1: *> \brief \b DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLARFGP + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarfgp.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarfgp.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarfgp.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLARFGP( 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: *> DLARFGP 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, beta is non-negative, and x is
44: *> an (n-1)-element real 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: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] N
60: *> \verbatim
61: *> N is INTEGER
62: *> The order of the elementary reflector.
63: *> \endverbatim
64: *>
65: *> \param[in,out] ALPHA
66: *> \verbatim
67: *> ALPHA is DOUBLE PRECISION
68: *> On entry, the value alpha.
69: *> On exit, it is overwritten with the value beta.
70: *> \endverbatim
71: *>
72: *> \param[in,out] X
73: *> \verbatim
74: *> X is DOUBLE PRECISION array, dimension
75: *> (1+(N-2)*abs(INCX))
76: *> On entry, the vector x.
77: *> On exit, it is overwritten with the vector v.
78: *> \endverbatim
79: *>
80: *> \param[in] INCX
81: *> \verbatim
82: *> INCX is INTEGER
83: *> The increment between elements of X. INCX > 0.
84: *> \endverbatim
85: *>
86: *> \param[out] TAU
87: *> \verbatim
88: *> TAU is DOUBLE PRECISION
89: *> The value tau.
90: *> \endverbatim
91: *
92: * Authors:
93: * ========
94: *
95: *> \author Univ. of Tennessee
96: *> \author Univ. of California Berkeley
97: *> \author Univ. of Colorado Denver
98: *> \author NAG Ltd.
99: *
100: *> \ingroup doubleOTHERauxiliary
101: *
102: * =====================================================================
103: SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
104: *
105: * -- LAPACK auxiliary routine --
106: * -- LAPACK is a software package provided by Univ. of Tennessee, --
107: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
108: *
109: * .. Scalar Arguments ..
110: INTEGER INCX, N
111: DOUBLE PRECISION ALPHA, TAU
112: * ..
113: * .. Array Arguments ..
114: DOUBLE PRECISION X( * )
115: * ..
116: *
117: * =====================================================================
118: *
119: * .. Parameters ..
120: DOUBLE PRECISION TWO, ONE, ZERO
121: PARAMETER ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
122: * ..
123: * .. Local Scalars ..
124: INTEGER J, KNT
125: DOUBLE PRECISION BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
126: * ..
127: * .. External Functions ..
128: DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
129: EXTERNAL DLAMCH, DLAPY2, DNRM2
130: * ..
131: * .. Intrinsic Functions ..
132: INTRINSIC ABS, SIGN
133: * ..
134: * .. External Subroutines ..
135: EXTERNAL DSCAL
136: * ..
137: * .. Executable Statements ..
138: *
139: IF( N.LE.0 ) THEN
140: TAU = ZERO
141: RETURN
142: END IF
143: *
144: XNORM = DNRM2( N-1, X, INCX )
145: *
146: IF( XNORM.EQ.ZERO ) THEN
147: *
148: * H = [+/-1, 0; I], sign chosen so ALPHA >= 0
149: *
150: IF( ALPHA.GE.ZERO ) THEN
151: * When TAU.eq.ZERO, the vector is special-cased to be
152: * all zeros in the application routines. We do not need
153: * to clear it.
154: TAU = ZERO
155: ELSE
156: * However, the application routines rely on explicit
157: * zero checks when TAU.ne.ZERO, and we must clear X.
158: TAU = TWO
159: DO J = 1, N-1
160: X( 1 + (J-1)*INCX ) = 0
161: END DO
162: ALPHA = -ALPHA
163: END IF
164: ELSE
165: *
166: * general case
167: *
168: BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
169: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'E' )
170: KNT = 0
171: IF( ABS( BETA ).LT.SMLNUM ) THEN
172: *
173: * XNORM, BETA may be inaccurate; scale X and recompute them
174: *
175: BIGNUM = ONE / SMLNUM
176: 10 CONTINUE
177: KNT = KNT + 1
178: CALL DSCAL( N-1, BIGNUM, X, INCX )
179: BETA = BETA*BIGNUM
180: ALPHA = ALPHA*BIGNUM
181: IF( (ABS( BETA ).LT.SMLNUM) .AND. (KNT .LT. 20) )
182: $ GO TO 10
183: *
184: * New BETA is at most 1, at least SMLNUM
185: *
186: XNORM = DNRM2( N-1, X, INCX )
187: BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
188: END IF
189: SAVEALPHA = ALPHA
190: ALPHA = ALPHA + BETA
191: IF( BETA.LT.ZERO ) THEN
192: BETA = -BETA
193: TAU = -ALPHA / BETA
194: ELSE
195: ALPHA = XNORM * (XNORM/ALPHA)
196: TAU = ALPHA / BETA
197: ALPHA = -ALPHA
198: END IF
199: *
200: IF ( ABS(TAU).LE.SMLNUM ) THEN
201: *
202: * In the case where the computed TAU ends up being a denormalized number,
203: * it loses relative accuracy. This is a BIG problem. Solution: flush TAU
204: * to ZERO. This explains the next IF statement.
205: *
206: * (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
207: * (Thanks Pat. Thanks MathWorks.)
208: *
209: IF( SAVEALPHA.GE.ZERO ) THEN
210: TAU = ZERO
211: ELSE
212: TAU = TWO
213: DO J = 1, N-1
214: X( 1 + (J-1)*INCX ) = 0
215: END DO
216: BETA = -SAVEALPHA
217: END IF
218: *
219: ELSE
220: *
221: * This is the general case.
222: *
223: CALL DSCAL( N-1, ONE / ALPHA, X, INCX )
224: *
225: END IF
226: *
227: * If BETA is subnormal, it may lose relative accuracy
228: *
229: DO 20 J = 1, KNT
230: BETA = BETA*SMLNUM
231: 20 CONTINUE
232: ALPHA = BETA
233: END IF
234: *
235: RETURN
236: *
237: * End of DLARFGP
238: *
239: END
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