1: *> \brief \b DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAGV2 + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagv2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagv2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
22: * CSR, SNR )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER LDA, LDB
26: * DOUBLE PRECISION CSL, CSR, SNL, SNR
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
30: * $ B( LDB, * ), BETA( 2 )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
40: *> matrix pencil (A,B) where B is upper triangular. This routine
41: *> computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
42: *> SNR such that
43: *>
44: *> 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
45: *> types), then
46: *>
47: *> [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
48: *> [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
49: *>
50: *> [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
51: *> [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],
52: *>
53: *> 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
54: *> then
55: *>
56: *> [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
57: *> [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
58: *>
59: *> [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
60: *> [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]
61: *>
62: *> where b11 >= b22 > 0.
63: *>
64: *> \endverbatim
65: *
66: * Arguments:
67: * ==========
68: *
69: *> \param[in,out] A
70: *> \verbatim
71: *> A is DOUBLE PRECISION array, dimension (LDA, 2)
72: *> On entry, the 2 x 2 matrix A.
73: *> On exit, A is overwritten by the ``A-part'' of the
74: *> generalized Schur form.
75: *> \endverbatim
76: *>
77: *> \param[in] LDA
78: *> \verbatim
79: *> LDA is INTEGER
80: *> THe leading dimension of the array A. LDA >= 2.
81: *> \endverbatim
82: *>
83: *> \param[in,out] B
84: *> \verbatim
85: *> B is DOUBLE PRECISION array, dimension (LDB, 2)
86: *> On entry, the upper triangular 2 x 2 matrix B.
87: *> On exit, B is overwritten by the ``B-part'' of the
88: *> generalized Schur form.
89: *> \endverbatim
90: *>
91: *> \param[in] LDB
92: *> \verbatim
93: *> LDB is INTEGER
94: *> THe leading dimension of the array B. LDB >= 2.
95: *> \endverbatim
96: *>
97: *> \param[out] ALPHAR
98: *> \verbatim
99: *> ALPHAR is DOUBLE PRECISION array, dimension (2)
100: *> \endverbatim
101: *>
102: *> \param[out] ALPHAI
103: *> \verbatim
104: *> ALPHAI is DOUBLE PRECISION array, dimension (2)
105: *> \endverbatim
106: *>
107: *> \param[out] BETA
108: *> \verbatim
109: *> BETA is DOUBLE PRECISION array, dimension (2)
110: *> (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
111: *> pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
112: *> be zero.
113: *> \endverbatim
114: *>
115: *> \param[out] CSL
116: *> \verbatim
117: *> CSL is DOUBLE PRECISION
118: *> The cosine of the left rotation matrix.
119: *> \endverbatim
120: *>
121: *> \param[out] SNL
122: *> \verbatim
123: *> SNL is DOUBLE PRECISION
124: *> The sine of the left rotation matrix.
125: *> \endverbatim
126: *>
127: *> \param[out] CSR
128: *> \verbatim
129: *> CSR is DOUBLE PRECISION
130: *> The cosine of the right rotation matrix.
131: *> \endverbatim
132: *>
133: *> \param[out] SNR
134: *> \verbatim
135: *> SNR is DOUBLE PRECISION
136: *> The sine of the right rotation matrix.
137: *> \endverbatim
138: *
139: * Authors:
140: * ========
141: *
142: *> \author Univ. of Tennessee
143: *> \author Univ. of California Berkeley
144: *> \author Univ. of Colorado Denver
145: *> \author NAG Ltd.
146: *
147: *> \date December 2016
148: *
149: *> \ingroup doubleOTHERauxiliary
150: *
151: *> \par Contributors:
152: * ==================
153: *>
154: *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
155: *
156: * =====================================================================
157: SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
158: $ CSR, SNR )
159: *
160: * -- LAPACK auxiliary routine (version 3.7.0) --
161: * -- LAPACK is a software package provided by Univ. of Tennessee, --
162: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
163: * December 2016
164: *
165: * .. Scalar Arguments ..
166: INTEGER LDA, LDB
167: DOUBLE PRECISION CSL, CSR, SNL, SNR
168: * ..
169: * .. Array Arguments ..
170: DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
171: $ B( LDB, * ), BETA( 2 )
172: * ..
173: *
174: * =====================================================================
175: *
176: * .. Parameters ..
177: DOUBLE PRECISION ZERO, ONE
178: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
179: * ..
180: * .. Local Scalars ..
181: DOUBLE PRECISION ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ,
182: $ R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1,
183: $ WR2
184: * ..
185: * .. External Subroutines ..
186: EXTERNAL DLAG2, DLARTG, DLASV2, DROT
187: * ..
188: * .. External Functions ..
189: DOUBLE PRECISION DLAMCH, DLAPY2
190: EXTERNAL DLAMCH, DLAPY2
191: * ..
192: * .. Intrinsic Functions ..
193: INTRINSIC ABS, MAX
194: * ..
195: * .. Executable Statements ..
