SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO ) * * -- LAPACK auxiliary routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. INTEGER I, INFO, N DOUBLE PRECISION RHO, SIGMA * .. * .. Array Arguments .. DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * ) * .. * * Purpose * ======= * * This subroutine computes the square root of the I-th updated * eigenvalue of a positive symmetric rank-one modification to * a positive diagonal matrix whose entries are given as the squares * of the corresponding entries in the array d, and that * * 0 <= D(i) < D(j) for i < j * * and that RHO > 0. This is arranged by the calling routine, and is * no loss in generality. The rank-one modified system is thus * * diag( D ) * diag( D ) + RHO * Z * Z_transpose. * * where we assume the Euclidean norm of Z is 1. * * The method consists of approximating the rational functions in the * secular equation by simpler interpolating rational functions. * * Arguments * ========= * * N (input) INTEGER * The length of all arrays. * * I (input) INTEGER * The index of the eigenvalue to be computed. 1 <= I <= N. * * D (input) DOUBLE PRECISION array, dimension ( N ) * The original eigenvalues. It is assumed that they are in * order, 0 <= D(I) < D(J) for I < J. * * Z (input) DOUBLE PRECISION array, dimension ( N ) * The components of the updating vector. * * DELTA (output) DOUBLE PRECISION array, dimension ( N ) * If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th * component. If N = 1, then DELTA(1) = 1. The vector DELTA * contains the information necessary to construct the * (singular) eigenvectors. * * RHO (input) DOUBLE PRECISION * The scalar in the symmetric updating formula. * * SIGMA (output) DOUBLE PRECISION * The computed sigma_I, the I-th updated eigenvalue. * * WORK (workspace) DOUBLE PRECISION array, dimension ( N ) * If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th * component. If N = 1, then WORK( 1 ) = 1. * * INFO (output) INTEGER * = 0: successful exit * > 0: if INFO = 1, the updating process failed. * * Internal Parameters * =================== * * Logical variable ORGATI (origin-at-i?) is used for distinguishing * whether D(i) or D(i+1) is treated as the origin. * * ORGATI = .true. origin at i * ORGATI = .false. origin at i+1 * * Logical variable SWTCH3 (switch-for-3-poles?) is for noting * if we are working with THREE poles! * * MAXIT is the maximum number of iterations allowed for each * eigenvalue. * * Further Details * =============== * * Based on contributions by * Ren-Cang Li, Computer Science Division, University of California * at Berkeley, USA * * ===================================================================== * * .. Parameters .. INTEGER MAXIT PARAMETER ( MAXIT = 20 ) DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, $ THREE = 3.0D+0, FOUR = 4.0D+0, EIGHT = 8.0D+0, $ TEN = 10.0D+0 ) * .. * .. Local Scalars .. LOGICAL ORGATI, SWTCH, SWTCH3 INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER DOUBLE PRECISION A, B, C, DELSQ, DELSQ2, DPHI, DPSI, DTIIM, $ DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS, $ ERRETM, ETA, PHI, PREW, PSI, RHOINV, SG2LB, $ SG2UB, TAU, TEMP, TEMP1, TEMP2, W * .. * .. Local Arrays .. DOUBLE PRECISION DD( 3 ), ZZ( 3 ) * .. * .. External Subroutines .. EXTERNAL DLAED6, DLASD5 * .. * .