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