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    1: *> \brief \b DGBCON
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DGBCON + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbcon.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbcon.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbcon.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
   22: *                          WORK, IWORK, INFO )
   23: *
   24: *       .. Scalar Arguments ..
   25: *       CHARACTER          NORM
   26: *       INTEGER            INFO, KL, KU, LDAB, N
   27: *       DOUBLE PRECISION   ANORM, RCOND
   28: *       ..
   29: *       .. Array Arguments ..
   30: *       INTEGER            IPIV( * ), IWORK( * )
   31: *       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
   32: *       ..
   33: *
   34: *
   35: *> \par Purpose:
   36: *  =============
   37: *>
   38: *> \verbatim
   39: *>
   40: *> DGBCON estimates the reciprocal of the condition number of a real
   41: *> general band matrix A, in either the 1-norm or the infinity-norm,
   42: *> using the LU factorization computed by DGBTRF.
   43: *>
   44: *> An estimate is obtained for norm(inv(A)), and the reciprocal of the
   45: *> condition number is computed as
   46: *>    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
   47: *> \endverbatim
   48: *
   49: *  Arguments:
   50: *  ==========
   51: *
   52: *> \param[in] NORM
   53: *> \verbatim
   54: *>          NORM is CHARACTER*1
   55: *>          Specifies whether the 1-norm condition number or the
   56: *>          infinity-norm condition number is required:
   57: *>          = '1' or 'O':  1-norm;
   58: *>          = 'I':         Infinity-norm.
   59: *> \endverbatim
   60: *>
   61: *> \param[in] N
   62: *> \verbatim
   63: *>          N is INTEGER
   64: *>          The order of the matrix A.  N >= 0.
   65: *> \endverbatim
   66: *>
   67: *> \param[in] KL
   68: *> \verbatim
   69: *>          KL is INTEGER
   70: *>          The number of subdiagonals within the band of A.  KL >= 0.
   71: *> \endverbatim
   72: *>
   73: *> \param[in] KU
   74: *> \verbatim
   75: *>          KU is INTEGER
   76: *>          The number of superdiagonals within the band of A.  KU >= 0.
   77: *> \endverbatim
   78: *>
   79: *> \param[in] AB
   80: *> \verbatim
   81: *>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
   82: *>          Details of the LU factorization of the band matrix A, as
   83: *>          computed by DGBTRF.  U is stored as an upper triangular band
   84: *>          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
   85: *>          the multipliers used during the factorization are stored in
   86: *>          rows KL+KU+2 to 2*KL+KU+1.
   87: *> \endverbatim
   88: *>
   89: *> \param[in] LDAB
   90: *> \verbatim
   91: *>          LDAB is INTEGER
   92: *>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
   93: *> \endverbatim
   94: *>
   95: *> \param[in] IPIV
   96: *> \verbatim
   97: *>          IPIV is INTEGER array, dimension (N)
   98: *>          The pivot indices; for 1 <= i <= N, row i of the matrix was
   99: *>          interchanged with row IPIV(i).
  100: *> \endverbatim
  101: *>
  102: *> \param[in] ANORM
  103: *> \verbatim
  104: *>          ANORM is DOUBLE PRECISION
  105: *>          If NORM = '1' or 'O', the 1-norm of the original matrix A.
  106: *>          If NORM = 'I', the infinity-norm of the original matrix A.
  107: *> \endverbatim
  108: *>
  109: *> \param[out] RCOND
  110: *> \verbatim
  111: *>          RCOND is DOUBLE PRECISION
  112: *>          The reciprocal of the condition number of the matrix A,
  113: *>          computed as RCOND = 1/(norm(A) * norm(inv(A))).
  114: *> \endverbatim
  115: *>
  116: *> \param[out] WORK
  117: *> \verbatim
  118: *>          WORK is DOUBLE PRECISION array, dimension (3*N)
  119: *> \endverbatim
  120: *>
  121: *> \param[out] IWORK
  122: *> \verbatim
  123: *>          IWORK is INTEGER array, dimension (N)
  124: *> \endverbatim
  125: *>
  126: *> \param[out] INFO
  127: *> \verbatim
  128: *>          INFO is INTEGER
  129: *>          = 0:  successful exit
  130: *>          < 0: if INFO = -i, the i-th argument had an illegal value
  131: *> \endverbatim
  132: *
  133: *  Authors:
  134: *  ========
  135: *
  136: *> \author Univ. of Tennessee
  137: *> \author Univ. of California Berkeley
  138: *> \author Univ. of Colorado Denver
  139: *> \author NAG Ltd.
  140: *
  141: *> \date December 2016
  142: *
  143: *> \ingroup doubleGBcomputational
  144: *
  145: *  =====================================================================
  146:       SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND,
  147:      $                   WORK, IWORK, INFO )
  148: *
  149: *  -- LAPACK computational routine (version 3.7.0) --
  150: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  151: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  152: *     December 2016
  153: *
  154: *     .. Scalar Arguments ..
  155:       CHARACTER          NORM
  156:       INTEGER            INFO, KL, KU, LDAB, N
  157:       DOUBLE PRECISION   ANORM, RCOND
  158: *     ..
  159: *     .. Array Arguments ..
  160:       INTEGER            IPIV( * ), IWORK( * )
  161:       DOUBLE PRECISION   AB( LDAB, * ), WORK( * )
  162: *     ..
  163: *
  164: *  =====================================================================
  165: *
  166: *     .. Parameters ..
