1*> \brief <b> CGBSVXX computes the solution to system of linear equations A * X = B for GB 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 CGBSVXX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgbsvxx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgbsvxx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgbsvxx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 22* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 23* RCOND, RPVGRW, BERR, N_ERR_BNDS, 24* ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 25* WORK, RWORK, INFO ) 26* 27* .. Scalar Arguments .. 28* CHARACTER EQUED, FACT, TRANS 29* INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, 30* $ N_ERR_BNDS 31* REAL RCOND, RPVGRW 32* .. 33* .. Array Arguments .. 34* INTEGER IPIV( * ) 35* COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), 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*> CGBSVXX 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. CGBSVXX 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*> CGBSVXX 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*> CGBSVXX 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 CGBSVXX 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 = 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] KL 158*> \verbatim 159*> KL is INTEGER 160*> The number of subdiagonals within the band of A. KL >= 0. 161*> \endverbatim 162*> 163*> \param[in] KU 164*> \verbatim 165*> KU is INTEGER 166*> The number of superdiagonals within the band of A. KU >= 0. 167*> \endverbatim 168*> 169*> \param[in] NRHS 170*> \verbatim 171*> NRHS is INTEGER 172*> The number of right hand sides, i.e., the number of columns 173*> of the matrices B and X. NRHS >= 0. 174*> \endverbatim 175*> 176*> \param[in,out] AB 177*> \verbatim 178*> AB is COMPLEX array, dimension (LDAB,N) 179*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. 180*> The j-th column of A is stored in the j-th column of the 181*> array AB as follows: 182*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) 183*> 184*> If FACT = 'F' and EQUED is not 'N', then AB must have been 185*> equilibrated by the scaling factors in R and/or C. AB is not 186*> modified if FACT = 'F' or 'N', or if FACT = 'E' and 187*> EQUED = 'N' on exit. 188*> 189*> On exit, if EQUED .ne. 'N', A is scaled as follows: 190*> EQUED = 'R': A := diag(R) * A 191*> EQUED = 'C': A := A * diag(C) 192*> EQUED = 'B': A := diag(R) * A * diag(C). 193*> \endverbatim 194*> 195*> \param[in] LDAB 196*> \verbatim 197*> LDAB is INTEGER 198*> The leading dimension of the array AB. LDAB >= KL+KU+1. 199*> \endverbatim 200*> 201*> \param[in,out] AFB 202*> \verbatim 203*> AFB is COMPLEX array, dimension (LDAFB,N) 204*> If FACT = 'F', then AFB is an input argument and on entry 205*> contains details of the LU factorization of the band matrix 206*> A, as computed by CGBTRF. U is stored as an upper triangular 207*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, 208*> and the multipliers used during the factorization are stored 209*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is 210*> the factored form of the equilibrated matrix A. 211*> 212*> If FACT = 'N', then AF is an output argument and on exit 213*> returns the factors L and U from the factorization A = P*L*U 214*> of the original matrix A. 215*> 216*> If FACT = 'E', then AF is an output argument and on exit 217*> returns the factors L and U from the factorization A = P*L*U 218*> of the equilibrated matrix A (see the description of A for 219*> the form of the equilibrated matrix). 220*> \endverbatim 221*> 222*> \param[in] LDAFB 223*> \verbatim 224*> LDAFB is INTEGER 225*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. 226*> \endverbatim 227*> 228*> \param[in,out] IPIV 229*> \verbatim 230*> IPIV is INTEGER array, dimension (N) 231*> If FACT = 'F', then IPIV is an input argument and on entry 232*> contains the pivot indices from the factorization A = P*L*U 233*> as computed by CGETRF; row i of the matrix was interchanged 234*> with row IPIV(i). 