1*> \brief <b> ZGBSVX 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 ZGBSVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbsvx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbsvx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbsvx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 22* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 23* RCOND, FERR, BERR, WORK, RWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER EQUED, FACT, TRANS 27* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS 28* DOUBLE PRECISION RCOND 29* .. 30* .. Array Arguments .. 31* INTEGER IPIV( * ) 32* DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ), 33* $ RWORK( * ) 34* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 35* $ WORK( * ), X( LDX, * ) 36* .. 37* 38* 39*> \par Purpose: 40* ============= 41*> 42*> \verbatim 43*> 44*> ZGBSVX uses the LU factorization to compute the solution to a complex 45*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B, 46*> where A is a band matrix of order N with KL subdiagonals and KU 47*> superdiagonals, and X and B are N-by-NRHS matrices. 48*> 49*> Error bounds on the solution and a condition estimate are also 50*> provided. 51*> \endverbatim 52* 53*> \par Description: 54* ================= 55*> 56*> \verbatim 57*> 58*> The following steps are performed by this subroutine: 59*> 60*> 1. If FACT = 'E', real scaling factors are computed to equilibrate 61*> the system: 62*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B 63*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B 64*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B 65*> Whether or not the system will be equilibrated depends on the 66*> scaling of the matrix A, but if equilibration is used, A is 67*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') 68*> or diag(C)*B (if TRANS = 'T' or 'C'). 69*> 70*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the 71*> matrix A (after equilibration if FACT = 'E') as 72*> A = L * U, 73*> where L is a product of permutation and unit lower triangular 74*> matrices with KL subdiagonals, and U is upper triangular with 75*> KL+KU superdiagonals. 76*> 77*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine 78*> returns with INFO = i. Otherwise, the factored form of A is used 79*> to estimate the condition number of the matrix A. If the 80*> reciprocal of the condition number is less than machine precision, 81*> INFO = N+1 is returned as a warning, but the routine still goes on 82*> to solve for X and compute error bounds as described below. 83*> 84*> 4. The system of equations is solved for X using the factored form 85*> of A. 86*> 87*> 5. Iterative refinement is applied to improve the computed solution 88*> matrix and calculate error bounds and backward error estimates 89*> for it. 90*> 91*> 6. If equilibration was used, the matrix X is premultiplied by 92*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so 93*> that it solves the original system before equilibration. 94*> \endverbatim 95* 96* Arguments: 97* ========== 98* 99*> \param[in] FACT 100*> \verbatim 101*> FACT is CHARACTER*1 102*> Specifies whether or not the factored form of the matrix A is 103*> supplied on entry, and if not, whether the matrix A should be 104*> equilibrated before it is factored. 105*> = 'F': On entry, AFB and IPIV contain the factored form of 106*> A. If EQUED is not 'N', the matrix A has been 107*> equilibrated with scaling factors given by R and C. 108*> AB, AFB, and IPIV are not modified. 109*> = 'N': The matrix A will be copied to AFB and factored. 110*> = 'E': The matrix A will be equilibrated if necessary, then 111*> copied to AFB and factored. 112*> \endverbatim 113*> 114*> \param[in] TRANS 115*> \verbatim 116*> TRANS is CHARACTER*1 117*> Specifies the form of the system of equations. 118*> = 'N': A * X = B (No transpose) 119*> = 'T': A**T * X = B (Transpose) 120*> = 'C': A**H * X = B (Conjugate transpose) 121*> \endverbatim 122*> 123*> \param[in] N 124*> \verbatim 125*> N is INTEGER 126*> The number of linear equations, i.e., the order of the 127*> matrix A. N >= 0. 