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. Executable Statements .. * * Since this routine is called in an inner loop, we do no argument * checking. * * Quick return for N=1 and 2. * INFO = 0 IF( N.EQ.1 ) THEN * * Presumably, I=1 upon entry * SIGMA = SQRT( D( 1 )*D( 1 )+RHO*Z( 1 )*Z( 1 ) ) DELTA( 1 ) = ONE WORK( 1 ) = ONE RETURN END IF IF( N.EQ.2 ) THEN CALL DLASD5( I, D, Z, DELTA, RHO, SIGMA, WORK ) RETURN END IF * * Compute machine epsilon * EPS = DLAMCH( 'Epsilon' ) RHOINV = ONE / RHO * * The case I = N * IF( I.EQ.N ) THEN * * Initialize some basic variables * II = N - 1 NITER = 1 * * Calculate initial guess * TEMP = RHO / TWO * * If ||Z||_2 is not one, then TEMP should be set to * RHO * ||Z||_2^2 / TWO * TEMP1 = TEMP / ( D( N )+SQRT( D( N )*D( N )+TEMP ) ) DO 10 J = 1, N WORK( J ) = D( J ) + D( N ) + TEMP1 DELTA( J ) = ( D( J )-D( N ) ) - TEMP1 10 CONTINUE * PSI = ZERO DO 20 J = 1, N - 2 PSI = PSI + Z( J )*Z( J ) / ( DELTA( J )*WORK( J ) ) 20 CONTINUE * C = RHOINV + PSI W = C + Z( II )*Z( II ) / ( DELTA( II )*WORK( II ) ) + $ Z( N )*Z( N ) / ( DELTA( N )*WORK( N ) ) * IF( W.LE.ZERO ) THEN TEMP1 = SQRT( D( N )*D( N )+RHO ) TEMP = Z( N-1 )*Z( N-1 ) / ( ( D( N-1 )+TEMP1 )* $ ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) + $ Z( N )*Z( N ) / RHO * * The following TAU is to approximate * SIGMA_n^2 - D( N )*D( N ) * IF( C.LE.TEMP ) THEN TAU = RHO ELSE DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) ) A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N ) B = Z( N )*Z( N )*DELSQ IF( A.LT.ZERO ) THEN TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A ) ELSE TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C ) END IF END IF * * It can be proved that * D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU <= D(N)^2+RHO * ELSE DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) ) A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N ) B = Z( N )*Z( N )*DELSQ * * The following TAU is to approximate * SIGMA_n^2 - D( N )*D( N ) * IF( A.LT.ZERO ) THEN TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A ) ELSE TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C ) END IF * * It can be proved that * D(N)^2 < D(N)^2+TAU < SIGMA(N)^2 < D(N)^2+RHO/2 * END IF * * The following ETA is to approximate SIGMA_n - D( N ) * ETA = TAU / ( D( N )+SQRT( D( N )*D( N )+TAU ) ) * SIGMA = D( N ) + ETA DO 30 J = 1, N DELTA( J ) = ( D( J )-D( I ) ) - ETA WORK( J ) = D( J ) + D( I ) + ETA 30 CONTINUE * * Evaluate PSI and the derivative DPSI * DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 40 J = 1, II TEMP = Z( J ) / ( DELTA( J )*WORK( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 40 CONTINUE ERRETM = ABS( ERRETM ) * * Evaluate PHI and the derivative DPHI * TEMP = Z( N ) / ( DELTA( N )*WORK( N ) ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + $ ABS( TAU )*( DPSI+DPHI ) * W = RHOINV + PHI + PSI * * Test for convergence * IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF * * Calculate the new step * NITER = NITER + 1 DTNSQ1 = WORK( N-1 )*DELTA( N-1 ) DTNSQ = WORK( N )*DELTA( N ) C = W - DTNSQ1*DPSI - DTNSQ*DPHI A = ( DTNSQ+DTNSQ1 )*W - DTNSQ*DTNSQ1*( DPSI+DPHI ) B = DTNSQ*DTNSQ1*W IF( C.LT.ZERO ) $ C = ABS( C ) IF( C.EQ.ZERO ) THEN ETA = RHO - SIGMA*SIGMA ELSE IF( A.GE.ZERO ) THEN ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF * * Note, eta should be positive if w is negative, and * eta should be negative otherwise. However, * if for some reason caused by roundoff, eta*w > 0, * we simply use one Newton step instead. This way * will guarantee eta*w < 0. * IF( W*ETA.GT.ZERO ) $ ETA = -W / ( DPSI+DPHI ) TEMP = ETA - DTNSQ IF( TEMP.GT.RHO ) $ ETA = RHO + DTNSQ * TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) ) DO 50 J = 1, N DELTA( J ) = DELTA( J ) - ETA WORK( J ) = WORK( J ) + ETA 50 CONTINUE * SIGMA = SIGMA + ETA * * Evaluate PSI and the derivative DPSI * DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 60 J = 1, II TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 60 CONTINUE ERRETM = ABS( ERRETM ) * * Evaluate PHI and the derivative DPHI * TEMP = Z( N ) / ( WORK( N )*DELTA( N ) ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + $ ABS( TAU )*( DPSI+DPHI ) * W = RHOINV + PHI + PSI * * Main loop to update the values of the array DELTA * ITER = NITER + 1 * DO 90 NITER = ITER, MAXIT * * Test for convergence * IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF * * Calculate the new step * DTNSQ1 = WORK( N-1 )*DELTA( N-1 ) DTNSQ = WORK( N )*DELTA( N ) C = W - DTNSQ1*DPSI - DTNSQ*DPHI A = ( DTNSQ+DTNSQ1 )*W - DTNSQ1*DTNSQ*( DPSI+DPHI ) B = DTNSQ1*DTNSQ*W IF( A.GE.ZERO ) THEN ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF * * Note, eta should be positive if w is negative, and * eta should be negative otherwise. However, * if for some reason caused by roundoff, eta*w > 0, * we simply use one Newton step instead. This way * will guarantee eta*w < 0. * IF( W*ETA.GT.ZERO ) $ ETA = -W / ( DPSI+DPHI ) TEMP = ETA - DTNSQ IF( TEMP.LE.ZERO ) $ ETA = ETA / TWO * TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) ) DO 70 J = 1, N DELTA( J ) = DELTA( J ) - ETA WORK( J ) = WORK( J ) + ETA 70 CONTINUE * SIGMA = SIGMA + ETA * * Evaluate PSI and the derivative DPSI * DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 80 J = 1, II TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 80 CONTINUE ERRETM = ABS( ERRETM ) * * Evaluate PHI and the derivative DPHI * TEMP = Z( N ) / ( WORK( N )*DELTA( N ) ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + $ ABS( TAU )*( DPSI+DPHI ) * W = RHOINV + PHI + PSI 90 CONTINUE * * Return with INFO = 1, NITER = MAXIT and not converged * INFO = 1 GO TO 240 * * End for the case I = N * ELSE * * The case for I < N * NITER = 1 IP1 = I + 1 * * Calculate initial guess * DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) ) DELSQ2 = DELSQ / TWO TEMP = DELSQ2 / ( D( I )+SQRT( D( I )*D( I )+DELSQ2 ) ) DO 100 J = 1, N WORK( J ) = D( J ) + D( I ) + TEMP DELTA( J ) = ( D( J )-D( I ) ) - TEMP 100 CONTINUE * PSI = ZERO DO 110 J = 1, I - 1 PSI = PSI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) ) 110 CONTINUE * PHI = ZERO DO 120 J = N, I + 2, -1 PHI = PHI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) ) 120 CONTINUE C = RHOINV + PSI + PHI W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) + $ Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) ) * IF( W.GT.ZERO ) THEN * * d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2 * * We choose d(i) as origin. * ORGATI = .TRUE. SG2LB = ZERO SG2UB = DELSQ2 A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 ) B = Z( I )*Z( I )*DELSQ IF( A.GT.ZERO ) THEN TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) ELSE TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) END IF * * TAU now is an estimation of SIGMA^2 - D( I )^2. The * following, however, is the corresponding estimation of * SIGMA - D( I ). * ETA = TAU / ( D( I )+SQRT( D( I )*D( I )+TAU ) ) ELSE * * (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2 * * We choose d(i+1) as origin. * ORGATI = .FALSE. SG2LB = -DELSQ2 SG2UB = ZERO A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 ) B = Z( IP1 )*Z( IP1 )*DELSQ IF( A.LT.ZERO ) THEN TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) ) ELSE TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C ) END IF * * TAU now is an estimation of SIGMA^2 - D( IP1 )^2. The * following, however, is the corresponding estimation of * SIGMA - D( IP1 ). * ETA = TAU / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+ $ TAU ) ) ) END IF * IF( ORGATI ) THEN II = I SIGMA = D( I ) + ETA DO 130 J = 1, N WORK( J ) = D( J ) + D( I ) + ETA DELTA( J ) = ( D( J )-D( I ) ) - ETA 130 CONTINUE ELSE II = I + 1 SIGMA = D( IP1 ) + ETA DO 140 J = 1, N WORK( J ) = D( J ) + D( IP1 ) + ETA DELTA( J ) = ( D( J )-D( IP1 ) ) - ETA 140 CONTINUE END IF IIM1 = II - 1 IIP1 = II + 1 * * Evaluate PSI and the derivative DPSI * DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 150 J = 