Annotation of rpl/lapack/lapack/zgbrfsx.f, revision 1.17

1.5       bertrand    1: *> \brief \b ZGBRFSX
                      2: *
                      3: *  =========== DOCUMENTATION ===========
                      4: *
1.13      bertrand    5: * Online html documentation available at
                      6: *            http://www.netlib.org/lapack/explore-html/
1.5       bertrand    7: *
                      8: *> \htmlonly
1.13      bertrand    9: *> Download ZGBRFSX + dependencies
                     10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbrfsx.f">
                     11: *> [TGZ]</a>
                     12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbrfsx.f">
                     13: *> [ZIP]</a>
                     14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbrfsx.f">
1.5       bertrand   15: *> [TXT]</a>
1.13      bertrand   16: *> \endhtmlonly
1.5       bertrand   17: *
                     18: *  Definition:
                     19: *  ===========
                     20: *
                     21: *       SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
                     22: *                           LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND,
                     23: *                           BERR, N_ERR_BNDS, ERR_BNDS_NORM,
                     24: *                           ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
                     25: *                           INFO )
1.13      bertrand   26: *
1.5       bertrand   27: *       .. Scalar Arguments ..
                     28: *       CHARACTER          TRANS, EQUED
                     29: *       INTEGER            INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
                     30: *      $                   NPARAMS, N_ERR_BNDS
                     31: *       DOUBLE PRECISION   RCOND
                     32: *       ..
                     33: *       .. Array Arguments ..
                     34: *       INTEGER            IPIV( * )
                     35: *       COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
                     36: *      $                   X( LDX , * ),WORK( * )
                     37: *       DOUBLE PRECISION   R( * ), C( * ), PARAMS( * ), BERR( * ),
                     38: *      $                   ERR_BNDS_NORM( NRHS, * ),
                     39: *      $                   ERR_BNDS_COMP( NRHS, * ), RWORK( * )
                     40: *       ..
1.13      bertrand   41: *
1.5       bertrand   42: *
                     43: *> \par Purpose:
                     44: *  =============
                     45: *>
                     46: *> \verbatim
                     47: *>
                     48: *>    ZGBRFSX improves the computed solution to a system of linear
                     49: *>    equations and provides error bounds and backward error estimates
                     50: *>    for the solution.  In addition to normwise error bound, the code
                     51: *>    provides maximum componentwise error bound if possible.  See
                     52: *>    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
                     53: *>    error bounds.
                     54: *>
                     55: *>    The original system of linear equations may have been equilibrated
                     56: *>    before calling this routine, as described by arguments EQUED, R
                     57: *>    and C below. In this case, the solution and error bounds returned
                     58: *>    are for the original unequilibrated system.
                     59: *> \endverbatim
                     60: *
                     61: *  Arguments:
                     62: *  ==========
                     63: *
                     64: *> \verbatim
                     65: *>     Some optional parameters are bundled in the PARAMS array.  These
                     66: *>     settings determine how refinement is performed, but often the
                     67: *>     defaults are acceptable.  If the defaults are acceptable, users
                     68: *>     can pass NPARAMS = 0 which prevents the source code from accessing
                     69: *>     the PARAMS argument.
                     70: *> \endverbatim
                     71: *>
                     72: *> \param[in] TRANS
                     73: *> \verbatim
                     74: *>          TRANS is CHARACTER*1
                     75: *>     Specifies the form of the system of equations:
                     76: *>       = 'N':  A * X = B     (No transpose)
                     77: *>       = 'T':  A**T * X = B  (Transpose)
1.16      bertrand   78: *>       = 'C':  A**H * X = B  (Conjugate transpose)
1.5       bertrand   79: *> \endverbatim
                     80: *>
                     81: *> \param[in] EQUED
                     82: *> \verbatim
                     83: *>          EQUED is CHARACTER*1
                     84: *>     Specifies the form of equilibration that was done to A
                     85: *>     before calling this routine. This is needed to compute
                     86: *>     the solution and error bounds correctly.
                     87: *>       = 'N':  No equilibration
                     88: *>       = 'R':  Row equilibration, i.e., A has been premultiplied by
                     89: *>               diag(R).
                     90: *>       = 'C':  Column equilibration, i.e., A has been postmultiplied
                     91: *>               by diag(C).
                     92: *>       = 'B':  Both row and column equilibration, i.e., A has been
                     93: *>               replaced by diag(R) * A * diag(C).
                     94: *>               The right hand side B has been changed accordingly.