  167:       DOUBLE PRECISION   ONE, ZERO
  168:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
  169: *     ..
  170: *     .. Local Scalars ..
  171:       LOGICAL            LNOTI, ONENRM
  172:       CHARACTER          NORMIN
  173:       INTEGER            IX, J, JP, KASE, KASE1, KD, LM
  174:       DOUBLE PRECISION   AINVNM, SCALE, SMLNUM, T
  175: *     ..
  176: *     .. Local Arrays ..
  177:       INTEGER            ISAVE( 3 )
  178: *     ..
  179: *     .. External Functions ..
  180:       LOGICAL            LSAME
  181:       INTEGER            IDAMAX
  182:       DOUBLE PRECISION   DDOT, DLAMCH
  183:       EXTERNAL           LSAME, IDAMAX, DDOT, DLAMCH
  184: *     ..
  185: *     .. External Subroutines ..
  186:       EXTERNAL           DAXPY, DLACN2, DLATBS, DRSCL, XERBLA
  187: *     ..
  188: *     .. Intrinsic Functions ..
  189:       INTRINSIC          ABS, MIN
  190: *     ..
  191: *     .. Executable Statements ..
  192: *
  193: *     Test the input parameters.
  194: *
  195:       INFO = 0
  196:       ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' )
  197:       IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN
  198:          INFO = -1
  199:       ELSE IF( N.LT.0 ) THEN
  200:          INFO = -2
  201:       ELSE IF( KL.LT.0 ) THEN
  202:          INFO = -3
  203:       ELSE IF( KU.LT.0 ) THEN
  204:          INFO = -4
  205:       ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
  206:          INFO = -6
  207:       ELSE IF( ANORM.LT.ZERO ) THEN
  208:          INFO = -8
  209:       END IF
  210:       IF( INFO.NE.0 ) THEN
  211:          CALL XERBLA( 'DGBCON', -INFO )
  212:          RETURN
  213:       END IF
  214: *
  215: *     Quick return if possible
  216: *
  217:       RCOND = ZERO
  218:       IF( N.EQ.0 ) THEN
  219:          RCOND = ONE
  220:          RETURN
  221:       ELSE IF( ANORM.EQ.ZERO ) THEN
  222:          RETURN
  223:       END IF
  224: *
  225:       SMLNUM = DLAMCH( 'Safe minimum' )
  226: *
  227: *     Estimate the norm of inv(A).
  228: *
  229:       AINVNM = ZERO
  230:       NORMIN = 'N'
  231:       IF( ONENRM ) THEN
  232:          KASE1 = 1
  233:       ELSE
  234:          KASE1 = 2
  235:       END IF
  236:       KD = KL + KU + 1
  237:       LNOTI = KL.GT.0
  238:       KASE = 0
  239:    10 CONTINUE
  240:       CALL DLACN2( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE, ISAVE )
  241:       IF( KASE.NE.0 ) THEN
  242:          IF( KASE.EQ.KASE1 ) THEN
  243: *
  244: *           Multiply by inv(L).
  245: *
  246:             IF( LNOTI ) THEN
  247:                DO 20 J = 1, N - 1
  248:                   LM = MIN( KL, N-J )
  249:                   JP = IPIV( J )
  250:                   T = WORK( JP )
  251:                   IF( JP.NE.J ) THEN
  252:                      WORK( JP ) = WORK( J )
  253:                      WORK( J ) = T
  254:                   END IF
  255:                   CALL DAXPY( LM, -T, AB( KD+1, J ), 1, WORK( J+1 ), 1 )
  256:    20          CONTINUE
  257:             END IF
  258: *
  259: *           Multiply by inv(U).
  260: *
  261:             CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N,
  262:      $                   KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
  263:      $                   INFO )
  264:          ELSE
  265: *
  266: *           Multiply by inv(U**T).
  267: *
  268:             CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N,
  269:      $                   KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ),
  270:      $                   INFO )
  271: *
  272: *           Multiply by inv(L**T).
  273: *
  274:             IF( LNOTI ) THEN
  275:                DO 30 J = N - 1, 1, -1
  276:                   LM = MIN( KL, N-J )
  277:                   WORK( J ) = WORK( J ) - DDOT( LM, AB( KD+1, J ), 1,
  278:      $                        WORK( J+1 ), 1 )
  279:                   JP = IPIV( J )
  280:                   IF( JP.NE.J ) THEN
  281:                      T = WORK( JP )
  282:                      WORK( JP ) = WORK( J )
  283:                      WORK( J ) = T
  284:                   END IF
  285:    30          CONTINUE
  286:             END IF
  287:          END IF
  288: *
  289: *        Divide X by 1/SCALE if doing so will not cause overflow.
  290: *
  291:          NORMIN = 'Y'
  292:          IF( SCALE.NE.ONE ) THEN
  293:             IX = IDAMAX( N, WORK, 1 )
  294:             IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO )
  295:      $         GO TO 40
  296:             CALL DRSCL( N, SCALE, WORK, 1 )
  297:          END IF
  298:          GO TO 10
  299:       END IF
  300: *
  301: *     Compute the estimate of the reciprocal condition number.
  302: *
  303:       IF( AINVNM.NE.ZERO )
  304:      $   RCOND = ( ONE / AINVNM ) / ANORM
  305: *
  306:    40 CONTINUE
  307:       RETURN
  308: *
  309: *     End of DGBCON
  310: *
  311:       END