numeric-linalg

001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
022
023
024
025
026
027
028
029
030
031
032
033
034
035
036
037
038
039
040
041
042
043
044
045
046
047
048
049
050
051
052
053
054
055
056
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
075
076
077
078
079
080
081
082
083
084
085
086
087
088
089
090
091
092
093
094
095
096
097
098
099
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162

*> \brief \b DLAPLL measures the linear dependence of two vectors.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAPLL + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlapll.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlapll.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlapll.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
*       .. Scalar Arguments ..
*       INTEGER            INCX, INCY, N
*       DOUBLE PRECISION   SSMIN
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> Given two column vectors X and Y, let
*>
*>                      A = ( X Y ).
*>
*> The subroutine first computes the QR factorization of A = Q*R,
*> and then computes the SVD of the 2-by-2 upper triangular matrix R.
*> The smaller singular value of R is returned in SSMIN, which is used
*> as the measurement of the linear dependency of the vectors X and Y.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The length of the vectors X and Y.
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCX)
*>          On entry, X contains the N-vector X.
*>          On exit, X is overwritten.
*> \endverbatim
*>
*> \param[in] INCX
*> \verbatim
*>          INCX is INTEGER
*>          The increment between successive elements of X. INCX > 0.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array,
*>                         dimension (1+(N-1)*INCY)
*>          On entry, Y contains the N-vector Y.
*>          On exit, Y is overwritten.
*> \endverbatim
*>
*> \param[in] INCY
*> \verbatim
*>          INCY is INTEGER
*>          The increment between successive elements of Y. INCY > 0.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*>          SSMIN is DOUBLE PRECISION
*>          The smallest singular value of the N-by-2 matrix A = ( X Y ).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup lapll
*
*  =====================================================================
      SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INCX, INCY, N
      DOUBLE PRECISION   SSMIN
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   X( * ), Y( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      DOUBLE PRECISION   A11, A12, A22, C, SSMAX, TAU
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DDOT
      EXTERNAL           DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DLARFG, DLAS2
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 ) THEN
         SSMIN = ZERO
         RETURN
      END IF
*
*     Compute the QR factorization of the N-by-2 matrix ( X Y )
*
      CALL DLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU )
      A11 = X( 1 )
      X( 1 ) = ONE
*
      C = -TAU*DDOT( N, X, INCX, Y, INCY )
      CALL DAXPY( N, C, X, INCX, Y, INCY )
*
      CALL DLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU )
*
      A12 = Y( 1 )
      A22 = Y( 1+INCY )
*
*     Compute the SVD of 2-by-2 Upper triangular matrix.
*
      CALL DLAS2( A11, A12, A22, SSMIN, SSMAX )
*
      RETURN
*
*     End of DLAPLL
*
      END