LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
dlapll.f
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1 *> \brief \b DLAPLL measures the linear dependence of two vectors.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INCX, INCY, N
25 * DOUBLE PRECISION SSMIN
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION X( * ), Y( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> Given two column vectors X and Y, let
38 *>
39 *> A = ( X Y ).
40 *>
41 *> The subroutine first computes the QR factorization of A = Q*R,
42 *> and then computes the SVD of the 2-by-2 upper triangular matrix R.
43 *> The smaller singular value of R is returned in SSMIN, which is used
44 *> as the measurement of the linear dependency of the vectors X and Y.
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] N
51 *> \verbatim
52 *> N is INTEGER
53 *> The length of the vectors X and Y.
54 *> \endverbatim
55 *>
56 *> \param[in,out] X
57 *> \verbatim
58 *> X is DOUBLE PRECISION array,
59 *> dimension (1+(N-1)*INCX)
60 *> On entry, X contains the N-vector X.
61 *> On exit, X is overwritten.
62 *> \endverbatim
63 *>
64 *> \param[in] INCX
65 *> \verbatim
66 *> INCX is INTEGER
67 *> The increment between successive elements of X. INCX > 0.
68 *> \endverbatim
69 *>
70 *> \param[in,out] Y
71 *> \verbatim
72 *> Y is DOUBLE PRECISION array,
73 *> dimension (1+(N-1)*INCY)
74 *> On entry, Y contains the N-vector Y.
75 *> On exit, Y is overwritten.
76 *> \endverbatim
77 *>
78 *> \param[in] INCY
79 *> \verbatim
80 *> INCY is INTEGER
81 *> The increment between successive elements of Y. INCY > 0.
82 *> \endverbatim
83 *>
84 *> \param[out] SSMIN
85 *> \verbatim
86 *> SSMIN is DOUBLE PRECISION
87 *> The smallest singular value of the N-by-2 matrix A = ( X Y ).
88 *> \endverbatim
89 *
90 * Authors:
91 * ========
92 *
93 *> \author Univ. of Tennessee
94 *> \author Univ. of California Berkeley
95 *> \author Univ. of Colorado Denver
96 *> \author NAG Ltd.
97 *
98 *> \date September 2012
99 *
100 *> \ingroup doubleOTHERauxiliary
101 *
102 * =====================================================================
103  SUBROUTINE dlapll( N, X, INCX, Y, INCY, SSMIN )
104 *
105 * -- LAPACK auxiliary routine (version 3.4.2) --
106 * -- LAPACK is a software package provided by Univ. of Tennessee, --
107 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
108 * September 2012
109 *
110 * .. Scalar Arguments ..
111  INTEGER incx, incy, n
112  DOUBLE PRECISION ssmin
113 * ..
114 * .. Array Arguments ..
115  DOUBLE PRECISION x( * ), y( * )
116 * ..
117 *
118 * =====================================================================
119 *
120 * .. Parameters ..
121  DOUBLE PRECISION zero, one
122  parameter( zero = 0.0d+0, one = 1.0d+0 )
123 * ..
124 * .. Local Scalars ..
125  DOUBLE PRECISION a11, a12, a22, c, ssmax, tau
126 * ..
127 * .. External Functions ..
128  DOUBLE PRECISION ddot
129  EXTERNAL ddot
130 * ..
131 * .. External Subroutines ..
132  EXTERNAL daxpy, dlarfg, dlas2
133 * ..
134 * .. Executable Statements ..
135 *
136 * Quick return if possible
137 *
138  IF( n.LE.1 ) THEN
139  ssmin = zero
140  return
141  END IF
142 *
143 * Compute the QR factorization of the N-by-2 matrix ( X Y )
144 *
145  CALL dlarfg( n, x( 1 ), x( 1+incx ), incx, tau )
146  a11 = x( 1 )
147  x( 1 ) = one
148 *
149  c = -tau*ddot( n, x, incx, y, incy )
150  CALL daxpy( n, c, x, incx, y, incy )
151 *
152  CALL dlarfg( n-1, y( 1+incy ), y( 1+2*incy ), incy, tau )
153 *
154  a12 = y( 1 )
155  a22 = y( 1+incy )
156 *
157 * Compute the SVD of 2-by-2 Upper triangular matrix.
158 *
159  CALL dlas2( a11, a12, a22, ssmin, ssmax )
160 *
161  return
162 *
163 * End of DLAPLL
164 *
165  END