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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, 00002 $ SNV, CSQ, SNQ ) 00003 * 00004 * -- LAPACK auxiliary routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 LOGICAL UPPER 00011 DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV 00012 COMPLEX*16 A2, B2, SNQ, SNU, SNV 00013 * .. 00014 * 00015 * Purpose 00016 * ======= 00017 * 00018 * ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such 00019 * that if ( UPPER ) then 00020 * 00021 * U'*A*Q = U'*( A1 A2 )*Q = ( x 0 ) 00022 * ( 0 A3 ) ( x x ) 00023 * and 00024 * V'*B*Q = V'*( B1 B2 )*Q = ( x 0 ) 00025 * ( 0 B3 ) ( x x ) 00026 * 00027 * or if ( .NOT.UPPER ) then 00028 * 00029 * U'*A*Q = U'*( A1 0 )*Q = ( x x ) 00030 * ( A2 A3 ) ( 0 x ) 00031 * and 00032 * V'*B*Q = V'*( B1 0 )*Q = ( x x ) 00033 * ( B2 B3 ) ( 0 x ) 00034 * where 00035 * 00036 * U = ( CSU SNU ), V = ( CSV SNV ), 00037 * ( -CONJG(SNU) CSU ) ( -CONJG(SNV) CSV ) 00038 * 00039 * Q = ( CSQ SNQ ) 00040 * ( -CONJG(SNQ) CSQ ) 00041 * 00042 * Z' denotes the conjugate transpose of Z. 00043 * 00044 * The rows of the transformed A and B are parallel. Moreover, if the 00045 * input 2-by-2 matrix A is not zero, then the transformed (1,1) entry 00046 * of A is not zero. If the input matrices A and B are both not zero, 00047 * then the transformed (2,2) element of B is not zero, except when the 00048 * first rows of input A and B are parallel and the second rows are 00049 * zero. 00050 * 00051 * Arguments 00052 * ========= 00053 * 00054 * UPPER (input) LOGICAL 00055 * = .TRUE.: the input matrices A and B are upper triangular. 00056 * = .FALSE.: the input matrices A and B are lower triangular. 00057 * 00058 * A1 (input) DOUBLE PRECISION 00059 * A2 (input) COMPLEX*16 00060 * A3 (input) DOUBLE PRECISION 00061 * On entry, A1, A2 and A3 are elements of the input 2-by-2 00062 * upper (lower) triangular matrix A. 00063 * 00064 * B1 (input) DOUBLE PRECISION 00065 * B2 (input) COMPLEX*16 00066 * B3 (input) DOUBLE PRECISION 00067 * On entry, B1, B2 and B3 are elements of the input 2-by-2 00068 * upper (lower) triangular matrix B. 00069 * 00070 * CSU (output) DOUBLE PRECISION 00071 * SNU (output) COMPLEX*16 00072 * The desired unitary matrix U. 00073 * 00074 * CSV (output) DOUBLE PRECISION 00075 * SNV (output) COMPLEX*16 00076 * The desired unitary matrix V. 00077 * 00078 * CSQ (output) DOUBLE PRECISION 00079 * SNQ (output) COMPLEX*16 00080 * The desired unitary matrix Q. 00081 * 00082 * ===================================================================== 00083 * 00084 * .. Parameters .. 00085 DOUBLE PRECISION ZERO, ONE 00086 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00087 * .. 00088 * .. Local Scalars .. 00089 DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB12, AVB11, 00090 $ AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2, 00091 $ SNL, SNR, UA11R, UA22R, VB11R, VB22R 00092 COMPLEX*16 B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11, 00093 $ VB12, VB21, VB22 00094 * .. 00095 * .. External Subroutines .. 00096 EXTERNAL DLASV2, ZLARTG 00097 * .. 00098 * .. Intrinsic Functions .. 00099 INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG 00100 * .. 00101 * .. Statement Functions .. 00102 DOUBLE PRECISION ABS1 00103 * .. 00104 * .. Statement Function definitions .. 00105 ABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) ) 00106 * .. 00107 * .. Executable Statements .. 00108 * 00109 IF( UPPER ) THEN 00110 * 00111 * Input matrices A and B are upper triangular matrices 00112 * 00113 * Form matrix C = A*adj(B) = ( a b ) 00114 * ( 0 d ) 00115 * 00116 A = A1*B3 00117 D = A3*B1 00118 B = A2*B1 - A1*B2 00119 FB = ABS( B ) 00120 * 00121 * Transform complex 2-by-2 matrix C to real matrix by unitary 00122 * diagonal matrix diag(1,D1). 00123 * 00124 D1 = ONE 00125 IF( FB.NE.ZERO ) 00126 $ D1 = B / FB 00127 * 00128 * The SVD of real 2 by 2 triangular C 00129 * 00130 * ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 ) 00131 * ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T ) 00132 * 00133 CALL DLASV2( A, FB, D, S1, S2, SNR, CSR, SNL, CSL ) 00134 * 00135 IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) ) 00136 $ THEN 00137 * 00138 * Compute the (1,1) and (1,2) elements of U'*A and V'*B, 00139 * and (1,2) element of |U|'*|A| and |V|'*|B|. 