1*> \brief \b SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.
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 SLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
22*                          SNV, CSQ, SNQ )
23*
24*       .. Scalar Arguments ..
25*       LOGICAL            UPPER
26*       REAL               A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
27*      $                   SNU, SNV
28*       ..
29*
30*
31*> \par Purpose:
32*  =============
33*>
34*> \verbatim
35*>
36*> SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
37*> that if ( UPPER ) then
38*>
39*>           U**T *A*Q = U**T *( A1 A2 )*Q = ( x  0  )
40*>                             ( 0  A3 )     ( x  x  )
41*> and
42*>           V**T*B*Q = V**T *( B1 B2 )*Q = ( x  0  )
43*>                            ( 0  B3 )     ( x  x  )
44*>
45*> or if ( .NOT.UPPER ) then
46*>
47*>           U**T *A*Q = U**T *( A1 0  )*Q = ( x  x  )
48*>                             ( A2 A3 )     ( 0  x  )
49*> and
50*>           V**T*B*Q = V**T*( B1 0  )*Q = ( x  x  )
51*>                           ( B2 B3 )     ( 0  x  )
52*>
53*> The rows of the transformed A and B are parallel, where
54*>
55*>   U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
56*>       ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )
57*>
58*> Z**T denotes the transpose of Z.
59*>
60*> \endverbatim
61*
62*  Arguments:
63*  ==========
64*
65*> \param[in] UPPER
66*> \verbatim
67*>          UPPER is LOGICAL
68*>          = .TRUE.: the input matrices A and B are upper triangular.
69*>          = .FALSE.: the input matrices A and B are lower triangular.
70*> \endverbatim
71*>
72*> \param[in] A1
73*> \verbatim
74*>          A1 is REAL
75*> \endverbatim
76*>
77*> \param[in] A2
78*> \verbatim
79*>          A2 is REAL
80*> \endverbatim
81*>
82*> \param[in] A3
83*> \verbatim
84*>          A3 is REAL
85*>          On entry, A1, A2 and A3 are elements of the input 2-by-2
86*>          upper (lower) triangular matrix A.
87*> \endverbatim
88*>
89*> \param[in] B1
90*> \verbatim
91*>          B1 is REAL
92*> \endverbatim
93*>
94*> \param[in] B2
95*> \verbatim
96*>          B2 is REAL
97*> \endverbatim
98*>
99*> \param[in] B3
100*> \verbatim
101*>          B3 is REAL
102*>          On entry, B1, B2 and B3 are elements of the input 2-by-2
103*>          upper (lower) triangular matrix B.
104*> \endverbatim
105*>
106*> \param[out] CSU
107*> \verbatim
108*>          CSU is REAL
109*> \endverbatim
110*>
111*> \param[out] SNU
112*> \verbatim
113*>          SNU is REAL
114*>          The desired orthogonal matrix U.
115*> \endverbatim
116*>
117*> \param[out] CSV
118*> \verbatim
119*>          CSV is REAL
120*> \endverbatim
121*>
122*> \param[out] SNV
123*> \verbatim
124*>          SNV is REAL
125*>          The desired orthogonal matrix V.
126*> \endverbatim
127*>
128*> \param[out] CSQ
129*> \verbatim
130*>          CSQ is REAL
131*> \endverbatim
132*>
133*> \param[out] SNQ
134*> \verbatim
135*>          SNQ is REAL
136*>          The desired orthogonal matrix Q.
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 realOTHERauxiliary
150*
151*  =====================================================================
152      SUBROUTINE SLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
153     $                   SNV, CSQ, SNQ )
154*
155*  -- LAPACK auxiliary routine (version 3.7.0) --
156*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
157*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
158*     December 2016
159*
160*     .. Scalar Arguments ..
161      LOGICAL            UPPER
162      REAL               A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
163     $                   SNU, SNV
164*     ..
165*
166*  =====================================================================
167*
168*     .. Parameters ..
169      REAL               ZERO
170      PARAMETER          ( ZERO = 0.0E+0 )
171*     ..
172*     .. Local Scalars ..
173      REAL               A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12,
174     $                   AVB21, AVB22, CSL, CSR, D, S1, S2, SNL,
175     $                   SNR, UA11R, UA22R, VB11R, VB22R, B, C, R, UA11,
176     $                   UA12, UA21, UA22, VB11, VB12, VB21, VB22
177*     ..
178*     .. External Subroutines ..
179      EXTERNAL           SLARTG, SLASV2
180*     ..
181*     .. Intrinsic Functions ..
182      INTRINSIC          ABS
183*     ..
184*     .. Executable Statements ..
185*
186      IF( UPPER ) THEN
187*
188*        Input matrices A and B are upper triangular matrices
189*
190*        Form matrix C = A*adj(B) = ( a b )
191*                                   ( 0 d )
192*
193         A = A1*B3
194         D = A3*B1
195         B = A2*B1 - A1*B2
196*
197*        The SVD of real 2-by-2 triangular C
198*
199*         ( CSL -SNL )*( A B )*(  CSR  SNR ) = ( R 0 )
200*         ( SNL  CSL ) ( 0 D ) ( -SNR  CSR )   ( 0 T )
201*
202         CALL SLASV2( A, B, D, S1, S2, SNR, CSR, SNL, CSL )
203*
204         IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
205     $        THEN
206*
207*           Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
208*           and (1,2) element of |U|**T *|A| and |V|**T *|B|.
