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    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: *> \date April 2012
  364: *
  365: *> \ingroup doubleGBsolve
  366: *
  367: *  =====================================================================
  368:       SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
  369:      $                   LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
  370:      $                   RCOND, FERR, BERR, WORK, IWORK, INFO )
  371: *
  372: *  -- LAPACK driver routine (version 3.4.1) --
  373: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  374: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  375: *     April 2012
  376: *
  377: *     .. Scalar Arguments ..
  378:       CHARACTER          EQUED, FACT, TRANS
  379:       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
  380:       DOUBLE PRECISION   RCOND
  381: *     ..
  382: *     .. Array Arguments ..
  383:       INTEGER            IPIV( * ), IWORK( * )
  384:       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
  385:      $                   BERR( * ), C( * ), FERR( * ), R( * ),
  386:      $                   WORK( * ), X( LDX, * )
  387: *     ..
  388: *
  389: *  =====================================================================
  390: *
  391: *     .. Parameters ..
  392:       DOUBLE PRECISION   ZERO, ONE
  393:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
  394: *     ..
  395: *     .. Local Scalars ..
  396:       LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
  397:       CHARACTER          NORM
  398:       INTEGER            I, INFEQU, J, J1, J2
  399:       DOUBLE PRECISION   AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
  400:      $                   ROWCND, RPVGRW, SMLNUM
  401: *     ..
  402: *     .. External Functions ..
  403:       LOGICAL            LSAME
  404:       DOUBLE PRECISION   DLAMCH, DLANGB, DLANTB
  405:       EXTERNAL           LSAME, DLAMCH, DLANGB, DLANTB
  406: *     ..
  407: *     .. External Subroutines ..
  408:       EXTERNAL           DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS,
  409:      $                   DLACPY, DLAQGB, XERBLA
  410: *     ..
  411: *     .. Intrinsic Functions ..
  412:       INTRINSIC          ABS, MAX, MIN
  413: *     ..
  414: *     .. Executable Statements ..
  415: *
  416:       INFO = 0
  417:       NOFACT = LSAME( FACT, 'N' )
  418:       EQUIL = LSAME( FACT, 'E' )
  419:       NOTRAN = LSAME( TRANS, 'N' )
  420:       IF( NOFACT .OR. EQUIL ) THEN
  421:          EQUED = 'N'
  422:          ROWEQU = .FALSE.
  423:          COLEQU = .FALSE.
  424:       ELSE
  425:          ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
  426:          COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
  427:          SMLNUM = DLAMCH( 'Safe minimum' )
  428:          BIGNUM = ONE / SMLNUM
  429:       END IF
  430: *
  431: *     Test the input parameters.
  432: *
  433:       IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
  434:      $     THEN
  435:          INFO = -1
  436:       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
  437:      $         LSAME( TRANS, 'C' ) ) THEN
  438:          INFO = -2
  439:       ELSE IF( N.LT.0 ) THEN
  440:          INFO = -3
  441:       ELSE IF( KL.LT.0 ) THEN
  442:          INFO = -4
  443:       ELSE IF( KU.LT.0 ) THEN
  444:          INFO = -5
  445:       ELSE IF( NRHS.LT.0 ) THEN
  446:          INFO = -6
  447:       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
  448:          INFO = -8
  449:       ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
  450:          INFO = -10
  451:       ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
  452:      $         ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
  453:          INFO = -12
  454:       ELSE
  455:          IF( ROWEQU ) THEN
  456:             RCMIN = BIGNUM
  457:             RCMAX = ZERO
  458:             DO 10 J = 1, N
  459:                RCMIN = MIN( RCMIN, R( J ) )
  460:                RCMAX = MAX( RCMAX, R( J ) )
  461:    10       CONTINUE
  462:             IF( RCMIN.LE.ZERO ) THEN
  463:                INFO = -13
  464:             ELSE IF( N.GT.0 ) THEN
  465:                ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
  466:             ELSE
  467:                ROWCND = ONE
  468:             END IF
  469:          END IF
  470:          IF( COLEQU .AND. INFO.EQ.0 ) THEN
  471:             RCMIN = BIGNUM
  472:             RCMAX = ZERO
  473:             DO 20 J = 1, N
  474:                RCMIN = MIN( RCMIN, C( J ) )
  475:                RCMAX = MAX( RCMAX, C( J ) )
  476:    20       CONTINUE
  477:             IF( RCMIN.LE.ZERO ) THEN
  478:                INFO = -14
  479:             ELSE IF( N.GT.0 ) THEN
  480:                COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
  481:             ELSE
  482:                COLCND = ONE
  483:             END IF
  484:          END IF
  485:          IF( INFO.EQ.0 ) THEN
  486:             IF( LDB.LT.MAX( 1, N ) ) THEN
  487:                INFO = -16
  488:             ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
  489:                INFO = -18
  490:             END IF
  491:          END IF
  492:       END IF
  493: *
  494:       IF( INFO.NE.0 ) THEN
  495:          CALL XERBLA( 'DGBSVX', -INFO )
  496:          RETURN
  497:       END IF
  498: *
  499:       IF( EQUIL ) THEN
  500: *
  501: *        Compute row and column scalings to equilibrate the matrix A.
