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SUBROUTINE DLARRE( N, D, E, TOL, NSPLIT, ISPLIT, M, W, WOFF,
$ GERSCH, WORK, INFO )
*
* -- LAPACK auxiliary routine (instru to count ops, version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER INFO, M, N, NSPLIT
DOUBLE PRECISION TOL
* ..
* .. Array Arguments ..
INTEGER ISPLIT( * )
DOUBLE PRECISION D( * ), E( * ), GERSCH( * ), W( * ), WOFF( * ),
$ WORK( * )
* ..
* Common block to return operation count
* .. Common blocks ..
COMMON / LATIME / OPS, ITCNT
* ..
* .. Scalars in Common ..
DOUBLE PRECISION ITCNT, OPS
* ..
*
* Purpose
* =======
*
* Given the tridiagonal matrix T, DLARRE sets "small" off-diagonal
* elements to zero, and for each unreduced block T_i, it finds
* (i) the numbers sigma_i
* (ii) the base T_i - sigma_i I = L_i D_i L_i^T representations and
* (iii) eigenvalues of each L_i D_i L_i^T.
* The representations and eigenvalues found are then used by
* DSTEGR to compute the eigenvectors of a symmetric tridiagonal
* matrix. Currently, the base representations are limited to being
* positive or negative definite, and the eigenvalues of the definite
* matrices are found by the dqds algorithm (subroutine DLASQ2). As
* an added benefit, DLARRE also outputs the n Gerschgorin
* intervals for each L_i D_i L_i^T.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix.
*
* D (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal
* matrix T.
* On exit, the n diagonal elements of the diagonal
* matrices D_i.
*
* E (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix T; E(N) need not be set.
* On exit, the subdiagonal elements of the unit bidiagonal
* matrices L_i.
*
* TOL (input) DOUBLE PRECISION
* The threshold for splitting. If on input |E(i)| < TOL, then
* the matrix T is split into smaller blocks.
*
* NSPLIT (input) INTEGER
* The number of blocks T splits into. 1 <= NSPLIT <= N.
*
* ISPLIT (output) INTEGER array, dimension (2*N)
* The splitting points, at which T breaks up into submatrices.
* The first submatrix consists of rows/columns 1 to ISPLIT(1),
* the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
* etc., and the NSPLIT-th consists of rows/columns
* ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
*
* M (output) INTEGER
* The total number of eigenvalues (of all the L_i D_i L_i^T)
* found.
*
* W (output) DOUBLE PRECISION array, dimension (N)
* The first M elements contain the eigenvalues. The
* eigenvalues of each of the blocks, L_i D_i L_i^T, are
* sorted in ascending order.
*
* WOFF (output) DOUBLE PRECISION array, dimension (N)
* The NSPLIT base points sigma_i.
*
* GERSCH (output) DOUBLE PRECISION array, dimension (2*N)
* The n Gerschgorin intervals.
*
* WORK (input) DOUBLE PRECISION array, dimension (4*N???)
* Workspace.
*
* INFO (output) INTEGER
* Output error code from DLASQ2
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, FOUR, FOURTH
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
$ FOUR = 4.0D0, FOURTH = ONE / FOUR )
* ..
* .. Local Scalars ..
