Annotation of rpl/lapack/lapack/zlarfg.f, revision 1.17
1.12 bertrand 1: *> \brief \b ZLARFG generates an elementary reflector (Householder matrix).
1.9 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
1.16 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.16 bertrand 9: *> Download ZLARFG + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlarfg.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlarfg.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlarfg.f">
1.9 bertrand 15: *> [TXT]</a>
1.16 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLARFG( N, ALPHA, X, INCX, TAU )
1.16 bertrand 22: *
1.9 bertrand 23: * .. Scalar Arguments ..
24: * INTEGER INCX, N
25: * COMPLEX*16 ALPHA, TAU
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 X( * )
29: * ..
1.16 bertrand 30: *
1.9 bertrand 31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> ZLARFG generates a complex elementary reflector H of order n, such
38: *> that
39: *>
40: *> H**H * ( alpha ) = ( beta ), H**H * H = I.
41: *> ( x ) ( 0 )
42: *>
43: *> where alpha and beta are scalars, with beta real, and x is an
44: *> (n-1)-element complex vector. H is represented in the form
45: *>
46: *> H = I - tau * ( 1 ) * ( 1 v**H ) ,
47: *> ( v )
48: *>
49: *> where tau is a complex scalar and v is a complex (n-1)-element
50: *> vector. Note that H is not hermitian.
51: *>
52: *> If the elements of x are all zero and alpha is real, then tau = 0
53: *> and H is taken to be the unit matrix.
54: *>
55: *> Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1 .
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 COMPLEX*16
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 COMPLEX*16 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 COMPLEX*16
91: *> The value tau.
92: *> \endverbatim
93: *
94: * Authors:
95: * ========
96: *
1.16 bertrand 97: *> \author Univ. of Tennessee
98: *> \author Univ. of California Berkeley
99: *> \author Univ. of Colorado Denver
100: *> \author NAG Ltd.
1.9 bertrand 101: *
1.16 bertrand 102: *> \date December 2016
1.9 bertrand 103: *
104: *> \ingroup complex16OTHERauxiliary
105: *
106: * =====================================================================
1.1 bertrand 107: SUBROUTINE ZLARFG( N, ALPHA, X, INCX, TAU )
108: *
1.16 bertrand 109: * -- LAPACK auxiliary routine (version 3.7.0) --
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..--
1.16 bertrand 112: * December 2016
1.1 bertrand 113: *
114: * .. Scalar Arguments ..
115: INTEGER INCX, N
116: COMPLEX*16 ALPHA, TAU
117: * ..
118: * .. Array Arguments ..
119: COMPLEX*16 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 ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM
131: * ..
132: * .. External Functions ..
133: DOUBLE PRECISION DLAMCH, DLAPY3, DZNRM2
134: COMPLEX*16 ZLADIV
135: EXTERNAL DLAMCH, DLAPY3, DZNRM2, ZLADIV
136: * ..
137: * .. Intrinsic Functions ..
138: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, SIGN
139: * ..
140: * .. External Subroutines ..
141: EXTERNAL ZDSCAL, ZSCAL
142: * ..
143: * .. Executable Statements ..
144: *
145: IF( N.LE.0 ) THEN
146: TAU = ZERO
147: RETURN
148: END IF
149: *
150: XNORM = DZNRM2( N-1, X, INCX )
151: ALPHR = DBLE( ALPHA )
152: ALPHI = DIMAG( ALPHA )
153: *
154: IF( XNORM.EQ.ZERO .AND. ALPHI.EQ.ZERO ) THEN
155: *
156: * H = I
157: *
158: TAU = ZERO
159: ELSE
160: *
161: * general case
162: *
163: BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
164: SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
165: RSAFMN = ONE / SAFMIN
166: *
167: KNT = 0
168: IF( ABS( BETA ).LT.SAFMIN ) THEN
169: *
170: * XNORM, BETA may be inaccurate; scale X and recompute them
171: *
172: 10 CONTINUE
173: KNT = KNT + 1
174: CALL ZDSCAL( N-1, RSAFMN, X, INCX )
175: BETA = BETA*RSAFMN
176: ALPHI = ALPHI*RSAFMN
177: ALPHR = ALPHR*RSAFMN
178: IF( ABS( BETA ).LT.SAFMIN )
179: $ GO TO 10
180: *
181: * New BETA is at most 1, at least SAFMIN
182: *
183: XNORM = DZNRM2( N-1, X, INCX )
184: ALPHA = DCMPLX( ALPHR, ALPHI )
185: BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
186: END IF
187: TAU = DCMPLX( ( BETA-ALPHR ) / BETA, -ALPHI / BETA )
188: ALPHA = ZLADIV( DCMPLX( ONE ), ALPHA-BETA )
189: CALL ZSCAL( N-1, ALPHA, X, INCX )
190: *
191: * If ALPHA is subnormal, it may lose relative accuracy
192: *
193: DO 20 J = 1, KNT
194: BETA = BETA*SAFMIN
195: 20 CONTINUE
196: ALPHA = BETA
197: END IF
198: *
199: RETURN
200: *
201: * End of ZLARFG
202: *
203: END
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