1: SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
2: $ CSR, SNR )
3: *
4: * -- LAPACK auxiliary routine (version 3.2.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * June 2010
8: *
9: * .. Scalar Arguments ..
10: INTEGER LDA, LDB
11: DOUBLE PRECISION CSL, CSR, SNL, SNR
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ),
15: $ B( LDB, * ), BETA( 2 )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
22: * matrix pencil (A,B) where B is upper triangular. This routine
23: * computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
24: * SNR such that
25: *
26: * 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
27: * types), then
28: *
29: * [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
30: * [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
31: *
32: * [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
33: * [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],
34: *
35: * 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
36: * then
37: *
38: * [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
39: * [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
40: *
41: * [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
42: * [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]
43: *
44: * where b11 >= b22 > 0.
45: *
46: *
47: * Arguments
48: * =========
49: *
50: * A (input/output) DOUBLE PRECISION array, dimension (LDA, 2)
51: * On entry, the 2 x 2 matrix A.
52: * On exit, A is overwritten by the ``A-part'' of the
53: * generalized Schur form.
54: *
55: * LDA (input) INTEGER
56: * THe leading dimension of the array A. LDA >= 2.
57: *
58: * B (input/output) DOUBLE PRECISION array, dimension (LDB, 2)
59: * On entry, the upper triangular 2 x 2 matrix B.
60: * On exit, B is overwritten by the ``B-part'' of the
61: * generalized Schur form.
62: *
63: * LDB (input) INTEGER
64: * THe leading dimension of the array B. LDB >= 2.
65: *
66: * ALPHAR (output) DOUBLE PRECISION array, dimension (2)
67: * ALPHAI (output) DOUBLE PRECISION array, dimension (2)
68: * BETA (output) DOUBLE PRECISION array, dimension (2)
69: * (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
70: * pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
71: * be zero.
72: *
73: * CSL (output) DOUBLE PRECISION
74: * The cosine of the left rotation matrix.
75: *
76: * SNL (output) DOUBLE PRECISION
77: * The sine of the left rotation matrix.
78: *
79: * CSR (output) DOUBLE PRECISION
80: * The cosine of the right rotation matrix.
81: *
82: * SNR (output) DOUBLE PRECISION
83: * The sine of the right rotation matrix.
84: *
85: * Further Details
86: * ===============
87: *
88: * Based on contributions by
89: * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
90: *
91: * =====================================================================
92: *
93: * .. Parameters ..
94: DOUBLE PRECISION ZERO, ONE
95: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
96: * ..
97: * .. Local Scalars ..
98: DOUBLE PRECISION ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ,
99: $ R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1,
100: $ WR2
101: * ..
102: * .. External Subroutines ..
103: EXTERNAL DLAG2, DLARTG, DLASV2, DROT
104: * ..
105: * .. External Functions ..
106: DOUBLE PRECISION DLAMCH, DLAPY2
107: EXTERNAL DLAMCH, DLAPY2
108: * ..
109: * .. Intrinsic Functions ..
110: INTRINSIC ABS, MAX
111: * ..
112: * .. Executable Statements ..