209*
210            UA11R = CSL*A1
211            UA12 = CSL*A2 + SNL*A3
212*
213            VB11R = CSR*B1
214            VB12 = CSR*B2 + SNR*B3
215*
216            AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 )
217            AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 )
218*
219*           zero (1,2) elements of U**T *A and V**T *B
220*
221            IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN
222               IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 /
223     $             ( ABS( VB11R )+ABS( VB12 ) ) ) THEN
224                  CALL SLARTG( -UA11R, UA12, CSQ, SNQ, R )
225               ELSE
226                  CALL SLARTG( -VB11R, VB12, CSQ, SNQ, R )
227               END IF
228            ELSE
229               CALL SLARTG( -VB11R, VB12, CSQ, SNQ, R )
230            END IF
231*
232            CSU = CSL
233            SNU = -SNL
234            CSV = CSR
235            SNV = -SNR
236*
237         ELSE
238*
239*           Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
240*           and (2,2) element of |U|**T *|A| and |V|**T *|B|.
241*
242            UA21 = -SNL*A1
243            UA22 = -SNL*A2 + CSL*A3
244*
245            VB21 = -SNR*B1
246            VB22 = -SNR*B2 + CSR*B3
247*
248            AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 )
249            AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 )
250*
251*           zero (2,2) elements of U**T*A and V**T*B, and then swap.
252*
253            IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN
254               IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 /
255     $             ( ABS( VB21 )+ABS( VB22 ) ) ) THEN
256                  CALL SLARTG( -UA21, UA22, CSQ, SNQ, R )
257               ELSE
258                  CALL SLARTG( -VB21, VB22, CSQ, SNQ, R )
259               END IF
260            ELSE
261               CALL SLARTG( -VB21, VB22, CSQ, SNQ, R )
262            END IF
263*
264            CSU = SNL
265            SNU = CSL
266            CSV = SNR
267            SNV = CSR
268*
269         END IF
270*
271      ELSE
272*
273*        Input matrices A and B are lower triangular matrices
274*
275*        Form matrix C = A*adj(B) = ( a 0 )
276*                                   ( c d )
277*
278         A = A1*B3
279         D = A3*B1
280         C = A2*B3 - A3*B2
281*
282*        The SVD of real 2-by-2 triangular C
283*
284*         ( CSL -SNL )*( A 0 )*(  CSR  SNR ) = ( R 0 )
285*         ( SNL  CSL ) ( C D ) ( -SNR  CSR )   ( 0 T )
286*
287         CALL SLASV2( A, C, D, S1, S2, SNR, CSR, SNL, CSL )
288*
289         IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
290     $        THEN
291*
292*           Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
293*           and (2,1) element of |U|**T *|A| and |V|**T *|B|.
294*
295            UA21 = -SNR*A1 + CSR*A2
296            UA22R = CSR*A3
297*
298            VB21 = -SNL*B1 + CSL*B2
299            VB22R = CSL*B3
300*
301            AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 )
302            AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 )
303*
304*           zero (2,1) elements of U**T *A and V**T *B.
305*
306            IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN
307               IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 /
308     $             ( ABS( VB21 )+ABS( VB22R ) ) ) THEN
309                  CALL SLARTG( UA22R, UA21, CSQ, SNQ, R )
310               ELSE
311                  CALL SLARTG( VB22R, VB21, CSQ, SNQ, R )
312               END IF
313            ELSE
314               CALL SLARTG( VB22R, VB21, CSQ, SNQ, R )
315            END IF
316*
317            CSU = CSR
318            SNU = -SNR
319            CSV = CSL
320            SNV = -SNL
321*
322         ELSE
323*
324*           Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
325*           and (1,1) element of |U|**T *|A| and |V|**T *|B|.
326*
327            UA11 = CSR*A1 + SNR*A2
328            UA12 = SNR*A3
329*
330            VB11 = CSL*B1 + SNL*B2
331            VB12 = SNL*B3
332*
333            AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 )
334            AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 )
335*
336*           zero (1,1) elements of U**T*A and V**T*B, and then swap.
337*
338            IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN
339               IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 /
340     $             ( ABS( VB11 )+ABS( VB12 ) ) ) THEN
341                  CALL SLARTG( UA12, UA11, CSQ, SNQ, R )
342               ELSE
343                  CALL SLARTG( VB12, VB11, CSQ, SNQ, R )
344               END IF
345            ELSE
346               CALL SLARTG( VB12, VB11, CSQ, SNQ, R )
347            END IF
348*
349            CSU = SNR
350            SNU = CSR
351            CSV = SNL
352            SNV = CSL
353*
354         END IF
355*
356      END IF
357*
358      RETURN
359*
360*     End of SLAGS2
361*
362      END
363