235*> 236*> If FACT = 'N', then IPIV is an output argument and on exit 237*> contains the pivot indices from the factorization A = P*L*U 238*> of the original matrix A. 239*> 240*> If FACT = 'E', then IPIV is an output argument and on exit 241*> contains the pivot indices from the factorization A = P*L*U 242*> of the equilibrated matrix A. 243*> \endverbatim 244*> 245*> \param[in,out] EQUED 246*> \verbatim 247*> EQUED is CHARACTER*1 248*> Specifies the form of equilibration that was done. 249*> = 'N': No equilibration (always true if FACT = 'N'). 250*> = 'R': Row equilibration, i.e., A has been premultiplied by 251*> diag(R). 252*> = 'C': Column equilibration, i.e., A has been postmultiplied 253*> by diag(C). 254*> = 'B': Both row and column equilibration, i.e., A has been 255*> replaced by diag(R) * A * diag(C). 256*> EQUED is an input argument if FACT = 'F'; otherwise, it is an 257*> output argument. 258*> \endverbatim 259*> 260*> \param[in,out] R 261*> \verbatim 262*> R is REAL array, dimension (N) 263*> The row scale factors for A. If EQUED = 'R' or 'B', A is 264*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R 265*> is not accessed. R is an input argument if FACT = 'F'; 266*> otherwise, R is an output argument. If FACT = 'F' and 267*> EQUED = 'R' or 'B', each element of R must be positive. 268*> If R is output, each element of R is a power of the radix. 269*> If R is input, each element of R should be a power of the radix 270*> to ensure a reliable solution and error estimates. Scaling by 271*> powers of the radix does not cause rounding errors unless the 272*> result underflows or overflows. Rounding errors during scaling 273*> lead to refining with a matrix that is not equivalent to the 274*> input matrix, producing error estimates that may not be 275*> reliable. 276*> \endverbatim 277*> 278*> \param[in,out] C 279*> \verbatim 280*> C is REAL array, dimension (N) 281*> The column scale factors for A. If EQUED = 'C' or 'B', A is 282*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C 283*> is not accessed. C is an input argument if FACT = 'F'; 284*> otherwise, C is an output argument. If FACT = 'F' and 285*> EQUED = 'C' or 'B', each element of C must be positive. 286*> If C is output, each element of C is a power of the radix. 287*> If C is input, each element of C should be a power of the radix 288*> to ensure a reliable solution and error estimates. Scaling by 289*> powers of the radix does not cause rounding errors unless the 290*> result underflows or overflows. Rounding errors during scaling 291*> lead to refining with a matrix that is not equivalent to the 292*> input matrix, producing error estimates that may not be 293*> reliable. 294*> \endverbatim 295*> 296*> \param[in,out] B 297*> \verbatim 298*> B is COMPLEX array, dimension (LDB,NRHS) 299*> On entry, the N-by-NRHS right hand side matrix B. 300*> On exit, 301*> if EQUED = 'N', B is not modified; 302*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by 303*> diag(R)*B; 304*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is 305*> overwritten by diag(C)*B. 306*> \endverbatim 307*> 308*> \param[in] LDB 309*> \verbatim 310*> LDB is INTEGER 311*> The leading dimension of the array B. LDB >= max(1,N). 312*> \endverbatim 313*> 314*> \param[out] X 315*> \verbatim 316*> X is COMPLEX array, dimension (LDX,NRHS) 317*> If INFO = 0, the N-by-NRHS solution matrix X to the original 318*> system of equations. Note that A and B are modified on exit 319*> if EQUED .ne. 'N', and the solution to the equilibrated system is 320*> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or 321*> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. 322*> \endverbatim 323*> 324*> \param[in] LDX 325*> \verbatim 326*> LDX is INTEGER 327*> The leading dimension of the array X. LDX >= max(1,N). 