1*> \brief <b> CGESVXX computes the solution to system of linear equations A * X = B for GE 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 CGESVXX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgesvxx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgesvxx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgesvxx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, 22* EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, 23* BERR, N_ERR_BNDS, ERR_BNDS_NORM, 24* ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, 25* INFO ) 26* 27* .. Scalar Arguments .. 28* CHARACTER EQUED, FACT, TRANS 29* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 30* $ N_ERR_BNDS 31* REAL RCOND, RPVGRW 32* .. 33* .. Array Arguments .. 34* INTEGER IPIV( * ) 35* COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 36* $ X( LDX , * ),WORK( * ) 37* REAL R( * ), C( * ), PARAMS( * ), BERR( * ), 38* $ ERR_BNDS_NORM( NRHS, * ), 39* $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) 40* .. 41* 42* 43*> \par Purpose: 44* ============= 45*> 46*> \verbatim 47*> 48*> CGESVXX uses the LU factorization to compute the solution to a 49*> complex system of linear equations A * X = B, where A is an 50*> N-by-N matrix and X and B are N-by-NRHS matrices. 51*> 52*> If requested, both normwise and maximum componentwise error bounds 53*> are returned. CGESVXX will return a solution with a tiny 54*> guaranteed error (O(eps) where eps is the working machine 55*> precision) unless the matrix is very ill-conditioned, in which 56*> case a warning is returned. Relevant condition numbers also are 57*> calculated and returned. 58*> 59*> CGESVXX accepts user-provided factorizations and equilibration 60*> factors; see the definitions of the FACT and EQUED options. 61*> Solving with refinement and using a factorization from a previous 62*> CGESVXX call will also produce a solution with either O(eps) 63*> errors or warnings, but we cannot make that claim for general 64*> user-provided factorizations and equilibration factors if they 65*> differ from what CGESVXX would itself produce. 66*> \endverbatim 67* 68*> \par Description: 69* ================= 70*> 71*> \verbatim 72*> 73*> The following steps are performed: 74*> 75*> 1. If FACT = 'E', real scaling factors are computed to equilibrate 76*> the system: 77*> 78*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B 79*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B 80*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B 81*> 82*> Whether or not the system will be equilibrated depends on the 83*> scaling of the matrix A, but if equilibration is used, A is 84*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') 85*> or diag(C)*B (if TRANS = 'T' or 'C'). 86*> 87*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor 88*> the matrix A (after equilibration if FACT = 'E') as 89*> 90*> A = P * L * U, 91*> 92*> where P is a permutation matrix, L is a unit lower triangular 93*> matrix, and U is upper triangular. 94*> 95*> 3. If some U(i,i)=0, so that U is exactly singular, then the 96*> routine returns with INFO = i. Otherwise, the factored form of A 97*> is used to estimate the condition number of the matrix A (see 98*> argument RCOND). If the reciprocal of the condition number is less 99*> than machine precision, the routine still goes on to solve for X 100*> and compute error bounds as described below. 101*> 102*> 4. The system of equations is solved for X using the factored form 103*> of A. 104*> 105*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 106*> the routine will use iterative refinement to try to get a small 107*> error and error bounds. Refinement calculates the residual to at 108*> least twice the working precision. 109*> 110*> 6. If equilibration was used, the matrix X is premultiplied by 111*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so 112*> that it solves the original system before equilibration. 