Annotation of rpl/lapack/lapack/dlag2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
! 2: $ WR2, WI )
! 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 SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
! 12: * ..
! 13: * .. Array Arguments ..
! 14: DOUBLE PRECISION A( LDA, * ), B( LDB, * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
! 21: * problem A - w B, with scaling as necessary to avoid over-/underflow.
! 22: *
! 23: * The scaling factor "s" results in a modified eigenvalue equation
! 24: *
! 25: * s A - w B
! 26: *
! 27: * where s is a non-negative scaling factor chosen so that w, w B,
! 28: * and s A do not overflow and, if possible, do not underflow, either.
! 29: *
! 30: * Arguments
! 31: * =========
! 32: *
! 33: * A (input) DOUBLE PRECISION array, dimension (LDA, 2)
! 34: * On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
! 35: * is less than 1/SAFMIN. Entries less than
! 36: * sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
! 37: *
! 38: * LDA (input) INTEGER
! 39: * The leading dimension of the array A. LDA >= 2.
! 40: *
! 41: * B (input) DOUBLE PRECISION array, dimension (LDB, 2)
! 42: * On entry, the 2 x 2 upper triangular matrix B. It is
! 43: * assumed that the one-norm of B is less than 1/SAFMIN. The
! 44: * diagonals should be at least sqrt(SAFMIN) times the largest
! 45: * element of B (in absolute value); if a diagonal is smaller
! 46: * than that, then +/- sqrt(SAFMIN) will be used instead of
! 47: * that diagonal.
! 48: *
! 49: * LDB (input) INTEGER
! 50: * The leading dimension of the array B. LDB >= 2.
! 51: *
! 52: * SAFMIN (input) DOUBLE PRECISION
! 53: * The smallest positive number s.t. 1/SAFMIN does not
! 54: * overflow. (This should always be DLAMCH('S') -- it is an
! 55: * argument in order to avoid having to call DLAMCH frequently.)
! 56: *
! 57: * SCALE1 (output) DOUBLE PRECISION
! 58: * A scaling factor used to avoid over-/underflow in the
! 59: * eigenvalue equation which defines the first eigenvalue. If
! 60: * the eigenvalues are complex, then the eigenvalues are
! 61: * ( WR1 +/- WI i ) / SCALE1 (which may lie outside the
! 62: * exponent range of the machine), SCALE1=SCALE2, and SCALE1
! 63: * will always be positive. If the eigenvalues are real, then
! 64: * the first (real) eigenvalue is WR1 / SCALE1 , but this may
! 65: * overflow or underflow, and in fact, SCALE1 may be zero or
! 66: * less than the underflow threshhold if the exact eigenvalue
! 67: * is sufficiently large.
! 68: *
! 69: * SCALE2 (output) DOUBLE PRECISION
! 70: * A scaling factor used to avoid over-/underflow in the
! 71: * eigenvalue equation which defines the second eigenvalue. If
! 72: * the eigenvalues are complex, then SCALE2=SCALE1. If the
! 73: * eigenvalues are real, then the second (real) eigenvalue is
! 74: * WR2 / SCALE2 , but this may overflow or underflow, and in
! 75: * fact, SCALE2 may be zero or less than the underflow
! 76: * threshhold if the exact eigenvalue is sufficiently large.
! 77: *
! 78: * WR1 (output) DOUBLE PRECISION
! 79: * If the eigenvalue is real, then WR1 is SCALE1 times the
! 80: * eigenvalue closest to the (2,2) element of A B**(-1). If the
! 81: * eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
! 82: * part of the eigenvalues.
! 83: *
! 84: * WR2 (output) DOUBLE PRECISION
! 85: * If the eigenvalue is real, then WR2 is SCALE2 times the
! 86: * other eigenvalue. If the eigenvalue is complex, then
! 87: * WR1=WR2 is SCALE1 times the real part of the eigenvalues.
! 88: *
! 89: * WI (output) DOUBLE PRECISION
! 90: * If the eigenvalue is real, then WI is zero. If the
! 91: * eigenvalue is complex, then WI is SCALE1 times the imaginary
! 92: * part of the eigenvalues. WI will always be non-negative.