328*> \endverbatim 329*> 330*> \param[out] RCOND 331*> \verbatim 332*> RCOND is REAL 333*> Reciprocal scaled condition number. This is an estimate of the 334*> reciprocal Skeel condition number of the matrix A after 335*> equilibration (if done). If this is less than the machine 336*> precision (in particular, if it is zero), the matrix is singular 337*> to working precision. Note that the error may still be small even 338*> if this number is very small and the matrix appears ill- 339*> conditioned. 340*> \endverbatim 341*> 342*> \param[out] RPVGRW 343*> \verbatim 344*> RPVGRW is REAL 345*> Reciprocal pivot growth. On exit, this contains the reciprocal 346*> pivot growth factor norm(A)/norm(U). The "max absolute element" 347*> norm is used. If this is much less than 1, then the stability of 348*> the LU factorization of the (equilibrated) matrix A could be poor. 349*> This also means that the solution X, estimated condition numbers, 350*> and error bounds could be unreliable. If factorization fails with 351*> 0<INFO<=N, then this contains the reciprocal pivot growth factor 352*> for the leading INFO columns of A. In CGESVX, this quantity is 353*> returned in RWORK(1). 354*> \endverbatim 355*> 356*> \param[out] BERR 357*> \verbatim 358*> BERR is REAL array, dimension (NRHS) 359*> Componentwise relative backward error. This is the 360*> componentwise relative backward error of each solution vector X(j) 361*> (i.e., the smallest relative change in any element of A or B that 362*> makes X(j) an exact solution). 363*> \endverbatim 364*> 365*> \param[in] N_ERR_BNDS 366*> \verbatim 367*> N_ERR_BNDS is INTEGER 368*> Number of error bounds to return for each right hand side 369*> and each type (normwise or componentwise). See ERR_BNDS_NORM and 370*> ERR_BNDS_COMP below. 371*> \endverbatim 372*> 373*> \param[out] ERR_BNDS_NORM 374*> \verbatim 375*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 376*> For each right-hand side, this array contains information about 377*> various error bounds and condition numbers corresponding to the 378*> normwise relative error, which is defined as follows: 379*> 380*> Normwise relative error in the ith solution vector: 381*> max_j (abs(XTRUE(j,i) - X(j,i))) 382*> ------------------------------ 383*> max_j abs(X(j,i)) 384*> 385*> The array is indexed by the type of error information as described 386*> below. There currently are up to three pieces of information 387*> returned. 388*> 389*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 390*> right-hand side. 391*> 392*> The second index in ERR_BNDS_NORM(:,err) contains the following 393*> three fields: 394*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 395*> reciprocal condition number is less than the threshold 396*> sqrt(n) * slamch('Epsilon'). 397*> 398*> err = 2 "Guaranteed" error bound: The estimated forward error, 399*> almost certainly within a factor of 10 of the true error 400*> so long as the next entry is greater than the threshold 401*> sqrt(n) * slamch('Epsilon'). This error bound should only 402*> be trusted if the previous boolean is true. 403*> 404*> err = 3 Reciprocal condition number: Estimated normwise 405*> reciprocal condition number. Compared with the threshold 406*> sqrt(n) * slamch('Epsilon') to determine if the error 407*> estimate is "guaranteed". These reciprocal condition 408*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 409*> appropriately scaled matrix Z. 410*> Let Z = S*A, where S scales each row by a power of the 411*> radix so all absolute row sums of Z are approximately 1. 412*> 413*> See Lapack Working Note 165 for further details and extra 414*> cautions. 415*> \endverbatim 416*> 417*> \param[out] ERR_BNDS_COMP 418*> \verbatim 419*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 420*> For each right-hand side, this array contains information about 421*> various error bounds and condition numbers corresponding to the 422*> componentwise relative error, which is defined as follows: 423*> 424*> Componentwise relative error in the ith solution vector: 425*> abs(XTRUE(j,i) - X(j,i)) 426*> max_j ---------------------- 427*> abs(X(j,i)) 428*> 429*> The array is indexed by the right-hand side i (on which the 430*> componentwise relative error depends), and the type of error 431*> information as described below. There currently are up to three 432*> pieces of information returned for each right-hand side. If 433*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 434*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most 435*> the first (:,N_ERR_BNDS) entries are returned. 