                     95: *> \endverbatim
                     96: *>
                     97: *> \param[in] N
                     98: *> \verbatim
                     99: *>          N is INTEGER
                    100: *>     The order of the matrix A.  N >= 0.
                    101: *> \endverbatim
                    102: *>
                    103: *> \param[in] KL
                    104: *> \verbatim
                    105: *>          KL is INTEGER
                    106: *>     The number of subdiagonals within the band of A.  KL >= 0.
                    107: *> \endverbatim
                    108: *>
                    109: *> \param[in] KU
                    110: *> \verbatim
                    111: *>          KU is INTEGER
                    112: *>     The number of superdiagonals within the band of A.  KU >= 0.
                    113: *> \endverbatim
                    114: *>
                    115: *> \param[in] NRHS
                    116: *> \verbatim
                    117: *>          NRHS is INTEGER
                    118: *>     The number of right hand sides, i.e., the number of columns
                    119: *>     of the matrices B and X.  NRHS >= 0.
                    120: *> \endverbatim
                    121: *>
                    122: *> \param[in] AB
                    123: *> \verbatim
1.7       bertrand  124: *>          AB is COMPLEX*16 array, dimension (LDAB,N)
1.5       bertrand  125: *>     The original band matrix A, stored in rows 1 to KL+KU+1.
                    126: *>     The j-th column of A is stored in the j-th column of the
                    127: *>     array AB as follows:
                    128: *>     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
                    129: *> \endverbatim
                    130: *>
                    131: *> \param[in] LDAB
                    132: *> \verbatim
                    133: *>          LDAB is INTEGER
                    134: *>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
                    135: *> \endverbatim
                    136: *>
                    137: *> \param[in] AFB
                    138: *> \verbatim
1.7       bertrand  139: *>          AFB is COMPLEX*16 array, dimension (LDAFB,N)
1.5       bertrand  140: *>     Details of the LU factorization of the band matrix A, as
1.17    ! bertrand  141: *>     computed by ZGBTRF.  U is stored as an upper triangular band
1.5       bertrand  142: *>     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
                    143: *>     the multipliers used during the factorization are stored in
                    144: *>     rows KL+KU+2 to 2*KL+KU+1.
                    145: *> \endverbatim
                    146: *>
                    147: *> \param[in] LDAFB
                    148: *> \verbatim
                    149: *>          LDAFB is INTEGER
                    150: *>     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
                    151: *> \endverbatim
                    152: *>
                    153: *> \param[in] IPIV
                    154: *> \verbatim
                    155: *>          IPIV is INTEGER array, dimension (N)
1.17    ! bertrand  156: *>     The pivot indices from ZGETRF; for 1<=i<=N, row i of the
1.5       bertrand  157: *>     matrix was interchanged with row IPIV(i).
                    158: *> \endverbatim
                    159: *>
                    160: *> \param[in,out] R
                    161: *> \verbatim
1.7       bertrand  162: *>          R is DOUBLE PRECISION array, dimension (N)
1.5       bertrand  163: *>     The row scale factors for A.  If EQUED = 'R' or 'B', A is
                    164: *>     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                    165: *>     is not accessed.  R is an input argument if FACT = 'F';
                    166: *>     otherwise, R is an output argument.  If FACT = 'F' and
                    167: *>     EQUED = 'R' or 'B', each element of R must be positive.
                    168: *>     If R is output, each element of R is a power of the radix.
                    169: *>     If R is input, each element of R should be a power of the radix
                    170: *>     to ensure a reliable solution and error estimates. Scaling by
                    171: *>     powers of the radix does not cause rounding errors unless the
                    172: *>     result underflows or overflows. Rounding errors during scaling
                    173: *>     lead to refining with a matrix that is not equivalent to the
                    174: *>     input matrix, producing error estimates that may not be
                    175: *>     reliable.
                    176: *> \endverbatim
                    177: *>
                    178: *> \param[in,out] C
                    179: *> \verbatim
1.7       bertrand  180: *>          C is DOUBLE PRECISION array, dimension (N)
1.5       bertrand  181: *>     The column scale factors for A.  If EQUED = 'C' or 'B', A is
                    182: *>     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                    183: *>     is not accessed.  C is an input argument if FACT = 'F';
                    184: *>     otherwise, C is an output argument.  If FACT = 'F' and
                    185: *>     EQUED = 'C' or 'B', each element of C must be positive.
                    186: *>     If C is output, each element of C is a power of the radix.