1, IIM1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 150 CONTINUE ERRETM = ABS( ERRETM ) * * Evaluate PHI and the derivative DPHI * DPHI = ZERO PHI = ZERO DO 160 J = N, IIP1, -1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 160 CONTINUE * W = RHOINV + PHI + PSI * * W is the value of the secular function with * its ii-th element removed. * SWTCH3 = .FALSE. IF( ORGATI ) THEN IF( W.LT.ZERO ) $ SWTCH3 = .TRUE. ELSE IF( W.GT.ZERO ) $ SWTCH3 = .TRUE. END IF IF( II.EQ.1 .OR. II.EQ.N ) $ SWTCH3 = .FALSE. * TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = W + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + $ THREE*ABS( TEMP ) + ABS( TAU )*DW * * Test for convergence * IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF * IF( W.LE.ZERO ) THEN SG2LB = MAX( SG2LB, TAU ) ELSE SG2UB = MIN( SG2UB, TAU ) END IF * * Calculate the new step * NITER = NITER + 1 IF( .NOT.SWTCH3 ) THEN DTIPSQ = WORK( IP1 )*DELTA( IP1 ) DTISQ = WORK( I )*DELTA( I ) IF( ORGATI ) THEN C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2 ELSE C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2 END IF A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW B = DTIPSQ*DTISQ*W IF( C.EQ.ZERO ) THEN IF( A.EQ.ZERO ) THEN IF( ORGATI ) THEN A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI ) ELSE A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI ) END IF END IF ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ELSE * * Interpolation using THREE most relevant poles * DTIIM = WORK( IIM1 )*DELTA( IIM1 ) DTIIP = WORK( IIP1 )*DELTA( IIP1 ) TEMP = RHOINV + PSI + PHI IF( ORGATI ) THEN TEMP1 = Z( IIM1 ) / DTIIM TEMP1 = TEMP1*TEMP1 C = ( TEMP - DTIIP*( DPSI+DPHI ) ) - $ ( D( IIM1 )-D( IIP1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1 ZZ( 1 ) = Z( IIM1 )*Z( IIM1 ) IF( DPSI.LT.TEMP1 ) THEN ZZ( 3 ) = DTIIP*DTIIP*DPHI ELSE ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI ) END IF ELSE TEMP1 = Z( IIP1 ) / DTIIP TEMP1 = TEMP1*TEMP1 C = ( TEMP - DTIIM*( DPSI+DPHI ) ) - $ ( D( IIP1 )-D( IIM1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1 IF( DPHI.LT.TEMP1 ) THEN ZZ( 1 ) = DTIIM*DTIIM*DPSI ELSE ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) ) END IF ZZ( 3 ) = Z( IIP1 )*Z( IIP1 ) END IF ZZ( 2 ) = Z( II )*Z( II ) DD( 1 ) = DTIIM DD( 2 ) = DELTA( II )*WORK( II ) DD( 3 ) = DTIIP CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO ) IF( INFO.NE.0 ) $ GO TO 240 END IF * * Note, eta should be positive if w is negative, and * eta should be negative otherwise. However, * if for some reason caused by roundoff, eta*w > 0, * we simply use one Newton step instead. This way * will guarantee eta*w < 0. * IF( W*ETA.GE.ZERO ) $ ETA = -W / DW IF( ORGATI ) THEN TEMP1 = WORK( I )*DELTA( I ) TEMP = ETA - TEMP1 ELSE TEMP1 = WORK( IP1 )*DELTA( IP1 ) TEMP = ETA - TEMP1 END IF IF( TEMP.GT.SG2UB .OR. TEMP.LT.SG2LB ) THEN IF( W.LT.ZERO ) THEN ETA = ( SG2UB-TAU ) / TWO ELSE ETA = ( SG2LB-TAU ) / TWO END IF END IF * TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) ) * PREW = W * SIGMA = SIGMA + ETA DO 170 J = 1, N WORK( J ) = WORK( J ) + ETA DELTA( J ) = DELTA( J ) - ETA 170 CONTINUE * * Evaluate PSI and the derivative DPSI * DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 180 J = 1, IIM1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 180 CONTINUE ERRETM = ABS( ERRETM ) * * Evaluate PHI and the derivative DPHI * DPHI = ZERO PHI = ZERO DO 190 J = N, IIP1, -1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 190 CONTINUE * TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = RHOINV + PHI + PSI + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + $ THREE*ABS( TEMP ) + ABS( TAU )*DW * IF( W.LE.ZERO ) THEN SG2LB = MAX( SG2LB, TAU ) ELSE SG2UB = MIN( SG2UB, TAU ) END IF * SWTCH = .FALSE. IF( ORGATI ) THEN