00140 * 00141 UA11R = CSL*A1 00142 UA12 = CSL*A2 + D1*SNL*A3 00143 * 00144 VB11R = CSR*B1 00145 VB12 = CSR*B2 + D1*SNR*B3 00146 * 00147 AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 ) 00148 AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 ) 00149 * 00150 * zero (1,2) elements of U'*A and V'*B 00151 * 00152 IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN 00153 CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ, 00154 $ R ) 00155 ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN 00156 CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ, 00157 $ R ) 00158 ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 / 00159 $ ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN 00160 CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ, 00161 $ R ) 00162 ELSE 00163 CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ, 00164 $ R ) 00165 END IF 00166 * 00167 CSU = CSL 00168 SNU = -D1*SNL 00169 CSV = CSR 00170 SNV = -D1*SNR 00171 * 00172 ELSE 00173 * 00174 * Compute the (2,1) and (2,2) elements of U'*A and V'*B, 00175 * and (2,2) element of |U|'*|A| and |V|'*|B|. 00176 * 00177 UA21 = -DCONJG( D1 )*SNL*A1 00178 UA22 = -DCONJG( D1 )*SNL*A2 + CSL*A3 00179 * 00180 VB21 = -DCONJG( D1 )*SNR*B1 00181 VB22 = -DCONJG( D1 )*SNR*B2 + CSR*B3 00182 * 00183 AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 ) 00184 AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 ) 00185 * 00186 * zero (2,2) elements of U'*A and V'*B, and then swap. 00187 * 00188 IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN 00189 CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ, 00190 $ R ) 00191 ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN 00192 CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ, 00193 $ R ) 00194 ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 / 00195 $ ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN 00196 CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ, 00197 $ R ) 00198 ELSE 00199 CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ, 00200 $ R ) 00201 END IF 00202 * 00203 CSU = SNL 00204 SNU = D1*CSL 00205 CSV = SNR 00206 SNV = D1*CSR 00207 * 00208 END IF 00209 * 00210 ELSE 00211 * 00212 * Input matrices A and B are lower triangular matrices 00213 * 00214 * Form matrix C = A*adj(B) = ( a 0 ) 00215 * ( c d ) 00216 * 00217 A = A1*B3 00218 D = A3*B1 00219 C = A2*B3 - A3*B2 00220 FC = ABS( C ) 00221 * 00222 * Transform complex 2-by-2 matrix C to real matrix by unitary 00223 * diagonal matrix diag(d1,1). 00224 * 00225 D1 = ONE 00226 IF( FC.NE.ZERO ) 00227 $ D1 = C / FC 00228 * 00229 * The SVD of real 2 by 2 triangular C 00230 * 00231 * ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 ) 00232 * ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T ) 00233 * 00234 CALL DLASV2( A, FC, D, S1, S2, SNR, CSR, SNL, CSL ) 00235 * 00236 IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) ) 00237 $ THEN 00238 * 00239 * Compute the (2,1) and (2,2) elements of U'*A and V'*B, 00240 * and (2,1) element of |U|'*|A| and |V|'*|B|. 00241 * 00242 UA21 = -D1*SNR*A1 + CSR*A2 00243 UA22R = CSR*A3 00244 * 00245 VB21 = -D1*SNL*B1 + CSL*B2 00246 VB22R = CSL*B3 00247 * 00248 AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 ) 00249 AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 ) 00250 * 00251 * zero (2,1) elements of U'*A and V'*B. 00252 * 00253 IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN 00254 CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R ) 00255 ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN 00256 CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R ) 00257 ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 / 00258 $ ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN 00259 CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R ) 00260 ELSE 00261 CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R ) 00262 END IF 00263 * 00264 CSU = CSR 00265 SNU = -DCONJG( D1 )*SNR 00266 CSV = CSL 00267 SNV = -DCONJG( D1 )*SNL 00268 * 00269 ELSE 00270 * 00271 * Compute the (1,1) and (1,2) elements of U'*A and V'*B, 00272 * and (1,1) element of |U|'*|A| and |V|'*|B|. 00273 * 00274 UA11 = CSR*A1 + DCONJG( D1 )*SNR*A2 00275 UA12 = DCONJG( D1 )*SNR*A3 00276 * 00277 VB11 = CSL*B1 + DCONJG( D1 )*SNL*B2 00278 VB12 = DCONJG( D1 )*SNL*B3 00279 * 00280 AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 ) 00281 AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 ) 00282 * 00283 * zero (1,1) elements of U'*A and V'*B, and then swap. 00284 * 00285 IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN 00286 CALL ZLARTG( VB12, VB11, CSQ, SNQ, R ) 00287 ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN 00288 CALL ZLARTG( UA12, UA11, CSQ, SNQ, R ) 00289 ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 / 00290 $ ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN 00291 CALL ZLARTG( UA12, UA11, CSQ, SNQ, R ) 00292 ELSE 00293 CALL ZLARTG( VB12, VB11, CSQ, SNQ, R ) 00294 END IF 00295 * 00296 CSU = SNR 00297 SNU = DCONJG( D1 )*CSR 00298 CSV = SNL 00299 SNV = DCONJG( D1 )*CSL 00300 * 00301 END IF 00302 * 00303 END IF 00304 * 00305 RETURN 00306 * 00307 * End of ZLAGS2 00308 * 00309 END
1.7.3 
