1*> \brief \b ZLAGS2
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
22*                          SNV, CSQ, SNQ )
23*
24*       .. Scalar Arguments ..
25*       LOGICAL            UPPER
26*       DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV
27*       COMPLEX*16         A2, B2, SNQ, SNU, SNV
28*       ..
29*
30*
31*> \par Purpose:
32*  =============
33*>
34*> \verbatim
35*>
36*> ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
37*> that if ( UPPER ) then
38*>
39*>           U**H *A*Q = U**H *( A1 A2 )*Q = ( x  0  )
40*>                             ( 0  A3 )     ( x  x  )
41*> and
42*>           V**H*B*Q = V**H *( B1 B2 )*Q = ( x  0  )
43*>                            ( 0  B3 )     ( x  x  )
44*>
45*> or if ( .NOT.UPPER ) then
46*>
47*>           U**H *A*Q = U**H *( A1 0  )*Q = ( x  x  )
48*>                             ( A2 A3 )     ( 0  x  )
49*> and
50*>           V**H *B*Q = V**H *( B1 0  )*Q = ( x  x  )
51*>                             ( B2 B3 )     ( 0  x  )
52*> where
53*>
54*>   U = (   CSU    SNU ), V = (  CSV    SNV ),
55*>       ( -SNU**H  CSU )      ( -SNV**H CSV )
56*>
57*>   Q = (   CSQ    SNQ )
58*>       ( -SNQ**H  CSQ )
59*>
60*> The rows of the transformed A and B are parallel. Moreover, if the
61*> input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
62*> of A is not zero. If the input matrices A and B are both not zero,
63*> then the transformed (2,2) element of B is not zero, except when the
64*> first rows of input A and B are parallel and the second rows are
65*> zero.
66*> \endverbatim
67*
68*  Arguments:
69*  ==========
70*
71*> \param[in] UPPER
72*> \verbatim
73*>          UPPER is LOGICAL
74*>          = .TRUE.: the input matrices A and B are upper triangular.
75*>          = .FALSE.: the input matrices A and B are lower triangular.
76*> \endverbatim
77*>
78*> \param[in] A1
79*> \verbatim
80*>          A1 is DOUBLE PRECISION
81*> \endverbatim
82*>
83*> \param[in] A2
84*> \verbatim
85*>          A2 is COMPLEX*16
86*> \endverbatim
87*>
88*> \param[in] A3
89*> \verbatim
90*>          A3 is DOUBLE PRECISION
91*>          On entry, A1, A2 and A3 are elements of the input 2-by-2
92*>          upper (lower) triangular matrix A.
93*> \endverbatim
94*>
95*> \param[in] B1
96*> \verbatim
97*>          B1 is DOUBLE PRECISION
98*> \endverbatim
99*>
100*> \param[in] B2
101*> \verbatim
102*>          B2 is COMPLEX*16
103*> \endverbatim
104*>
105*> \param[in] B3
106*> \verbatim
107*>          B3 is DOUBLE PRECISION
108*>          On entry, B1, B2 and B3 are elements of the input 2-by-2
109*>          upper (lower) triangular matrix B.
110*> \endverbatim
111*>
112*> \param[out] CSU
113*> \verbatim
114*>          CSU is DOUBLE PRECISION
115*> \endverbatim
116*>
117*> \param[out] SNU
118*> \verbatim
119*>          SNU is COMPLEX*16
120*>          The desired unitary matrix U.
121*> \endverbatim
122*>
123*> \param[out] CSV
124*> \verbatim
125*>          CSV is DOUBLE PRECISION
126*> \endverbatim
127*>
128*> \param[out] SNV
129*> \verbatim
130*>          SNV is COMPLEX*16
131*>          The desired unitary matrix V.
132*> \endverbatim
133*>
134*> \param[out] CSQ
135*> \verbatim
136*>          CSQ is DOUBLE PRECISION
137*> \endverbatim
138*>
139*> \param[out] SNQ
140*> \verbatim
141*>          SNQ is COMPLEX*16
142*>          The desired unitary matrix Q.
143*> \endverbatim
144*
145*  Authors:
146*  ========
147*
148*> \author Univ. of Tennessee
149*> \author Univ. of California Berkeley
150*> \author Univ. of Colorado Denver
151*> \author NAG Ltd.
152*
153*> \ingroup complex16OTHERauxiliary
154*
155*  =====================================================================
156      SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
157     $                   SNV, CSQ, SNQ )
158*
159*  -- LAPACK auxiliary routine --
160*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
161*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
162*
163*     .. Scalar Arguments ..
164      LOGICAL            UPPER
165      DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV
166      COMPLEX*16         A2, B2, SNQ, SNU, SNV
167*     ..
168*
169*  =====================================================================
170*
171*     .. Parameters ..
172      DOUBLE PRECISION   ZERO, ONE
173      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
174*     ..
175*     .. Local Scalars ..
176      DOUBLE PRECISION   A, AUA11, AUA12, AUA21, AUA22, AVB12, AVB11,
177     $                   AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2,
178     $                   SNL, SNR, UA11R, UA22R, VB11R, VB22R
179      COMPLEX*16         B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11,
180     $                   VB12, VB21, VB22
181*     ..
182*     .. External Subroutines ..
183      EXTERNAL           DLASV2, ZLARTG
184*     ..
185*     .. Intrinsic Functions ..
186      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG
187*     ..
188*     .. Statement Functions ..
189      DOUBLE PRECISION   ABS1
190*     ..
191*     .. Statement Function definitions ..
