Annotation of rpl/lapack/lapack/zgbsvx.f, revision 1.1

1.1     ! bertrand    1:       SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
        !             2:      $                   LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
        !             3:      $                   RCOND, FERR, BERR, WORK, RWORK, INFO )
        !             4: *
        !             5: *  -- LAPACK driver routine (version 3.2) --
        !             6: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
        !             7: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
        !             8: *     November 2006
        !             9: *
        !            10: *     .. Scalar Arguments ..
        !            11:       CHARACTER          EQUED, FACT, TRANS
        !            12:       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
        !            13:       DOUBLE PRECISION   RCOND
        !            14: *     ..
        !            15: *     .. Array Arguments ..
        !            16:       INTEGER            IPIV( * )
        !            17:       DOUBLE PRECISION   BERR( * ), C( * ), FERR( * ), R( * ),
        !            18:      $                   RWORK( * )
        !            19:       COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
        !            20:      $                   WORK( * ), X( LDX, * )
        !            21: *     ..
        !            22: *
        !            23: *  Purpose
        !            24: *  =======
        !            25: *
        !            26: *  ZGBSVX uses the LU factorization to compute the solution to a complex
        !            27: *  system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
        !            28: *  where A is a band matrix of order N with KL subdiagonals and KU
        !            29: *  superdiagonals, and X and B are N-by-NRHS matrices.
        !            30: *
        !            31: *  Error bounds on the solution and a condition estimate are also
        !            32: *  provided.
        !            33: *
        !            34: *  Description
        !            35: *  ===========
        !            36: *
        !            37: *  The following steps are performed by this subroutine:
        !            38: *
        !            39: *  1. If FACT = 'E', real scaling factors are computed to equilibrate
        !            40: *     the system:
        !            41: *        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
        !            42: *        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
        !            43: *        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
        !            44: *     Whether or not the system will be equilibrated depends on the
        !            45: *     scaling of the matrix A, but if equilibration is used, A is
        !            46: *     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
        !            47: *     or diag(C)*B (if TRANS = 'T' or 'C').
        !            48: *
        !            49: *  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
        !            50: *     matrix A (after equilibration if FACT = 'E') as
        !            51: *        A = L * U,
        !            52: *     where L is a product of permutation and unit lower triangular
        !            53: *     matrices with KL subdiagonals, and U is upper triangular with
        !            54: *     KL+KU superdiagonals.
        !            55: *
        !            56: *  3. If some U(i,i)=0, so that U is exactly singular, then the routine
        !            57: *     returns with INFO = i. Otherwise, the factored form of A is used
        !            58: *     to estimate the condition number of the matrix A.  If the
        !            59: *     reciprocal of the condition number is less than machine precision,
        !            60: *     INFO = N+1 is returned as a warning, but the routine still goes on
        !            61: *     to solve for X and compute error bounds as described below.
        !            62: *
        !            63: *  4. The system of equations is solved for X using the factored form
        !            64: *     of A.
        !            65: *
        !            66: *  5. Iterative refinement is applied to improve the computed solution
        !            67: *     matrix and calculate error bounds and backward error estimates
        !            68: *     for it.
        !            69: *
        !            70: *  6. If equilibration was used, the matrix X is premultiplied by
        !            71: *     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
        !            72: *     that it solves the original system before equilibration.
        !            73: *
        !            74: *  Arguments
        !            75: *  =========
        !            76: *
        !            77: *  FACT    (input) CHARACTER*1
        !            78: *          Specifies whether or not the factored form of the matrix A is
        !            79: *          supplied on entry, and if not, whether the matrix A should be
        !            80: *          equilibrated before it is factored.
        !            81: *          = 'F':  On entry, AFB and IPIV contain the factored form of
        !            82: *                  A.  If EQUED is not 'N', the matrix A has been
        !            83: *                  equilibrated with scaling factors given by R and C.
        !            84: *                  AB, AFB, and IPIV are not modified.
        !            85: *          = 'N':  The matrix A will be copied to AFB and factored.