! 93: *
! 94: * =====================================================================
! 95: *
! 96: * .. Parameters ..
! 97: DOUBLE PRECISION ZERO, ONE, TWO
! 98: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
! 99: DOUBLE PRECISION HALF
! 100: PARAMETER ( HALF = ONE / TWO )
! 101: DOUBLE PRECISION FUZZY1
! 102: PARAMETER ( FUZZY1 = ONE+1.0D-5 )
! 103: * ..
! 104: * .. Local Scalars ..
! 105: DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
! 106: $ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
! 107: $ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
! 108: $ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
! 109: $ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
! 110: $ WSCALE, WSIZE, WSMALL
! 111: * ..
! 112: * .. Intrinsic Functions ..
! 113: INTRINSIC ABS, MAX, MIN, SIGN, SQRT
! 114: * ..
! 115: * .. Executable Statements ..
! 116: *
! 117: RTMIN = SQRT( SAFMIN )
! 118: RTMAX = ONE / RTMIN
! 119: SAFMAX = ONE / SAFMIN
! 120: *
! 121: * Scale A
! 122: *
! 123: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
! 124: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
! 125: ASCALE = ONE / ANORM
! 126: A11 = ASCALE*A( 1, 1 )
! 127: A21 = ASCALE*A( 2, 1 )
! 128: A12 = ASCALE*A( 1, 2 )
! 129: A22 = ASCALE*A( 2, 2 )
! 130: *
! 131: * Perturb B if necessary to insure non-singularity
! 132: *
! 133: B11 = B( 1, 1 )
! 134: B12 = B( 1, 2 )
! 135: B22 = B( 2, 2 )
! 136: BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
! 137: IF( ABS( B11 ).LT.BMIN )
! 138: $ B11 = SIGN( BMIN, B11 )
! 139: IF( ABS( B22 ).LT.BMIN )
! 140: $ B22 = SIGN( BMIN, B22 )
! 141: *
! 142: * Scale B
! 143: *
! 144: BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
! 145: BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
! 146: BSCALE = ONE / BSIZE
! 147: B11 = B11*BSCALE
! 148: B12 = B12*BSCALE
! 149: B22 = B22*BSCALE
! 150: *
! 151: * Compute larger eigenvalue by method described by C. van Loan
! 152: *
! 153: * ( AS is A shifted by -SHIFT*B )
! 154: *
! 155: BINV11 = ONE / B11
! 156: BINV22 = ONE / B22
! 157: S1 = A11*BINV11
! 158: S2 = A22*BINV22
! 159: IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
! 160: AS12 = A12 - S1*B12
! 161: AS22 = A22 - S1*B22
! 162: SS = A21*( BINV11*BINV22 )
! 163: ABI22 = AS22*BINV22 - SS*B12
! 164: PP = HALF*ABI22
! 165: SHIFT = S1
! 166: ELSE
! 167: AS12 = A12 - S2*B12
! 168: AS11 = A11 - S2*B11
! 169: SS = A21*( BINV11*BINV22 )
! 170: ABI22 = -SS*B12
! 171: PP = HALF*( AS11*BINV11+ABI22 )
! 172: SHIFT = S2
! 173: END IF
! 174: QQ = SS*AS12
! 175: IF( ABS( PP*RTMIN ).GE.ONE ) THEN
! 176: DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
! 177: R = SQRT( ABS( DISCR ) )*RTMAX
! 178: ELSE
! 179: IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
! 180: DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
! 181: R = SQRT( ABS( DISCR ) )*RTMIN
! 182: ELSE
! 183: DISCR = PP**2 + QQ
! 184: R = SQRT( ABS( DISCR ) )
! 185: END IF
! 186: END IF
! 187: *
! 188: * Note: the test of R in the following IF is to cover the case when
! 189: * DISCR is small and negative and is flushed to zero during
! 190: * the calculation of R. On machines which have a consistent
! 191: * flush-to-zero threshhold and handle numbers above that
! 192: * threshhold correctly, it would not be necessary.