196: *
197: SAFMIN = DLAMCH( 'S' )
198: ULP = DLAMCH( 'P' )
199: *
200: * Scale A
201: *
202: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
203: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
204: ASCALE = ONE / ANORM
205: A( 1, 1 ) = ASCALE*A( 1, 1 )
206: A( 1, 2 ) = ASCALE*A( 1, 2 )
207: A( 2, 1 ) = ASCALE*A( 2, 1 )
208: A( 2, 2 ) = ASCALE*A( 2, 2 )
209: *
210: * Scale B
211: *
212: BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
213: $ SAFMIN )
214: BSCALE = ONE / BNORM
215: B( 1, 1 ) = BSCALE*B( 1, 1 )
216: B( 1, 2 ) = BSCALE*B( 1, 2 )
217: B( 2, 2 ) = BSCALE*B( 2, 2 )
218: *
219: * Check if A can be deflated
220: *
221: IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN
222: CSL = ONE
223: SNL = ZERO
224: CSR = ONE
225: SNR = ZERO
226: A( 2, 1 ) = ZERO
227: B( 2, 1 ) = ZERO
228: WI = ZERO
229: *
230: * Check if B is singular
231: *
232: ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
233: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
234: CSR = ONE
235: SNR = ZERO
236: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
237: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
238: A( 2, 1 ) = ZERO
239: B( 1, 1 ) = ZERO
240: B( 2, 1 ) = ZERO
241: WI = ZERO
242: *
243: ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN
244: CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T )
245: SNR = -SNR
246: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
247: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
248: CSL = ONE
249: SNL = ZERO
250: A( 2, 1 ) = ZERO
251: B( 2, 1 ) = ZERO
252: B( 2, 2 ) = ZERO
253: WI = ZERO
254: *
255: ELSE
256: *
257: * B is nonsingular, first compute the eigenvalues of (A,B)
258: *
259: CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,
260: $ WI )
261: *
262: IF( WI.EQ.ZERO ) THEN
263: *
264: * two real eigenvalues, compute s*A-w*B
265: *
266: H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 )
267: H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 )
268: H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 )
269: *
270: RR = DLAPY2( H1, H2 )
271: QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
272: *
273: IF( RR.GT.QQ ) THEN
274: *
275: * find right rotation matrix to zero 1,1 element of
276: * (sA - wB)
277: *
278: CALL DLARTG( H2, H1, CSR, SNR, T )
279: *
280: ELSE
281: *
282: * find right rotation matrix to zero 2,1 element of
283: * (sA - wB)
284: *
285: CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
286: *
287: END IF
288: *
289: SNR = -SNR
290: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
291: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
292: *
293: * compute inf norms of A and B
294: *
295: H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ),
296: $ ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) )
297: H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
298: $ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
299: *
300: IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
301: *
302: * find left rotation matrix Q to zero out B(2,1)
303: *
304: CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
305: *
306: ELSE
307: *
308: * find left rotation matrix Q to zero out A(2,1)
309: *
310: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
311: *
312: END IF
313: *
314: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
315: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
316: *
317: A( 2, 1 ) = ZERO
318: B( 2, 1 ) = ZERO
319: *
320: ELSE
321: *
322: * a pair of complex conjugate eigenvalues
323: * first compute the SVD of the matrix B
324: *
325: CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
326: $ CSR, SNL, CSL )
327: *
328: * Form (A,B) := Q(A,B)Z**T where Q is left rotation matrix and
329: * Z is right rotation matrix computed from DLASV2
330: *
331: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
332: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
333: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
334: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
335: *
336: B( 2, 1 ) = ZERO
337: B( 1, 2 ) = ZERO
338: *
339: END IF
340: *
341: END IF
342: *
343: * Unscaling
344: *
345: A( 1, 1 ) = ANORM*A( 1, 1 )
346: A( 2, 1 ) = ANORM*A( 2, 1 )
347: A( 1, 2 ) = ANORM*A( 1, 2 )
348: A( 2, 2 ) = ANORM*A( 2, 2 )
349: B( 1, 1 ) = BNORM*B( 1, 1 )
350: B( 2, 1 ) = BNORM*B( 2, 1 )
351: B( 1, 2 ) = BNORM*B( 1, 2 )
352: B( 2, 2 ) = BNORM*B( 2, 2 )
353: *
354: IF( WI.EQ.ZERO ) THEN
355: ALPHAR( 1 ) = A( 1, 1 )
356: ALPHAR( 2 ) = A( 2, 2 )
357: ALPHAI( 1 ) = ZERO
358: ALPHAI( 2 ) = ZERO
359: BETA( 1 ) = B( 1, 1 )
360: BETA( 2 ) = B( 2, 2 )
361: ELSE
362: ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
363: ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
364: ALPHAR( 2 ) = ALPHAR( 1 )
365: ALPHAI( 2 ) = -ALPHAI( 1 )
366: BETA( 1 ) = ONE
367: BETA( 2 ) = ONE
368: END IF
369: *
370: RETURN
371: *
372: * End of DLAGV2
373: *
374: END
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