113*> \endverbatim 114* 115* Arguments: 116* ========== 117* 118*> \verbatim 119*> Some optional parameters are bundled in the PARAMS array. These 120*> settings determine how refinement is performed, but often the 121*> defaults are acceptable. If the defaults are acceptable, users 122*> can pass NPARAMS = 0 which prevents the source code from accessing 123*> the PARAMS argument. 124*> \endverbatim 125*> 126*> \param[in] FACT 127*> \verbatim 128*> FACT is CHARACTER*1 129*> Specifies whether or not the factored form of the matrix A is 130*> supplied on entry, and if not, whether the matrix A should be 131*> equilibrated before it is factored. 132*> = 'F': On entry, AF and IPIV contain the factored form of A. 133*> If EQUED is not 'N', the matrix A has been 134*> equilibrated with scaling factors given by R and C. 135*> A, AF, and IPIV are not modified. 136*> = 'N': The matrix A will be copied to AF and factored. 137*> = 'E': The matrix A will be equilibrated if necessary, then 138*> copied to AF and factored. 139*> \endverbatim 140*> 141*> \param[in] TRANS 142*> \verbatim 143*> TRANS is CHARACTER*1 144*> Specifies the form of the system of equations: 145*> = 'N': A * X = B (No transpose) 146*> = 'T': A**T * X = B (Transpose) 147*> = 'C': A**H * X = B (Conjugate Transpose) 148*> \endverbatim 149*> 150*> \param[in] N 151*> \verbatim 152*> N is INTEGER 153*> The number of linear equations, i.e., the order of the 154*> matrix A. N >= 0. 155*> \endverbatim 156*> 157*> \param[in] NRHS 158*> \verbatim 159*> NRHS is INTEGER 160*> The number of right hand sides, i.e., the number of columns 161*> of the matrices B and X. NRHS >= 0. 162*> \endverbatim 163*> 164*> \param[in,out] A 165*> \verbatim 166*> A is COMPLEX array, dimension (LDA,N) 167*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is 168*> not 'N', then A must have been equilibrated by the scaling 169*> factors in R and/or C. A is not modified if FACT = 'F' or 170*> 'N', or if FACT = 'E' and EQUED = 'N' on exit. 171*> 172*> On exit, if EQUED .ne. 'N', A is scaled as follows: 173*> EQUED = 'R': A := diag(R) * A 174*> EQUED = 'C': A := A * diag(C) 175*> EQUED = 'B': A := diag(R) * A * diag(C). 176*> \endverbatim 177*> 178*> \param[in] LDA 179*> \verbatim 180*> LDA is INTEGER 181*> The leading dimension of the array A. LDA >= max(1,N). 182*> \endverbatim 183*> 184*> \param[in,out] AF 185*> \verbatim 186*> AF is COMPLEX array, dimension (LDAF,N) 187*> If FACT = 'F', then AF is an input argument and on entry 188*> contains the factors L and U from the factorization 189*> A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then 190*> AF is the factored form of the equilibrated matrix A. 191*> 192*> If FACT = 'N', then AF is an output argument and on exit 193*> returns the factors L and U from the factorization A = P*L*U 194*> of the original matrix A. 195*> 196*> If FACT = 'E', then AF is an output argument and on exit 197*> returns the factors L and U from the factorization A = P*L*U 198*> of the equilibrated matrix A (see the description of A for 199*> the form of the 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] IPIV 209*> \verbatim 210*> IPIV is INTEGER array, dimension (N) 211*> If FACT = 'F', then IPIV is an input argument and on entry 212*> contains the pivot indices from the factorization A = P*L*U 213*> as computed by CGETRF; row i of the matrix was interchanged 214*> with row IPIV(i). 215*> 216*> If FACT = 'N', then IPIV is an output argument and on exit 217*> contains the pivot indices from the factorization A = P*L*U 218*> of the original matrix A. 219*> 220*> If FACT = 'E', then IPIV is an output argument and on exit 221*> contains the pivot indices from the factorization A = P*L*U 222*> of the equilibrated matrix A. 223*> \endverbatim 224*> 225*> \param[in,out] EQUED 226*> \verbatim 227*> EQUED is CHARACTER*1 228*> Specifies the form of equilibration that was done. 229*> = 'N': No equilibration (always true if FACT = 'N'). 