LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ slags2()

 subroutine slags2 ( logical UPPER, real A1, real A2, real A3, real B1, real B2, real B3, real CSU, real SNU, real CSV, real SNV, real CSQ, real SNQ )

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.

Purpose:
``` SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
that if ( UPPER ) then

U**T *A*Q = U**T *( A1 A2 )*Q = ( x  0  )
( 0  A3 )     ( x  x  )
and
V**T*B*Q = V**T *( B1 B2 )*Q = ( x  0  )
( 0  B3 )     ( x  x  )

or if ( .NOT.UPPER ) then

U**T *A*Q = U**T *( A1 0  )*Q = ( x  x  )
( A2 A3 )     ( 0  x  )
and
V**T*B*Q = V**T*( B1 0  )*Q = ( x  x  )
( B2 B3 )     ( 0  x  )

The rows of the transformed A and B are parallel, where

U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )

Z**T denotes the transpose of Z.```
Parameters
 [in] UPPER ``` UPPER is LOGICAL = .TRUE.: the input matrices A and B are upper triangular. = .FALSE.: the input matrices A and B are lower triangular.``` [in] A1 ` A1 is REAL` [in] A2 ` A2 is REAL` [in] A3 ``` A3 is REAL On entry, A1, A2 and A3 are elements of the input 2-by-2 upper (lower) triangular matrix A.``` [in] B1 ` B1 is REAL` [in] B2 ` B2 is REAL` [in] B3 ``` B3 is REAL On entry, B1, B2 and B3 are elements of the input 2-by-2 upper (lower) triangular matrix B.``` [out] CSU ` CSU is REAL` [out] SNU ``` SNU is REAL The desired orthogonal matrix U.``` [out] CSV ` CSV is REAL` [out] SNV ``` SNV is REAL The desired orthogonal matrix V.``` [out] CSQ ` CSQ is REAL` [out] SNQ ``` SNQ is REAL The desired orthogonal matrix Q.```

Definition at line 150 of file slags2.f.

152*
153* -- LAPACK auxiliary routine --
154* -- LAPACK is a software package provided by Univ. of Tennessee, --
155* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
156*
157* .. Scalar Arguments ..
158 LOGICAL UPPER
159 REAL A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ,
160 \$ SNU, SNV
161* ..
162*
163* =====================================================================
164*
165* .. Parameters ..
166 REAL ZERO
167 parameter( zero = 0.0e+0 )
168* ..
169* .. Local Scalars ..
170 REAL A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12,
171 \$ AVB21, AVB22, CSL, CSR, D, S1, S2, SNL,
172 \$ SNR, UA11R, UA22R, VB11R, VB22R, B, C, R, UA11,
173 \$ UA12, UA21, UA22, VB11, VB12, VB21, VB22
174* ..
175* .. External Subroutines ..
176 EXTERNAL slartg, slasv2
177* ..
178* .. Intrinsic Functions ..
179 INTRINSIC abs
180* ..
181* .. Executable Statements ..
182*
183 IF( upper ) THEN
184*
185* Input matrices A and B are upper triangular matrices
186*
187* Form matrix C = A*adj(B) = ( a b )
188* ( 0 d )
189*
190 a = a1*b3
191 d = a3*b1
192 b = a2*b1 - a1*b2
193*
194* The SVD of real 2-by-2 triangular C
195*
196* ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 )
197* ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T )
198*
199 CALL slasv2( a, b, d, s1, s2, snr, csr, snl, csl )
200*
201 IF( abs( csl ).GE.abs( snl ) .OR. abs( csr ).GE.abs( snr ) )
202 \$ THEN
203*
204* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
205* and (1,2) element of |U|**T *|A| and |V|**T *|B|.
206*
207 ua11r = csl*a1
208 ua12 = csl*a2 + snl*a3
209*
210 vb11r = csr*b1
211 vb12 = csr*b2 + snr*b3
212*
213 aua12 = abs( csl )*abs( a2 ) + abs( snl )*abs( a3 )
214 avb12 = abs( csr )*abs( b2 ) + abs( snr )*abs( b3 )
215*
216* zero (1,2) elements of U**T *A and V**T *B
217*
218 IF( ( abs( ua11r )+abs( ua12 ) ).NE.zero ) THEN
219 IF( aua12 / ( abs( ua11r )+abs( ua12 ) ).LE.avb12 /
220 \$ ( abs( vb11r )+abs( vb12 ) ) ) THEN