                    187: *>     If C is input, each element of C should be a power of the radix
                    188: *>     to ensure a reliable solution and error estimates. Scaling by
                    189: *>     powers of the radix does not cause rounding errors unless the
                    190: *>     result underflows or overflows. Rounding errors during scaling
                    191: *>     lead to refining with a matrix that is not equivalent to the
                    192: *>     input matrix, producing error estimates that may not be
                    193: *>     reliable.
                    194: *> \endverbatim
                    195: *>
                    196: *> \param[in] B
                    197: *> \verbatim
1.11      bertrand  198: *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
1.5       bertrand  199: *>     The right hand side matrix B.
                    200: *> \endverbatim
                    201: *>
                    202: *> \param[in] LDB
                    203: *> \verbatim
                    204: *>          LDB is INTEGER
                    205: *>     The leading dimension of the array B.  LDB >= max(1,N).
                    206: *> \endverbatim
                    207: *>
                    208: *> \param[in,out] X
                    209: *> \verbatim
1.7       bertrand  210: *>          X is COMPLEX*16 array, dimension (LDX,NRHS)
1.17    ! bertrand  211: *>     On entry, the solution matrix X, as computed by ZGETRS.
1.5       bertrand  212: *>     On exit, the improved solution matrix X.
                    213: *> \endverbatim
                    214: *>
                    215: *> \param[in] LDX
                    216: *> \verbatim
                    217: *>          LDX is INTEGER
                    218: *>     The leading dimension of the array X.  LDX >= max(1,N).
                    219: *> \endverbatim
                    220: *>
                    221: *> \param[out] RCOND
                    222: *> \verbatim
                    223: *>          RCOND is DOUBLE PRECISION
                    224: *>     Reciprocal scaled condition number.  This is an estimate of the
                    225: *>     reciprocal Skeel condition number of the matrix A after
                    226: *>     equilibration (if done).  If this is less than the machine
                    227: *>     precision (in particular, if it is zero), the matrix is singular
                    228: *>     to working precision.  Note that the error may still be small even
                    229: *>     if this number is very small and the matrix appears ill-
                    230: *>     conditioned.
                    231: *> \endverbatim
                    232: *>
                    233: *> \param[out] BERR
                    234: *> \verbatim
1.11      bertrand  235: *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
1.5       bertrand  236: *>     Componentwise relative backward error.  This is the
                    237: *>     componentwise relative backward error of each solution vector X(j)
                    238: *>     (i.e., the smallest relative change in any element of A or B that
                    239: *>     makes X(j) an exact solution).
                    240: *> \endverbatim
                    241: *>
                    242: *> \param[in] N_ERR_BNDS
                    243: *> \verbatim
                    244: *>          N_ERR_BNDS is INTEGER
                    245: *>     Number of error bounds to return for each right hand side
                    246: *>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
                    247: *>     ERR_BNDS_COMP below.
                    248: *> \endverbatim
                    249: *>
                    250: *> \param[out] ERR_BNDS_NORM
                    251: *> \verbatim
                    252: *>          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                    253: *>     For each right-hand side, this array contains information about
                    254: *>     various error bounds and condition numbers corresponding to the
                    255: *>     normwise relative error, which is defined as follows:
                    256: *>
                    257: *>     Normwise relative error in the ith solution vector:
                    258: *>             max_j (abs(XTRUE(j,i) - X(j,i)))
                    259: *>            ------------------------------
                    260: *>                  max_j abs(X(j,i))
                    261: *>
                    262: *>     The array is indexed by the type of error information as described
                    263: *>     below. There currently are up to three pieces of information
                    264: *>     returned.
                    265: *>
                    266: *>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                    267: *>     right-hand side.
                    268: *>
                    269: *>     The second index in ERR_BNDS_NORM(:,err) contains the following
                    270: *>     three fields:
                    271: *>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
                    272: *>              reciprocal condition number is less than the threshold
                    273: *>              sqrt(n) * dlamch('Epsilon').
                    274: *>
                    275: *>     err = 2 "Guaranteed" error bound: The estimated forward error,
                    276: *>              almost certainly within a factor of 10 of the true error
                    277: *>              so long as the next entry is greater than the threshold
                    278: *>              sqrt(n) * dlamch('Epsilon'). This error bound should only
                    279: *>              be trusted if the previous boolean is true.
                    280: *>
                    281: *>     err = 3  Reciprocal condition number: Estimated normwise
                    282: *>              reciprocal condition number.  Compared with the threshold
                    283: *>              sqrt(n) * dlamch('Epsilon') to determine if the error
                    284: *>              estimate is "guaranteed". These reciprocal condition
                    285: *>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                    286: *>              appropriately scaled matrix Z.