436*> 437*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 438*> right-hand side. 439*> 440*> The second index in ERR_BNDS_COMP(:,err) contains the following 441*> three fields: 442*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 443*> reciprocal condition number is less than the threshold 444*> sqrt(n) * slamch('Epsilon'). 445*> 446*> err = 2 "Guaranteed" error bound: The estimated forward error, 447*> almost certainly within a factor of 10 of the true error 448*> so long as the next entry is greater than the threshold 449*> sqrt(n) * slamch('Epsilon'). This error bound should only 450*> be trusted if the previous boolean is true. 451*> 452*> err = 3 Reciprocal condition number: Estimated componentwise 453*> reciprocal condition number. Compared with the threshold 454*> sqrt(n) * slamch('Epsilon') to determine if the error 455*> estimate is "guaranteed". These reciprocal condition 456*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 457*> appropriately scaled matrix Z. 458*> Let Z = S*(A*diag(x)), where x is the solution for the 459*> current right-hand side and S scales each row of 460*> A*diag(x) by a power of the radix so all absolute row 461*> sums of Z are approximately 1. 462*> 463*> See Lapack Working Note 165 for further details and extra 464*> cautions. 465*> \endverbatim 466*> 467*> \param[in] NPARAMS 468*> \verbatim 469*> NPARAMS is INTEGER 470*> Specifies the number of parameters set in PARAMS. If <= 0, the 471*> PARAMS array is never referenced and default values are used. 472*> \endverbatim 473*> 474*> \param[in,out] PARAMS 475*> \verbatim 476*> PARAMS is REAL array, dimension NPARAMS 477*> Specifies algorithm parameters. If an entry is < 0.0, then 478*> that entry will be filled with default value used for that 479*> parameter. Only positions up to NPARAMS are accessed; defaults 480*> are used for higher-numbered parameters. 481*> 482*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 483*> refinement or not. 484*> Default: 1.0 485*> = 0.0: No refinement is performed, and no error bounds are 486*> computed. 487*> = 1.0: Use the double-precision refinement algorithm, 488*> possibly with doubled-single computations if the 489*> compilation environment does not support DOUBLE 490*> PRECISION. 491*> (other values are reserved for future use) 492*> 493*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 494*> computations allowed for refinement. 495*> Default: 10 496*> Aggressive: Set to 100 to permit convergence using approximate 497*> factorizations or factorizations other than LU. If 498*> the factorization uses a technique other than 499*> Gaussian elimination, the guarantees in 500*> err_bnds_norm and err_bnds_comp may no longer be 501*> trustworthy. 502*> 503*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 504*> will attempt to find a solution with small componentwise 505*> relative error in the double-precision algorithm. Positive 506*> is true, 0.0 is false. 507*> Default: 1.0 (attempt componentwise convergence) 508*> \endverbatim 509*> 510*> \param[out] WORK 511*> \verbatim 512*> WORK is COMPLEX array, dimension (2*N) 513*> \endverbatim 514*> 515*> \param[out] RWORK 516*> \verbatim 517*> RWORK is REAL array, dimension (2*N) 518*> \endverbatim 519*> 520*> \param[out] INFO 521*> \verbatim 522*> INFO is INTEGER 523*> = 0: Successful exit. The solution to every right-hand side is 524*> guaranteed. 525*> < 0: If INFO = -i, the i-th argument had an illegal value 526*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 527*> has been completed, but the factor U is exactly singular, so 528*> the solution and error bounds could not be computed. RCOND = 0 529*> is returned. 