221 CALL slartg( -ua11r, ua12, csq, snq, r )
222 ELSE
223 CALL slartg( -vb11r, vb12, csq, snq, r )
224 END IF
225 ELSE
226 CALL slartg( -vb11r, vb12, csq, snq, r )
227 END IF
228*
229 csu = csl
230 snu = -snl
231 csv = csr
232 snv = -snr
233*
234 ELSE
235*
236* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
237* and (2,2) element of |U|**T *|A| and |V|**T *|B|.
238*
239 ua21 = -snl*a1
240 ua22 = -snl*a2 + csl*a3
241*
242 vb21 = -snr*b1
243 vb22 = -snr*b2 + csr*b3
244*
245 aua22 = abs( snl )*abs( a2 ) + abs( csl )*abs( a3 )
246 avb22 = abs( snr )*abs( b2 ) + abs( csr )*abs( b3 )
247*
248* zero (2,2) elements of U**T*A and V**T*B, and then swap.
249*
250 IF( ( abs( ua21 )+abs( ua22 ) ).NE.zero ) THEN
251 IF( aua22 / ( abs( ua21 )+abs( ua22 ) ).LE.avb22 /
252 \$ ( abs( vb21 )+abs( vb22 ) ) ) THEN
253 CALL slartg( -ua21, ua22, csq, snq, r )
254 ELSE
255 CALL slartg( -vb21, vb22, csq, snq, r )
256 END IF
257 ELSE
258 CALL slartg( -vb21, vb22, csq, snq, r )
259 END IF
260*
261 csu = snl
262 snu = csl
263 csv = snr
264 snv = csr
265*
266 END IF
267*
268 ELSE
269*
270* Input matrices A and B are lower triangular matrices
271*
272* Form matrix C = A*adj(B) = ( a 0 )
273* ( c d )
274*
275 a = a1*b3
276 d = a3*b1
277 c = a2*b3 - a3*b2
278*
279* The SVD of real 2-by-2 triangular C
280*
281* ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 )
282* ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T )
283*
284 CALL slasv2( a, c, d, s1, s2, snr, csr, snl, csl )
285*
286 IF( abs( csr ).GE.abs( snr ) .OR. abs( csl ).GE.abs( snl ) )
287 \$ THEN
288*
289* Compute the (2,1) and (2,2) elements of U**T *A and V**T *B,
290* and (2,1) element of |U|**T *|A| and |V|**T *|B|.
291*
292 ua21 = -snr*a1 + csr*a2
293 ua22r = csr*a3
294*
295 vb21 = -snl*b1 + csl*b2
296 vb22r = csl*b3
297*
298 aua21 = abs( snr )*abs( a1 ) + abs( csr )*abs( a2 )
299 avb21 = abs( snl )*abs( b1 ) + abs( csl )*abs( b2 )
300*
301* zero (2,1) elements of U**T *A and V**T *B.
302*
303 IF( ( abs( ua21 )+abs( ua22r ) ).NE.zero ) THEN
304 IF( aua21 / ( abs( ua21 )+abs( ua22r ) ).LE.avb21 /
305 \$ ( abs( vb21 )+abs( vb22r ) ) ) THEN
306 CALL slartg( ua22r, ua21, csq, snq, r )
307 ELSE
308 CALL slartg( vb22r, vb21, csq, snq, r )
309 END IF
310 ELSE
311 CALL slartg( vb22r, vb21, csq, snq, r )
312 END IF
313*
314 csu = csr
315 snu = -snr
316 csv = csl
317 snv = -snl
318*
319 ELSE
320*
321* Compute the (1,1) and (1,2) elements of U**T *A and V**T *B,
322* and (1,1) element of |U|**T *|A| and |V|**T *|B|.
323*
324 ua11 = csr*a1 + snr*a2
325 ua12 = snr*a3
326*
327 vb11 = csl*b1 + snl*b2
328 vb12 = snl*b3
329*
330 aua11 = abs( csr )*abs( a1 ) + abs( snr )*abs( a2 )
331 avb11 = abs( csl )*abs( b1 ) + abs( snl )*abs( b2 )
332*
333* zero (1,1) elements of U**T*A and V**T*B, and then swap.
334*
335 IF( ( abs( ua11 )+abs( ua12 ) ).NE.zero ) THEN
336 IF( aua11 / ( abs( ua11 )+abs( ua12 ) ).LE.avb11 /
337 \$ ( abs( vb11 )+abs( vb12 ) ) ) THEN
338 CALL slartg( ua12, ua11, csq, snq, r )
339 ELSE
340 CALL slartg( vb12, vb11, csq, snq, r )
341 END IF
342 ELSE
343 CALL slartg( vb12, vb11, csq, snq, r )
344 END IF
345*
346 csu = snr
347 snu = csr
348 csv = snl
349 snv = csl
350*
351 END IF
352*
353 END IF
354*
355 RETURN
356*
357* End of SLAGS2
358*
subroutine slasv2(F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL)
SLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
Definition: slasv2.f:138
subroutine slartg(f, g, c, s, r)
SLARTG generates a plane rotation with real cosine and real sine.
Definition: slartg.f90:111
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