SUBROUTINE SLAED5( I, D, Z, DELTA, RHO, DLAM ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER I REAL DLAM, RHO * .. * .. Array Arguments .. REAL D( 2 ), DELTA( 2 ), Z( 2 ) * .. * * Purpose * ======= * * This subroutine computes the I-th eigenvalue of a symmetric rank-one * modification of a 2-by-2 diagonal matrix * * diag( D ) + RHO * Z * transpose(Z) . * * The diagonal elements in the array D are assumed to satisfy * * D(i) < D(j) for i < j . * * We also assume RHO > 0 and that the Euclidean norm of the vector * Z is one. * * Arguments * ========= * * I (input) INTEGER * The index of the eigenvalue to be computed. I = 1 or I = 2. * * D (input) REAL array, dimension (2) * The original eigenvalues. We assume D(1) < D(2). * * Z (input) REAL array, dimension (2) * The components of the updating vector. * * DELTA (output) REAL array, dimension (2) * The vector DELTA contains the information necessary * to construct the eigenvectors. * * RHO (input) REAL * The scalar in the symmetric updating formula. * * DLAM (output) REAL * The computed lambda_I, the I-th updated eigenvalue. * * Further Details * =============== * * Based on contributions by * Ren-Cang Li, Computer Science Division, University of California * at Berkeley, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO, FOUR PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0, \$ FOUR = 4.0E0 ) * .. * .. Local Scalars .. REAL B, C, DEL, TAU, TEMP, W * .. * .. Intrinsic Functions .. INTRINSIC ABS, SQRT * .. * .. Executable Statements .. * DEL = D( 2 ) - D( 1 ) IF( I.EQ.1 ) THEN W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL IF( W.GT.ZERO ) THEN B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 1 )*Z( 1 )*DEL * * B > ZERO, always * TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) ) DLAM = D( 1 ) + TAU DELTA( 1 ) = -Z( 1 ) / TAU DELTA( 2 ) = Z( 2 ) / ( DEL-TAU ) ELSE B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DEL IF( B.GT.ZERO ) THEN TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) ) ELSE TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO END IF DLAM = D( 2 ) + TAU DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) DELTA( 2 ) = -Z( 2 ) / TAU END IF TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) DELTA( 1 ) = DELTA( 1 ) / TEMP DELTA( 2 ) = DELTA( 2 ) / TEMP ELSE * * Now I=2 * B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DEL IF( B.GT.ZERO ) THEN TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO ELSE TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) ) END IF DLAM = D( 2 ) + TAU DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) DELTA( 2 ) = -Z( 2 ) / TAU TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) DELTA( 1 ) = DELTA( 1 ) / TEMP DELTA( 2 ) = DELTA( 2 ) / TEMP END IF RETURN * * End OF SLAED5 * END