Annotation of rpl/lapack/lapack/zlags2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
! 2: $ SNV, CSQ, SNQ )
! 3: *
! 4: * -- LAPACK auxiliary routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: LOGICAL UPPER
! 11: DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV
! 12: COMPLEX*16 A2, B2, SNQ, SNU, SNV
! 13: * ..
! 14: *
! 15: * Purpose
! 16: * =======
! 17: *
! 18: * ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
! 19: * that if ( UPPER ) then
! 20: *
! 21: * U'*A*Q = U'*( A1 A2 )*Q = ( x 0 )
! 22: * ( 0 A3 ) ( x x )
! 23: * and
! 24: * V'*B*Q = V'*( B1 B2 )*Q = ( x 0 )
! 25: * ( 0 B3 ) ( x x )
! 26: *
! 27: * or if ( .NOT.UPPER ) then
! 28: *
! 29: * U'*A*Q = U'*( A1 0 )*Q = ( x x )
! 30: * ( A2 A3 ) ( 0 x )
! 31: * and
! 32: * V'*B*Q = V'*( B1 0 )*Q = ( x x )
! 33: * ( B2 B3 ) ( 0 x )
! 34: * where
! 35: *
! 36: * U = ( CSU SNU ), V = ( CSV SNV ),
! 37: * ( -CONJG(SNU) CSU ) ( -CONJG(SNV) CSV )
! 38: *
! 39: * Q = ( CSQ SNQ )
! 40: * ( -CONJG(SNQ) CSQ )
! 41: *
! 42: * Z' denotes the conjugate transpose of Z.
! 43: *
! 44: * The rows of the transformed A and B are parallel. Moreover, if the
! 45: * input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
! 46: * of A is not zero. If the input matrices A and B are both not zero,
! 47: * then the transformed (2,2) element of B is not zero, except when the
! 48: * first rows of input A and B are parallel and the second rows are
! 49: * zero.
! 50: *
! 51: * Arguments
! 52: * =========
! 53: *
! 54: * UPPER (input) LOGICAL
! 55: * = .TRUE.: the input matrices A and B are upper triangular.
! 56: * = .FALSE.: the input matrices A and B are lower triangular.
! 57: *
! 58: * A1 (input) DOUBLE PRECISION
! 59: * A2 (input) COMPLEX*16
! 60: * A3 (input) DOUBLE PRECISION
! 61: * On entry, A1, A2 and A3 are elements of the input 2-by-2
! 62: * upper (lower) triangular matrix A.
! 63: *
! 64: * B1 (input) DOUBLE PRECISION
! 65: * B2 (input) COMPLEX*16
! 66: * B3 (input) DOUBLE PRECISION
! 67: * On entry, B1, B2 and B3 are elements of the input 2-by-2
! 68: * upper (lower) triangular matrix B.
! 69: *
! 70: * CSU (output) DOUBLE PRECISION
! 71: * SNU (output) COMPLEX*16
! 72: * The desired unitary matrix U.
! 73: *
! 74: * CSV (output) DOUBLE PRECISION
! 75: * SNV (output) COMPLEX*16
! 76: * The desired unitary matrix V.
! 77: *
! 78: * CSQ (output) DOUBLE PRECISION
! 79: * SNQ (output) COMPLEX*16
! 80: * The desired unitary matrix Q.
! 81: *
! 82: * =====================================================================
! 83: *
! 84: * .. Parameters ..
! 85: DOUBLE PRECISION ZERO, ONE
! 86: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 87: * ..
! 88: * .. Local Scalars ..
! 89: DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB12, AVB11,
! 90: $ AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2,
! 91: $ SNL, SNR, UA11R, UA22R, VB11R, VB22R
! 92: COMPLEX*16 B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11,
! 93: $ VB12, VB21, VB22
! 94: * ..
! 95: * .. External Subroutines ..
! 96: EXTERNAL DLASV2, ZLARTG
! 97: * ..
! 98: * .. Intrinsic Functions ..
! 99: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG
! 100: * ..
! 101: * .. Statement Functions ..
! 102: DOUBLE PRECISION ABS1
! 103: * ..
! 104: * .. Statement Function definitions ..
! 105: ABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
! 106: * ..
! 107: * .. Executable Statements ..
! 108: *
! 109: IF( UPPER ) THEN
! 110: *
! 111: * Input matrices A and B are upper triangular matrices
! 112: *
! 113: * Form matrix C = A*adj(B) = ( a b )
! 114: * ( 0 d )
! 115: *
! 116: A = A1*B3
! 117: D = A3*B1
! 118: B = A2*B1 - A1*B2
! 119: FB = ABS( B )
! 120: *
! 121: * Transform complex 2-by-2 matrix C to real matrix by unitary
! 122: * diagonal matrix diag(1,D1).
! 123: *
! 124: D1 = ONE
! 125: IF( FB.NE.ZERO )
! 126: $ D1 = B / FB
! 127: *
! 128: * The SVD of real 2 by 2 triangular C
! 129: *
! 130: * ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 )
! 131: * ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T )
! 132: *
! 133: CALL DLASV2( A, FB, D, S1, S2, SNR, CSR, SNL, CSL )
! 134: *
! 135: IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
! 136: $ THEN
! 137: *
! 138: * Compute the (1,1) and (1,2) elements of U'*A and V'*B,
! 139: * and (1,2) element of |U|'*|A| and |V|'*|B|.
