1: *> \brief \b DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
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: *> \ingroup doubleOTHERauxiliary
113: *
114: *> \par Further Details:
115: * =====================
116: *>
117: *> \verbatim
118: *>
119: *> Modified by V. Sima, Research Institute for Informatics, Bucharest,
120: *> Romania, to reduce the risk of cancellation errors,
121: *> when computing real eigenvalues, and to ensure, if possible, that
122: *> abs(RT1R) >= abs(RT2R).
123: *> \endverbatim
124: *>
125: * =====================================================================
126: SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
127: *
128: * -- LAPACK auxiliary routine --
129: * -- LAPACK is a software package provided by Univ. of Tennessee, --
130: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
131: *
132: * .. Scalar Arguments ..
133: DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
134: * ..
135: *
136: * =====================================================================
137: *
138: * .. Parameters ..
139: DOUBLE PRECISION ZERO, HALF, ONE, TWO
140: PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
141: $ TWO = 2.0D0 )
142: DOUBLE PRECISION MULTPL
143: PARAMETER ( MULTPL = 4.0D+0 )
144: * ..
145: * .. Local Scalars ..
146: DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
147: $ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z, SAFMIN,
148: $ SAFMN2, SAFMX2
149: INTEGER COUNT
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: SAFMIN = DLAMCH( 'S' )
161: EPS = DLAMCH( 'P' )
162: SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
163: $ LOG( DLAMCH( 'B' ) ) / TWO )
164: SAFMX2 = ONE / SAFMN2
165: IF( C.EQ.ZERO ) THEN
166: CS = ONE
167: SN = ZERO
168: *
169: ELSE IF( B.EQ.ZERO ) THEN
170: *
171: * Swap rows and columns
172: *
173: CS = ZERO
174: SN = ONE
175: TEMP = D
176: D = A
177: A = TEMP
178: B = -C
179: C = ZERO
180: *
181: ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
182: $ THEN
183: CS = ONE
184: SN = ZERO
185: *
186: ELSE
187: *
188: TEMP = A - D
189: P = HALF*TEMP
190: BCMAX = MAX( ABS( B ), ABS( C ) )
191: BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
192: SCALE = MAX( ABS( P ), BCMAX )
193: Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
194: *
195: * If Z is of the order of the machine accuracy, postpone the
196: * decision on the nature of eigenvalues
197: *
198: IF( Z.GE.MULTPL*EPS ) THEN
199: *
200: * Real eigenvalues. Compute A and D.
201: *
202: Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
203: A = D + Z
204: D = D - ( BCMAX / Z )*BCMIS
205: *
206: * Compute B and the rotation matrix
207: *
208: TAU = DLAPY2( C, Z )
209: CS = Z / TAU
210: SN = C / TAU
211: B = B - C
212: C = ZERO
213: *
214: ELSE
215: *
216: * Complex eigenvalues, or real (almost) equal eigenvalues.
217: * Make diagonal elements equal.
218: *
219: COUNT = 0
220: SIGMA = B + C
221: 10 CONTINUE
222: COUNT = COUNT + 1
223: SCALE = MAX( ABS(TEMP), ABS(SIGMA) )
224: IF( SCALE.GE.SAFMX2 ) THEN
225: SIGMA = SIGMA * SAFMN2
226: TEMP = TEMP * SAFMN2
227: IF (COUNT .LE. 20)
228: $ GOTO 10
229: END IF
230: IF( SCALE.LE.SAFMN2 ) THEN
231: SIGMA = SIGMA * SAFMX2
232: TEMP = TEMP * SAFMX2
233: IF (COUNT .LE. 20)
234: $ GOTO 10
235: END IF
236: P = HALF*TEMP
237: TAU = DLAPY2( SIGMA, TEMP )
238: CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
239: SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
240: *
241: * Compute [ AA BB ] = [ A B ] [ CS -SN ]
242: * [ CC DD ] [ C D ] [ SN CS ]
243: *
244: AA = A*CS + B*SN
245: BB = -A*SN + B*CS
246: CC = C*CS + D*SN
247: DD = -C*SN + D*CS
248: *
249: * Compute [ A B ] = [ CS SN ] [ AA BB ]
250: * [ C D ] [-SN CS ] [ CC DD ]
251: *
252: A = AA*CS + CC*SN
253: B = BB*CS + DD*SN
254: C = -AA*SN + CC*CS
255: D = -BB*SN + DD*CS
256: *
257: TEMP = HALF*( A+D )
258: A = TEMP
259: D = TEMP
260: *
261: IF( C.NE.ZERO ) THEN
262: IF( B.NE.ZERO ) THEN
263: IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
264: *
265: * Real eigenvalues: reduce to upper triangular form
266: *
267: SAB = SQRT( ABS( B ) )
268: SAC = SQRT( ABS( C ) )
269: P = SIGN( SAB*SAC, C )
270: TAU = ONE / SQRT( ABS( B+C ) )
271: A = TEMP + P
272: D = TEMP - P
273: B = B - C
274: C = ZERO
275: CS1 = SAB*TAU
276: SN1 = SAC*TAU
277: TEMP = CS*CS1 - SN*SN1
278: SN = CS*SN1 + SN*CS1
279: CS = TEMP
280: END IF
281: ELSE
282: B = -C
283: C = ZERO
284: TEMP = CS
285: CS = -SN
286: SN = TEMP
287: END IF
288: END IF
289: END IF
290: *
291: END IF
292: *
293: * Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
294: *
295: RT1R = A
296: RT2R = D
297: IF( C.EQ.ZERO ) THEN
298: RT1I = ZERO
299: RT2I = ZERO
300: ELSE
301: RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
302: RT2I = -RT1I
303: END IF
304: RETURN
305: *
306: * End of DLANV2
307: *
308: END
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