  502: *
  503:          CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
  504:      $                AMAX, INFEQU )
  505:          IF( INFEQU.EQ.0 ) THEN
  506: *
  507: *           Equilibrate the matrix.
  508: *
  509:             CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
  510:      $                   AMAX, EQUED )
  511:             ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
  512:             COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
  513:          END IF
  514:       END IF
  515: *
  516: *     Scale the right hand side.
  517: *
  518:       IF( NOTRAN ) THEN
  519:          IF( ROWEQU ) THEN
  520:             DO 40 J = 1, NRHS
  521:                DO 30 I = 1, N
  522:                   B( I, J ) = R( I )*B( I, J )
  523:    30          CONTINUE
  524:    40       CONTINUE
  525:          END IF
  526:       ELSE IF( COLEQU ) THEN
  527:          DO 60 J = 1, NRHS
  528:             DO 50 I = 1, N
  529:                B( I, J ) = C( I )*B( I, J )
  530:    50       CONTINUE
  531:    60    CONTINUE
  532:       END IF
  533: *
  534:       IF( NOFACT .OR. EQUIL ) THEN
  535: *
  536: *        Compute the LU factorization of the band matrix A.
  537: *
  538:          DO 70 J = 1, N
  539:             J1 = MAX( J-KU, 1 )
  540:             J2 = MIN( J+KL, N )
  541:             CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
  542:      $                  AFB( KL+KU+1-J+J1, J ), 1 )
  543:    70    CONTINUE
  544: *
  545:          CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
  546: *
  547: *        Return if INFO is non-zero.
  548: *
  549:          IF( INFO.GT.0 ) THEN
  550: *
  551: *           Compute the reciprocal pivot growth factor of the
  552: *           leading rank-deficient INFO columns of A.
  553: *
  554:             ANORM = ZERO
  555:             DO 90 J = 1, INFO
  556:                DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
  557:                   ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
  558:    80          CONTINUE
  559:    90       CONTINUE
  560:             RPVGRW = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
  561:      $                       AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
  562:      $                       WORK )
  563:             IF( RPVGRW.EQ.ZERO ) THEN
  564:                RPVGRW = ONE
  565:             ELSE
  566:                RPVGRW = ANORM / RPVGRW
  567:             END IF
  568:             WORK( 1 ) = RPVGRW
  569:             RCOND = ZERO
  570:             RETURN
  571:          END IF
  572:       END IF
  573: *
  574: *     Compute the norm of the matrix A and the
  575: *     reciprocal pivot growth factor RPVGRW.
  576: *
  577:       IF( NOTRAN ) THEN
  578:          NORM = '1'
  579:       ELSE
  580:          NORM = 'I'
  581:       END IF
  582:       ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
  583:       RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
  584:       IF( RPVGRW.EQ.ZERO ) THEN
  585:          RPVGRW = ONE
  586:       ELSE
  587:          RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
  588:       END IF
  589: *
  590: *     Compute the reciprocal of the condition number of A.
  591: *
  592:       CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
  593:      $             WORK, IWORK, INFO )
  594: *
  595: *     Compute the solution matrix X.
  596: *
  597:       CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
  598:       CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
  599:      $             INFO )
  600: *
  601: *     Use iterative refinement to improve the computed solution and
  602: *     compute error bounds and backward error estimates for it.
  603: *
  604:       CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
  605:      $             B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
  606: *
  607: *     Transform the solution matrix X to a solution of the original
  608: *     system.
  609: *
  610:       IF( NOTRAN ) THEN
  611:          IF( COLEQU ) THEN
  612:             DO 110 J = 1, NRHS
  613:                DO 100 I = 1, N
  614:                   X( I, J ) = C( I )*X( I, J )
  615:   100          CONTINUE
  616:   110       CONTINUE
  617:             DO 120 J = 1, NRHS
  618:                FERR( J ) = FERR( J ) / COLCND
  619:   120       CONTINUE
  620:          END IF
  621:       ELSE IF( ROWEQU ) THEN
  622:          DO 140 J = 1, NRHS
  623:             DO 130 I = 1, N
  624:                X( I, J ) = R( I )*X( I, J )
  625:   130       CONTINUE
  626:   140    CONTINUE
  627:          DO 150 J = 1, NRHS
  628:             FERR( J ) = FERR( J ) / ROWCND
  629:   150    CONTINUE
  630:       END IF
  631: *
  632: *     Set INFO = N+1 if the matrix is singular to working precision.
  633: *
  634:       IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
  635:      $   INFO = N + 1
  636: *
  637:       WORK( 1 ) = RPVGRW
  638:       RETURN
  639: *
  640: *     End of DGBSVX
  641: *
  642:       END