Annotation of rpl/lapack/lapack/dlarfgp.f, revision 1.8
1.6 bertrand 1: *> \brief \b DLARFGP
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: *> \date November 2011
101: *
102: *> \ingroup doubleOTHERauxiliary
103: *
104: * =====================================================================
1.1 bertrand 105: SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
106: *
1.6 bertrand 107: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 108: * -- LAPACK is a software package provided by Univ. of Tennessee, --
109: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.6 bertrand 110: * November 2011
1.1 bertrand 111: *
112: * .. Scalar Arguments ..
113: INTEGER INCX, N
114: DOUBLE PRECISION ALPHA, TAU
115: * ..
116: * .. Array Arguments ..
117: DOUBLE PRECISION X( * )
118: * ..
119: *
120: * =====================================================================
121: *
122: * .. Parameters ..
123: DOUBLE PRECISION TWO, ONE, ZERO
124: PARAMETER ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
125: * ..
126: * .. Local Scalars ..
127: INTEGER J, KNT
128: DOUBLE PRECISION BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
129: * ..
130: * .. External Functions ..
131: DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
132: EXTERNAL DLAMCH, DLAPY2, DNRM2
133: * ..
134: * .. Intrinsic Functions ..
135: INTRINSIC ABS, SIGN
136: * ..
137: * .. External Subroutines ..
138: EXTERNAL DSCAL
139: * ..
140: * .. Executable Statements ..
141: *
142: IF( N.LE.0 ) THEN
143: TAU = ZERO
144: RETURN
145: END IF
146: *
147: XNORM = DNRM2( N-1, X, INCX )
148: *
149: IF( XNORM.EQ.ZERO ) THEN
150: *
151: * H = [+/-1, 0; I], sign chosen so ALPHA >= 0
152: *
153: IF( ALPHA.GE.ZERO ) THEN
154: * When TAU.eq.ZERO, the vector is special-cased to be
155: * all zeros in the application routines. We do not need
156: * to clear it.
157: TAU = ZERO
158: ELSE
159: * However, the application routines rely on explicit
160: * zero checks when TAU.ne.ZERO, and we must clear X.
161: TAU = TWO
162: DO J = 1, N-1
163: X( 1 + (J-1)*INCX ) = 0
164: END DO
165: ALPHA = -ALPHA
166: END IF
167: ELSE
168: *
169: * general case
170: *
171: BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
172: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'E' )
173: KNT = 0
174: IF( ABS( BETA ).LT.SMLNUM ) THEN
175: *
176: * XNORM, BETA may be inaccurate; scale X and recompute them
177: *
178: BIGNUM = ONE / SMLNUM
179: 10 CONTINUE
180: KNT = KNT + 1
181: CALL DSCAL( N-1, BIGNUM, X, INCX )
182: BETA = BETA*BIGNUM
183: ALPHA = ALPHA*BIGNUM
184: IF( ABS( BETA ).LT.SMLNUM )
185: $ GO TO 10
186: *
187: * New BETA is at most 1, at least SMLNUM
188: *
189: XNORM = DNRM2( N-1, X, INCX )
190: BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
191: END IF
192: SAVEALPHA = ALPHA
193: ALPHA = ALPHA + BETA
194: IF( BETA.LT.ZERO ) THEN
195: BETA = -BETA
196: TAU = -ALPHA / BETA
197: ELSE
198: ALPHA = XNORM * (XNORM/ALPHA)
199: TAU = ALPHA / BETA
200: ALPHA = -ALPHA
201: END IF
202: *
203: IF ( ABS(TAU).LE.SMLNUM ) THEN
204: *
205: * In the case where the computed TAU ends up being a denormalized number,
206: * it loses relative accuracy. This is a BIG problem. Solution: flush TAU
207: * to ZERO. This explains the next IF statement.
208: *
209: * (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
210: * (Thanks Pat. Thanks MathWorks.)
211: *
212: IF( SAVEALPHA.GE.ZERO ) THEN
213: TAU = ZERO
214: ELSE
215: TAU = TWO
216: DO J = 1, N-1
217: X( 1 + (J-1)*INCX ) = 0
218: END DO
219: BETA = -SAVEALPHA
220: END IF
221: *
222: ELSE
223: *
224: * This is the general case.
225: *
226: CALL DSCAL( N-1, ONE / ALPHA, X, INCX )
227: *
228: END IF
229: *
230: * If BETA is subnormal, it may lose relative accuracy
231: *
232: DO 20 J = 1, KNT
233: BETA = BETA*SMLNUM
234: 20 CONTINUE
235: ALPHA = BETA
236: END IF
237: *
238: RETURN
239: *
240: * End of DLARFGP
241: *
242: END
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