1: *> \brief \b DLANV2
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLANV2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlanv2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanv2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanv2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
22: *
23: * .. Scalar Arguments ..
24: * DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
25: * ..
26: *
27: *
28: *> \par Purpose:
29: * =============
30: *>
31: *> \verbatim
32: *>
33: *> DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
34: *> matrix in standard form:
35: *>
36: *> [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
37: *> [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
38: *>
39: *> where either
40: *> 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
41: *> 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
42: *> conjugate eigenvalues.
43: *> \endverbatim
44: *
45: * Arguments:
46: * ==========
47: *
48: *> \param[in,out] A
49: *> \verbatim
50: *> A is DOUBLE PRECISION
51: *> \endverbatim
52: *>
53: *> \param[in,out] B
54: *> \verbatim
55: *> B is DOUBLE PRECISION
56: *> \endverbatim
57: *>
58: *> \param[in,out] C
59: *> \verbatim
60: *> C is DOUBLE PRECISION
61: *> \endverbatim
62: *>
63: *> \param[in,out] D
64: *> \verbatim
65: *> D is DOUBLE PRECISION
66: *> On entry, the elements of the input matrix.
67: *> On exit, they are overwritten by the elements of the
68: *> standardised Schur form.
69: *> \endverbatim
70: *>
71: *> \param[out] RT1R
72: *> \verbatim
73: *> RT1R is DOUBLE PRECISION
74: *> \endverbatim
75: *>
76: *> \param[out] RT1I
77: *> \verbatim
78: *> RT1I is DOUBLE PRECISION
79: *> \endverbatim
80: *>
81: *> \param[out] RT2R
82: *> \verbatim
83: *> RT2R is DOUBLE PRECISION
84: *> \endverbatim
85: *>
86: *> \param[out] RT2I
87: *> \verbatim
88: *> RT2I is DOUBLE PRECISION
89: *> The real and imaginary parts of the eigenvalues. If the
90: *> eigenvalues are a complex conjugate pair, RT1I > 0.
91: *> \endverbatim
92: *>
93: *> \param[out] CS
94: *> \verbatim
95: *> CS is DOUBLE PRECISION
96: *> \endverbatim
97: *>
98: *> \param[out] SN
99: *> \verbatim
100: *> SN is DOUBLE PRECISION
101: *> Parameters of the rotation matrix.
102: *> \endverbatim
103: *
104: * Authors:
105: * ========
106: *
107: *> \author Univ. of Tennessee
108: *> \author Univ. of California Berkeley
109: *> \author Univ. of Colorado Denver
110: *> \author NAG Ltd.
111: *
112: *> \date November 2011
113: *
114: *> \ingroup doubleOTHERauxiliary
115: *
116: *> \par Further Details:
117: * =====================
118: *>
119: *> \verbatim
120: *>
121: *> Modified by V. Sima, Research Institute for Informatics, Bucharest,
122: *> Romania, to reduce the risk of cancellation errors,
123: *> when computing real eigenvalues, and to ensure, if possible, that
124: *> abs(RT1R) >= abs(RT2R).
125: *> \endverbatim
126: *>
127: * =====================================================================
128: SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
129: *
130: * -- LAPACK auxiliary routine (version 3.4.0) --
131: * -- LAPACK is a software package provided by Univ. of Tennessee, --
132: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133: * November 2011
134: *
135: * .. Scalar Arguments ..
136: DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
137: * ..
138: *
139: * =====================================================================
140: *
141: * .. Parameters ..
142: DOUBLE PRECISION ZERO, HALF, ONE
143: PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
144: DOUBLE PRECISION MULTPL
145: PARAMETER ( MULTPL = 4.0D+0 )
146: * ..
147: * .. Local Scalars ..
148: DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
149: $ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
150: * ..
151: * .. External Functions ..
152: DOUBLE PRECISION DLAMCH, DLAPY2
153: EXTERNAL DLAMCH, DLAPY2
154: * ..
