LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zlags2.f
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00001 *> \brief \b ZLAGS2
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
00022 *                          SNV, CSQ, SNQ )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       LOGICAL            UPPER
00026 *       DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV
00027 *       COMPLEX*16         A2, B2, SNQ, SNU, SNV
00028 *       ..
00029 *  
00030 *
00031 *> \par Purpose:
00032 *  =============
00033 *>
00034 *> \verbatim
00035 *>
00036 *> ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
00037 *> that if ( UPPER ) then
00038 *>
00039 *>           U**H *A*Q = U**H *( A1 A2 )*Q = ( x  0  )
00040 *>                             ( 0  A3 )     ( x  x  )
00041 *> and
00042 *>           V**H*B*Q = V**H *( B1 B2 )*Q = ( x  0  )
00043 *>                            ( 0  B3 )     ( x  x  )
00044 *>
00045 *> or if ( .NOT.UPPER ) then
00046 *>
00047 *>           U**H *A*Q = U**H *( A1 0  )*Q = ( x  x  )
00048 *>                             ( A2 A3 )     ( 0  x  )
00049 *> and
00050 *>           V**H *B*Q = V**H *( B1 0  )*Q = ( x  x  )
00051 *>                             ( B2 B3 )     ( 0  x  )
00052 *> where
00053 *>
00054 *>   U = (   CSU    SNU ), V = (  CSV    SNV ),
00055 *>       ( -SNU**H  CSU )      ( -SNV**H CSV )
00056 *>
00057 *>   Q = (   CSQ    SNQ )
00058 *>       ( -SNQ**H  CSQ )
00059 *>
00060 *> The rows of the transformed A and B are parallel. Moreover, if the
00061 *> input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
00062 *> of A is not zero. If the input matrices A and B are both not zero,
00063 *> then the transformed (2,2) element of B is not zero, except when the
00064 *> first rows of input A and B are parallel and the second rows are
00065 *> zero.
00066 *> \endverbatim
00067 *
00068 *  Arguments:
00069 *  ==========
00070 *
00071 *> \param[in] UPPER
00072 *> \verbatim
00073 *>          UPPER is LOGICAL
00074 *>          = .TRUE.: the input matrices A and B are upper triangular.
00075 *>          = .FALSE.: the input matrices A and B are lower triangular.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] A1
00079 *> \verbatim
00080 *>          A1 is DOUBLE PRECISION
00081 *> \endverbatim
00082 *>
00083 *> \param[in] A2
00084 *> \verbatim
00085 *>          A2 is COMPLEX*16
00086 *> \endverbatim
00087 *>
00088 *> \param[in] A3
00089 *> \verbatim
00090 *>          A3 is DOUBLE PRECISION
00091 *>          On entry, A1, A2 and A3 are elements of the input 2-by-2
00092 *>          upper (lower) triangular matrix A.
00093 *> \endverbatim
00094 *>
00095 *> \param[in] B1
00096 *> \verbatim
00097 *>          B1 is DOUBLE PRECISION
00098 *> \endverbatim
00099 *>
00100 *> \param[in] B2
00101 *> \verbatim
00102 *>          B2 is COMPLEX*16
00103 *> \endverbatim
00104 *>
00105 *> \param[in] B3
00106 *> \verbatim
00107 *>          B3 is DOUBLE PRECISION
00108 *>          On entry, B1, B2 and B3 are elements of the input 2-by-2
00109 *>          upper (lower) triangular matrix B.
00110 *> \endverbatim
00111 *>
00112 *> \param[out] CSU
00113 *> \verbatim
00114 *>          CSU is DOUBLE PRECISION
00115 *> \endverbatim
00116 *>
00117 *> \param[out] SNU
00118 *> \verbatim
00119 *>          SNU is COMPLEX*16
00120 *>          The desired unitary matrix U.
00121 *> \endverbatim
00122 *>
00123 *> \param[out] CSV
00124 *> \verbatim
00125 *>          CSV is DOUBLE PRECISION
00126 *> \endverbatim
00127 *>
00128 *> \param[out] SNV
00129 *> \verbatim
00130 *>          SNV is COMPLEX*16
00131 *>          The desired unitary matrix V.
00132 *> \endverbatim
00133 *>
00134 *> \param[out] CSQ
00135 *> \verbatim
00136 *>          CSQ is DOUBLE PRECISION
00137 *> \endverbatim
00138 *>
00139 *> \param[out] SNQ
00140 *> \verbatim
00141 *>          SNQ is COMPLEX*16
00142 *>          The desired unitary matrix Q.
00143 *> \endverbatim
00144 *
00145 *  Authors:
00146 *  ========
00147 *
00148 *> \author Univ. of Tennessee 
00149 *> \author Univ. of California Berkeley 
00150 *> \author Univ. of Colorado Denver 
00151 *> \author NAG Ltd. 
00152 *
00153 *> \date November 2011
00154 *
00155 *> \ingroup complex16OTHERauxiliary
00156 *
00157 *  =====================================================================
00158       SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
00159      $                   SNV, CSQ, SNQ )
00160 *
00161 *  -- LAPACK auxiliary routine (version 3.4.0) --
00162 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00163 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00164 *     November 2011
00165 *
00166 *     .. Scalar Arguments ..
00167       LOGICAL            UPPER
00168       DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV
00169       COMPLEX*16         A2, B2, SNQ, SNU, SNV
00170 *     ..
00171 *
00172 *  =====================================================================
00173 *
00174 *     .. Parameters ..
00175       DOUBLE PRECISION   ZERO, ONE
00176       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00177 *     ..
00178 *     .. Local Scalars ..
00179       DOUBLE PRECISION   A, AUA11, AUA12, AUA21, AUA22, AVB12, AVB11, 
00180      $                   AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2, 
00181      $                   SNL, SNR, UA11R, UA22R, VB11R, VB22R
00182       COMPLEX*16         B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11,
00183      $                   VB12, VB21, VB22
00184 *     ..
00185 *     .. External Subroutines ..
00186       EXTERNAL           DLASV2, ZLARTG
00187 *     ..
00188 *     .. Intrinsic Functions ..
00189       INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG
00190 *     ..
00191 *     .. Statement Functions ..
00192       DOUBLE PRECISION   ABS1
00193 *     ..
00194 *     .. Statement Function definitions ..
00195       ABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
00196 *     ..
00197 *     .. Executable Statements ..
00198 *
00199       IF( UPPER ) THEN
00200 *
00201 *        Input matrices A and B are upper triangular matrices
00202 *
00203 *        Form matrix C = A*adj(B) = ( a b )
00204 *                                   ( 0 d )
00205 *
00206          A = A1*B3
00207          D = A3*B1
00208          B = A2*B1 - A1*B2
00209          FB = ABS( B )
00210 *
00211 *        Transform complex 2-by-2 matrix C to real matrix by unitary
00212 *        diagonal matrix diag(1,D1).
00213 *
00214          D1 = ONE
00215          IF( FB.NE.ZERO )
00216      $      D1 = B / FB
00217 *
00218 *        The SVD of real 2 by 2 triangular C
00219 *
00220 *         ( CSL -SNL )*( A B )*(  CSR  SNR ) = ( R 0 )
00221 *         ( SNL  CSL ) ( 0 D ) ( -SNR  CSR )   ( 0 T )
00222 *
00223          CALL DLASV2( A, FB, D, S1, S2, SNR, CSR, SNL, CSL )
00224 *
00225          IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
00226      $        THEN
00227 *
00228 *           Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
00229 *           and (1,2) element of |U|**H *|A| and |V|**H *|B|.
00230 *
00231             UA11R = CSL*A1
00232             UA12 = CSL*A2 + D1*SNL*A3
00233 *
00234             VB11R = CSR*B1
00235             VB12 = CSR*B2 + D1*SNR*B3
00236 *
00237             AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 )
00238             AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 )
00239 *
00240 *           zero (1,2) elements of U**H *A and V**H *B
00241 *
00242             IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN
00243                CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
00244      $                      R )
00245             ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN
00246                CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
00247      $                      R )
00248             ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 /
00249      $               ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN
00250                CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
00251      $                      R )
00252             ELSE
00253                CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
00254      $                      R )
00255             END IF
00256 *
00257             CSU = CSL
00258             SNU = -D1*SNL
00259             CSV = CSR
00260             SNV = -D1*SNR
00261 *
00262          ELSE
00263 *
00264 *           Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
00265 *           and (2,2) element of |U|**H *|A| and |V|**H *|B|.