128*> \endverbatim 129*> 130*> \param[in] KL 131*> \verbatim 132*> KL is INTEGER 133*> The number of subdiagonals within the band of A. KL >= 0. 134*> \endverbatim 135*> 136*> \param[in] KU 137*> \verbatim 138*> KU is INTEGER 139*> The number of superdiagonals within the band of A. KU >= 0. 140*> \endverbatim 141*> 142*> \param[in] NRHS 143*> \verbatim 144*> NRHS is INTEGER 145*> The number of right hand sides, i.e., the number of columns 146*> of the matrices B and X. NRHS >= 0. 147*> \endverbatim 148*> 149*> \param[in,out] AB 150*> \verbatim 151*> AB is COMPLEX*16 array, dimension (LDAB,N) 152*> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. 153*> The j-th column of A is stored in the j-th column of the 154*> array AB as follows: 155*> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) 156*> 157*> If FACT = 'F' and EQUED is not 'N', then A must have been 158*> equilibrated by the scaling factors in R and/or C. AB is not 159*> modified if FACT = 'F' or 'N', or if FACT = 'E' and 160*> EQUED = 'N' on exit. 161*> 162*> On exit, if EQUED .ne. 'N', A is scaled as follows: 163*> EQUED = 'R': A := diag(R) * A 164*> EQUED = 'C': A := A * diag(C) 165*> EQUED = 'B': A := diag(R) * A * diag(C). 166*> \endverbatim 167*> 168*> \param[in] LDAB 169*> \verbatim 170*> LDAB is INTEGER 171*> The leading dimension of the array AB. LDAB >= KL+KU+1. 172*> \endverbatim 173*> 174*> \param[in,out] AFB 175*> \verbatim 176*> AFB is COMPLEX*16 array, dimension (LDAFB,N) 177*> If FACT = 'F', then AFB is an input argument and on entry 178*> contains details of the LU factorization of the band matrix 179*> A, as computed by ZGBTRF. U is stored as an upper triangular 180*> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, 181*> and the multipliers used during the factorization are stored 182*> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is 183*> the factored form of the equilibrated matrix A. 184*> 185*> If FACT = 'N', then AFB is an output argument and on exit 186*> returns details of the LU factorization of A. 187*> 188*> If FACT = 'E', then AFB is an output argument and on exit 189*> returns details of the LU factorization of the equilibrated 190*> matrix A (see the description of AB for the form of the 191*> equilibrated matrix). 192*> \endverbatim 193*> 194*> \param[in] LDAFB 195*> \verbatim 196*> LDAFB is INTEGER 197*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. 198*> \endverbatim 199*> 200*> \param[in,out] IPIV 201*> \verbatim 202*> IPIV is INTEGER array, dimension (N) 203*> If FACT = 'F', then IPIV is an input argument and on entry 204*> contains the pivot indices from the factorization A = L*U 205*> as computed by ZGBTRF; row i of the matrix was interchanged 206*> with row IPIV(i). 207*> 208*> If FACT = 'N', then IPIV is an output argument and on exit 209*> contains the pivot indices from the factorization A = L*U 210*> of the original matrix A. 211*> 212*> If FACT = 'E', then IPIV is an output argument and on exit 213*> contains the pivot indices from the factorization A = L*U 214*> of the equilibrated matrix A. 215*> \endverbatim 216*> 217*> \param[in,out] EQUED 218*> \verbatim 219*> EQUED is CHARACTER*1 220*> Specifies the form of equilibration that was done. 221*> = 'N': No equilibration (always true if FACT = 'N'). 222*> = 'R': Row equilibration, i.e., A has been premultiplied by 223*> diag(R). 224*> = 'C': Column equilibration, i.e., A has been postmultiplied 225*> by diag(C). 226*> = 'B': Both row and column equilibration, i.e., A has been 227*> replaced by diag(R) * A * diag(C). 228*> EQUED is an input argument if FACT = 'F'; otherwise, it is an 229*> output argument. 230*> \endverbatim 231*> 232*> \param[in,out] R 233*> \verbatim 234*> R is DOUBLE PRECISION array, dimension (N) 235*> The row scale factors for A. If EQUED = 'R' or 'B', A is 236*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R 237*> is not accessed. R is an input argument if FACT = 'F'; 238*> otherwise, R is an output argument. If FACT = 'F' and 239*> EQUED = 'R' or 'B', each element of R must be positive. 