        !            86: *          = 'E':  The matrix A will be equilibrated if necessary, then
        !            87: *                  copied to AFB and factored.
        !            88: *
        !            89: *  TRANS   (input) CHARACTER*1
        !            90: *          Specifies the form of the system of equations.
        !            91: *          = 'N':  A * X = B     (No transpose)
        !            92: *          = 'T':  A**T * X = B  (Transpose)
        !            93: *          = 'C':  A**H * X = B  (Conjugate transpose)
        !            94: *
        !            95: *  N       (input) INTEGER
        !            96: *          The number of linear equations, i.e., the order of the
        !            97: *          matrix A.  N >= 0.
        !            98: *
        !            99: *  KL      (input) INTEGER
        !           100: *          The number of subdiagonals within the band of A.  KL >= 0.
        !           101: *
        !           102: *  KU      (input) INTEGER
        !           103: *          The number of superdiagonals within the band of A.  KU >= 0.
        !           104: *
        !           105: *  NRHS    (input) INTEGER
        !           106: *          The number of right hand sides, i.e., the number of columns
        !           107: *          of the matrices B and X.  NRHS >= 0.
        !           108: *
        !           109: *  AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
        !           110: *          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
        !           111: *          The j-th column of A is stored in the j-th column of the
        !           112: *          array AB as follows:
        !           113: *          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
        !           114: *
        !           115: *          If FACT = 'F' and EQUED is not 'N', then A must have been
        !           116: *          equilibrated by the scaling factors in R and/or C.  AB is not
        !           117: *          modified if FACT = 'F' or 'N', or if FACT = 'E' and
        !           118: *          EQUED = 'N' on exit.
        !           119: *
        !           120: *          On exit, if EQUED .ne. 'N', A is scaled as follows:
        !           121: *          EQUED = 'R':  A := diag(R) * A
        !           122: *          EQUED = 'C':  A := A * diag(C)
        !           123: *          EQUED = 'B':  A := diag(R) * A * diag(C).
        !           124: *
        !           125: *  LDAB    (input) INTEGER
        !           126: *          The leading dimension of the array AB.  LDAB >= KL+KU+1.
        !           127: *
        !           128: *  AFB     (input or output) COMPLEX*16 array, dimension (LDAFB,N)
        !           129: *          If FACT = 'F', then AFB is an input argument and on entry
        !           130: *          contains details of the LU factorization of the band matrix
        !           131: *          A, as computed by ZGBTRF.  U is stored as an upper triangular
        !           132: *          band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
        !           133: *          and the multipliers used during the factorization are stored
        !           134: *          in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
        !           135: *          the factored form of the equilibrated matrix A.
        !           136: *
        !           137: *          If FACT = 'N', then AFB is an output argument and on exit
        !           138: *          returns details of the LU factorization of A.
        !           139: *
        !           140: *          If FACT = 'E', then AFB is an output argument and on exit
        !           141: *          returns details of the LU factorization of the equilibrated
        !           142: *          matrix A (see the description of AB for the form of the
        !           143: *          equilibrated matrix).
        !           144: *
        !           145: *  LDAFB   (input) INTEGER
        !           146: *          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
        !           147: *
        !           148: *  IPIV    (input or output) INTEGER array, dimension (N)
        !           149: *          If FACT = 'F', then IPIV is an input argument and on entry
        !           150: *          contains the pivot indices from the factorization A = L*U
        !           151: *          as computed by ZGBTRF; row i of the matrix was interchanged
        !           152: *          with row IPIV(i).
        !           153: *
        !           154: *          If FACT = 'N', then IPIV is an output argument and on exit
        !           155: *          contains the pivot indices from the factorization A = L*U
        !           156: *          of the original matrix A.
        !           157: *
        !           158: *          If FACT = 'E', then IPIV is an output argument and on exit
        !           159: *          contains the pivot indices from the factorization A = L*U
        !           160: *          of the equilibrated matrix A.