230*> = 'R': Row equilibration, i.e., A has been premultiplied by 231*> diag(R). 232*> = 'C': Column equilibration, i.e., A has been postmultiplied 233*> by diag(C). 234*> = 'B': Both row and column equilibration, i.e., A has been 235*> replaced by diag(R) * A * diag(C). 236*> EQUED is an input argument if FACT = 'F'; otherwise, it is an 237*> output argument. 238*> \endverbatim 239*> 240*> \param[in,out] R 241*> \verbatim 242*> R is REAL array, dimension (N) 243*> The row scale factors for A. If EQUED = 'R' or 'B', A is 244*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R 245*> is not accessed. R is an input argument if FACT = 'F'; 246*> otherwise, R is an output argument. If FACT = 'F' and 247*> EQUED = 'R' or 'B', each element of R must be positive. 248*> If R is output, each element of R is a power of the radix. 249*> If R is input, each element of R should be a power of the radix 250*> to ensure a reliable solution and error estimates. Scaling by 251*> powers of the radix does not cause rounding errors unless the 252*> result underflows or overflows. Rounding errors during scaling 253*> lead to refining with a matrix that is not equivalent to the 254*> input matrix, producing error estimates that may not be 255*> reliable. 256*> \endverbatim 257*> 258*> \param[in,out] C 259*> \verbatim 260*> C is REAL array, dimension (N) 261*> The column scale factors for A. If EQUED = 'C' or 'B', A is 262*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C 263*> is not accessed. C is an input argument if FACT = 'F'; 264*> otherwise, C is an output argument. If FACT = 'F' and 265*> EQUED = 'C' or 'B', each element of C must be positive. 266*> If C is output, each element of C is a power of the radix. 267*> If C is input, each element of C should be a power of the radix 268*> to ensure a reliable solution and error estimates. Scaling by 269*> powers of the radix does not cause rounding errors unless the 270*> result underflows or overflows. Rounding errors during scaling 271*> lead to refining with a matrix that is not equivalent to the 272*> input matrix, producing error estimates that may not be 273*> reliable. 274*> \endverbatim 275*> 276*> \param[in,out] B 277*> \verbatim 278*> B is COMPLEX array, dimension (LDB,NRHS) 279*> On entry, the N-by-NRHS right hand side matrix B. 280*> On exit, 281*> if EQUED = 'N', B is not modified; 282*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by 283*> diag(R)*B; 284*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is 285*> overwritten by diag(C)*B. 286*> \endverbatim 287*> 288*> \param[in] LDB 289*> \verbatim 290*> LDB is INTEGER 291*> The leading dimension of the array B. LDB >= max(1,N). 292*> \endverbatim 293*> 294*> \param[out] X 295*> \verbatim 296*> X is COMPLEX array, dimension (LDX,NRHS) 297*> If INFO = 0, the N-by-NRHS solution matrix X to the original 298*> system of equations. Note that A and B are modified on exit 299*> if EQUED .ne. 'N', and the solution to the equilibrated system is 300*> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or 301*> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. 302*> \endverbatim 303*> 304*> \param[in] LDX 305*> \verbatim 306*> LDX is INTEGER 307*> The leading dimension of the array X. LDX >= max(1,N). 308*> \endverbatim 309*> 310*> \param[out] RCOND 311*> \verbatim 312*> RCOND is REAL 313*> Reciprocal scaled condition number. This is an estimate of the 314*> reciprocal Skeel condition number of the matrix A after 315*> equilibration (if done). If this is less than the machine 316*> precision (in particular, if it is zero), the matrix is singular 317*> to working precision. Note that the error may still be small even 318*> if this number is very small and the matrix appears ill- 319*> conditioned. 