LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ slapll()

 subroutine slapll ( integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real SSMIN )

SLAPLL measures the linear dependence of two vectors.

Purpose:
``` 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.```
Parameters
 [in] N ``` N is INTEGER The length of the vectors X and Y.``` [in,out] X ``` X is REAL array, dimension (1+(N-1)*INCX) On entry, X contains the N-vector X. On exit, X is overwritten.``` [in] INCX ``` INCX is INTEGER The increment between successive elements of X. INCX > 0.``` [in,out] Y ``` Y is REAL array, dimension (1+(N-1)*INCY) On entry, Y contains the N-vector Y. On exit, Y is overwritten.``` [in] INCY ``` INCY is INTEGER The increment between successive elements of Y. INCY > 0.``` [out] SSMIN ``` SSMIN is REAL The smallest singular value of the N-by-2 matrix A = ( X Y ).```

Definition at line 101 of file slapll.f.

102*
103* -- LAPACK auxiliary routine --
104* -- LAPACK is a software package provided by Univ. of Tennessee, --
105* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
106*
107* .. Scalar Arguments ..
108 INTEGER INCX, INCY, N
109 REAL SSMIN
110* ..
111* .. Array Arguments ..
112 REAL X( * ), Y( * )
113* ..
114*
115* =====================================================================
116*
117* .. Parameters ..
118 REAL ZERO, ONE
119 parameter( zero = 0.0e+0, one = 1.0e+0 )
120* ..
121* .. Local Scalars ..
122 REAL A11, A12, A22, C, SSMAX, TAU
123* ..
124* .. External Functions ..
125 REAL SDOT
126 EXTERNAL sdot
127* ..
128* .. External Subroutines ..
129 EXTERNAL saxpy, slarfg, slas2
130* ..
131* .. Executable Statements ..
132*
133* Quick return if possible
134*
135 IF( n.LE.1 ) THEN
136 ssmin = zero
137 RETURN
138 END IF
139*
140* Compute the QR factorization of the N-by-2 matrix ( X Y )
141*
142 CALL slarfg( n, x( 1 ), x( 1+incx ), incx, tau )
143 a11 = x( 1 )
144 x( 1 ) = one
145*
146 c = -tau*sdot( n, x, incx, y, incy )
147 CALL saxpy( n, c, x, incx, y, incy )
148*
149 CALL slarfg( n-1, y( 1+incy ), y( 1+2*incy ), incy, tau )
150*
151 a12 = y( 1 )
152 a22 = y( 1+incy )
153*
154* Compute the SVD of 2-by-2 Upper triangular matrix.
155*
156 CALL slas2( a11, a12, a22, ssmin, ssmax )
157*
158 RETURN
159*
160* End of SLAPLL
161*
subroutine slas2(F, G, H, SSMIN, SSMAX)
SLAS2 computes singular values of a 2-by-2 triangular matrix.
Definition: slas2.f:107
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
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