113: *
114: SAFMIN = DLAMCH( 'S' )
115: ULP = DLAMCH( 'P' )
116: *
117: * Scale A
118: *
119: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
120: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
121: ASCALE = ONE / ANORM
122: A( 1, 1 ) = ASCALE*A( 1, 1 )
123: A( 1, 2 ) = ASCALE*A( 1, 2 )
124: A( 2, 1 ) = ASCALE*A( 2, 1 )
125: A( 2, 2 ) = ASCALE*A( 2, 2 )
126: *
127: * Scale B
128: *
129: BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
130: $ SAFMIN )
131: BSCALE = ONE / BNORM
132: B( 1, 1 ) = BSCALE*B( 1, 1 )
133: B( 1, 2 ) = BSCALE*B( 1, 2 )
134: B( 2, 2 ) = BSCALE*B( 2, 2 )
135: *
136: * Check if A can be deflated
137: *
138: IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN
139: CSL = ONE
140: SNL = ZERO
141: CSR = ONE
142: SNR = ZERO
143: A( 2, 1 ) = ZERO
144: B( 2, 1 ) = ZERO
145: WI = ZERO
146: *
147: * Check if B is singular
148: *
149: ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
150: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
151: CSR = ONE
152: SNR = ZERO
153: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
154: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
155: A( 2, 1 ) = ZERO
156: B( 1, 1 ) = ZERO
157: B( 2, 1 ) = ZERO
158: WI = ZERO
159: *
160: ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN
161: CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T )
162: SNR = -SNR
163: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
164: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
165: CSL = ONE
166: SNL = ZERO
167: A( 2, 1 ) = ZERO
168: B( 2, 1 ) = ZERO
169: B( 2, 2 ) = ZERO
170: WI = ZERO
171: *
172: ELSE
173: *
174: * B is nonsingular, first compute the eigenvalues of (A,B)
175: *
176: CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,
177: $ WI )
178: *
179: IF( WI.EQ.ZERO ) THEN
180: *
181: * two real eigenvalues, compute s*A-w*B
182: *
183: H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 )
184: H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 )
185: H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 )
186: *
187: RR = DLAPY2( H1, H2 )
188: QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
189: *
190: IF( RR.GT.QQ ) THEN
191: *
192: * find right rotation matrix to zero 1,1 element of
193: * (sA - wB)
194: *
195: CALL DLARTG( H2, H1, CSR, SNR, T )
196: *
197: ELSE
198: *
199: * find right rotation matrix to zero 2,1 element of
200: * (sA - wB)
201: *
202: CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
203: *
204: END IF
205: *
206: SNR = -SNR
207: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
208: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
209: *
210: * compute inf norms of A and B
211: *
212: H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ),
213: $ ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) )
214: H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
215: $ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
216: *
217: IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
218: *
219: * find left rotation matrix Q to zero out B(2,1)
220: *
221: CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
222: *
223: ELSE
224: *
225: * find left rotation matrix Q to zero out A(2,1)
226: *
227: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
228: *
229: END IF
230: *
231: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
232: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
233: *
234: A( 2, 1 ) = ZERO
235: B( 2, 1 ) = ZERO
236: *
237: ELSE
238: *
239: * a pair of complex conjugate eigenvalues
240: * first compute the SVD of the matrix B
241: *
242: CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
243: $ CSR, SNL, CSL )
244: *
245: * Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and
246: * Z is right rotation matrix computed from DLASV2
247: *
248: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
249: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
250: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
251: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
252: *
253: B( 2, 1 ) = ZERO
254: B( 1, 2 ) = ZERO
255: *
256: END IF
257: *
258: END IF
259: *
260: * Unscaling
261: *
262: A( 1, 1 ) = ANORM*A( 1, 1 )
263: A( 2, 1 ) = ANORM*A( 2, 1 )
264: A( 1, 2 ) = ANORM*A( 1, 2 )
265: A( 2, 2 ) = ANORM*A( 2, 2 )
266: B( 1, 1 ) = BNORM*B( 1, 1 )
267: B( 2, 1 ) = BNORM*B( 2, 1 )
268: B( 1, 2 ) = BNORM*B( 1, 2 )
269: B( 2, 2 ) = BNORM*B( 2, 2 )
270: *
271: IF( WI.EQ.ZERO ) THEN
272: ALPHAR( 1 ) = A( 1, 1 )
273: ALPHAR( 2 ) = A( 2, 2 )
274: ALPHAI( 1 ) = ZERO
275: ALPHAI( 2 ) = ZERO
276: BETA( 1 ) = B( 1, 1 )
277: BETA( 2 ) = B( 2, 2 )
278: ELSE
279: ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
280: ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
281: ALPHAR( 2 ) = ALPHAR( 1 )
282: ALPHAI( 2 ) = -ALPHAI( 1 )
283: BETA( 1 ) = ONE
284: BETA( 2 ) = ONE
285: END IF
286: *
287: RETURN
288: *
289: * End of DLAGV2
290: *
291: END
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