320*> \endverbatim 321*> 322*> \param[out] RPVGRW 323*> \verbatim 324*> RPVGRW is REAL 325*> Reciprocal pivot growth. On exit, this contains the reciprocal 326*> pivot growth factor norm(A)/norm(U). The "max absolute element" 327*> norm is used. If this is much less than 1, then the stability of 328*> the LU factorization of the (equilibrated) matrix A could be poor. 329*> This also means that the solution X, estimated condition numbers, 330*> and error bounds could be unreliable. If factorization fails with 331*> 0<INFO<=N, then this contains the reciprocal pivot growth factor 332*> for the leading INFO columns of A. In CGESVX, this quantity is 333*> returned in WORK(1). 334*> \endverbatim 335*> 336*> \param[out] BERR 337*> \verbatim 338*> BERR is REAL array, dimension (NRHS) 339*> Componentwise relative backward error. This is the 340*> componentwise relative backward error of each solution vector X(j) 341*> (i.e., the smallest relative change in any element of A or B that 342*> makes X(j) an exact solution). 343*> \endverbatim 344*> 345*> \param[in] N_ERR_BNDS 346*> \verbatim 347*> N_ERR_BNDS is INTEGER 348*> Number of error bounds to return for each right hand side 349*> and each type (normwise or componentwise). See ERR_BNDS_NORM and 350*> ERR_BNDS_COMP below. 351*> \endverbatim 352*> 353*> \param[out] ERR_BNDS_NORM 354*> \verbatim 355*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 356*> For each right-hand side, this array contains information about 357*> various error bounds and condition numbers corresponding to the 358*> normwise relative error, which is defined as follows: 359*> 360*> Normwise relative error in the ith solution vector: 361*> max_j (abs(XTRUE(j,i) - X(j,i))) 362*> ------------------------------ 363*> max_j abs(X(j,i)) 364*> 365*> The array is indexed by the type of error information as described 366*> below. There currently are up to three pieces of information 367*> returned. 368*> 369*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 370*> right-hand side. 371*> 372*> The second index in ERR_BNDS_NORM(:,err) contains the following 373*> three fields: 374*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 375*> reciprocal condition number is less than the threshold 376*> sqrt(n) * slamch('Epsilon'). 377*> 378*> err = 2 "Guaranteed" error bound: The estimated forward error, 379*> almost certainly within a factor of 10 of the true error 380*> so long as the next entry is greater than the threshold 381*> sqrt(n) * slamch('Epsilon'). This error bound should only 382*> be trusted if the previous boolean is true. 383*> 384*> err = 3 Reciprocal condition number: Estimated normwise 385*> reciprocal condition number. Compared with the threshold 386*> sqrt(n) * slamch('Epsilon') to determine if the error 387*> estimate is "guaranteed". These reciprocal condition 388*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 389*> appropriately scaled matrix Z. 390*> Let Z = S*A, where S scales each row by a power of the 391*> radix so all absolute row sums of Z are approximately 1. 392*> 393*> See Lapack Working Note 165 for further details and extra 394*> cautions. 395*> \endverbatim 396*> 397*> \param[out] ERR_BNDS_COMP 398*> \verbatim 399*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 400*> For each right-hand side, this array contains information about 401*> various error bounds and condition numbers corresponding to the 402*> componentwise relative error, which is defined as follows: 403*> 404*> Componentwise relative error in the ith solution vector: 405*> abs(XTRUE(j,i) - X(j,i)) 406*> max_j ---------------------- 407*> abs(X(j,i)) 408*> 409*> The array is indexed by the right-hand side i (on which the 410*> componentwise relative error depends), and the type of error 411*> information as described below. There currently are up to three 412*> pieces of information returned for each right-hand side. If 413*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 414*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most 415*> the first (:,N_ERR_BNDS) entries are returned. 