! 193: *
! 194: IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
! 195: SUM = PP + SIGN( R, PP )
! 196: DIFF = PP - SIGN( R, PP )
! 197: WBIG = SHIFT + SUM
! 198: *
! 199: * Compute smaller eigenvalue
! 200: *
! 201: WSMALL = SHIFT + DIFF
! 202: IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
! 203: WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
! 204: WSMALL = WDET / WBIG
! 205: END IF
! 206: *
! 207: * Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
! 208: * for WR1.
! 209: *
! 210: IF( PP.GT.ABI22 ) THEN
! 211: WR1 = MIN( WBIG, WSMALL )
! 212: WR2 = MAX( WBIG, WSMALL )
! 213: ELSE
! 214: WR1 = MAX( WBIG, WSMALL )
! 215: WR2 = MIN( WBIG, WSMALL )
! 216: END IF
! 217: WI = ZERO
! 218: ELSE
! 219: *
! 220: * Complex eigenvalues
! 221: *
! 222: WR1 = SHIFT + PP
! 223: WR2 = WR1
! 224: WI = R
! 225: END IF
! 226: *
! 227: * Further scaling to avoid underflow and overflow in computing
! 228: * SCALE1 and overflow in computing w*B.
! 229: *
! 230: * This scale factor (WSCALE) is bounded from above using C1 and C2,
! 231: * and from below using C3 and C4.
! 232: * C1 implements the condition s A must never overflow.
! 233: * C2 implements the condition w B must never overflow.
! 234: * C3, with C2,
! 235: * implement the condition that s A - w B must never overflow.
! 236: * C4 implements the condition s should not underflow.
! 237: * C5 implements the condition max(s,|w|) should be at least 2.
! 238: *
! 239: C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
! 240: C2 = SAFMIN*MAX( ONE, BNORM )
! 241: C3 = BSIZE*SAFMIN
! 242: IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
! 243: C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
! 244: ELSE
! 245: C4 = ONE
! 246: END IF
! 247: IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
! 248: C5 = MIN( ONE, ASCALE*BSIZE )
! 249: ELSE
! 250: C5 = ONE
! 251: END IF
! 252: *
! 253: * Scale first eigenvalue
! 254: *
! 255: WABS = ABS( WR1 ) + ABS( WI )
! 256: WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
! 257: $ MIN( C4, HALF*MAX( WABS, C5 ) ) )
! 258: IF( WSIZE.NE.ONE ) THEN
! 259: WSCALE = ONE / WSIZE
! 260: IF( WSIZE.GT.ONE ) THEN
! 261: SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
! 262: $ MIN( ASCALE, BSIZE )
! 263: ELSE
! 264: SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
! 265: $ MAX( ASCALE, BSIZE )
! 266: END IF
! 267: WR1 = WR1*WSCALE
! 268: IF( WI.NE.ZERO ) THEN
! 269: WI = WI*WSCALE
! 270: WR2 = WR1
! 271: SCALE2 = SCALE1
! 272: END IF
! 273: ELSE
! 274: SCALE1 = ASCALE*BSIZE
! 275: SCALE2 = SCALE1
! 276: END IF
! 277: *
! 278: * Scale second eigenvalue (if real)
! 279: *
! 280: IF( WI.EQ.ZERO ) THEN
! 281: WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
! 282: $ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
! 283: IF( WSIZE.NE.ONE ) THEN
! 284: WSCALE = ONE / WSIZE
! 285: IF( WSIZE.GT.ONE ) THEN
! 286: SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
! 287: $ MIN( ASCALE, BSIZE )
! 288: ELSE
! 289: SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
! 290: $ MAX( ASCALE, BSIZE )
! 291: END IF
! 292: WR2 = WR2*WSCALE
! 293: ELSE
! 294: SCALE2 = ASCALE*BSIZE
! 295: END IF
! 296: END IF
! 297: *
! 298: * End of DLAG2
! 299: *
! 300: RETURN
! 301: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to dlag2.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack