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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15*> [TXT]</a>
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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*> \date November 2011
154*
155*> \ingroup complex16OTHERauxiliary
156*
157*  =====================================================================
158      SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
159     $                   SNV, CSQ, SNQ )
160*
161*  -- LAPACK auxiliary routine (version 3.4.0) --
162*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
163*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
164*     November 2011
165*
166*     .. Scalar Arguments ..
167      LOGICAL            UPPER
168      DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV
169      COMPLEX*16         A2, B2, SNQ, SNU, SNV
170*     ..
171*
172*  =====================================================================
173*
174*     .. Parameters ..
175      DOUBLE PRECISION   ZERO, ONE
176      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
177*     ..
178*     .. Local Scalars ..
179      DOUBLE PRECISION   A, AUA11, AUA12, AUA21, AUA22, AVB12, AVB11,
180     $                   AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2,
181     $                   SNL, SNR, UA11R, UA22R, VB11R, VB22R
182      COMPLEX*16         B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11,
183     $                   VB12, VB21, VB22
184*     ..
185*     .. External Subroutines ..
186      EXTERNAL           DLASV2, ZLARTG
187*     ..
188*     .. Intrinsic Functions ..
189      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG
190*     ..
191*     .. Statement Functions ..
192      DOUBLE PRECISION   ABS1
193*     ..
194*     .. Statement Function definitions ..
195      ABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
196*     ..
197*     .. Executable Statements ..
198*
199      IF( UPPER ) THEN
200*
201*        Input matrices A and B are upper triangular matrices
202*
203*        Form matrix C = A*adj(B) = ( a b )
204*                                   ( 0 d )
205*
206         A = A1*B3
207         D = A3*B1
208         B = A2*B1 - A1*B2
209         FB = ABS( B )
210*
211*        Transform complex 2-by-2 matrix C to real matrix by unitary
212*        diagonal matrix diag(1,D1).
213*
214         D1 = ONE
215         IF( FB.NE.ZERO )
216     $      D1 = B / FB
217*
218*        The SVD of real 2 by 2 triangular C
219*
220*         ( CSL -SNL )*( A B )*(  CSR  SNR ) = ( R 0 )
221*         ( SNL  CSL ) ( 0 D ) ( -SNR  CSR )   ( 0 T )
222*
223         CALL DLASV2( A, FB, D, S1, S2, SNR, CSR, SNL, CSL )
224*
225         IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
226     $        THEN
227*
228*           Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
229*           and (1,2) element of |U|**H *|A| and |V|**H *|B|.
230*
231            UA11R = CSL*A1
232            UA12 = CSL*A2 + D1*SNL*A3
233*
234            VB11R = CSR*B1
235            VB12 = CSR*B2 + D1*SNR*B3
236*
237            AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 )
238            AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 )
239*
240*           zero (1,2) elements of U**H *A and V**H *B
241*
242            IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN
243               CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
244     $                      R )
245            ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN
246               CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
247     $                      R )
248            ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 /
249     $               ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN
250               CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
251     $                      R )
252            ELSE
253               CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
254     $                      R )
255            END IF
256*
257            CSU = CSL
258            SNU = -D1*SNL
259            CSV = CSR
260            SNV = -D1*SNR
261*
262         ELSE
263*
264*           Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
265*           and (2,2) element of |U|**H *|A| and |V|**H *|B|.
266*
267            UA21 = -DCONJG( D1 )*SNL*A1
268            UA22 = -DCONJG( D1 )*SNL*A2 + CSL*A3
269*
270            VB21 = -DCONJG( D1 )*SNR*B1
271            VB22 = -DCONJG( D1 )*SNR*B2 + CSR*B3
272*
273            AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 )
274            AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 )
275*
276*           zero (2,2) elements of U**H *A and V**H *B, and then swap.
277*
278            IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN
279               CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
280     $                      R )
281            ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN
282               CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
283     $                      R )
284            ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 /
285     $               ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN
286               CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
287     $                      R )
288            ELSE
289               CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
290     $                      R )
291            END IF
292*
293            CSU = SNL
294            SNU = D1*CSL
295            CSV = SNR
296            SNV = D1*CSR
297*
298         END IF
299*
300      ELSE
301*
302*        Input matrices A and B are lower triangular matrices
303*
304*        Form matrix C = A*adj(B) = ( a 0 )
305*                                   ( c d )
306*
307         A = A1*B3
308         D = A3*B1
309         C = A2*B3 - A3*B2
310         FC = ABS( C )
311*
312*        Transform complex 2-by-2 matrix C to real matrix by unitary
313*        diagonal matrix diag(d1,1).
314*
315         D1 = ONE
316         IF( FC.NE.ZERO )
317     $      D1 = C / FC
318*
319*        The SVD of real 2 by 2 triangular C
320*
321*         ( CSL -SNL )*( A 0 )*(  CSR  SNR ) = ( R 0 )
322*         ( SNL  CSL ) ( C D ) ( -SNR  CSR )   ( 0 T )
323*
324         CALL DLASV2( A, FC, D, S1, S2, SNR, CSR, SNL, CSL )
325*
326         IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
327     $        THEN
328*
329*           Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
330*           and (2,1) element of |U|**H *|A| and |V|**H *|B|.
331*
332            UA21 = -D1*SNR*A1 + CSR*A2
333            UA22R = CSR*A3
334*
335            VB21 = -D1*SNL*B1 + CSL*B2
336            VB22R = CSL*B3
337*
338            AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 )
339            AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 )
340*
341*           zero (2,1) elements of U**H *A and V**H *B.
342*
343            IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN
344               CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
345            ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN
346               CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
347            ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 /
348     $               ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN
349               CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
350            ELSE
351               CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
352            END IF
353*
354            CSU = CSR
355            SNU = -DCONJG( D1 )*SNR
356            CSV = CSL
357            SNV = -DCONJG( D1 )*SNL
358*
359         ELSE
360*
361*           Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
362*           and (1,1) element of |U|**H *|A| and |V|**H *|B|.
363*
364            UA11 = CSR*A1 + DCONJG( D1 )*SNR*A2
365            UA12 = DCONJG( D1 )*SNR*A3
366*
367            VB11 = CSL*B1 + DCONJG( D1 )*SNL*B2
368            VB12 = DCONJG( D1 )*SNL*B3
369*
370            AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 )
371            AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 )
372*
373*           zero (1,1) elements of U**H *A and V**H *B, and then swap.
374*
375            IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN
376               CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
377            ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN
378               CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
379            ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 /
380     $               ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN
381               CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
382            ELSE
383               CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
384            END IF
385*
386            CSU = SNR
387            SNU = DCONJG( D1 )*CSR
388            CSV = SNL
389            SNV = DCONJG( D1 )*CSL
390*
391         END IF
392*
393      END IF
394*
395      RETURN
396*
397*     End of ZLAGS2
398*
399      END
400