--- rpl/lapack/lapack/dlasd4.f 2010/01/26 15:22:46 1.1
+++ rpl/lapack/lapack/dlasd4.f 2016/08/27 15:34:31 1.17
@@ -1,9 +1,162 @@
+*> \brief \b DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DLASD4 + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
+*
+* .. Scalar Arguments ..
+* INTEGER I, INFO, N
+* DOUBLE PRECISION RHO, SIGMA
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The length of all arrays.
+*> \endverbatim
+*>
+*> \param[in] I
+*> \verbatim
+*> I is INTEGER
+*> The index of the eigenvalue to be computed. 1 <= I <= N.
+*> \endverbatim
+*>
+*> \param[in] D
+*> \verbatim
+*> D is DOUBLE PRECISION array, dimension ( N )
+*> The original eigenvalues. It is assumed that they are in
+*> order, 0 <= D(I) < D(J) for I < J.
+*> \endverbatim
+*>
+*> \param[in] Z
+*> \verbatim
+*> Z is DOUBLE PRECISION array, dimension ( N )
+*> The components of the updating vector.
+*> \endverbatim
+*>
+*> \param[out] DELTA
+*> \verbatim
+*> DELTA is 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.
+*> \endverbatim
+*>
+*> \param[in] RHO
+*> \verbatim
+*> RHO is DOUBLE PRECISION
+*> The scalar in the symmetric updating formula.
+*> \endverbatim
+*>
+*> \param[out] SIGMA
+*> \verbatim
+*> SIGMA is DOUBLE PRECISION
+*> The computed sigma_I, the I-th updated eigenvalue.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is 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.
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit
+*> > 0: if INFO = 1, the updating process failed.
+*> \endverbatim
+*
+*> \par Internal Parameters:
+* =========================
+*>
+*> \verbatim
+*> 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.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2013
+*
+*> \ingroup auxOTHERauxiliary
+*
+*> \par Contributors:
+* ==================
+*>
+*> Ren-Cang Li, Computer Science Division, University of California
+*> at Berkeley, USA
+*>
+* =====================================================================
SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
-* -- LAPACK auxiliary routine (version 3.2) --
+* -- LAPACK auxiliary routine (version 3.5.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* November 2013
*
* .. Scalar Arguments ..
INTEGER I, INFO, N
@@ -13,101 +166,23 @@
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 )
+ PARAMETER ( MAXIT = 400 )
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
+ LOGICAL ORGATI, SWTCH, SWTCH3, GEOMAVG
INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER
- DOUBLE PRECISION A, B, C, DELSQ, DELSQ2, DPHI, DPSI, DTIIM,
+ DOUBLE PRECISION A, B, C, DELSQ, DELSQ2, SQ2, DPHI, DPSI, DTIIM,
$ DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS,
- $ ERRETM, ETA, PHI, PREW, PSI, RHOINV, SG2LB,
- $ SG2UB, TAU, TEMP, TEMP1, TEMP2, W
+ $ ERRETM, ETA, PHI, PREW, PSI, RHOINV, SGLB,
+ $ SGUB, TAU, TAU2, TEMP, TEMP1, TEMP2, W
* ..
* .. Local Arrays ..
DOUBLE PRECISION DD( 3 ), ZZ( 3 )
@@ -148,6 +223,7 @@
*
EPS = DLAMCH( 'Epsilon' )
RHOINV = ONE / RHO
+ TAU2= ZERO
*
* The case I = N
*
@@ -186,7 +262,7 @@
$ ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) +
$ Z( N )*Z( N ) / RHO
*
-* The following TAU is to approximate
+* The following TAU2 is to approximate
* SIGMA_n^2 - D( N )*D( N )
*
IF( C.LE.TEMP ) THEN
@@ -196,42 +272,45 @@
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 )
+ TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
- TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
+ TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
+ TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
END IF
*
* It can be proved that
-* D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU <= D(N)^2+RHO
+* D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU2 <= 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
+* The following TAU2 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 )
+ TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
- TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
+ TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
+ TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
+
*
* It can be proved that
-* D(N)^2 < D(N)^2+TAU < SIGMA(N)^2 < D(N)^2+RHO/2
+* D(N)^2 < D(N)^2+TAU2 < SIGMA(N)^2 < D(N)^2+RHO/2
*
END IF
*
-* The following ETA is to approximate SIGMA_n - D( N )
+* The following TAU is to approximate SIGMA_n - D( N )
*
- ETA = TAU / ( D( N )+SQRT( D( N )*D( N )+TAU ) )
+* TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
- SIGMA = D( N ) + ETA
+ SIGMA = D( N ) + TAU
DO 30 J = 1, N
