Annotation of rpl/lapack/lapack/dlarfg.f, revision 1.15
1.12 bertrand 1: *> \brief \b DLARFG generates an elementary reflector (Householder matrix).
1.9 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLARFG + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfg.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfg.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfg.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INCX, N
25: * DOUBLE PRECISION ALPHA, TAU
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION X( * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DLARFG generates a real elementary reflector H of order n, such
38: *> that
39: *>
40: *> H * ( alpha ) = ( beta ), H**T * H = I.
41: *> ( x ) ( 0 )
42: *>
43: *> where alpha and beta are scalars, and x is an (n-1)-element real
44: *> vector. H is represented in the form
45: *>
46: *> H = I - tau * ( 1 ) * ( 1 v**T ) ,
47: *> ( v )
48: *>
49: *> where tau is a real scalar and v is a real (n-1)-element
50: *> vector.
51: *>
52: *> If the elements of x are all zero, then tau = 0 and H is taken to be
53: *> the unit matrix.
54: *>
55: *> Otherwise 1 <= tau <= 2.
56: *> \endverbatim
57: *
58: * Arguments:
59: * ==========
60: *
61: *> \param[in] N
62: *> \verbatim
63: *> N is INTEGER
64: *> The order of the elementary reflector.
65: *> \endverbatim
66: *>
67: *> \param[in,out] ALPHA
68: *> \verbatim
69: *> ALPHA is DOUBLE PRECISION
70: *> On entry, the value alpha.
71: *> On exit, it is overwritten with the value beta.
72: *> \endverbatim
73: *>
74: *> \param[in,out] X
75: *> \verbatim
76: *> X is DOUBLE PRECISION array, dimension
77: *> (1+(N-2)*abs(INCX))
78: *> On entry, the vector x.
79: *> On exit, it is overwritten with the vector v.
80: *> \endverbatim
81: *>
82: *> \param[in] INCX
83: *> \verbatim
84: *> INCX is INTEGER
85: *> The increment between elements of X. INCX > 0.
86: *> \endverbatim
87: *>
88: *> \param[out] TAU
89: *> \verbatim
90: *> TAU is DOUBLE PRECISION
91: *> The value tau.
92: *> \endverbatim
93: *
94: * Authors:
95: * ========
96: *
97: *> \author Univ. of Tennessee
98: *> \author Univ. of California Berkeley
99: *> \author Univ. of Colorado Denver
100: *> \author NAG Ltd.
101: *
1.12 bertrand 102: *> \date September 2012
1.9 bertrand 103: *
104: *> \ingroup doubleOTHERauxiliary
105: *
106: * =====================================================================
1.1 bertrand 107: SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU )
108: *
1.12 bertrand 109: * -- LAPACK auxiliary routine (version 3.4.2) --
1.1 bertrand 110: * -- LAPACK is a software package provided by Univ. of Tennessee, --
111: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.12 bertrand 112: * September 2012
1.1 bertrand 113: *
114: * .. Scalar Arguments ..
115: INTEGER INCX, N
116: DOUBLE PRECISION ALPHA, TAU
117: * ..
118: * .. Array Arguments ..
119: DOUBLE PRECISION X( * )
120: * ..
121: *
122: * =====================================================================
123: *
124: * .. Parameters ..
125: DOUBLE PRECISION ONE, ZERO
126: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
127: * ..
128: * .. Local Scalars ..
129: INTEGER J, KNT
130: DOUBLE PRECISION BETA, RSAFMN, SAFMIN, XNORM
131: * ..
132: * .. External Functions ..
133: DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
134: EXTERNAL DLAMCH, DLAPY2, DNRM2
135: * ..
136: * .. Intrinsic Functions ..
137: INTRINSIC ABS, SIGN
138: * ..
139: * .. External Subroutines ..
140: EXTERNAL DSCAL
141: * ..
142: * .. Executable Statements ..
143: *
144: IF( N.LE.1 ) THEN
145: TAU = ZERO
146: RETURN
147: END IF
148: *
149: XNORM = DNRM2( N-1, X, INCX )
150: *
151: IF( XNORM.EQ.ZERO ) THEN
152: *
153: * H = I
154: *
155: TAU = ZERO
156: ELSE
157: *
158: * general case
159: *
160: BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
161: SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
162: KNT = 0
163: IF( ABS( BETA ).LT.SAFMIN ) THEN
164: *
165: * XNORM, BETA may be inaccurate; scale X and recompute them
166: *
167: RSAFMN = ONE / SAFMIN
168: 10 CONTINUE
169: KNT = KNT + 1
170: CALL DSCAL( N-1, RSAFMN, X, INCX )
171: BETA = BETA*RSAFMN
172: ALPHA = ALPHA*RSAFMN
173: IF( ABS( BETA ).LT.SAFMIN )
174: $ GO TO 10
175: *
176: * New BETA is at most 1, at least SAFMIN
177: *
178: XNORM = DNRM2( N-1, X, INCX )
179: BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
180: END IF
181: TAU = ( BETA-ALPHA ) / BETA
182: CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX )
183: *
184: * If ALPHA is subnormal, it may lose relative accuracy
185: *
186: DO 20 J = 1, KNT
187: BETA = BETA*SAFMIN
188: 20 CONTINUE
189: ALPHA = BETA
190: END IF
191: *
192: RETURN
193: *
194: * End of DLARFG
195: *
196: END
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