INTEGER CNT, I, IBEGIN, IEND, IN, J, JBLK, MAXCNT
DOUBLE PRECISION DELTA, EPS, GL, GU, NRM, OFFD, S, SGNDEF,
$ SIGMA, TAU, TMP1, WIDTH
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLASQ2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
EPS = DLAMCH( 'Precision' )
*
* Compute Splitting Points
*
NSPLIT = 1
DO 10 I = 1, N - 1
IF( ABS( E( I ) ).LE.TOL ) THEN
ISPLIT( NSPLIT ) = I
NSPLIT = NSPLIT + 1
END IF
10 CONTINUE
ISPLIT( NSPLIT ) = N
*
IBEGIN = 1
DO 170 JBLK = 1, NSPLIT
IEND = ISPLIT( JBLK )
IF( IBEGIN.EQ.IEND ) THEN
W( IBEGIN ) = D( IBEGIN )
WOFF( JBLK ) = ZERO
IBEGIN = IEND + 1
GO TO 170
END IF
IN = IEND - IBEGIN + 1
*
* Form the n Gerschgorin intervals
*
OPS = OPS + DBLE( 4 )
GL = D( IBEGIN ) - ABS( E( IBEGIN ) )
GU = D( IBEGIN ) + ABS( E( IBEGIN ) )
GERSCH( 2*IBEGIN-1 ) = GL
GERSCH( 2*IBEGIN ) = GU
GERSCH( 2*IEND-1 ) = D( IEND ) - ABS( E( IEND-1 ) )
GERSCH( 2*IEND ) = D( IEND ) + ABS( E( IEND-1 ) )
GL = MIN( GERSCH( 2*IEND-1 ), GL )
GU = MAX( GERSCH( 2*IEND ), GU )
DO 20 I = IBEGIN + 1, IEND - 1
OPS = OPS + DBLE( 3 )
OFFD = ABS( E( I-1 ) ) + ABS( E( I ) )
GERSCH( 2*I-1 ) = D( I ) - OFFD
GL = MIN( GERSCH( 2*I-1 ), GL )
GERSCH( 2*I ) = D( I ) + OFFD
GU = MAX( GERSCH( 2*I ), GU )
20 CONTINUE
NRM = MAX( ABS( GL ), ABS( GU ) )
*
* Find the number SIGMA where the base representation
* T - sigma I = L D L^T is to be formed.
*
WIDTH = GU - GL
DO 30 I = IBEGIN, IEND - 1
OPS = OPS + DBLE( 1 )
WORK( I ) = E( I )*E( I )
30 CONTINUE
OPS = OPS + DBLE( 6 )
DO 50 J = 1, 2
IF( J.EQ.1 ) THEN
TAU = GL + FOURTH*WIDTH
ELSE
TAU = GU - FOURTH*WIDTH
END IF
TMP1 = D( IBEGIN ) - TAU
IF( TMP1.LT.ZERO ) THEN
CNT = 1
ELSE
CNT = 0
END IF
DO 40 I = IBEGIN + 1, IEND
OPS = OPS + DBLE( 3 )
TMP1 = D( I ) - TAU - WORK( I-1 ) / TMP1
IF( TMP1.LT.ZERO )
$ CNT = CNT + 1
40 CONTINUE
IF( CNT.EQ.0 ) THEN
GL = TAU
ELSE IF( CNT.EQ.IN ) THEN
GU = TAU
END IF
IF( J.EQ.1 ) THEN
MAXCNT = CNT
SIGMA = GL
SGNDEF = ONE
ELSE
IF( IN-CNT.GT.MAXCNT ) THEN
SIGMA = GU
SGNDEF = -ONE
END IF
END IF
50 CONTINUE
*
* Find the base L D L^T representation
*
OPS = OPS + DBLE( 1 )
WORK( 3*IN ) = ONE
DELTA = EPS
TAU = SGNDEF*NRM
60 CONTINUE
OPS = OPS + DBLE( 3+5*( IN-1 ) )
SIGMA = SIGMA - DELTA*TAU
WORK( 1 ) = D( IBEGIN ) - SIGMA
J = IBEGIN
DO 70 I = 1, IN - 1
WORK( 2*IN+I ) = ONE / WORK( 2*I-1 )
TMP1 = E( J )*WORK( 2*IN+I )
WORK( 2*I+1 ) = ( D( J+1 )-SIGMA ) - TMP1*E( J )
WORK( 2*I ) = TMP1
J = J + 1
70 CONTINUE
OPS = OPS + DBLE( IN )
DO 80 I = IN, 1, -1
TMP1 = SGNDEF*WORK( 2*I-1 )
IF( TMP1.LT.ZERO .OR. WORK( 2*IN+I ).EQ.ZERO .OR. .NOT.