416*> 417*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 418*> right-hand side. 419*> 420*> The second index in ERR_BNDS_COMP(:,err) contains the following 421*> three fields: 422*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 423*> reciprocal condition number is less than the threshold 424*> sqrt(n) * slamch('Epsilon'). 425*> 426*> err = 2 "Guaranteed" error bound: The estimated forward error, 427*> almost certainly within a factor of 10 of the true error 428*> so long as the next entry is greater than the threshold 429*> sqrt(n) * slamch('Epsilon'). This error bound should only 430*> be trusted if the previous boolean is true. 431*> 432*> err = 3 Reciprocal condition number: Estimated componentwise 433*> reciprocal condition number. Compared with the threshold 434*> sqrt(n) * slamch('Epsilon') to determine if the error 435*> estimate is "guaranteed". These reciprocal condition 436*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 437*> appropriately scaled matrix Z. 438*> Let Z = S*(A*diag(x)), where x is the solution for the 439*> current right-hand side and S scales each row of 440*> A*diag(x) by a power of the radix so all absolute row 441*> sums of Z are approximately 1. 442*> 443*> See Lapack Working Note 165 for further details and extra 444*> cautions. 445*> \endverbatim 446*> 447*> \param[in] NPARAMS 448*> \verbatim 449*> NPARAMS is INTEGER 450*> Specifies the number of parameters set in PARAMS. If <= 0, the 451*> PARAMS array is never referenced and default values are used. 452*> \endverbatim 453*> 454*> \param[in,out] PARAMS 455*> \verbatim 456*> PARAMS is REAL array, dimension NPARAMS 457*> Specifies algorithm parameters. If an entry is < 0.0, then 458*> that entry will be filled with default value used for that 459*> parameter. Only positions up to NPARAMS are accessed; defaults 460*> are used for higher-numbered parameters. 461*> 462*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 463*> refinement or not. 464*> Default: 1.0 465*> = 0.0: No refinement is performed, and no error bounds are 466*> computed. 467*> = 1.0: Use the double-precision refinement algorithm, 468*> possibly with doubled-single computations if the 469*> compilation environment does not support DOUBLE 470*> PRECISION. 471*> (other values are reserved for future use) 472*> 473*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 474*> computations allowed for refinement. 475*> Default: 10 476*> Aggressive: Set to 100 to permit convergence using approximate 477*> factorizations or factorizations other than LU. If 478*> the factorization uses a technique other than 479*> Gaussian elimination, the guarantees in 480*> err_bnds_norm and err_bnds_comp may no longer be 481*> trustworthy. 482*> 483*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 484*> will attempt to find a solution with small componentwise 485*> relative error in the double-precision algorithm. Positive 486*> is true, 0.0 is false. 487*> Default: 1.0 (attempt componentwise convergence) 488*> \endverbatim 489*> 490*> \param[out] WORK 491*> \verbatim 492*> WORK is COMPLEX array, dimension (2*N) 493*> \endverbatim 494*> 495*> \param[out] RWORK 496*> \verbatim 497*> RWORK is REAL array, dimension (2*N) 498*> \endverbatim 499*> 500*> \param[out] INFO 501*> \verbatim 502*> INFO is INTEGER 503*> = 0: Successful exit. The solution to every right-hand side is 504*> guaranteed. 505*> < 0: If INFO = -i, the i-th argument had an illegal value 506*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 507*> has been completed, but the factor U is exactly singular, so 508*> the solution and error bounds could not be computed. RCOND = 0 509*> is returned. 