240*> \endverbatim 241*> 242*> \param[in,out] C 243*> \verbatim 244*> C is DOUBLE PRECISION array, dimension (N) 245*> The column scale factors for A. If EQUED = 'C' or 'B', A is 246*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C 247*> is not accessed. C is an input argument if FACT = 'F'; 248*> otherwise, C is an output argument. If FACT = 'F' and 249*> EQUED = 'C' or 'B', each element of C must be positive. 250*> \endverbatim 251*> 252*> \param[in,out] B 253*> \verbatim 254*> B is COMPLEX*16 array, dimension (LDB,NRHS) 255*> On entry, the right hand side matrix B. 256*> On exit, 257*> if EQUED = 'N', B is not modified; 258*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by 259*> diag(R)*B; 260*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is 261*> overwritten by diag(C)*B. 262*> \endverbatim 263*> 264*> \param[in] LDB 265*> \verbatim 266*> LDB is INTEGER 267*> The leading dimension of the array B. LDB >= max(1,N). 268*> \endverbatim 269*> 270*> \param[out] X 271*> \verbatim 272*> X is COMPLEX*16 array, dimension (LDX,NRHS) 273*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X 274*> to the original system of equations. Note that A and B are 275*> modified on exit if EQUED .ne. 'N', and the solution to the 276*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and 277*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' 278*> and EQUED = 'R' or 'B'. 279*> \endverbatim 280*> 281*> \param[in] LDX 282*> \verbatim 283*> LDX is INTEGER 284*> The leading dimension of the array X. LDX >= max(1,N). 285*> \endverbatim 286*> 287*> \param[out] RCOND 288*> \verbatim 289*> RCOND is DOUBLE PRECISION 290*> The estimate of the reciprocal condition number of the matrix 291*> A after equilibration (if done). If RCOND is less than the 292*> machine precision (in particular, if RCOND = 0), the matrix 293*> is singular to working precision. This condition is 294*> indicated by a return code of INFO > 0. 295*> \endverbatim 296*> 297*> \param[out] FERR 298*> \verbatim 299*> FERR is DOUBLE PRECISION array, dimension (NRHS) 300*> The estimated forward error bound for each solution vector 301*> X(j) (the j-th column of the solution matrix X). 302*> If XTRUE is the true solution corresponding to X(j), FERR(j) 303*> is an estimated upper bound for the magnitude of the largest 304*> element in (X(j) - XTRUE) divided by the magnitude of the 305*> largest element in X(j). The estimate is as reliable as 306*> the estimate for RCOND, and is almost always a slight 307*> overestimate of the true error. 308*> \endverbatim 309*> 310*> \param[out] BERR 311*> \verbatim 312*> BERR is DOUBLE PRECISION array, dimension (NRHS) 313*> The componentwise relative backward error of each solution 314*> vector X(j) (i.e., the smallest relative change in 315*> any element of A or B that makes X(j) an exact solution). 316*> \endverbatim 317*> 318*> \param[out] WORK 319*> \verbatim 320*> WORK is COMPLEX*16 array, dimension (2*N) 321*> \endverbatim 322*> 323*> \param[out] RWORK 324*> \verbatim 325*> RWORK is DOUBLE PRECISION array, dimension (N) 326*> On exit, RWORK(1) contains the reciprocal pivot growth 327*> factor norm(A)/norm(U). The "max absolute element" norm is 328*> used. If RWORK(1) is much less than 1, then the stability 329*> of the LU factorization of the (equilibrated) matrix A 330*> could be poor. This also means that the solution X, condition 331*> estimator RCOND, and forward error bound FERR could be 332*> unreliable. If factorization fails with 0<INFO<=N, then 333*> RWORK(1) contains the reciprocal pivot growth factor for the 334*> leading INFO columns of A. 335*> \endverbatim 336*> 337*> \param[out] INFO 338*> \verbatim 339*> INFO is INTEGER 340*> = 0: successful exit 341*> < 0: if INFO = -i, the i-th argument had an illegal value 342*> > 0: if INFO = i, and i is 343*> <= N: U(i,i) is exactly zero. The factorization 344*> has been completed, but the factor U is exactly 345*> singular, so the solution and error bounds 346*> could not be computed. RCOND = 0 is returned. 