$ ( TMP1.GT.ZERO .OR. TMP1.LT.ONE ) ) THEN
OPS = OPS + DBLE( 1 )
DELTA = TWO*DELTA
GO TO 60
END IF
J = J - 1
80 CONTINUE
*
OPS = OPS + DBLE( IN-1 )
J = IBEGIN
D( IBEGIN ) = WORK( 1 )
WORK( 1 ) = ABS( WORK( 1 ) )
DO 90 I = 1, IN - 1
TMP1 = E( J )
E( J ) = WORK( 2*I )
WORK( 2*I ) = ABS( TMP1*WORK( 2*I ) )
J = J + 1
D( J ) = WORK( 2*I+1 )
WORK( 2*I+1 ) = ABS( WORK( 2*I+1 ) )
90 CONTINUE
*
CALL DLASQ2( IN, WORK, INFO )
*
OPS = OPS + DBLE( 2 )
TAU = SGNDEF*WORK( IN )
WORK( 3*IN ) = ONE
DELTA = TWO*EPS
100 CONTINUE
OPS = OPS + DBLE( 2 )
TAU = TAU*( ONE-DELTA )
*
OPS = OPS + DBLE( 9*( IN-1 )+1 )
S = -TAU
J = IBEGIN
DO 110 I = 1, IN - 1
WORK( I ) = D( J ) + S
WORK( 2*IN+I ) = ONE / WORK( I )
* WORK( N+I ) = ( E( I ) * D( I ) ) / WORK( I )
WORK( IN+I ) = ( E( J )*D( J ) )*WORK( 2*IN+I )
S = S*WORK( IN+I )*E( J ) - TAU
J = J + 1
110 CONTINUE
WORK( IN ) = D( IEND ) + S
*
* Checking to see if all the diagonal elements of the new
* L D L^T representation have the same sign
*
OPS = OPS + DBLE( IN+1 )
DO 120 I = IN, 1, -1
TMP1 = SGNDEF*WORK( I )
IF( TMP1.LT.ZERO .OR. WORK( 2*IN+I ).EQ.ZERO .OR. .NOT.
$ ( TMP1.GT.ZERO .OR. TMP1.LT.ONE ) ) THEN
OPS = OPS + DBLE( 1 )
DELTA = TWO*DELTA
GO TO 100
END IF
120 CONTINUE
*
SIGMA = SIGMA + TAU
CALL DCOPY( IN, WORK, 1, D( IBEGIN ), 1 )
CALL DCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 )
WOFF( JBLK ) = SIGMA
*
* Update the n Gerschgorin intervals
*
OPS = OPS + DBLE( 2 )
DO 130 I = IBEGIN, IEND
GERSCH( 2*I-1 ) = GERSCH( 2*I-1 ) - SIGMA
GERSCH( 2*I ) = GERSCH( 2*I ) - SIGMA
130 CONTINUE
*
* Compute the eigenvalues of L D L^T.
*
J = IBEGIN
OPS = OPS + DBLE( 2*( IN-1 ) )
DO 140 I = 1, IN - 1
WORK( 2*I-1 ) = ABS( D( J ) )
WORK( 2*I ) = E( J )*E( J )*WORK( 2*I-1 )
J = J + 1
140 CONTINUE
WORK( 2*IN-1 ) = ABS( D( IEND ) )
*
CALL DLASQ2( IN, WORK, INFO )
*
J = IBEGIN
IF( SGNDEF.GT.ZERO ) THEN
DO 150 I = 1, IN
W( J ) = WORK( IN-I+1 )
J = J + 1
150 CONTINUE
ELSE
DO 160 I = 1, IN
W( J ) = -WORK( I )
J = J + 1
160 CONTINUE
END IF
IBEGIN = IEND + 1
170 CONTINUE
M = N
*
RETURN
*
* End of DLARRE
*
END
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