00266 *
00267             UA21 = -DCONJG( D1 )*SNL*A1
00268             UA22 = -DCONJG( D1 )*SNL*A2 + CSL*A3
00269 *
00270             VB21 = -DCONJG( D1 )*SNR*B1
00271             VB22 = -DCONJG( D1 )*SNR*B2 + CSR*B3
00272 *
00273             AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 )
00274             AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 )
00275 *
00276 *           zero (2,2) elements of U**H *A and V**H *B, and then swap.
00277 *
00278             IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN
00279                CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
00280      $                      R )
00281             ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN
00282                CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
00283      $                      R )
00284             ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 /
00285      $               ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN
00286                CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
00287      $                      R )
00288             ELSE
00289                CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
00290      $                      R )
00291             END IF
00292 *
00293             CSU = SNL
00294             SNU = D1*CSL
00295             CSV = SNR
00296             SNV = D1*CSR
00297 *
00298          END IF
00299 *
00300       ELSE
00301 *
00302 *        Input matrices A and B are lower triangular matrices
00303 *
00304 *        Form matrix C = A*adj(B) = ( a 0 )
00305 *                                   ( c d )
00306 *
00307          A = A1*B3
00308          D = A3*B1
00309          C = A2*B3 - A3*B2
00310          FC = ABS( C )
00311 *
00312 *        Transform complex 2-by-2 matrix C to real matrix by unitary
00313 *        diagonal matrix diag(d1,1).
00314 *
00315          D1 = ONE
00316          IF( FC.NE.ZERO )
00317      $      D1 = C / FC
00318 *
00319 *        The SVD of real 2 by 2 triangular C
00320 *
00321 *         ( CSL -SNL )*( A 0 )*(  CSR  SNR ) = ( R 0 )
00322 *         ( SNL  CSL ) ( C D ) ( -SNR  CSR )   ( 0 T )
00323 *
00324          CALL DLASV2( A, FC, D, S1, S2, SNR, CSR, SNL, CSL )
00325 *
00326          IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
00327      $        THEN
00328 *
00329 *           Compute the (2,1) and (2,2) elements of U**H *A and V**H *B,
00330 *           and (2,1) element of |U|**H *|A| and |V|**H *|B|.
00331 *
00332             UA21 = -D1*SNR*A1 + CSR*A2
00333             UA22R = CSR*A3
00334 *
00335             VB21 = -D1*SNL*B1 + CSL*B2
00336             VB22R = CSL*B3
00337 *
00338             AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 )
00339             AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 )
00340 *
00341 *           zero (2,1) elements of U**H *A and V**H *B.
00342 *
00343             IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN
00344                CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
00345             ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN
00346                CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
00347             ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 /
00348      $               ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN
00349                CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
00350             ELSE
00351                CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
00352             END IF
00353 *
00354             CSU = CSR
00355             SNU = -DCONJG( D1 )*SNR
00356             CSV = CSL
00357             SNV = -DCONJG( D1 )*SNL
00358 *
00359          ELSE
00360 *
00361 *           Compute the (1,1) and (1,2) elements of U**H *A and V**H *B,
00362 *           and (1,1) element of |U|**H *|A| and |V|**H *|B|.
00363 *
00364             UA11 = CSR*A1 + DCONJG( D1 )*SNR*A2
00365             UA12 = DCONJG( D1 )*SNR*A3
00366 *
00367             VB11 = CSL*B1 + DCONJG( D1 )*SNL*B2
00368             VB12 = DCONJG( D1 )*SNL*B3
00369 *
00370             AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 )
00371             AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 )
00372 *
00373 *           zero (1,1) elements of U**H *A and V**H *B, and then swap.
00374 *
00375             IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN
00376                CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
00377             ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN
00378                CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
00379             ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 /
00380      $               ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN
00381                CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
00382             ELSE
00383                CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
00384             END IF
00385 *
00386             CSU = SNR
00387             SNU = DCONJG( D1 )*CSR
00388             CSV = SNL
00389             SNV = DCONJG( D1 )*CSL
00390 *
00391          END IF
00392 *
00393       END IF
00394 *
00395       RETURN
00396 *
00397 *     End of ZLAGS2
00398 *
00399       END
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