        !           161: *
        !           162: *  EQUED   (input or output) CHARACTER*1
        !           163: *          Specifies the form of equilibration that was done.
        !           164: *          = 'N':  No equilibration (always true if FACT = 'N').
        !           165: *          = 'R':  Row equilibration, i.e., A has been premultiplied by
        !           166: *                  diag(R).
        !           167: *          = 'C':  Column equilibration, i.e., A has been postmultiplied
        !           168: *                  by diag(C).
        !           169: *          = 'B':  Both row and column equilibration, i.e., A has been
        !           170: *                  replaced by diag(R) * A * diag(C).
        !           171: *          EQUED is an input argument if FACT = 'F'; otherwise, it is an
        !           172: *          output argument.
        !           173: *
        !           174: *  R       (input or output) DOUBLE PRECISION array, dimension (N)
        !           175: *          The row scale factors for A.  If EQUED = 'R' or 'B', A is
        !           176: *          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
        !           177: *          is not accessed.  R is an input argument if FACT = 'F';
        !           178: *          otherwise, R is an output argument.  If FACT = 'F' and
        !           179: *          EQUED = 'R' or 'B', each element of R must be positive.
        !           180: *
        !           181: *  C       (input or output) DOUBLE PRECISION array, dimension (N)
        !           182: *          The column scale factors for A.  If EQUED = 'C' or 'B', A is
        !           183: *          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
        !           184: *          is not accessed.  C is an input argument if FACT = 'F';
        !           185: *          otherwise, C is an output argument.  If FACT = 'F' and
        !           186: *          EQUED = 'C' or 'B', each element of C must be positive.
        !           187: *
        !           188: *  B       (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
        !           189: *          On entry, the right hand side matrix B.
        !           190: *          On exit,
        !           191: *          if EQUED = 'N', B is not modified;
        !           192: *          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
        !           193: *          diag(R)*B;
        !           194: *          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
        !           195: *          overwritten by diag(C)*B.
        !           196: *
        !           197: *  LDB     (input) INTEGER
        !           198: *          The leading dimension of the array B.  LDB >= max(1,N).
        !           199: *
        !           200: *  X       (output) COMPLEX*16 array, dimension (LDX,NRHS)
        !           201: *          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
        !           202: *          to the original system of equations.  Note that A and B are
        !           203: *          modified on exit if EQUED .ne. 'N', and the solution to the
        !           204: *          equilibrated system is inv(diag(C))*X if TRANS = 'N' and
        !           205: *          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
        !           206: *          and EQUED = 'R' or 'B'.
        !           207: *
        !           208: *  LDX     (input) INTEGER
        !           209: *          The leading dimension of the array X.  LDX >= max(1,N).
        !           210: *
        !           211: *  RCOND   (output) DOUBLE PRECISION
        !           212: *          The estimate of the reciprocal condition number of the matrix
        !           213: *          A after equilibration (if done).  If RCOND is less than the
        !           214: *          machine precision (in particular, if RCOND = 0), the matrix
        !           215: *          is singular to working precision.  This condition is
        !           216: *          indicated by a return code of INFO > 0.
        !           217: *
        !           218: *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
        !           219: *          The estimated forward error bound for each solution vector
        !           220: *          X(j) (the j-th column of the solution matrix X).
        !           221: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
        !           222: *          is an estimated upper bound for the magnitude of the largest
        !           223: *          element in (X(j) - XTRUE) divided by the magnitude of the
        !           224: *          largest element in X(j).  The estimate is as reliable as
        !           225: *          the estimate for RCOND, and is almost always a slight
        !           226: *          overestimate of the true error.
        !           227: *
        !           228: *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
        !           229: *          The componentwise relative backward error of each solution
        !           230: *          vector X(j) (i.e., the smallest relative change in
        !           231: *          any element of A or B that makes X(j) an exact solution).