                    287: *>              Let Z = S*A, where S scales each row by a power of the
                    288: *>              radix so all absolute row sums of Z are approximately 1.
                    289: *>
                    290: *>     See Lapack Working Note 165 for further details and extra
                    291: *>     cautions.
                    292: *> \endverbatim
                    293: *>
                    294: *> \param[out] ERR_BNDS_COMP
                    295: *> \verbatim
                    296: *>          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                    297: *>     For each right-hand side, this array contains information about
                    298: *>     various error bounds and condition numbers corresponding to the
                    299: *>     componentwise relative error, which is defined as follows:
                    300: *>
                    301: *>     Componentwise relative error in the ith solution vector:
                    302: *>                    abs(XTRUE(j,i) - X(j,i))
                    303: *>             max_j ----------------------
                    304: *>                         abs(X(j,i))
                    305: *>
                    306: *>     The array is indexed by the right-hand side i (on which the
                    307: *>     componentwise relative error depends), and the type of error
                    308: *>     information as described below. There currently are up to three
                    309: *>     pieces of information returned for each right-hand side. If
                    310: *>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
1.16      bertrand  311: *>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
1.5       bertrand  312: *>     the first (:,N_ERR_BNDS) entries are returned.
                    313: *>
                    314: *>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                    315: *>     right-hand side.
                    316: *>
                    317: *>     The second index in ERR_BNDS_COMP(:,err) contains the following
                    318: *>     three fields:
                    319: *>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
                    320: *>              reciprocal condition number is less than the threshold
                    321: *>              sqrt(n) * dlamch('Epsilon').
                    322: *>
                    323: *>     err = 2 "Guaranteed" error bound: The estimated forward error,
                    324: *>              almost certainly within a factor of 10 of the true error
                    325: *>              so long as the next entry is greater than the threshold
                    326: *>              sqrt(n) * dlamch('Epsilon'). This error bound should only
                    327: *>              be trusted if the previous boolean is true.
                    328: *>
                    329: *>     err = 3  Reciprocal condition number: Estimated componentwise
                    330: *>              reciprocal condition number.  Compared with the threshold
                    331: *>              sqrt(n) * dlamch('Epsilon') to determine if the error
                    332: *>              estimate is "guaranteed". These reciprocal condition
                    333: *>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                    334: *>              appropriately scaled matrix Z.
                    335: *>              Let Z = S*(A*diag(x)), where x is the solution for the
                    336: *>              current right-hand side and S scales each row of
                    337: *>              A*diag(x) by a power of the radix so all absolute row
                    338: *>              sums of Z are approximately 1.
                    339: *>
                    340: *>     See Lapack Working Note 165 for further details and extra
                    341: *>     cautions.
                    342: *> \endverbatim
                    343: *>
                    344: *> \param[in] NPARAMS
                    345: *> \verbatim
                    346: *>          NPARAMS is INTEGER
1.16      bertrand  347: *>     Specifies the number of parameters set in PARAMS.  If <= 0, the
1.5       bertrand  348: *>     PARAMS array is never referenced and default values are used.
                    349: *> \endverbatim
                    350: *>
                    351: *> \param[in,out] PARAMS
                    352: *> \verbatim
1.7       bertrand  353: *>          PARAMS is DOUBLE PRECISION array, dimension NPARAMS
1.16      bertrand  354: *>     Specifies algorithm parameters.  If an entry is < 0.0, then
1.5       bertrand  355: *>     that entry will be filled with default value used for that
                    356: *>     parameter.  Only positions up to NPARAMS are accessed; defaults
                    357: *>     are used for higher-numbered parameters.
                    358: *>
                    359: *>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                    360: *>            refinement or not.
                    361: *>         Default: 1.0D+0
1.16      bertrand  362: *>            = 0.0:  No refinement is performed, and no error bounds are
1.5       bertrand  363: *>                    computed.
1.16      bertrand  364: *>            = 1.0:  Use the double-precision refinement algorithm,
1.5       bertrand  365: *>                    possibly with doubled-single computations if the
                    366: *>                    compilation environment does not support DOUBLE
                    367: *>                    PRECISION.
                    368: *>              (other values are reserved for future use)
                    369: *>
                    370: *>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                    371: *>            computations allowed for refinement.
