1: *> \brief \b DLAS2 computes singular values of a 2-by-2 triangular matrix.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAS2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlas2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlas2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlas2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
22: *
23: * .. Scalar Arguments ..
24: * DOUBLE PRECISION F, G, H, SSMAX, SSMIN
25: * ..
26: *
27: *
28: *> \par Purpose:
29: * =============
30: *>
31: *> \verbatim
32: *>
33: *> DLAS2 computes the singular values of the 2-by-2 matrix
34: *> [ F G ]
35: *> [ 0 H ].
36: *> On return, SSMIN is the smaller singular value and SSMAX is the
37: *> larger singular value.
38: *> \endverbatim
39: *
40: * Arguments:
41: * ==========
42: *
43: *> \param[in] F
44: *> \verbatim
45: *> F is DOUBLE PRECISION
46: *> The (1,1) element of the 2-by-2 matrix.
47: *> \endverbatim
48: *>
49: *> \param[in] G
50: *> \verbatim
51: *> G is DOUBLE PRECISION
52: *> The (1,2) element of the 2-by-2 matrix.
53: *> \endverbatim
54: *>
55: *> \param[in] H
56: *> \verbatim
57: *> H is DOUBLE PRECISION
58: *> The (2,2) element of the 2-by-2 matrix.
59: *> \endverbatim
60: *>
61: *> \param[out] SSMIN
62: *> \verbatim
63: *> SSMIN is DOUBLE PRECISION
64: *> The smaller singular value.
65: *> \endverbatim
66: *>
67: *> \param[out] SSMAX
68: *> \verbatim
69: *> SSMAX is DOUBLE PRECISION
70: *> The larger singular value.
71: *> \endverbatim
72: *
73: * Authors:
74: * ========
75: *
76: *> \author Univ. of Tennessee
77: *> \author Univ. of California Berkeley
78: *> \author Univ. of Colorado Denver
79: *> \author NAG Ltd.
80: *
81: *> \date September 2012
82: *
83: *> \ingroup auxOTHERauxiliary
84: *
85: *> \par Further Details:
86: * =====================
87: *>
88: *> \verbatim
89: *>
90: *> Barring over/underflow, all output quantities are correct to within
91: *> a few units in the last place (ulps), even in the absence of a guard
92: *> digit in addition/subtraction.
93: *>
94: *> In IEEE arithmetic, the code works correctly if one matrix element is
95: *> infinite.
96: *>
97: *> Overflow will not occur unless the largest singular value itself
98: *> overflows, or is within a few ulps of overflow. (On machines with
99: *> partial overflow, like the Cray, overflow may occur if the largest
100: *> singular value is within a factor of 2 of overflow.)
101: *>
102: *> Underflow is harmless if underflow is gradual. Otherwise, results
103: *> may correspond to a matrix modified by perturbations of size near
104: *> the underflow threshold.
105: *> \endverbatim
106: *>
107: * =====================================================================
108: SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX )
109: *
110: * -- LAPACK auxiliary routine (version 3.4.2) --
111: * -- LAPACK is a software package provided by Univ. of Tennessee, --
112: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
113: * September 2012
114: *
115: * .. Scalar Arguments ..
116: DOUBLE PRECISION F, G, H, SSMAX, SSMIN
117: * ..
118: *
119: * ====================================================================
120: *
121: * .. Parameters ..
122: DOUBLE PRECISION ZERO
123: PARAMETER ( ZERO = 0.0D0 )
124: DOUBLE PRECISION ONE
125: PARAMETER ( ONE = 1.0D0 )
126: DOUBLE PRECISION TWO
127: PARAMETER ( TWO = 2.0D0 )
128: * ..
129: * .. Local Scalars ..
130: DOUBLE PRECISION AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
131: * ..
132: * .. Intrinsic Functions ..
133: INTRINSIC ABS, MAX, MIN, SQRT
134: * ..
135: * .. Executable Statements ..
136: *
137: FA = ABS( F )
138: GA = ABS( G )
139: HA = ABS( H )
140: FHMN = MIN( FA, HA )
141: FHMX = MAX( FA, HA )
142: IF( FHMN.EQ.ZERO ) THEN
143: SSMIN = ZERO
144: IF( FHMX.EQ.ZERO ) THEN
145: SSMAX = GA
146: ELSE
147: SSMAX = MAX( FHMX, GA )*SQRT( ONE+
148: $ ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
149: END IF
150: ELSE
151: IF( GA.LT.FHMX ) THEN
152: AS = ONE + FHMN / FHMX
153: AT = ( FHMX-FHMN ) / FHMX
154: AU = ( GA / FHMX )**2
155: C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
156: SSMIN = FHMN*C
157: SSMAX = FHMX / C
158: ELSE
159: AU = FHMX / GA
160: IF( AU.EQ.ZERO ) THEN
161: *
162: * Avoid possible harmful underflow if exponent range
163: * asymmetric (true SSMIN may not underflow even if
164: * AU underflows)
165: *
166: SSMIN = ( FHMN*FHMX ) / GA
167: SSMAX = GA
168: ELSE
169: AS = ONE + FHMN / FHMX
170: AT = ( FHMX-FHMN ) / FHMX
171: C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
172: $ SQRT( ONE+( AT*AU )**2 ) )
173: SSMIN = ( FHMN*C )*AU
174: SSMIN = SSMIN + SSMIN
175: SSMAX = GA / ( C+C )
176: END IF
177: END IF
178: END IF
179: RETURN
180: *
181: * End of DLAS2
182: *
183: END
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