 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zlags2()

 subroutine zlags2 ( logical UPPER, double precision A1, complex*16 A2, double precision A3, double precision B1, complex*16 B2, double precision B3, double precision CSU, complex*16 SNU, double precision CSV, complex*16 SNV, double precision CSQ, complex*16 SNQ )

ZLAGS2

Purpose:
``` ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
that if ( UPPER ) then

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

or if ( .NOT.UPPER ) then

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

U = (   CSU    SNU ), V = (  CSV    SNV ),
( -SNU**H  CSU )      ( -SNV**H CSV )

Q = (   CSQ    SNQ )
( -SNQ**H  CSQ )

The rows of the transformed A and B are parallel. Moreover, if the
input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
of A is not zero. If the input matrices A and B are both not zero,
then the transformed (2,2) element of B is not zero, except when the
first rows of input A and B are parallel and the second rows are
zero.```
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 DOUBLE PRECISION` [in] A2 ` A2 is COMPLEX*16` [in] A3 ``` A3 is DOUBLE PRECISION On entry, A1, A2 and A3 are elements of the input 2-by-2 upper (lower) triangular matrix A.``` [in] B1 ` B1 is DOUBLE PRECISION` [in] B2 ` B2 is COMPLEX*16` [in] B3 ``` B3 is DOUBLE PRECISION On entry, B1, B2 and B3 are elements of the input 2-by-2 upper (lower) triangular matrix B.``` [out] CSU ` CSU is DOUBLE PRECISION` [out] SNU ``` SNU is COMPLEX*16 The desired unitary matrix U.``` [out] CSV ` CSV is DOUBLE PRECISION` [out] SNV ``` SNV is COMPLEX*16 The desired unitary matrix V.``` [out] CSQ ` CSQ is DOUBLE PRECISION` [out] SNQ ``` SNQ is COMPLEX*16 The desired unitary matrix Q.```
Date
December 2016

Definition at line 160 of file zlags2.f.

160 *
161 * -- LAPACK auxiliary routine (version 3.7.0) --
162 * -- LAPACK is a software package provided by Univ. of Tennessee, --
163 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
164 * December 2016
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 *
subroutine zlartg(F, G, CS, SN, R)
ZLARTG generates a plane rotation with real cosine and complex sine.
Definition: zlartg.f:105
subroutine dlasv2(F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL)
DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
Definition: dlasv2.f:140
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