Annotation of rpl/lapack/lapack/dlagv2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL,
! 2: $ CSR, SNR )
! 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: 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: *
! 146: * Check if B is singular
! 147: *
! 148: ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN
! 149: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
! 150: CSR = ONE
! 151: SNR = ZERO
! 152: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
! 153: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
! 154: A( 2, 1 ) = ZERO
! 155: B( 1, 1 ) = ZERO
! 156: B( 2, 1 ) = ZERO
! 157: *
! 158: ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN
! 159: CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T )
! 160: SNR = -SNR
! 161: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
! 162: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
! 163: CSL = ONE
! 164: SNL = ZERO
! 165: A( 2, 1 ) = ZERO
! 166: B( 2, 1 ) = ZERO
! 167: B( 2, 2 ) = ZERO
! 168: *
! 169: ELSE
! 170: *
! 171: * B is nonsingular, first compute the eigenvalues of (A,B)
! 172: *
! 173: CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2,
! 174: $ WI )
! 175: *
! 176: IF( WI.EQ.ZERO ) THEN
! 177: *
! 178: * two real eigenvalues, compute s*A-w*B
! 179: *
! 180: H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 )
! 181: H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 )
! 182: H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 )
! 183: *
! 184: RR = DLAPY2( H1, H2 )
! 185: QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 )
! 186: *
! 187: IF( RR.GT.QQ ) THEN
! 188: *
! 189: * find right rotation matrix to zero 1,1 element of
! 190: * (sA - wB)
! 191: *
! 192: CALL DLARTG( H2, H1, CSR, SNR, T )
! 193: *
! 194: ELSE
! 195: *
! 196: * find right rotation matrix to zero 2,1 element of
! 197: * (sA - wB)
! 198: *
! 199: CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T )
! 200: *
! 201: END IF
! 202: *
! 203: SNR = -SNR
! 204: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
! 205: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
! 206: *
! 207: * compute inf norms of A and B
! 208: *
! 209: H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ),
! 210: $ ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) )
! 211: H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
! 212: $ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
! 213: *
! 214: IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN
! 215: *
! 216: * find left rotation matrix Q to zero out B(2,1)
! 217: *
! 218: CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R )
! 219: *
! 220: ELSE
! 221: *
! 222: * find left rotation matrix Q to zero out A(2,1)
! 223: *
! 224: CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R )
! 225: *
! 226: END IF
! 227: *
! 228: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
! 229: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
! 230: *
! 231: A( 2, 1 ) = ZERO
! 232: B( 2, 1 ) = ZERO
! 233: *
! 234: ELSE
! 235: *
! 236: * a pair of complex conjugate eigenvalues
! 237: * first compute the SVD of the matrix B
! 238: *
! 239: CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR,
! 240: $ CSR, SNL, CSL )
! 241: *
! 242: * Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and
! 243: * Z is right rotation matrix computed from DLASV2
! 244: *
! 245: CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL )
! 246: CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL )
! 247: CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR )
! 248: CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR )
! 249: *
! 250: B( 2, 1 ) = ZERO
! 251: B( 1, 2 ) = ZERO
! 252: *
! 253: END IF
! 254: *
! 255: END IF
! 256: *
! 257: * Unscaling
! 258: *
! 259: A( 1, 1 ) = ANORM*A( 1, 1 )
! 260: A( 2, 1 ) = ANORM*A( 2, 1 )
! 261: A( 1, 2 ) = ANORM*A( 1, 2 )
! 262: A( 2, 2 ) = ANORM*A( 2, 2 )
! 263: B( 1, 1 ) = BNORM*B( 1, 1 )
! 264: B( 2, 1 ) = BNORM*B( 2, 1 )
! 265: B( 1, 2 ) = BNORM*B( 1, 2 )
! 266: B( 2, 2 ) = BNORM*B( 2, 2 )
! 267: *
! 268: IF( WI.EQ.ZERO ) THEN
! 269: ALPHAR( 1 ) = A( 1, 1 )
! 270: ALPHAR( 2 ) = A( 2, 2 )
! 271: ALPHAI( 1 ) = ZERO
! 272: ALPHAI( 2 ) = ZERO
! 273: BETA( 1 ) = B( 1, 1 )
! 274: BETA( 2 ) = B( 2, 2 )
! 275: ELSE
! 276: ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM
! 277: ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM
! 278: ALPHAR( 2 ) = ALPHAR( 1 )
! 279: ALPHAI( 2 ) = -ALPHAI( 1 )
! 280: BETA( 1 ) = ONE
! 281: BETA( 2 ) = ONE
! 282: END IF
! 283: *
! 284: RETURN
! 285: *
! 286: * End of DLAGV2
! 287: *
! 288: END
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