510*> = N+J: The solution corresponding to the Jth right-hand side is 511*> not guaranteed. The solutions corresponding to other right- 512*> hand sides K with K > J may not be guaranteed as well, but 513*> only the first such right-hand side is reported. If a small 514*> componentwise error is not requested (PARAMS(3) = 0.0) then 515*> the Jth right-hand side is the first with a normwise error 516*> bound that is not guaranteed (the smallest J such 517*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 518*> the Jth right-hand side is the first with either a normwise or 519*> componentwise error bound that is not guaranteed (the smallest 520*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 521*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 522*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 523*> about all of the right-hand sides check ERR_BNDS_NORM or 524*> ERR_BNDS_COMP. 525*> \endverbatim 526* 527* Authors: 528* ======== 529* 530*> \author Univ. of Tennessee 531*> \author Univ. of California Berkeley 532*> \author Univ. of Colorado Denver 533*> \author NAG Ltd. 534* 535*> \date April 2012 536* 537*> \ingroup complexGEsolve 538* 539* ===================================================================== 540 SUBROUTINE CGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, 541 $ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, 542 $ BERR, N_ERR_BNDS, ERR_BNDS_NORM, 543 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, 544 $ INFO ) 545* 546* -- LAPACK driver routine (version 3.7.0) -- 547* -- LAPACK is a software package provided by Univ. of Tennessee, -- 548* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 549* April 2012 550* 551* .. Scalar Arguments .. 552 CHARACTER EQUED, FACT, TRANS 553 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 554 $ N_ERR_BNDS 555 REAL RCOND, RPVGRW 556* .. 557* .. Array Arguments .. 558 INTEGER IPIV( * ) 559 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 560 $ X( LDX , * ),WORK( * ) 561 REAL R( * ), C( * ), PARAMS( * ), BERR( * ), 562 $ ERR_BNDS_NORM( NRHS, * ), 563 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) 564* .. 565* 566* ================================================================== 567* 568* .. Parameters .. 569 REAL ZERO, ONE 570 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 571 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 572 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 573 INTEGER CMP_ERR_I, PIV_GROWTH_I 574 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 575 $ BERR_I = 3 ) 576 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 577 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 578 $ PIV_GROWTH_I = 9 ) 579* .. 580* .. Local Scalars .. 581 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 582 INTEGER INFEQU, J 583 REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN, 584 $ ROWCND, SMLNUM 585* .. 586* .. External Functions .. 587 EXTERNAL LSAME, SLAMCH, CLA_GERPVGRW 588 LOGICAL LSAME 589 REAL SLAMCH, CLA_GERPVGRW 590* .. 591* .. External Subroutines .. 592 EXTERNAL CGEEQUB, CGETRF, CGETRS, CLACPY, CLAQGE, 593 $ XERBLA, CLASCL2, CGERFSX 594* .. 595* .. Intrinsic Functions .. 596 INTRINSIC MAX, MIN 597* .. 598* .. Executable Statements .. 599* 600 INFO = 0 601 NOFACT = LSAME( FACT, 'N' ) 602 EQUIL = LSAME( FACT, 'E' ) 603 NOTRAN = LSAME( TRANS, 'N' ) 604 SMLNUM = SLAMCH( 'Safe minimum' ) 605 BIGNUM = ONE / SMLNUM 606 IF( NOFACT .OR. EQUIL ) THEN 607 EQUED = 'N' 608 ROWEQU = .FALSE. 609 COLEQU = .FALSE. 610 ELSE 611 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 612 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 613 END IF 614* 615* Default is failure. If an input parameter is wrong or 616* factorization fails, make everything look horrible. Only the 617* pivot growth is set here, the rest is initialized in CGERFSX. 