                    372: *>         Default: 10
                    373: *>         Aggressive: Set to 100 to permit convergence using approximate
                    374: *>                     factorizations or factorizations other than LU. If
                    375: *>                     the factorization uses a technique other than
                    376: *>                     Gaussian elimination, the guarantees in
                    377: *>                     err_bnds_norm and err_bnds_comp may no longer be
                    378: *>                     trustworthy.
                    379: *>
                    380: *>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                    381: *>            will attempt to find a solution with small componentwise
                    382: *>            relative error in the double-precision algorithm.  Positive
                    383: *>            is true, 0.0 is false.
                    384: *>         Default: 1.0 (attempt componentwise convergence)
                    385: *> \endverbatim
                    386: *>
                    387: *> \param[out] WORK
                    388: *> \verbatim
                    389: *>          WORK is COMPLEX*16 array, dimension (2*N)
                    390: *> \endverbatim
                    391: *>
                    392: *> \param[out] RWORK
                    393: *> \verbatim
                    394: *>          RWORK is DOUBLE PRECISION array, dimension (2*N)
                    395: *> \endverbatim
                    396: *>
                    397: *> \param[out] INFO
                    398: *> \verbatim
                    399: *>          INFO is INTEGER
                    400: *>       = 0:  Successful exit. The solution to every right-hand side is
                    401: *>         guaranteed.
                    402: *>       < 0:  If INFO = -i, the i-th argument had an illegal value
                    403: *>       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
                    404: *>         has been completed, but the factor U is exactly singular, so
                    405: *>         the solution and error bounds could not be computed. RCOND = 0
                    406: *>         is returned.
                    407: *>       = N+J: The solution corresponding to the Jth right-hand side is
                    408: *>         not guaranteed. The solutions corresponding to other right-
                    409: *>         hand sides K with K > J may not be guaranteed as well, but
                    410: *>         only the first such right-hand side is reported. If a small
                    411: *>         componentwise error is not requested (PARAMS(3) = 0.0) then
                    412: *>         the Jth right-hand side is the first with a normwise error
                    413: *>         bound that is not guaranteed (the smallest J such
                    414: *>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
                    415: *>         the Jth right-hand side is the first with either a normwise or
                    416: *>         componentwise error bound that is not guaranteed (the smallest
                    417: *>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
                    418: *>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
                    419: *>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
                    420: *>         about all of the right-hand sides check ERR_BNDS_NORM or
                    421: *>         ERR_BNDS_COMP.
                    422: *> \endverbatim
                    423: *
                    424: *  Authors:
                    425: *  ========
                    426: *
1.13      bertrand  427: *> \author Univ. of Tennessee
                    428: *> \author Univ. of California Berkeley
                    429: *> \author Univ. of Colorado Denver
                    430: *> \author NAG Ltd.
1.5       bertrand  431: *
                    432: *> \ingroup complex16GBcomputational
                    433: *
                    434: *  =====================================================================
1.1       bertrand  435:       SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
                    436:      $                    LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND,
                    437:      $                    BERR, N_ERR_BNDS, ERR_BNDS_NORM,
                    438:      $                    ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
                    439:      $                    INFO )
                    440: *
1.17    ! bertrand  441: *  -- LAPACK computational routine --
1.5       bertrand  442: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    443: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.1       bertrand  444: *
                    445: *     .. Scalar Arguments ..
                    446:       CHARACTER          TRANS, EQUED
                    447:       INTEGER            INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
                    448:      $                   NPARAMS, N_ERR_BNDS
                    449:       DOUBLE PRECISION   RCOND
                    450: *     ..
                    451: *     .. Array Arguments ..
                    452:       INTEGER            IPIV( * )
                    453:       COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
                    454:      $                   X( LDX , * ),WORK( * )
                    455:       DOUBLE PRECISION   R( * ), C( * ), PARAMS( * ), BERR( * ),
                    456:      $                   ERR_BNDS_NORM( NRHS, * ),
                    457:      $                   ERR_BNDS_COMP( NRHS, * ), RWORK( * )
                    458: *     ..
                    459: *
1.5       bertrand  460: *  ==================================================================
1.1       bertrand  461: *
                    462: *     .. Parameters ..