IF( -W.GT.ABS( PREW ) / TEN ) $ SWTCH = .TRUE. ELSE IF( W.GT.ABS( PREW ) / TEN ) $ SWTCH = .TRUE. END IF * * Main loop to update the values of the array DELTA and WORK * ITER = NITER + 1 * DO 230 NITER = ITER, MAXIT * * Test for convergence * IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF * * Calculate the new step * IF( .NOT.SWTCH3 ) THEN DTIPSQ = WORK( IP1 )*DELTA( IP1 ) DTISQ = WORK( I )*DELTA( I ) IF( .NOT.SWTCH ) THEN IF( ORGATI ) THEN C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2 ELSE C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2 END IF ELSE TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) IF( ORGATI ) THEN DPSI = DPSI + TEMP*TEMP ELSE DPHI = DPHI + TEMP*TEMP END IF C = W - DTISQ*DPSI - DTIPSQ*DPHI END IF A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW B = DTIPSQ*DTISQ*W IF( C.EQ.ZERO ) THEN IF( A.EQ.ZERO ) THEN IF( .NOT.SWTCH ) THEN IF( ORGATI ) THEN A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ* $ ( DPSI+DPHI ) ELSE A = Z( IP1 )*Z( IP1 ) + $ DTISQ*DTISQ*( DPSI+DPHI ) END IF ELSE A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI END IF END IF ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ELSE * * Interpolation using THREE most relevant poles * DTIIM = WORK( IIM1 )*DELTA( IIM1 ) DTIIP = WORK( IIP1 )*DELTA( IIP1 ) TEMP = RHOINV + PSI + PHI IF( SWTCH ) THEN C = TEMP - DTIIM*DPSI - DTIIP*DPHI ZZ( 1 ) = DTIIM*DTIIM*DPSI ZZ( 3 ) = DTIIP*DTIIP*DPHI ELSE IF( ORGATI ) THEN TEMP1 = Z( IIM1 ) / DTIIM TEMP1 = TEMP1*TEMP1 TEMP2 = ( D( IIM1 )-D( IIP1 ) )* $ ( D( IIM1 )+D( IIP1 ) )*TEMP1 C = TEMP - DTIIP*( DPSI+DPHI ) - TEMP2 ZZ( 1 ) = Z( IIM1 )*Z( IIM1 ) IF( DPSI.LT.TEMP1 ) THEN ZZ( 3 ) = DTIIP*DTIIP*DPHI ELSE ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI ) END IF ELSE TEMP1 = Z( IIP1 ) / DTIIP TEMP1 = TEMP1*TEMP1 TEMP2 = ( D( IIP1 )-D( IIM1 ) )* $ ( D( IIM1 )+D( IIP1 ) )*TEMP1 C = TEMP - DTIIM*( DPSI+DPHI ) - TEMP2 IF( DPHI.LT.TEMP1 ) THEN ZZ( 1 ) = DTIIM*DTIIM*DPSI ELSE ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) ) END IF ZZ( 3 ) = Z( IIP1 )*Z( IIP1 ) END IF END IF DD( 1 ) = DTIIM DD( 2 ) = DELTA( II )*WORK( II ) DD( 3 ) = DTIIP CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO ) IF( INFO.NE.0 ) $ GO TO 240 END IF * * Note, eta should be positive if w is negative, and * eta should be negative otherwise. However, * if for some reason caused by roundoff, eta*w > 0, * we simply use one Newton step instead. This way * will guarantee eta*w < 0. * IF( W*ETA.GE.ZERO ) $ ETA = -W / DW IF( ORGATI ) THEN TEMP1 = WORK( I )*DELTA( I ) TEMP = ETA - TEMP1 ELSE TEMP1 = WORK( IP1 )*DELTA( IP1 ) TEMP = ETA - TEMP1 END IF IF( TEMP.GT.SG2UB .OR. TEMP.LT.SG2LB ) THEN IF( W.LT.ZERO ) THEN ETA = ( SG2UB-TAU ) / TWO ELSE ETA = ( SG2LB-TAU ) / TWO END IF END IF * TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) ) * SIGMA = SIGMA + ETA DO 200 J = 1, N WORK( J ) = WORK( J ) + ETA DELTA( J ) = DELTA( J ) - ETA 200 CONTINUE * PREW = W * * Evaluate PSI and the derivative DPSI * DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 210 J = 1, IIM1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 210 CONTINUE ERRETM = ABS( ERRETM ) * * Evaluate PHI and the derivative DPHI * DPHI = ZERO PHI = ZERO DO 220 J = N, IIP1, -1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 220 CONTINUE * TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = RHOINV + PHI + PSI + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + $ THREE*ABS( TEMP ) + ABS( TAU )*DW IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN ) $ SWTCH = .NOT.SWTCH * IF( W.LE.ZERO ) THEN SG2LB = MAX( SG2LB, TAU ) ELSE SG2UB = MIN( SG2UB, TAU ) END IF * 230 CONTINUE * * Return with INFO = 1, NITER = MAXIT and not converged * INFO = 1 * END IF * 240 CONTINUE RETURN * * End of DLASD4 * END