        !           232: *
        !           233: *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
        !           234: *
        !           235: *  RWORK   (workspace/output) DOUBLE PRECISION array, dimension (N)
        !           236: *          On exit, RWORK(1) contains the reciprocal pivot growth
        !           237: *          factor norm(A)/norm(U). The "max absolute element" norm is
        !           238: *          used. If RWORK(1) is much less than 1, then the stability
        !           239: *          of the LU factorization of the (equilibrated) matrix A
        !           240: *          could be poor. This also means that the solution X, condition
        !           241: *          estimator RCOND, and forward error bound FERR could be
        !           242: *          unreliable. If factorization fails with 0<INFO<=N, then
        !           243: *          RWORK(1) contains the reciprocal pivot growth factor for the
        !           244: *          leading INFO columns of A.
        !           245: *
        !           246: *  INFO    (output) INTEGER
        !           247: *          = 0:  successful exit
        !           248: *          < 0:  if INFO = -i, the i-th argument had an illegal value
        !           249: *          > 0:  if INFO = i, and i is
        !           250: *                <= N:  U(i,i) is exactly zero.  The factorization
        !           251: *                       has been completed, but the factor U is exactly
        !           252: *                       singular, so the solution and error bounds
        !           253: *                       could not be computed. RCOND = 0 is returned.
        !           254: *                = N+1: U is nonsingular, but RCOND is less than machine
        !           255: *                       precision, meaning that the matrix is singular
        !           256: *                       to working precision.  Nevertheless, the
        !           257: *                       solution and error bounds are computed because
        !           258: *                       there are a number of situations where the
        !           259: *                       computed solution can be more accurate than the
        !           260: *                       value of RCOND would suggest.
        !           261: *
        !           262: *  =====================================================================
        !           263: *  Moved setting of INFO = N+1 so INFO does not subsequently get
        !           264: *  overwritten.  Sven, 17 Mar 05. 
        !           265: *  =====================================================================
        !           266: *
        !           267: *     .. Parameters ..
        !           268:       DOUBLE PRECISION   ZERO, ONE
        !           269:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
        !           270: *     ..
        !           271: *     .. Local Scalars ..
        !           272:       LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
        !           273:       CHARACTER          NORM
        !           274:       INTEGER            I, INFEQU, J, J1, J2
        !           275:       DOUBLE PRECISION   AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
        !           276:      $                   ROWCND, RPVGRW, SMLNUM
        !           277: *     ..
        !           278: *     .. External Functions ..
        !           279:       LOGICAL            LSAME
        !           280:       DOUBLE PRECISION   DLAMCH, ZLANGB, ZLANTB
        !           281:       EXTERNAL           LSAME, DLAMCH, ZLANGB, ZLANTB
        !           282: *     ..
        !           283: *     .. External Subroutines ..
        !           284:       EXTERNAL           XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF,
        !           285:      $                   ZGBTRS, ZLACPY, ZLAQGB
        !           286: *     ..
        !           287: *     .. Intrinsic Functions ..
        !           288:       INTRINSIC          ABS, MAX, MIN
        !           289: *     ..
        !           290: *     .. Executable Statements ..
        !           291: *
        !           292:       INFO = 0
        !           293:       NOFACT = LSAME( FACT, 'N' )
        !           294:       EQUIL = LSAME( FACT, 'E' )
        !           295:       NOTRAN = LSAME( TRANS, 'N' )
        !           296:       IF( NOFACT .OR. EQUIL ) THEN
        !           297:          EQUED = 'N'
        !           298:          ROWEQU = .FALSE.
        !           299:          COLEQU = .FALSE.
        !           300:       ELSE
        !           301:          ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
        !           302:          COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
        !           303:          SMLNUM = DLAMCH( 'Safe minimum' )
        !           304:          BIGNUM = ONE / SMLNUM
        !           305:       END IF
        !           306: *
        !           307: *     Test the input parameters.