347*> = N+1: U is nonsingular, but RCOND is less than machine 348*> precision, meaning that the matrix is singular 349*> to working precision. Nevertheless, the 350*> solution and error bounds are computed because 351*> there are a number of situations where the 352*> computed solution can be more accurate than the 353*> value of RCOND would suggest. 354*> \endverbatim 355* 356* Authors: 357* ======== 358* 359*> \author Univ. of Tennessee 360*> \author Univ. of California Berkeley 361*> \author Univ. of Colorado Denver 362*> \author NAG Ltd. 363* 364*> \ingroup complex16GBsolve 365* 366* ===================================================================== 367 SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 368 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 369 $ RCOND, FERR, BERR, WORK, RWORK, INFO ) 370* 371* -- LAPACK driver routine -- 372* -- LAPACK is a software package provided by Univ. of Tennessee, -- 373* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 374* 375* .. Scalar Arguments .. 376 CHARACTER EQUED, FACT, TRANS 377 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS 378 DOUBLE PRECISION RCOND 379* .. 380* .. Array Arguments .. 381 INTEGER IPIV( * ) 382 DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ), 383 $ RWORK( * ) 384 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 385 $ WORK( * ), X( LDX, * ) 386* .. 387* 388* ===================================================================== 389* Moved setting of INFO = N+1 so INFO does not subsequently get 390* overwritten. Sven, 17 Mar 05. 391* ===================================================================== 392* 393* .. Parameters .. 394 DOUBLE PRECISION ZERO, ONE 395 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 396* .. 397* .. Local Scalars .. 398 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 399 CHARACTER NORM 400 INTEGER I, INFEQU, J, J1, J2 401 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, 402 $ ROWCND, RPVGRW, SMLNUM 403* .. 404* .. External Functions .. 405 LOGICAL LSAME 406 DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB 407 EXTERNAL LSAME, DLAMCH, ZLANGB, ZLANTB 408* .. 409* .. External Subroutines .. 410 EXTERNAL XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF, 411 $ ZGBTRS, ZLACPY, ZLAQGB 412* .. 413* .. Intrinsic Functions .. 414 INTRINSIC ABS, MAX, MIN 415* .. 416* .. Executable Statements .. 417* 418 INFO = 0 419 NOFACT = LSAME( FACT, 'N' ) 420 EQUIL = LSAME( FACT, 'E' ) 421 NOTRAN = LSAME( TRANS, 'N' ) 422 IF( NOFACT .OR. EQUIL ) THEN 423 EQUED = 'N' 424 ROWEQU = .FALSE. 425 COLEQU = .FALSE. 426 ELSE 427 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 428 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 429 SMLNUM = DLAMCH( 'Safe minimum' ) 430 BIGNUM = ONE / SMLNUM 431 END IF 432* 433* Test the input parameters. 434* 435 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 436 $ THEN 437 INFO = -1 438 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 439 $ LSAME( TRANS, 'C' ) ) THEN 440 INFO = -2 441 ELSE IF( N.LT.0 ) THEN 442 INFO = -3 443 ELSE IF( KL.LT.0 ) THEN 444 INFO = -4 445 ELSE IF( KU.LT.0 ) THEN 446 INFO = -5 447 ELSE IF( NRHS.LT.0 ) THEN 448 INFO = -6 449 ELSE IF( LDAB.LT.KL+KU+1 ) THEN 450 INFO = -8 451 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN 452 INFO = -10 453 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 454 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 455 INFO = -12 456 ELSE 457 IF( ROWEQU ) THEN 458 RCMIN = BIGNUM 459 RCMAX = ZERO 460 DO 10 J = 1, N 461 RCMIN = MIN( RCMIN, R( J ) ) 462 RCMAX = MAX( RCMAX, R( J ) ) 463 10 CONTINUE 464 IF( RCMIN.LE.ZERO ) THEN 465 INFO = -13 466 ELSE IF( N.GT.0 ) THEN 467 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 468 ELSE 469 ROWCND = ONE 470 END IF 471 END IF 472 IF( COLEQU .AND. INFO.EQ.0 ) THEN 473 RCMIN = BIGNUM 474 RCMAX = ZERO 475 DO 20 J = 1, N 476 RCMIN = MIN( RCMIN, C( J ) ) 477 RCMAX = MAX( RCMAX, C( J ) ) 478 20 CONTINUE 479 IF( RCMIN.LE.ZERO ) THEN 480 INFO = -14 481 ELSE IF( N.GT.0 ) THEN 482 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 483 ELSE 484 COLCND = ONE 485 END IF 486 END IF 487 IF( INFO.EQ.0 ) THEN 488 IF( LDB.LT.MAX( 1, N ) ) THEN 489 INFO = -16 490 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 491 INFO = -18 492 END IF 493 END IF 494 END IF 495* 496 IF( INFO.NE.0 ) THEN 497 CALL XERBLA( 'ZGBSVX', -INFO ) 498 RETURN 499 END IF 500* 501 IF( EQUIL ) THEN 502* 503* Compute row and column scalings to equilibrate the matrix A. 504* 505 CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 506 $ AMAX, INFEQU ) 507 IF( INFEQU.EQ.0 ) THEN 508* 509* Equilibrate the matrix. 510* 511 CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 512 $ AMAX, EQUED ) 513 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 514 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 515 END IF 516 END IF 517* 518* Scale the right hand side. 519* 520 IF( NOTRAN ) THEN 521 IF( ROWEQU ) THEN 522 DO 40 J = 1, NRHS 523 DO 30 I = 1, N 524 B( I, J ) = R( I )*B( I, J ) 525 30 CONTINUE 526 40 CONTINUE 527 END IF 528 ELSE IF( COLEQU ) THEN 529 DO 60 J = 1, NRHS 530 DO 50 I = 1, N 531 B( I, J ) = C( I )*B( I, J ) 532 50 CONTINUE 533 60 CONTINUE 534 END IF 535* 536 IF( NOFACT .OR. EQUIL ) THEN 537* 538* Compute the LU factorization of the band matrix A. 539* 540 DO 70 J = 1, N 541 J1 = MAX( J-KU, 1 ) 542 J2 = MIN( J+KL, N ) 543 CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1, 544 $ AFB( KL+KU+1-J+J1, J ), 1 ) 545 70 CONTINUE 546* 547 CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) 548* 549* Return if INFO is non-zero. 550* 551 IF( INFO.GT.0 ) THEN 552* 553* Compute the reciprocal pivot growth factor of the 554* leading rank-deficient INFO columns of A. 555* 556 ANORM = ZERO 557 DO 90 J = 1, INFO 558 DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 ) 559 ANORM = MAX( ANORM, ABS( AB( I, J ) ) ) 560 80 CONTINUE 561 90 CONTINUE 562 RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ), 563 $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB, 564 $ RWORK ) 565 IF( RPVGRW.EQ.ZERO ) THEN 566 RPVGRW = ONE 567 ELSE 568 RPVGRW = ANORM / RPVGRW 569 END IF 570 RWORK( 1 ) = RPVGRW 571 RCOND = ZERO 572 RETURN 573 END IF 574 END IF 575* 576* Compute the norm of the matrix A and the 577* reciprocal pivot growth factor RPVGRW. 578* 579 IF( NOTRAN ) THEN 580 NORM = '1' 581 ELSE 582 NORM = 'I' 583 END IF 584 ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK ) 585 RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK ) 586 IF( RPVGRW.EQ.ZERO ) THEN 587 RPVGRW = ONE 588 ELSE 589 RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW 590 END IF 591* 592* Compute the reciprocal of the condition number of A. 593* 594 CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND, 595 $ WORK, RWORK, INFO ) 596* 597* Compute the solution matrix X. 598* 599 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 600 CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, 601 $ INFO ) 602* 603* Use iterative refinement to improve the computed solution and 604* compute error bounds and backward error estimates for it. 605* 606 CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, 607 $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO ) 608* 609* Transform the solution matrix X to a solution of the original 610* system. 611* 612 IF( NOTRAN ) THEN 613 IF( COLEQU ) THEN 614 DO 110 J = 1, NRHS 615 DO 100 I = 1, N 616 X( I, J ) = C( I )*X( I, J ) 617 100 CONTINUE 618 110 CONTINUE 619 DO 120 J = 1, NRHS 620 FERR( J ) = FERR( J ) / COLCND 621 120 CONTINUE 622 END IF 623 ELSE IF( ROWEQU ) THEN 624 DO 140 J = 1, NRHS 625 DO 130 I = 1, N 626 X( I, J ) = R( I )*X( I, J ) 627 130 CONTINUE 628 140 CONTINUE 629 DO 150 J = 1, NRHS 630 FERR( J ) = FERR( J ) / ROWCND 631 150 CONTINUE 632 END IF 633* 634* Set INFO = N+1 if the matrix is singular to working precision. 635* 636 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 637 $ INFO = N + 1 638* 639 RWORK( 1 ) = RPVGRW 640 RETURN 641* 642* End of ZGBSVX 643* 644 END 645