- DELTA( J ) = ( D( J )-D( I ) ) - ETA
- WORK( J ) = D( J ) + D( I ) + ETA
+ DELTA( J ) = ( D( J )-D( N ) ) - TAU
+ WORK( J ) = D( J ) + D( N ) + TAU
30 CONTINUE
*
* Evaluate PSI and the derivative DPSI
@@ -252,8 +331,8 @@
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 )
+ ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
+* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
@@ -293,15 +372,15 @@
IF( TEMP.GT.RHO )
$ ETA = RHO + DTNSQ
*
- TAU = TAU + ETA
ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
+ TAU = TAU + ETA
+ SIGMA = SIGMA + ETA
+*
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
@@ -317,11 +396,12 @@
*
* Evaluate PHI and the derivative DPHI
*
- TEMP = Z( N ) / ( WORK( N )*DELTA( N ) )
+ TAU2 = WORK( N )*DELTA( N )
+ TEMP = Z( N ) / TAU2
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
- ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
- $ ABS( TAU )*( DPSI+DPHI )
+ ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
+* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
@@ -362,15 +442,15 @@
IF( TEMP.LE.ZERO )
$ ETA = ETA / TWO
*
- TAU = TAU + ETA
ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
+ TAU = TAU + ETA
+ SIGMA = SIGMA + ETA
+*
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
@@ -386,11 +466,12 @@
*
* Evaluate PHI and the derivative DPHI
*
- TEMP = Z( N ) / ( WORK( N )*DELTA( N ) )
+ TAU2 = WORK( N )*DELTA( N )
+ TEMP = Z( N ) / TAU2
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
- ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV +
- $ ABS( TAU )*( DPSI+DPHI )
+ ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
+* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
90 CONTINUE
@@ -413,7 +494,8 @@
*
DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) )
DELSQ2 = DELSQ / TWO
- TEMP = DELSQ2 / ( D( I )+SQRT( D( I )*D( I )+DELSQ2 ) )
+ SQ2=SQRT( ( D( I )*D( I )+D( IP1 )*D( IP1 ) ) / TWO )
+ TEMP = DELSQ2 / ( D( I )+SQ2 )
DO 100 J = 1, N
WORK( J ) = D( J ) + D( I ) + TEMP
DELTA( J ) = ( D( J )-D( I ) ) - TEMP
@@ -432,6 +514,7 @@
W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) +
$ Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) )
*
+ GEOMAVG = .FALSE.
IF( W.GT.ZERO ) THEN
*
* d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2
@@ -439,21 +522,28 @@
* We choose d(i) as origin.
*
ORGATI = .TRUE.
- SG2LB = ZERO
- SG2UB = DELSQ2
+ II = I
+ SGLB = ZERO
+ SGUB = DELSQ2 / ( D( I )+SQ2 )
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 ) ) )
+ TAU2 = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
ELSE
- TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
+ TAU2 = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
END IF
*
-* TAU now is an estimation of SIGMA^2 - D( I )^2. The
+* TAU2 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 ) )
+ TAU = TAU2 / ( D( I )+SQRT( D( I )*D( I )+TAU2 ) )
+ TEMP = SQRT(EPS)
+ IF( (D(I).LE.TEMP*D(IP1)).AND.(ABS(Z(I)).LE.TEMP)
+ $ .AND.(D(I).GT.ZERO) ) THEN
+ TAU = MIN( TEN*D(I), SGUB )
+ GEOMAVG = .TRUE.
+ END IF
ELSE
*
* (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2
@@ -461,39 +551,30 @@
* We choose d(i+1) as origin.
*
ORGATI = .FALSE.
- SG2LB = -DELSQ2
- SG2UB = ZERO
+ II = IP1
+ SGLB = -DELSQ2 / ( D( II )+SQ2 )
+ SGUB = 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 ) ) )
+ TAU2 = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
ELSE
- TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
+ TAU2 = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
END IF
*
-* TAU now is an estimation of SIGMA^2 - D( IP1 )^2. The
+* TAU2 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 ) ) )
+ TAU = TAU2 / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+
+ $ TAU2 ) ) )
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
+ SIGMA = D( II ) + TAU
+ DO 130 J = 1, N
+ WORK( J ) = D( J ) + D( II ) + TAU
+ DELTA( J ) = ( D( J )-D( II ) ) - TAU
+ 130 CONTINUE
IIM1 = II - 1
IIP1 = II + 1
*
@@ -541,8 +622,9 @@
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
+ ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
+ $ + THREE*ABS( TEMP )
+* $ + ABS( TAU2 )*DW
*
* Test for convergence
*
@@ -551,9 +633,9 @@
END IF
*
IF( W.LE.ZERO ) THEN
- SG2LB = MAX( SG2LB, TAU )
+ SGLB = MAX( SGLB, TAU )
ELSE
- SG2UB = MIN( SG2UB, TAU )
+ SGUB = MIN( SGUB, TAU )
END IF
*
* Calculate the new step
@@ -618,8 +700,38 @@
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
+*
+ IF( INFO.NE.0 ) THEN
+*
+* If INFO is not 0, i.e., DLAED6 failed, switch back
+* to 2 pole interpolation.