! 140: *
! 141: UA11R = CSL*A1
! 142: UA12 = CSL*A2 + D1*SNL*A3
! 143: *
! 144: VB11R = CSR*B1
! 145: VB12 = CSR*B2 + D1*SNR*B3
! 146: *
! 147: AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 )
! 148: AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 )
! 149: *
! 150: * zero (1,2) elements of U'*A and V'*B
! 151: *
! 152: IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN
! 153: CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
! 154: $ R )
! 155: ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN
! 156: CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
! 157: $ R )
! 158: ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 /
! 159: $ ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN
! 160: CALL ZLARTG( -DCMPLX( UA11R ), DCONJG( UA12 ), CSQ, SNQ,
! 161: $ R )
! 162: ELSE
! 163: CALL ZLARTG( -DCMPLX( VB11R ), DCONJG( VB12 ), CSQ, SNQ,
! 164: $ R )
! 165: END IF
! 166: *
! 167: CSU = CSL
! 168: SNU = -D1*SNL
! 169: CSV = CSR
! 170: SNV = -D1*SNR
! 171: *
! 172: ELSE
! 173: *
! 174: * Compute the (2,1) and (2,2) elements of U'*A and V'*B,
! 175: * and (2,2) element of |U|'*|A| and |V|'*|B|.
! 176: *
! 177: UA21 = -DCONJG( D1 )*SNL*A1
! 178: UA22 = -DCONJG( D1 )*SNL*A2 + CSL*A3
! 179: *
! 180: VB21 = -DCONJG( D1 )*SNR*B1
! 181: VB22 = -DCONJG( D1 )*SNR*B2 + CSR*B3
! 182: *
! 183: AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 )
! 184: AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 )
! 185: *
! 186: * zero (2,2) elements of U'*A and V'*B, and then swap.
! 187: *
! 188: IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN
! 189: CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
! 190: $ R )
! 191: ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN
! 192: CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
! 193: $ R )
! 194: ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 /
! 195: $ ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN
! 196: CALL ZLARTG( -DCONJG( UA21 ), DCONJG( UA22 ), CSQ, SNQ,
! 197: $ R )
! 198: ELSE
! 199: CALL ZLARTG( -DCONJG( VB21 ), DCONJG( VB22 ), CSQ, SNQ,
! 200: $ R )
! 201: END IF
! 202: *
! 203: CSU = SNL
! 204: SNU = D1*CSL
! 205: CSV = SNR
! 206: SNV = D1*CSR
! 207: *
! 208: END IF
! 209: *
! 210: ELSE
! 211: *
! 212: * Input matrices A and B are lower triangular matrices
! 213: *
! 214: * Form matrix C = A*adj(B) = ( a 0 )
! 215: * ( c d )
! 216: *
! 217: A = A1*B3
! 218: D = A3*B1
! 219: C = A2*B3 - A3*B2
! 220: FC = ABS( C )
! 221: *
! 222: * Transform complex 2-by-2 matrix C to real matrix by unitary
! 223: * diagonal matrix diag(d1,1).
! 224: *
! 225: D1 = ONE
! 226: IF( FC.NE.ZERO )
! 227: $ D1 = C / FC
! 228: *
! 229: * The SVD of real 2 by 2 triangular C
! 230: *
! 231: * ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 )
! 232: * ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T )
! 233: *
! 234: CALL DLASV2( A, FC, D, S1, S2, SNR, CSR, SNL, CSL )
! 235: *
! 236: IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
! 237: $ THEN
! 238: *
! 239: * Compute the (2,1) and (2,2) elements of U'*A and V'*B,
! 240: * and (2,1) element of |U|'*|A| and |V|'*|B|.
! 241: *
! 242: UA21 = -D1*SNR*A1 + CSR*A2
! 243: UA22R = CSR*A3
! 244: *
! 245: VB21 = -D1*SNL*B1 + CSL*B2
! 246: VB22R = CSL*B3
! 247: *
! 248: AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 )
! 249: AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 )
! 250: *
! 251: * zero (2,1) elements of U'*A and V'*B.
! 252: *
! 253: IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN
! 254: CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
! 255: ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN
! 256: CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
! 257: ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 /
! 258: $ ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN
! 259: CALL ZLARTG( DCMPLX( UA22R ), UA21, CSQ, SNQ, R )
! 260: ELSE
! 261: CALL ZLARTG( DCMPLX( VB22R ), VB21, CSQ, SNQ, R )
! 262: END IF
! 263: *
! 264: CSU = CSR
! 265: SNU = -DCONJG( D1 )*SNR
! 266: CSV = CSL
! 267: SNV = -DCONJG( D1 )*SNL
! 268: *
! 269: ELSE
! 270: *
! 271: * Compute the (1,1) and (1,2) elements of U'*A and V'*B,
! 272: * and (1,1) element of |U|'*|A| and |V|'*|B|.
! 273: *
! 274: UA11 = CSR*A1 + DCONJG( D1 )*SNR*A2
! 275: UA12 = DCONJG( D1 )*SNR*A3
! 276: *
! 277: VB11 = CSL*B1 + DCONJG( D1 )*SNL*B2
! 278: VB12 = DCONJG( D1 )*SNL*B3
! 279: *
! 280: AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 )
! 281: AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 )
! 282: *
! 283: * zero (1,1) elements of U'*A and V'*B, and then swap.
! 284: *
! 285: IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN
! 286: CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
! 287: ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN
! 288: CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
! 289: ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 /
! 290: $ ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN
! 291: CALL ZLARTG( UA12, UA11, CSQ, SNQ, R )
! 292: ELSE
! 293: CALL ZLARTG( VB12, VB11, CSQ, SNQ, R )
! 294: END IF
! 295: *
! 296: CSU = SNR
! 297: SNU = DCONJG( D1 )*CSR
! 298: CSV = SNL
! 299: SNV = DCONJG( D1 )*CSL
! 300: *
! 301: END IF
! 302: *
! 303: END IF
! 304: *
! 305: RETURN
! 306: *
! 307: * End of ZLAGS2
! 308: *
! 309: END
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