155: * .. Intrinsic Functions ..
156: INTRINSIC ABS, MAX, MIN, SIGN, SQRT
157: * ..
158: * .. Executable Statements ..
159: *
160: EPS = DLAMCH( 'P' )
161: IF( C.EQ.ZERO ) THEN
162: CS = ONE
163: SN = ZERO
164: GO TO 10
165: *
166: ELSE IF( B.EQ.ZERO ) THEN
167: *
168: * Swap rows and columns
169: *
170: CS = ZERO
171: SN = ONE
172: TEMP = D
173: D = A
174: A = TEMP
175: B = -C
176: C = ZERO
177: GO TO 10
178: ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
179: $ THEN
180: CS = ONE
181: SN = ZERO
182: GO TO 10
183: ELSE
184: *
185: TEMP = A - D
186: P = HALF*TEMP
187: BCMAX = MAX( ABS( B ), ABS( C ) )
188: BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
189: SCALE = MAX( ABS( P ), BCMAX )
190: Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
191: *
192: * If Z is of the order of the machine accuracy, postpone the
193: * decision on the nature of eigenvalues
194: *
195: IF( Z.GE.MULTPL*EPS ) THEN
196: *
197: * Real eigenvalues. Compute A and D.
198: *
199: Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
200: A = D + Z
201: D = D - ( BCMAX / Z )*BCMIS
202: *
203: * Compute B and the rotation matrix
204: *
205: TAU = DLAPY2( C, Z )
206: CS = Z / TAU
207: SN = C / TAU
208: B = B - C
209: C = ZERO
210: ELSE
211: *
212: * Complex eigenvalues, or real (almost) equal eigenvalues.
213: * Make diagonal elements equal.
214: *
215: SIGMA = B + C
216: TAU = DLAPY2( SIGMA, TEMP )
217: CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
218: SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
219: *
220: * Compute [ AA BB ] = [ A B ] [ CS -SN ]
221: * [ CC DD ] [ C D ] [ SN CS ]
222: *
223: AA = A*CS + B*SN
224: BB = -A*SN + B*CS
225: CC = C*CS + D*SN
226: DD = -C*SN + D*CS
227: *
228: * Compute [ A B ] = [ CS SN ] [ AA BB ]
229: * [ C D ] [-SN CS ] [ CC DD ]
230: *
231: A = AA*CS + CC*SN
232: B = BB*CS + DD*SN
233: C = -AA*SN + CC*CS
234: D = -BB*SN + DD*CS
235: *
236: TEMP = HALF*( A+D )
237: A = TEMP
238: D = TEMP
239: *
240: IF( C.NE.ZERO ) THEN
241: IF( B.NE.ZERO ) THEN
242: IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
243: *
244: * Real eigenvalues: reduce to upper triangular form
245: *
246: SAB = SQRT( ABS( B ) )
247: SAC = SQRT( ABS( C ) )
248: P = SIGN( SAB*SAC, C )
249: TAU = ONE / SQRT( ABS( B+C ) )
250: A = TEMP + P
251: D = TEMP - P
252: B = B - C
253: C = ZERO
254: CS1 = SAB*TAU
255: SN1 = SAC*TAU
256: TEMP = CS*CS1 - SN*SN1
257: SN = CS*SN1 + SN*CS1
258: CS = TEMP
259: END IF
260: ELSE
261: B = -C
262: C = ZERO
263: TEMP = CS
264: CS = -SN
265: SN = TEMP
266: END IF
267: END IF
268: END IF
269: *
270: END IF
271: *
272: 10 CONTINUE
273: *
274: * Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
275: *
276: RT1R = A
277: RT2R = D
278: IF( C.EQ.ZERO ) THEN
279: RT1I = ZERO
280: RT2I = ZERO
281: ELSE
282: RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
283: RT2I = -RT1I
284: END IF
285: RETURN
286: *
287: * End of DLANV2
288: *
289: END
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