Annotation of rpl/lapack/lapack/zpbequ.f, revision 1.8

1.8     ! bertrand    1: *> \brief \b ZPBEQU
        !             2: *
        !             3: *  =========== DOCUMENTATION ===========
        !             4: *
        !             5: * Online html documentation available at 
        !             6: *            http://www.netlib.org/lapack/explore-html/ 
        !             7: *
        !             8: *> \htmlonly
        !             9: *> Download ZPBEQU + dependencies 
        !            10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpbequ.f"> 
        !            11: *> [TGZ]</a> 
        !            12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbequ.f"> 
        !            13: *> [ZIP]</a> 
        !            14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbequ.f"> 
        !            15: *> [TXT]</a>
        !            16: *> \endhtmlonly 
        !            17: *
        !            18: *  Definition:
        !            19: *  ===========
        !            20: *
        !            21: *       SUBROUTINE ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
        !            22: * 
        !            23: *       .. Scalar Arguments ..
        !            24: *       CHARACTER          UPLO
        !            25: *       INTEGER            INFO, KD, LDAB, N
        !            26: *       DOUBLE PRECISION   AMAX, SCOND
        !            27: *       ..
        !            28: *       .. Array Arguments ..
        !            29: *       DOUBLE PRECISION   S( * )
        !            30: *       COMPLEX*16         AB( LDAB, * )
        !            31: *       ..
        !            32: *  
        !            33: *
        !            34: *> \par Purpose:
        !            35: *  =============
        !            36: *>
        !            37: *> \verbatim
        !            38: *>
        !            39: *> ZPBEQU computes row and column scalings intended to equilibrate a
        !            40: *> Hermitian positive definite band matrix A and reduce its condition
        !            41: *> number (with respect to the two-norm).  S contains the scale factors,
        !            42: *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
        !            43: *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
        !            44: *> choice of S puts the condition number of B within a factor N of the
        !            45: *> smallest possible condition number over all possible diagonal
        !            46: *> scalings.
        !            47: *> \endverbatim
        !            48: *
        !            49: *  Arguments:
        !            50: *  ==========
        !            51: *
        !            52: *> \param[in] UPLO
        !            53: *> \verbatim
        !            54: *>          UPLO is CHARACTER*1
        !            55: *>          = 'U':  Upper triangular of A is stored;
        !            56: *>          = 'L':  Lower triangular of A is stored.
        !            57: *> \endverbatim
        !            58: *>
        !            59: *> \param[in] N
        !            60: *> \verbatim
        !            61: *>          N is INTEGER
        !            62: *>          The order of the matrix A.  N >= 0.
        !            63: *> \endverbatim
        !            64: *>
        !            65: *> \param[in] KD
        !            66: *> \verbatim
        !            67: *>          KD is INTEGER
        !            68: *>          The number of superdiagonals of the matrix A if UPLO = 'U',
        !            69: *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
        !            70: *> \endverbatim
        !            71: *>
        !            72: *> \param[in] AB
        !            73: *> \verbatim
        !            74: *>          AB is COMPLEX*16 array, dimension (LDAB,N)
        !            75: *>          The upper or lower triangle of the Hermitian band matrix A,
        !            76: *>          stored in the first KD+1 rows of the array.  The j-th column
        !            77: *>          of A is stored in the j-th column of the array AB as follows:
        !            78: *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
        !            79: *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
        !            80: *> \endverbatim
        !            81: *>
        !            82: *> \param[in] LDAB
        !            83: *> \verbatim
        !            84: *>          LDAB is INTEGER
        !            85: *>          The leading dimension of the array A.  LDAB >= KD+1.
        !            86: *> \endverbatim
        !            87: *>
        !            88: *> \param[out] S
        !            89: *> \verbatim
        !            90: *>          S is DOUBLE PRECISION array, dimension (N)
        !            91: *>          If INFO = 0, S contains the scale factors for A.
        !            92: *> \endverbatim
        !            93: *>
        !            94: *> \param[out] SCOND
        !            95: *> \verbatim
        !            96: *>          SCOND is DOUBLE PRECISION
        !            97: *>          If INFO = 0, S contains the ratio of the smallest S(i) to
        !            98: *>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
        !            99: *>          large nor too small, it is not worth scaling by S.
        !           100: *> \endverbatim
        !           101: *>
        !           102: *> \param[out] AMAX
        !           103: *> \verbatim
        !           104: *>          AMAX is DOUBLE PRECISION
        !           105: *>          Absolute value of largest matrix element.  If AMAX is very
        !           106: *>          close to overflow or very close to underflow, the matrix
        !           107: *>          should be scaled.
        !           108: *> \endverbatim
        !           109: *>
        !           110: *> \param[out] INFO
        !           111: *> \verbatim
        !           112: *>          INFO is INTEGER
        !           113: *>          = 0:  successful exit
        !           114: *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
        !           115: *>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
        !           116: *> \endverbatim
        !           117: *
        !           118: *  Authors:
        !           119: *  ========
        !           120: *
        !           121: *> \author Univ. of Tennessee 
        !           122: *> \author Univ. of California Berkeley 
        !           123: *> \author Univ. of Colorado Denver 
        !           124: *> \author NAG Ltd. 
        !           125: *
        !           126: *> \date November 2011
        !           127: *
        !           128: *> \ingroup complex16OTHERcomputational
        !           129: *
        !           130: *  =====================================================================
1.1       bertrand  131:       SUBROUTINE ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
                    132: *
1.8     ! bertrand  133: *  -- LAPACK computational routine (version 3.4.0) --
1.1       bertrand  134: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                    135: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8     ! bertrand  136: *     November 2011
1.1       bertrand  137: *
                    138: *     .. Scalar Arguments ..
                    139:       CHARACTER          UPLO
                    140:       INTEGER            INFO, KD, LDAB, N
                    141:       DOUBLE PRECISION   AMAX, SCOND
                    142: *     ..
                    143: *     .. Array Arguments ..
                    144:       DOUBLE PRECISION   S( * )
                    145:       COMPLEX*16         AB( LDAB, * )
                    146: *     ..
                    147: *
                    148: *  =====================================================================
                    149: *
                    150: *     .. Parameters ..
                    151:       DOUBLE PRECISION   ZERO, ONE
                    152:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
                    153: *     ..
                    154: *     .. Local Scalars ..
                    155:       LOGICAL            UPPER
                    156:       INTEGER            I, J
                    157:       DOUBLE PRECISION   SMIN
                    158: *     ..
                    159: *     .. External Functions ..
                    160:       LOGICAL            LSAME
                    161:       EXTERNAL           LSAME
                    162: *     ..
                    163: *     .. External Subroutines ..
                    164:       EXTERNAL           XERBLA
                    165: *     ..
                    166: *     .. Intrinsic Functions ..
                    167:       INTRINSIC          DBLE, MAX, MIN, SQRT
                    168: *     ..
                    169: *     .. Executable Statements ..
                    170: *
                    171: *     Test the input parameters.
                    172: *
                    173:       INFO = 0
                    174:       UPPER = LSAME( UPLO, 'U' )
                    175:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
                    176:          INFO = -1
                    177:       ELSE IF( N.LT.0 ) THEN
                    178:          INFO = -2
                    179:       ELSE IF( KD.LT.0 ) THEN
                    180:          INFO = -3
                    181:       ELSE IF( LDAB.LT.KD+1 ) THEN
                    182:          INFO = -5
                    183:       END IF
                    184:       IF( INFO.NE.0 ) THEN
                    185:          CALL XERBLA( 'ZPBEQU', -INFO )
                    186:          RETURN
                    187:       END IF
                    188: *
                    189: *     Quick return if possible
                    190: *
                    191:       IF( N.EQ.0 ) THEN
                    192:          SCOND = ONE
                    193:          AMAX = ZERO
                    194:          RETURN
                    195:       END IF
                    196: *
                    197:       IF( UPPER ) THEN
                    198:          J = KD + 1
                    199:       ELSE
                    200:          J = 1
                    201:       END IF
                    202: *
                    203: *     Initialize SMIN and AMAX.
                    204: *
                    205:       S( 1 ) = DBLE( AB( J, 1 ) )
                    206:       SMIN = S( 1 )
                    207:       AMAX = S( 1 )
                    208: *
                    209: *     Find the minimum and maximum diagonal elements.
                    210: *
                    211:       DO 10 I = 2, N
                    212:          S( I ) = DBLE( AB( J, I ) )
                    213:          SMIN = MIN( SMIN, S( I ) )
                    214:          AMAX = MAX( AMAX, S( I ) )
                    215:    10 CONTINUE
                    216: *
                    217:       IF( SMIN.LE.ZERO ) THEN
                    218: *
                    219: *        Find the first non-positive diagonal element and return.
                    220: *
                    221:          DO 20 I = 1, N
                    222:             IF( S( I ).LE.ZERO ) THEN
                    223:                INFO = I
                    224:                RETURN
                    225:             END IF
                    226:    20    CONTINUE
                    227:       ELSE
                    228: *
                    229: *        Set the scale factors to the reciprocals
                    230: *        of the diagonal elements.
                    231: *
                    232:          DO 30 I = 1, N
                    233:             S( I ) = ONE / SQRT( S( I ) )
                    234:    30    CONTINUE
                    235: *
                    236: *        Compute SCOND = min(S(I)) / max(S(I))
                    237: *
                    238:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
                    239:       END IF
                    240:       RETURN
                    241: *
                    242: *     End of ZPBEQU
                    243: *
                    244:       END

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