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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, 2: $ CSR, SNR ) 3: * 4: * -- LAPACK auxiliary routine (version 3.2.2) -- 5: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 7: * June 2010 8: * 9: * .. Scalar Arguments .. 10: INTEGER LDA, LDB 11: DOUBLE PRECISION CSL, CSR, SNL, SNR 12: * .. 13: * .. Array Arguments .. 14: DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ), 15: $ B( LDB, * ), BETA( 2 ) 16: * .. 17: * 18: * Purpose 19: * ======= 20: * 21: * DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 22: * matrix pencil (A,B) where B is upper triangular. This routine 23: * computes orthogonal (rotation) matrices given by CSL, SNL and CSR, 24: * SNR such that 25: * 26: * 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0 27: * types), then 28: * 29: * [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] 30: * [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] 31: * 32: * [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] 33: * [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ], 34: * 35: * 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues, 36: * then 37: * 38: * [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] 39: * [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] 40: * 41: * [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] 42: * [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ] 43: * 44: * where b11 >= b22 > 0. 45: * 46: * 47: * Arguments 48: * ========= 49: * 50: * A (input/output) DOUBLE PRECISION array, dimension (LDA, 2) 51: * On entry, the 2 x 2 matrix A. 52: * On exit, A is overwritten by the ``A-part'' of the 53: * generalized Schur form. 54: * 55: * LDA (input) INTEGER 56: * THe leading dimension of the array A. LDA >= 2. 57: * 58: * B (input/output) DOUBLE PRECISION array, dimension (LDB, 2) 59: * On entry, the upper triangular 2 x 2 matrix B. 60: * On exit, B is overwritten by the ``B-part'' of the 61: * generalized Schur form. 62: * 63: * LDB (input) INTEGER 64: * THe leading dimension of the array B. LDB >= 2. 65: * 66: * ALPHAR (output) DOUBLE PRECISION array, dimension (2) 67: * ALPHAI (output) DOUBLE PRECISION array, dimension (2) 68: * BETA (output) DOUBLE PRECISION array, dimension (2) 69: * (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the 70: * pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may 71: * be zero. 72: * 73: * CSL (output) DOUBLE PRECISION 74: * The cosine of the left rotation matrix. 75: * 76: * SNL (output) DOUBLE PRECISION 77: * The sine of the left rotation matrix. 78: * 79: * CSR (output) DOUBLE PRECISION 80: * The cosine of the right rotation matrix. 81: * 82: * SNR (output) DOUBLE PRECISION 83: * The sine of the right rotation matrix. 84: * 85: * Further Details 86: * =============== 87: * 88: * Based on contributions by 89: * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 90: * 91: * ===================================================================== 92: * 93: * .. Parameters .. 94: DOUBLE PRECISION ZERO, ONE 95: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 96: * .. 97: * .. Local Scalars .. 98: DOUBLE PRECISION ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ, 99: $ R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1, 100: $ WR2 101: * .. 102: * .. External Subroutines .. 103: EXTERNAL DLAG2, DLARTG, DLASV2, DROT 104: * .. 105: * .. External Functions .. 106: DOUBLE PRECISION DLAMCH, DLAPY2 107: EXTERNAL DLAMCH, DLAPY2 108: * .. 109: * .. Intrinsic Functions .. 110: INTRINSIC ABS, MAX 111: * .. 112: * .. Executable Statements .. 113: * 114: SAFMIN = DLAMCH( 'S' ) 115: ULP = DLAMCH( 'P' ) 116: * 117: * Scale A 118: * 119: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ), 120: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN ) 121: ASCALE = ONE / ANORM 122: A( 1, 1 ) = ASCALE*A( 1, 1 ) 123: A( 1, 2 ) = ASCALE*A( 1, 2 ) 124: A( 2, 1 ) = ASCALE*A( 2, 1 ) 125: A( 2, 2 ) = ASCALE*A( 2, 2 ) 126: * 127: * Scale B 128: * 129: BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ), 130: $ SAFMIN ) 131: BSCALE = ONE / BNORM 132: B( 1, 1 ) = BSCALE*B( 1, 1 ) 133: B( 1, 2 ) = BSCALE*B( 1, 2 ) 134: B( 2, 2 ) = BSCALE*B( 2, 2 ) 135: * 136: * Check if A can be deflated 137: * 138: IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN 139: CSL = ONE 140: SNL = ZERO 141: CSR = ONE 142: SNR = ZERO 143: A( 2, 1 ) = ZERO 144: B( 2, 1 ) = ZERO 145: WI = ZERO 146: * 147: * Check if B is singular 148: * 149: ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN 150: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R ) 151: CSR = ONE 152: SNR = ZERO 153: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) 154: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) 155: A( 2, 1 ) = ZERO 156: B( 1, 1 ) = ZERO 157: B( 2, 1 ) = ZERO 158: WI = ZERO 159: * 160: ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN 161: CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T ) 162: SNR = -SNR 163: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) 164: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR ) 165: CSL = ONE 166: SNL = ZERO 167: A( 2, 1 ) = ZERO 168: B( 2, 1 ) = ZERO 169: B( 2, 2 ) = ZERO 170: WI = ZERO 171: * 172: ELSE 173: * 174: * B is nonsingular, first compute the eigenvalues of (A,B) 175: * 176: CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, 177: $ WI ) 178: * 179: IF( WI.EQ.ZERO ) THEN 180: * 181: * two real eigenvalues, compute s*A-w*B 182: * 183: H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 ) 184: H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 ) 185: H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 ) 186: * 187: RR = DLAPY2( H1, H2 ) 188: QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 ) 189: * 190: IF( RR.GT.QQ ) THEN 191: * 192: * find right rotation matrix to zero 1,1 element of 193: * (sA - wB) 194: * 195: CALL DLARTG( H2, H1, CSR, SNR, T ) 196: * 197: ELSE 198: * 199: * find right rotation matrix to zero 2,1 element of 200: * (sA - wB) 201: * 202: CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T ) 203: * 204: END IF 205: * 206: SNR = -SNR 207: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) 208: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR ) 209: * 210: * compute inf norms of A and B 211: * 212: H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ), 213: $ ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) ) 214: H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ), 215: $ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) ) 216: * 217: IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN 218: * 219: * find left rotation matrix Q to zero out B(2,1) 220: * 221: CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R ) 222: * 223: ELSE 224: * 225: * find left rotation matrix Q to zero out A(2,1) 226: * 227: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R ) 228: * 229: END IF 230: * 231: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) 232: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) 233: * 234: A( 2, 1 ) = ZERO 235: B( 2, 1 ) = ZERO 236: * 237: ELSE 238: * 239: * a pair of complex conjugate eigenvalues 240: * first compute the SVD of the matrix B 241: * 242: CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR, 243: $ CSR, SNL, CSL ) 244: * 245: * Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and 246: * Z is right rotation matrix computed from DLASV2 247: * 248: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) 249: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) 250: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) 251: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR ) 252: * 253: B( 2, 1 ) = ZERO 254: B( 1, 2 ) = ZERO 255: * 256: END IF 257: * 258: END IF 259: * 260: * Unscaling 261: * 262: A( 1, 1 ) = ANORM*A( 1, 1 ) 263: A( 2, 1 ) = ANORM*A( 2, 1 ) 264: A( 1, 2 ) = ANORM*A( 1, 2 ) 265: A( 2, 2 ) = ANORM*A( 2, 2 ) 266: B( 1, 1 ) = BNORM*B( 1, 1 ) 267: B( 2, 1 ) = BNORM*B( 2, 1 ) 268: B( 1, 2 ) = BNORM*B( 1, 2 ) 269: B( 2, 2 ) = BNORM*B( 2, 2 ) 270: * 271: IF( WI.EQ.ZERO ) THEN 272: ALPHAR( 1 ) = A( 1, 1 ) 273: ALPHAR( 2 ) = A( 2, 2 ) 274: ALPHAI( 1 ) = ZERO 275: ALPHAI( 2 ) = ZERO 276: BETA( 1 ) = B( 1, 1 ) 277: BETA( 2 ) = B( 2, 2 ) 278: ELSE 279: ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM 280: ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM 281: ALPHAR( 2 ) = ALPHAR( 1 ) 282: ALPHAI( 2 ) = -ALPHAI( 1 ) 283: BETA( 1 ) = ONE 284: BETA( 2 ) = ONE 285: END IF 286: * 287: RETURN 288: * 289: * End of DLAGV2 290: * 291: END