+*
+ SWTCH3 = .FALSE.
+ INFO = 0
+ 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
+ END IF
END IF
*
* Note, eta should be positive if w is negative, and
@@ -630,27 +742,33 @@
*
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
+*
+ ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
+ TEMP = TAU + ETA
+ IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
IF( W.LT.ZERO ) THEN
- ETA = ( SG2UB-TAU ) / TWO
+ ETA = ( SGUB-TAU ) / TWO
ELSE
- ETA = ( SG2LB-TAU ) / TWO
+ ETA = ( SGLB-TAU ) / TWO
+ END IF
+ IF( GEOMAVG ) THEN
+ IF( W .LT. ZERO ) THEN
+ IF( TAU .GT. ZERO ) THEN
+ ETA = SQRT(SGUB*TAU)-TAU
+ END IF
+ ELSE
+ IF( SGLB .GT. ZERO ) THEN
+ ETA = SQRT(SGLB*TAU)-TAU
+ END IF
+ END IF
END IF
END IF
*
- TAU = TAU + ETA
- ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
-*
PREW = W
*
+ TAU = TAU + ETA
SIGMA = SIGMA + ETA
+*
DO 170 J = 1, N
WORK( J ) = WORK( J ) + ETA
DELTA( J ) = DELTA( J ) - ETA
@@ -680,18 +798,14 @@
ERRETM = ERRETM + PHI
190 CONTINUE
*
- TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
+ TAU2 = WORK( II )*DELTA( II )
+ TEMP = Z( II ) / TAU2
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
+ ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
+ $ + THREE*ABS( TEMP )
+* $ + ABS( TAU2 )*DW
*
SWTCH = .FALSE.
IF( ORGATI ) THEN
@@ -711,9 +825,16 @@
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
+* $ .OR. (SGUB-SGLB).LE.EIGHT*ABS(SGUB+SGLB) ) THEN
GO TO 240
END IF
*
+ IF( W.LE.ZERO ) THEN
+ SGLB = MAX( SGLB, TAU )
+ ELSE
+ SGUB = MIN( SGUB, TAU )
+ END IF
+*
* Calculate the new step
*
IF( .NOT.SWTCH3 ) THEN
@@ -798,8 +919,54 @@
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
+*
+ IF( INFO.NE.0 ) THEN
+*
+* If INFO is not 0, i.e., DLAED6 failed, switch
+* back to two pole interpolation
+*
+ SWTCH3 = .FALSE.
+ INFO = 0
+ 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
+ END IF
END IF
*
* Note, eta should be positive if w is negative, and
@@ -810,32 +977,38 @@
*
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
+*
+ ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
+ TEMP=TAU+ETA
+ IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
IF( W.LT.ZERO ) THEN
- ETA = ( SG2UB-TAU ) / TWO
+ ETA = ( SGUB-TAU ) / TWO
ELSE
- ETA = ( SG2LB-TAU ) / TWO
+ ETA = ( SGLB-TAU ) / TWO
+ END IF
+ IF( GEOMAVG ) THEN
+ IF( W .LT. ZERO ) THEN
+ IF( TAU .GT. ZERO ) THEN
+ ETA = SQRT(SGUB*TAU)-TAU
+ END IF
+ ELSE
+ IF( SGLB .GT. ZERO ) THEN
+ ETA = SQRT(SGLB*TAU)-TAU
+ END IF
+ END IF
END IF
END IF
*
- TAU = TAU + ETA
- ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
+ PREW = W
*
+ TAU = TAU + 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
@@ -860,21 +1033,18 @@
ERRETM = ERRETM + PHI
220 CONTINUE
*
- TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
+ TAU2 = WORK( II )*DELTA( II )
+ TEMP = Z( II ) / TAU2
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
+ ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
+ $ + THREE*ABS( TEMP )
+* $ + ABS( TAU2 )*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