192      ABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
193*     ..
194*     .. Executable Statements ..
195*
196      IF( UPPER ) THEN
197*
198*        Input matrices A and B are upper triangular matrices
199*
200*        Form matrix C = A*adj(B) = ( a b )
201*                                   ( 0 d )
202*
203         A = A1*B3
204         D = A3*B1
205         B = A2*B1 - A1*B2
206         FB = ABS( B )
207*
208*        Transform complex 2-by-2 matrix C to real matrix by unitary
209*        diagonal matrix diag(1,D1).
210*
211         D1 = ONE
212         IF( FB.NE.ZERO )
213     $      D1 = B / FB
214*
215*        The SVD of real 2 by 2 triangular C
216*
217*         ( CSL -SNL )*( A B )*(  CSR  SNR ) = ( R 0 )
218*         ( SNL  CSL ) ( 0 D ) ( -SNR  CSR )   ( 0 T )
219*
220         CALL DLASV2( A, FB, D, S1, S2, SNR, CSR, SNL, CSL )
221*
222         IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
223     $        THEN
224*
225*           Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
226*           and (1,2) element of |U|**H *|A| and |V|**H *|B|.
227*
228            UA11R = CSL*A1
229            UA12 = CSL*A2 + D1*SNL*A3
230*
231            VB11R = CSR*B1
232            VB12 = CSR*B2 + D1*SNR*B3
233*
234            AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 )
235            AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 )
236*
237*           zero (1,2) elements of U**H *A and V**H *B
238*
239            IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN
240               CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
241     $                      R )
242            ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN
243               CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
244     $                      R )
245            ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 /
246     $               ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN
247               CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
248     $                      R )
249            ELSE
250               CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
251     $                      R )
252            END IF
253*
254            CSU = CSL
255            SNU = -D1*SNL
256            CSV = CSR
257            SNV = -D1*SNR
258*
259         ELSE
260*
261*           Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
262*           and (2,2) element of |U|**H *|A| and |V|**H *|B|.
263*
264            UA21 = -DCONJG( D1 )*SNL*A1
265            UA22 = -DCONJG( D1 )*SNL*A2 + CSL*A3
266*
267            VB21 = -DCONJG( D1 )*SNR*B1
268            VB22 = -DCONJG( D1 )*SNR*B2 + CSR*B3
269*
270            AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 )
271            AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 )
272*
273*           zero (2,2) elements of U**H *A and V**H *B, and then swap.
274*
275            IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN
276               CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
277     $                      R )
278            ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN
279               CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
280     $                      R )
281            ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 /
282     $               ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN
283               CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
284     $                      R )
285            ELSE
286               CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
287     $                      R )
288            END IF
289*
290            CSU = SNL
291            SNU = D1*CSL
292            CSV = SNR
293            SNV = D1*CSR
294*
295         END IF
296*
297      ELSE
298*
299*        Input matrices A and B are lower triangular matrices
300*
301*        Form matrix C = A*adj(B) = ( a 0 )
302*                                   ( c d )
303*
304         A = A1*B3
305         D = A3*B1
306         C = A2*B3 - A3*B2
307         FC = ABS( C )
308*
309*        Transform complex 2-by-2 matrix C to real matrix by unitary
310*        diagonal matrix diag(d1,1).
311*
312         D1 = ONE
313         IF( FC.NE.ZERO )
314     $      D1 = C / FC
315*
316*        The SVD of real 2 by 2 triangular C
317*
318*         ( CSL -SNL )*( A 0 )*(  CSR  SNR ) = ( R 0 )
319*         ( SNL  CSL ) ( C D ) ( -SNR  CSR )   ( 0 T )
320*
321         CALL DLASV2( A, FC, D, S1, S2, SNR, CSR, SNL, CSL )
322*
323         IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
324     $        THEN
325*
326*           Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
327*           and (2,1) element of |U|**H *|A| and |V|**H *|B|.
328*
329            UA21 = -D1*SNR*A1 + CSR*A2
330            UA22R = CSR*A3
331*
332            VB21 = -D1*SNL*B1 + CSL*B2
333            VB22R = CSL*B3
334*
335            AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 )
336            AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 )
337*
338*           zero (2,1) elements of U**H *A and V**H *B.
339*
340            IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN
341               CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
342            ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN
343               CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
344            ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 /
345     $               ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN
346               CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
347            ELSE
348               CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
349            END IF
350*
351            CSU = CSR
352            SNU = -DCONJG( D1 )*SNR
353            CSV = CSL
354            SNV = -DCONJG( D1 )*SNL
355*
356         ELSE
357*
358*           Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
359*           and (1,1) element of |U|**H *|A| and |V|**H *|B|.
360*
361            UA11 = CSR*A1 + DCONJG( D1 )*SNR*A2
362            UA12 = DCONJG( D1 )*SNR*A3
363*
364            VB11 = CSL*B1 + DCONJG( D1 )*SNL*B2
365            VB12 = DCONJG( D1 )*SNL*B3
366*
367            AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 )
368            AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 )
369*
370*           zero (1,1) elements of U**H *A and V**H *B, and then swap.
371*
372            IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN
373               CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
374            ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN
375               CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
376            ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 /
377     $               ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN
378               CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
379            ELSE
380               CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
381            END IF
382*
383            CSU = SNR
384            SNU = DCONJG( D1 )*CSR
385            CSV = SNL
386            SNV = DCONJG( D1 )*CSL
387*
388         END IF
389*
390      END IF
391*
392      RETURN
393*
394*     End of ZLAGS2
395*
396      END
397