530*> = N+J: The solution corresponding to the Jth right-hand side is 531*> not guaranteed. The solutions corresponding to other right- 532*> hand sides K with K > J may not be guaranteed as well, but 533*> only the first such right-hand side is reported. If a small 534*> componentwise error is not requested (PARAMS(3) = 0.0) then 535*> the Jth right-hand side is the first with a normwise error 536*> bound that is not guaranteed (the smallest J such 537*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 538*> the Jth right-hand side is the first with either a normwise or 539*> componentwise error bound that is not guaranteed (the smallest 540*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 541*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 542*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 543*> about all of the right-hand sides check ERR_BNDS_NORM or 544*> ERR_BNDS_COMP. 545*> \endverbatim 546* 547* Authors: 548* ======== 549* 550*> \author Univ. of Tennessee 551*> \author Univ. of California Berkeley 552*> \author Univ. of Colorado Denver 553*> \author NAG Ltd. 554* 555*> \ingroup complexGBsolve 556* 557* ===================================================================== 558 SUBROUTINE CGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 559 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 560 $ RCOND, RPVGRW, BERR, N_ERR_BNDS, 561 $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 562 $ WORK, RWORK, INFO ) 563* 564* -- LAPACK driver routine -- 565* -- LAPACK is a software package provided by Univ. of Tennessee, -- 566* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 567* 568* .. Scalar Arguments .. 569 CHARACTER EQUED, FACT, TRANS 570 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, 571 $ N_ERR_BNDS 572 REAL RCOND, RPVGRW 573* .. 574* .. Array Arguments .. 575 INTEGER IPIV( * ) 576 COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 577 $ X( LDX , * ),WORK( * ) 578 REAL R( * ), C( * ), PARAMS( * ), BERR( * ), 579 $ ERR_BNDS_NORM( NRHS, * ), 580 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) 581* .. 582* 583* ================================================================== 584* 585* .. Parameters .. 586 REAL ZERO, ONE 587 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 588 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 589 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 590 INTEGER CMP_ERR_I, PIV_GROWTH_I 591 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 592 $ BERR_I = 3 ) 593 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 594 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 595 $ PIV_GROWTH_I = 9 ) 596* .. 597* .. Local Scalars .. 598 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 599 INTEGER INFEQU, I, J, KL, KU 600 REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN, 601 $ ROWCND, SMLNUM 602* .. 603* .. External Functions .. 604 EXTERNAL LSAME, SLAMCH, CLA_GBRPVGRW 605 LOGICAL LSAME 606 REAL SLAMCH, CLA_GBRPVGRW 607* .. 608* .. External Subroutines .. 609 EXTERNAL CGBEQUB, CGBTRF, CGBTRS, CLACPY, CLAQGB, 610 $ XERBLA, CLASCL2, CGBRFSX 611* .. 612* .. Intrinsic Functions .. 613 INTRINSIC MAX, MIN 614* .. 615* .. Executable Statements .. 616* 617 INFO = 0 618 NOFACT = LSAME( FACT, 'N' ) 619 EQUIL = LSAME( FACT, 'E' ) 620 NOTRAN = LSAME( TRANS, 'N' ) 621 SMLNUM = SLAMCH( 'Safe minimum' ) 622 BIGNUM = ONE / SMLNUM 623 IF( NOFACT .OR. EQUIL ) THEN 624 EQUED = 'N' 625 ROWEQU = .FALSE. 626 COLEQU = .FALSE. 627 ELSE 628 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 629 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 630 END IF 631* 632* Default is failure. If an input parameter is wrong or 633* factorization fails, make everything look horrible. Only the 634* pivot growth is set here, the rest is initialized in CGBRFSX. 635* 636 RPVGRW = ZERO 637* 638* Test the input parameters. PARAMS is not tested until CGERFSX. 