                    463:       DOUBLE PRECISION   ZERO, ONE
                    464:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
                    465:       DOUBLE PRECISION   ITREF_DEFAULT, ITHRESH_DEFAULT
                    466:       DOUBLE PRECISION   COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT
                    467:       DOUBLE PRECISION   DZTHRESH_DEFAULT
                    468:       PARAMETER          ( ITREF_DEFAULT = 1.0D+0 )
                    469:       PARAMETER          ( ITHRESH_DEFAULT = 10.0D+0 )
                    470:       PARAMETER          ( COMPONENTWISE_DEFAULT = 1.0D+0 )
                    471:       PARAMETER          ( RTHRESH_DEFAULT = 0.5D+0 )
                    472:       PARAMETER          ( DZTHRESH_DEFAULT = 0.25D+0 )
                    473:       INTEGER            LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
                    474:      $                   LA_LINRX_CWISE_I
                    475:       PARAMETER          ( LA_LINRX_ITREF_I = 1,
                    476:      $                   LA_LINRX_ITHRESH_I = 2 )
                    477:       PARAMETER          ( LA_LINRX_CWISE_I = 3 )
                    478:       INTEGER            LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
                    479:      $                   LA_LINRX_RCOND_I
                    480:       PARAMETER          ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
                    481:       PARAMETER          ( LA_LINRX_RCOND_I = 3 )
                    482: *     ..
                    483: *     .. Local Scalars ..
                    484:       CHARACTER(1)       NORM
                    485:       LOGICAL            ROWEQU, COLEQU, NOTRAN, IGNORE_CWISE
                    486:       INTEGER            J, TRANS_TYPE, PREC_TYPE, REF_TYPE, N_NORMS,
                    487:      $                   ITHRESH
                    488:       DOUBLE PRECISION   ANORM, RCOND_TMP, ILLRCOND_THRESH, ERR_LBND,
                    489:      $                   CWISE_WRONG, RTHRESH, UNSTABLE_THRESH
                    490: *     ..
                    491: *     .. External Subroutines ..
                    492:       EXTERNAL           XERBLA, ZGBCON, ZLA_GBRFSX_EXTENDED
                    493: *     ..
                    494: *     .. Intrinsic Functions ..
                    495:       INTRINSIC          MAX, SQRT, TRANSFER
                    496: *     ..
                    497: *     .. External Functions ..
1.13      bertrand  498:       EXTERNAL           LSAME, ILAPREC
1.1       bertrand  499:       EXTERNAL           DLAMCH, ZLANGB, ZLA_GBRCOND_X, ZLA_GBRCOND_C
                    500:       DOUBLE PRECISION   DLAMCH, ZLANGB, ZLA_GBRCOND_X, ZLA_GBRCOND_C
                    501:       LOGICAL            LSAME
                    502:       INTEGER            ILATRANS, ILAPREC
                    503: *     ..
                    504: *     .. Executable Statements ..
                    505: *
                    506: *     Check the input parameters.
                    507: *
                    508:       INFO = 0
                    509:       TRANS_TYPE = ILATRANS( TRANS )
                    510:       REF_TYPE = INT( ITREF_DEFAULT )
                    511:       IF ( NPARAMS .GE. LA_LINRX_ITREF_I ) THEN
                    512:          IF ( PARAMS( LA_LINRX_ITREF_I ) .LT. 0.0D+0 ) THEN
                    513:             PARAMS( LA_LINRX_ITREF_I ) = ITREF_DEFAULT
                    514:          ELSE
                    515:             REF_TYPE = PARAMS( LA_LINRX_ITREF_I )
                    516:          END IF
                    517:       END IF
                    518: *
                    519: *     Set default parameters.
                    520: *
                    521:       ILLRCOND_THRESH = DBLE( N ) * DLAMCH( 'Epsilon' )
                    522:       ITHRESH = INT( ITHRESH_DEFAULT )
                    523:       RTHRESH = RTHRESH_DEFAULT
                    524:       UNSTABLE_THRESH = DZTHRESH_DEFAULT
                    525:       IGNORE_CWISE = COMPONENTWISE_DEFAULT .EQ. 0.0D+0
                    526: *
                    527:       IF ( NPARAMS.GE.LA_LINRX_ITHRESH_I ) THEN
                    528:          IF ( PARAMS( LA_LINRX_ITHRESH_I ).LT.0.0D+0 ) THEN
                    529:             PARAMS( LA_LINRX_ITHRESH_I ) = ITHRESH
                    530:          ELSE
                    531:             ITHRESH = INT( PARAMS( LA_LINRX_ITHRESH_I ) )
                    532:          END IF
                    533:       END IF
                    534:       IF ( NPARAMS.GE.LA_LINRX_CWISE_I ) THEN
                    535:          IF ( PARAMS( LA_LINRX_CWISE_I ).LT.0.0D+0 ) THEN
                    536:             IF ( IGNORE_CWISE ) THEN
                    537:                PARAMS( LA_LINRX_CWISE_I ) = 0.0D+0
                    538:             ELSE
                    539:                PARAMS( LA_LINRX_CWISE_I ) = 1.0D+0
                    540:             END IF
                    541:          ELSE
                    542:             IGNORE_CWISE = PARAMS( LA_LINRX_CWISE_I ) .EQ. 0.0D+0
                    543:          END IF
                    544:       END IF
                    545:       IF ( REF_TYPE .EQ. 0 .OR. N_ERR_BNDS .EQ. 0 ) THEN
                    546:          N_NORMS = 0
                    547:       ELSE IF ( IGNORE_CWISE ) THEN
                    548:          N_NORMS = 1
                    549:       ELSE
                    550:          N_NORMS = 2
                    551:       END IF
                    552: *
                    553:       NOTRAN = LSAME( TRANS, 'N' )
                    554:       ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
                    555:       COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
                    556: *
                    557: *     Test input parameters.