        !           308: *
        !           309:       IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
        !           310:      $     THEN
        !           311:          INFO = -1
        !           312:       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
        !           313:      $         LSAME( TRANS, 'C' ) ) THEN
        !           314:          INFO = -2
        !           315:       ELSE IF( N.LT.0 ) THEN
        !           316:          INFO = -3
        !           317:       ELSE IF( KL.LT.0 ) THEN
        !           318:          INFO = -4
        !           319:       ELSE IF( KU.LT.0 ) THEN
        !           320:          INFO = -5
        !           321:       ELSE IF( NRHS.LT.0 ) THEN
        !           322:          INFO = -6
        !           323:       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
        !           324:          INFO = -8
        !           325:       ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
        !           326:          INFO = -10
        !           327:       ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
        !           328:      $         ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
        !           329:          INFO = -12
        !           330:       ELSE
        !           331:          IF( ROWEQU ) THEN
        !           332:             RCMIN = BIGNUM
        !           333:             RCMAX = ZERO
        !           334:             DO 10 J = 1, N
        !           335:                RCMIN = MIN( RCMIN, R( J ) )
        !           336:                RCMAX = MAX( RCMAX, R( J ) )
        !           337:    10       CONTINUE
        !           338:             IF( RCMIN.LE.ZERO ) THEN
        !           339:                INFO = -13
        !           340:             ELSE IF( N.GT.0 ) THEN
        !           341:                ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
        !           342:             ELSE
        !           343:                ROWCND = ONE
        !           344:             END IF
        !           345:          END IF
        !           346:          IF( COLEQU .AND. INFO.EQ.0 ) THEN
        !           347:             RCMIN = BIGNUM
        !           348:             RCMAX = ZERO
        !           349:             DO 20 J = 1, N
        !           350:                RCMIN = MIN( RCMIN, C( J ) )
        !           351:                RCMAX = MAX( RCMAX, C( J ) )
        !           352:    20       CONTINUE
        !           353:             IF( RCMIN.LE.ZERO ) THEN
        !           354:                INFO = -14
        !           355:             ELSE IF( N.GT.0 ) THEN
        !           356:                COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
        !           357:             ELSE
        !           358:                COLCND = ONE
        !           359:             END IF
        !           360:          END IF
        !           361:          IF( INFO.EQ.0 ) THEN
        !           362:             IF( LDB.LT.MAX( 1, N ) ) THEN
        !           363:                INFO = -16
        !           364:             ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
        !           365:                INFO = -18
        !           366:             END IF
        !           367:          END IF
        !           368:       END IF
        !           369: *
        !           370:       IF( INFO.NE.0 ) THEN
        !           371:          CALL XERBLA( 'ZGBSVX', -INFO )
        !           372:          RETURN
        !           373:       END IF
        !           374: *
        !           375:       IF( EQUIL ) THEN
        !           376: *
        !           377: *        Compute row and column scalings to equilibrate the matrix A.
        !           378: *
        !           379:          CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
        !           380:      $                AMAX, INFEQU )
        !           381:          IF( INFEQU.EQ.0 ) THEN
        !           382: *
        !           383: *           Equilibrate the matrix.
        !           384: *
        !           385:             CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
        !           386:      $                   AMAX, EQUED )
        !           387:             ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
        !           388:             COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
        !           389:          END IF
        !           390:       END IF
        !           391: *
        !           392: *     Scale the right hand side.
        !           393: *
        !           394:       IF( NOTRAN ) THEN
        !           395:          IF( ROWEQU ) THEN
        !           396:             DO 40 J = 1, NRHS
        !           397:                DO 30 I = 1, N
        !           398:                   B( I, J ) = R( I )*B( I, J )
        !           399:    30          CONTINUE
        !           400:    40       CONTINUE
        !           401:          END IF
        !           402:       ELSE IF( COLEQU ) THEN
        !           403:          DO 60 J = 1, NRHS
        !           404:             DO 50 I = 1, N
        !           405:                B( I, J ) = C( I )*B( I, J )
        !           406:    50       CONTINUE
        !           407:    60    CONTINUE
        !           408:       END IF
        !           409: *
        !           410:       IF( NOFACT .OR. EQUIL ) THEN
        !           411: *
        !           412: *        Compute the LU factorization of the band matrix A.