618* 619 RPVGRW = ZERO 620* 621* Test the input parameters. PARAMS is not tested until CGERFSX. 622* 623 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 624 $ LSAME( FACT, 'F' ) ) THEN 625 INFO = -1 626 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 627 $ LSAME( TRANS, 'C' ) ) THEN 628 INFO = -2 629 ELSE IF( N.LT.0 ) THEN 630 INFO = -3 631 ELSE IF( NRHS.LT.0 ) THEN 632 INFO = -4 633 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 634 INFO = -6 635 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 636 INFO = -8 637 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 638 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 639 INFO = -10 640 ELSE 641 IF( ROWEQU ) THEN 642 RCMIN = BIGNUM 643 RCMAX = ZERO 644 DO 10 J = 1, N 645 RCMIN = MIN( RCMIN, R( J ) ) 646 RCMAX = MAX( RCMAX, R( J ) ) 647 10 CONTINUE 648 IF( RCMIN.LE.ZERO ) THEN 649 INFO = -11 650 ELSE IF( N.GT.0 ) THEN 651 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 652 ELSE 653 ROWCND = ONE 654 END IF 655 END IF 656 IF( COLEQU .AND. INFO.EQ.0 ) THEN 657 RCMIN = BIGNUM 658 RCMAX = ZERO 659 DO 20 J = 1, N 660 RCMIN = MIN( RCMIN, C( J ) ) 661 RCMAX = MAX( RCMAX, C( J ) ) 662 20 CONTINUE 663 IF( RCMIN.LE.ZERO ) THEN 664 INFO = -12 665 ELSE IF( N.GT.0 ) THEN 666 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 667 ELSE 668 COLCND = ONE 669 END IF 670 END IF 671 IF( INFO.EQ.0 ) THEN 672 IF( LDB.LT.MAX( 1, N ) ) THEN 673 INFO = -14 674 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 675 INFO = -16 676 END IF 677 END IF 678 END IF 679* 680 IF( INFO.NE.0 ) THEN 681 CALL XERBLA( 'CGESVXX', -INFO ) 682 RETURN 683 END IF 684* 685 IF( EQUIL ) THEN 686* 687* Compute row and column scalings to equilibrate the matrix A. 688* 689 CALL CGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, 690 $ INFEQU ) 691 IF( INFEQU.EQ.0 ) THEN 692* 693* Equilibrate the matrix. 694* 695 CALL CLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, 696 $ EQUED ) 697 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 698 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 699 END IF 700* 701* If the scaling factors are not applied, set them to 1.0. 702* 703 IF ( .NOT.ROWEQU ) THEN 704 DO J = 1, N 705 R( J ) = 1.0 706 END DO 707 END IF 708 IF ( .NOT.COLEQU ) THEN 709 DO J = 1, N 710 C( J ) = 1.0 711 END DO 712 END IF 713 END IF 714* 715* Scale the right-hand side. 716* 717 IF( NOTRAN ) THEN 718 IF( ROWEQU ) CALL CLASCL2( N, NRHS, R, B, LDB ) 719 ELSE 720 IF( COLEQU ) CALL CLASCL2( N, NRHS, C, B, LDB ) 721 END IF 722* 723 IF( NOFACT .OR. EQUIL ) THEN 724* 725* Compute the LU factorization of A. 726* 727 CALL CLACPY( 'Full', N, N, A, LDA, AF, LDAF ) 728 CALL CGETRF( N, N, AF, LDAF, IPIV, INFO ) 729* 730* Return if INFO is non-zero. 731* 732 IF( INFO.GT.0 ) THEN 733* 734* Pivot in column INFO is exactly 0 735* Compute the reciprocal pivot growth factor of the 736* leading rank-deficient INFO columns of A. 737* 738 RPVGRW = CLA_GERPVGRW( N, INFO, A, LDA, AF, LDAF ) 739 RETURN 740 END IF 741 END IF 742* 743* Compute the reciprocal pivot growth factor RPVGRW. 744* 745 RPVGRW = CLA_GERPVGRW( N, N, A, LDA, AF, LDAF ) 746* 747* Compute the solution matrix X. 748* 749 CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 750 CALL CGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 751* 752* Use iterative refinement to improve the computed solution and 753* compute error bounds and backward error estimates for it. 754* 755 CALL CGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, 756 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, 757 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 758 $ WORK, RWORK, INFO ) 759* 760* Scale solutions. 761* 762 IF ( COLEQU .AND. NOTRAN ) THEN 763 CALL CLASCL2 ( N, NRHS, C, X, LDX ) 764 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN 765 CALL CLASCL2 ( N, NRHS, R, X, LDX ) 766 END IF 767* 768 RETURN 769* 770* End of CGESVXX 771* 772 END 773