                    558: *
                    559:       IF( TRANS_TYPE.EQ.-1 ) THEN
                    560:         INFO = -1
                    561:       ELSE IF( .NOT.ROWEQU .AND. .NOT.COLEQU .AND.
                    562:      $         .NOT.LSAME( EQUED, 'N' ) ) THEN
                    563:         INFO = -2
                    564:       ELSE IF( N.LT.0 ) THEN
                    565:         INFO = -3
                    566:       ELSE IF( KL.LT.0 ) THEN
                    567:         INFO = -4
                    568:       ELSE IF( KU.LT.0 ) THEN
                    569:         INFO = -5
                    570:       ELSE IF( NRHS.LT.0 ) THEN
                    571:         INFO = -6
                    572:       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
                    573:         INFO = -8
                    574:       ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
                    575:         INFO = -10
                    576:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
                    577:         INFO = -13
                    578:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
                    579:         INFO = -15
                    580:       END IF
                    581:       IF( INFO.NE.0 ) THEN
                    582:         CALL XERBLA( 'ZGBRFSX', -INFO )
                    583:         RETURN
                    584:       END IF
                    585: *
                    586: *     Quick return if possible.
                    587: *
                    588:       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
                    589:          RCOND = 1.0D+0
                    590:          DO J = 1, NRHS
                    591:             BERR( J ) = 0.0D+0
                    592:             IF ( N_ERR_BNDS .GE. 1 ) THEN
                    593:                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
                    594:                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
                    595:             END IF
                    596:             IF ( N_ERR_BNDS .GE. 2 ) THEN
                    597:                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 0.0D+0
                    598:                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 0.0D+0
                    599:             END IF
                    600:             IF ( N_ERR_BNDS .GE. 3 ) THEN
                    601:                ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 1.0D+0
                    602:                ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 1.0D+0
                    603:             END IF
                    604:          END DO
                    605:          RETURN
                    606:       END IF
                    607: *
                    608: *     Default to failure.
                    609: *
                    610:       RCOND = 0.0D+0
                    611:       DO J = 1, NRHS
                    612:          BERR( J ) = 1.0D+0
                    613:          IF ( N_ERR_BNDS .GE. 1 ) THEN
                    614:             ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
                    615:             ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
                    616:          END IF
                    617:          IF ( N_ERR_BNDS .GE. 2 ) THEN
                    618:             ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
                    619:             ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
                    620:          END IF
                    621:          IF ( N_ERR_BNDS .GE. 3 ) THEN
                    622:             ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 0.0D+0
                    623:             ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 0.0D+0
                    624:          END IF
                    625:       END DO
                    626: *
                    627: *     Compute the norm of A and the reciprocal of the condition
                    628: *     number of A.