        !           413: *
        !           414:          DO 70 J = 1, N
        !           415:             J1 = MAX( J-KU, 1 )
        !           416:             J2 = MIN( J+KL, N )
        !           417:             CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
        !           418:      $                  AFB( KL+KU+1-J+J1, J ), 1 )
        !           419:    70    CONTINUE
        !           420: *
        !           421:          CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
        !           422: *
        !           423: *        Return if INFO is non-zero.
        !           424: *
        !           425:          IF( INFO.GT.0 ) THEN
        !           426: *
        !           427: *           Compute the reciprocal pivot growth factor of the
        !           428: *           leading rank-deficient INFO columns of A.
        !           429: *
        !           430:             ANORM = ZERO
        !           431:             DO 90 J = 1, INFO
        !           432:                DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
        !           433:                   ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
        !           434:    80          CONTINUE
        !           435:    90       CONTINUE
        !           436:             RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
        !           437:      $                       AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
        !           438:      $                       RWORK )
        !           439:             IF( RPVGRW.EQ.ZERO ) THEN
        !           440:                RPVGRW = ONE
        !           441:             ELSE
        !           442:                RPVGRW = ANORM / RPVGRW
        !           443:             END IF
        !           444:             RWORK( 1 ) = RPVGRW
        !           445:             RCOND = ZERO
        !           446:             RETURN
        !           447:          END IF
        !           448:       END IF
        !           449: *
        !           450: *     Compute the norm of the matrix A and the
        !           451: *     reciprocal pivot growth factor RPVGRW.
        !           452: *
        !           453:       IF( NOTRAN ) THEN
        !           454:          NORM = '1'
        !           455:       ELSE
        !           456:          NORM = 'I'
        !           457:       END IF
        !           458:       ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
        !           459:       RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
        !           460:       IF( RPVGRW.EQ.ZERO ) THEN
        !           461:          RPVGRW = ONE
        !           462:       ELSE
        !           463:          RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
        !           464:       END IF
        !           465: *
        !           466: *     Compute the reciprocal of the condition number of A.
        !           467: *
        !           468:       CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
        !           469:      $             WORK, RWORK, INFO )
        !           470: *
        !           471: *     Compute the solution matrix X.
        !           472: *
        !           473:       CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
        !           474:       CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
        !           475:      $             INFO )
        !           476: *
        !           477: *     Use iterative refinement to improve the computed solution and
        !           478: *     compute error bounds and backward error estimates for it.
        !           479: *
        !           480:       CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
        !           481:      $             B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
        !           482: *
        !           483: *     Transform the solution matrix X to a solution of the original
        !           484: *     system.
        !           485: *
        !           486:       IF( NOTRAN ) THEN
        !           487:          IF( COLEQU ) THEN
        !           488:             DO 110 J = 1, NRHS
        !           489:                DO 100 I = 1, N
        !           490:                   X( I, J ) = C( I )*X( I, J )
        !           491:   100          CONTINUE
        !           492:   110       CONTINUE
        !           493:             DO 120 J = 1, NRHS
        !           494:                FERR( J ) = FERR( J ) / COLCND
        !           495:   120       CONTINUE
        !           496:          END IF
        !           497:       ELSE IF( ROWEQU ) THEN
        !           498:          DO 140 J = 1, NRHS
        !           499:             DO 130 I = 1, N
        !           500:                X( I, J ) = R( I )*X( I, J )
        !           501:   130       CONTINUE
        !           502:   140    CONTINUE
        !           503:          DO 150 J = 1, NRHS
        !           504:             FERR( J ) = FERR( J ) / ROWCND
        !           505:   150    CONTINUE
        !           506:       END IF
        !           507: *
        !           508: *     Set INFO = N+1 if the matrix is singular to working precision.
        !           509: *
        !           510:       IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
        !           511:      $   INFO = N + 1
        !           512: *
        !           513:       RWORK( 1 ) = RPVGRW
        !           514:       RETURN
        !           515: *
        !           516: *     End of ZGBSVX
        !           517: *
        !           518:       END

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