix A. 615* 616 CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) 617 IF( INFEQU.EQ.0 ) THEN 618* 619* Equilibrate the matrix. 620* 621 CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 622 RCEQU = LSAME( EQUED, 'Y' ) 623 END IF 624 END IF 625* 626* Scale the right-hand side. 627* 628 IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB ) 629* 630 IF( NOFACT .OR. EQUIL ) THEN 631* 632* Compute the Cholesky factorization of A. 633* 634 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 635 CALL ZPOTRF( UPLO, N, AF, LDAF, INFO ) 636* 637* Return if INFO is non-zero. 638* 639 IF( INFO.GT.0 ) THEN 640* 641* Pivot in column INFO is exactly 0 642* Compute the reciprocal pivot growth factor of the 643* leading rank-deficient INFO columns of A. 644* 645 RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) 646 RETURN 647 END IF 648 END IF 649* 650* Compute the reciprocal pivot growth factor RPVGRW. 651* 652 RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) 653* 654* Compute the solution matrix X. 655* 656 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 657 CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 658* 659* Use iterative refinement to improve the computed solution and 660* compute error bounds and backward error estimates for it. 661* 662 CALL ZPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, 663 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 664 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) 665 666* 667* Scale solutions. 668* 669 IF ( RCEQU ) THEN 670 CALL ZLASCL2( N, NRHS, S, X, LDX ) 671 END IF 672* 673 RETURN 674* 675* End of ZPOSVXX 676* 677 END 678