                    629: *
                    630:       IF( NOTRAN ) THEN
                    631:          NORM = 'I'
                    632:       ELSE
                    633:          NORM = '1'
                    634:       END IF
                    635:       ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
                    636:       CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
                    637:      $     WORK, RWORK, INFO )
                    638: *
                    639: *     Perform refinement on each right-hand side
                    640: *
1.11      bertrand  641:       IF ( REF_TYPE .NE. 0 .AND. INFO .EQ. 0 ) THEN
1.1       bertrand  642: 
                    643:          PREC_TYPE = ILAPREC( 'E' )
                    644: 
                    645:          IF ( NOTRAN ) THEN
                    646:             CALL ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE,  N, KL, KU,
                    647:      $           NRHS, AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B,
                    648:      $           LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
                    649:      $           ERR_BNDS_COMP, WORK, RWORK, WORK(N+1),
                    650:      $           TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N),
                    651:      $           RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE,
                    652:      $           INFO )
                    653:          ELSE
                    654:             CALL ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE,  N, KL, KU,
                    655:      $           NRHS, AB, LDAB, AFB, LDAFB, IPIV, ROWEQU, R, B,
                    656:      $           LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
                    657:      $           ERR_BNDS_COMP, WORK, RWORK, WORK(N+1),
                    658:      $           TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N),
                    659:      $           RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE,
                    660:      $           INFO )
                    661:          END IF
                    662:       END IF
                    663: 
                    664:       ERR_LBND = MAX( 10.0D+0, SQRT( DBLE( N ) ) ) * DLAMCH( 'Epsilon' )
                    665:       IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 1) THEN
                    666: *
                    667: *     Compute scaled normwise condition number cond(A*C).
                    668: *
                    669:          IF ( COLEQU .AND. NOTRAN ) THEN
                    670:             RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, AFB,
                    671:      $           LDAFB, IPIV, C, .TRUE., INFO, WORK, RWORK )
                    672:          ELSE IF ( ROWEQU .AND. .NOT. NOTRAN ) THEN
                    673:             RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, AFB,
                    674:      $           LDAFB, IPIV, R, .TRUE., INFO, WORK, RWORK )
                    675:          ELSE
                    676:             RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, AFB,
                    677:      $           LDAFB, IPIV, C, .FALSE., INFO, WORK, RWORK )
                    678:          END IF
                    679:          DO J = 1, NRHS
                    680: *
                    681: *     Cap the error at 1.0.
                    682: *
                    683:             IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
                    684:      $           .AND. ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .GT. 1.0D+0)
                    685:      $           ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
                    686: *
                    687: *     Threshold the error (see LAWN).
                    688: *
                    689:             IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
                    690:                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
                    691:                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 0.0D+0
                    692:                IF ( INFO .LE. N ) INFO = N + J
                    693:             ELSE IF ( ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .LT. ERR_LBND )
                    694:      $              THEN
                    695:                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = ERR_LBND
                    696:                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
                    697:             END IF
                    698: *
                    699: *     Save the condition number.
                    700: *
                    701:             IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
                    702:                ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = RCOND_TMP
                    703:             END IF
                    704: 
                    705:          END DO
                    706:       END IF
                    707: 
                    708:       IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 2) THEN
                    709: *
                    710: *     Compute componentwise condition number cond(A*diag(Y(:,J))) for
                    711: *     each right-hand side using the current solution as an estimate of
                    712: *     the true solution.  If the componentwise error estimate is too
                    713: *     large, then the solution is a lousy estimate of truth and the
                    714: *     estimated RCOND may be too optimistic.  To avoid misleading users,
                    715: *     the inverse condition number is set to 0.0 when the estimated
                    716: *     cwise error is at least CWISE_WRONG.
                    717: *
                    718:          CWISE_WRONG = SQRT( DLAMCH( 'Epsilon' ) )
                    719:          DO J = 1, NRHS
                    720:             IF (ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .LT. CWISE_WRONG )
                    721:      $     THEN
                    722:                RCOND_TMP = ZLA_GBRCOND_X( TRANS, N, KL, KU, AB, LDAB,
                    723:      $              AFB, LDAFB, IPIV, X( 1, J ), INFO, WORK, RWORK )
                    724:             ELSE
                    725:                RCOND_TMP = 0.0D+0
                    726:             END IF
                    727: *
                    728: *     Cap the error at 1.0.
                    729: *
                    730:             IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
                    731:      $           .AND. ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .GT. 1.0D+0 )
                    732:      $           ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
                    733: *
                    734: *     Threshold the error (see LAWN).
                    735: *
                    736:             IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
                    737:                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
                    738:                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 0.0D+0
                    739:                IF ( PARAMS( LA_LINRX_CWISE_I ) .EQ. 1.0D+0
                    740:      $              .AND. INFO.LT.N + J ) INFO = N + J
                    741:             ELSE IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I )
                    742:      $              .LT. ERR_LBND ) THEN
                    743:                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = ERR_LBND
                    744:                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
                    745:             END IF
                    746: *
                    747: *     Save the condition number.
                    748: *
                    749:             IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
                    750:                ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = RCOND_TMP
                    751:             END IF
                    752: 
                    753:          END DO
                    754:       END IF
                    755: *
                    756:       RETURN
                    757: *
                    758: *     End of ZGBRFSX
                    759: *
                    760:       END

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