639* 640 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 641 $ LSAME( FACT, 'F' ) ) THEN 642 INFO = -1 643 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 644 $ LSAME( TRANS, 'C' ) ) THEN 645 INFO = -2 646 ELSE IF( N.LT.0 ) THEN 647 INFO = -3 648 ELSE IF( KL.LT.0 ) THEN 649 INFO = -4 650 ELSE IF( KU.LT.0 ) THEN 651 INFO = -5 652 ELSE IF( NRHS.LT.0 ) THEN 653 INFO = -6 654 ELSE IF( LDAB.LT.KL+KU+1 ) THEN 655 INFO = -8 656 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN 657 INFO = -10 658 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 659 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 660 INFO = -12 661 ELSE 662 IF( ROWEQU ) THEN 663 RCMIN = BIGNUM 664 RCMAX = ZERO 665 DO 10 J = 1, N 666 RCMIN = MIN( RCMIN, R( J ) ) 667 RCMAX = MAX( RCMAX, R( J ) ) 668 10 CONTINUE 669 IF( RCMIN.LE.ZERO ) THEN 670 INFO = -13 671 ELSE IF( N.GT.0 ) THEN 672 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 673 ELSE 674 ROWCND = ONE 675 END IF 676 END IF 677 IF( COLEQU .AND. INFO.EQ.0 ) THEN 678 RCMIN = BIGNUM 679 RCMAX = ZERO 680 DO 20 J = 1, N 681 RCMIN = MIN( RCMIN, C( J ) ) 682 RCMAX = MAX( RCMAX, C( J ) ) 683 20 CONTINUE 684 IF( RCMIN.LE.ZERO ) THEN 685 INFO = -14 686 ELSE IF( N.GT.0 ) THEN 687 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 688 ELSE 689 COLCND = ONE 690 END IF 691 END IF 692 IF( INFO.EQ.0 ) THEN 693 IF( LDB.LT.MAX( 1, N ) ) THEN 694 INFO = -15 695 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 696 INFO = -16 697 END IF 698 END IF 699 END IF 700* 701 IF( INFO.NE.0 ) THEN 702 CALL XERBLA( 'CGBSVXX', -INFO ) 703 RETURN 704 END IF 705* 706 IF( EQUIL ) THEN 707* 708* Compute row and column scalings to equilibrate the matrix A. 709* 710 CALL CGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 711 $ AMAX, INFEQU ) 712 IF( INFEQU.EQ.0 ) THEN 713* 714* Equilibrate the matrix. 715* 716 CALL CLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 717 $ AMAX, EQUED ) 718 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 719 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 720 END IF 721* 722* If the scaling factors are not applied, set them to 1.0. 723* 724 IF ( .NOT.ROWEQU ) THEN 725 DO J = 1, N 726 R( J ) = 1.0 727 END DO 728 END IF 729 IF ( .NOT.COLEQU ) THEN 730 DO J = 1, N 731 C( J ) = 1.0 732 END DO 733 END IF 734 END IF 735* 736* Scale the right-hand side. 737* 738 IF( NOTRAN ) THEN 739 IF( ROWEQU ) CALL CLASCL2( N, NRHS, R, B, LDB ) 740 ELSE 741 IF( COLEQU ) CALL CLASCL2( N, NRHS, C, B, LDB ) 742 END IF 743* 744 IF( NOFACT .OR. EQUIL ) THEN 745* 746* Compute the LU factorization of A. 747* 748 DO 40, J = 1, N 749 DO 30, I = KL+1, 2*KL+KU+1 750 AFB( I, J ) = AB( I-KL, J ) 751 30 CONTINUE 752 40 CONTINUE 753 CALL CGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) 754* 755* Return if INFO is non-zero. 756* 757 IF( INFO.GT.0 ) THEN 758* 759* Pivot in column INFO is exactly 0 760* Compute the reciprocal pivot growth factor of the 761* leading rank-deficient INFO columns of A. 762* 763 RPVGRW = CLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB, 764 $ LDAFB ) 765 RETURN 766 END IF 767 END IF 768* 769* Compute the reciprocal pivot growth factor RPVGRW. 770* 771 RPVGRW = CLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB ) 772* 773* Compute the solution matrix X. 774* 775 CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 776 CALL CGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, 777 $ INFO ) 778* 779* Use iterative refinement to improve the computed solution and 780* compute error bounds and backward error estimates for it. 781* 782 CALL CGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 783 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, 784 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 785 $ WORK, RWORK, INFO ) 786 787* 788* Scale solutions. 789* 790 IF ( COLEQU .AND. NOTRAN ) THEN 791 CALL CLASCL2( N, NRHS, C, X, LDX ) 792 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN 793 CALL CLASCL2( N, NRHS, R, X, LDX ) 794 END IF 795* 796 RETURN 797* 798* End of CGBSVXX 799* 800 END 801