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SUBROUTINE CLAR1V( N, B1, BN, SIGMA, D, L, LD, LLD, GERSCH, Z,
$ ZTZ, MINGMA, R, ISUPPZ, WORK )
*
* -- 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 B1, BN, N, R
REAL MINGMA, SIGMA, ZTZ
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * )
REAL D( * ), GERSCH( * ), L( * ), LD( * ), LLD( * ),
$ WORK( * )
COMPLEX Z( * )
* ..
* Common block to return operation count
* .. Common blocks ..
COMMON / LATIME / OPS, ITCNT
* ..
* .. Scalars in Common ..
REAL ITCNT, OPS
* ..
*
* Purpose
* =======
*
* CLAR1V computes the (scaled) r-th column of the inverse of
* the sumbmatrix in rows B1 through BN of the tridiagonal matrix
* L D L^T - sigma I. The following steps accomplish this computation :
* (a) Stationary qd transform, L D L^T - sigma I = L(+) D(+) L(+)^T,
* (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T,
* (c) Computation of the diagonal elements of the inverse of
* L D L^T - sigma I by combining the above transforms, and choosing
* r as the index where the diagonal of the inverse is (one of the)
* largest in magnitude.
* (d) Computation of the (scaled) r-th column of the inverse using the
* twisted factorization obtained by combining the top part of the
* the stationary and the bottom part of the progressive transform.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix L D L^T.
*
* B1 (input) INTEGER
* First index of the submatrix of L D L^T.
*
* BN (input) INTEGER
* Last index of the submatrix of L D L^T.
*
* SIGMA (input) REAL
* The shift. Initially, when R = 0, SIGMA should be a good
* approximation to an eigenvalue of L D L^T.
*
* L (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the unit bidiagonal matrix
* L, in elements 1 to N-1.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the diagonal matrix D.
*
* LD (input) REAL array, dimension (N-1)
* The n-1 elements L(i)*D(i).
*
* LLD (input) REAL array, dimension (N-1)
* The n-1 elements L(i)*L(i)*D(i).
*
* GERSCH (input) REAL array, dimension (2*N)
* The n Gerschgorin intervals. These are used to restrict
* the initial search for R, when R is input as 0.
*
* Z (output) COMPLEX array, dimension (N)
* The (scaled) r-th column of the inverse. Z(R) is returned
* to be 1.
*
* ZTZ (output) REAL
* The square of the norm of Z.
*
* MINGMA (output) REAL
* The reciprocal of the largest (in magnitude) diagonal
* element of the inverse of L D L^T - sigma I.
*
* R (input/output) INTEGER
* Initially, R should be input to be 0 and is then output as
* the index where the diagonal element of the inverse is
* largest in magnitude. In later iterations, this same value
* of R should be input.
*
* ISUPPZ (output) INTEGER array, dimension (2)
* The support of the vector in Z, i.e., the vector Z is
* nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
*
* WORK (workspace) REAL array, dimension (4*N)
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
* Ken Stanley, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLKSIZ
PARAMETER ( BLKSIZ = 32 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
COMPLEX CONE
PARAMETER ( CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
LOGICAL SAWNAN
INTEGER FROM, I, INDP, INDS, INDUMN, J, R1, R2, TO
REAL DMINUS, DPLUS, EPS, S, TMP
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
EPS = SLAMCH( 'Precision' )
IF( R.EQ.0 ) THEN
*
* Eliminate the top and bottom indices from the possible values
* of R where the desired eigenvector is largest in magnitude.
*
R1 = B1
DO 10 I = B1, BN
IF( SIGMA.GE.GERSCH( 2*I-1 ) .OR. SIGMA.LE.GERSCH( 2*I ) )
$ THEN
R1 = I
GO TO 20
END IF
10 CONTINUE
20 CONTINUE
R2 = BN
DO 30 I = BN, B1, -1
IF( SIGMA.GE.GERSCH( 2*I-1 ) .OR. SIGMA.LE.GERSCH( 2*I ) )
$ THEN
R2 = I
GO TO 40
END IF
30 CONTINUE
40 CONTINUE
ELSE
R1 = R
R2 = R
END IF
*
INDUMN = N
INDS = 2*N + 1
INDP = 3*N + 1
SAWNAN = .FALSE.
*
* Compute the stationary transform (using the differential form)
* untill the index R2
*
IF( B1.EQ.1 ) THEN
WORK( INDS ) = ZERO
ELSE
WORK( INDS ) = LLD( B1-1 )
END IF
OPS = OPS + REAL( 1 )
S = WORK( INDS ) - SIGMA
DO 50 I = B1, R2 - 1
OPS = OPS + REAL( 5 )
DPLUS = D( I ) + S
WORK( I ) = LD( I ) / DPLUS
WORK( INDS+I ) = S*WORK( I )*L( I )
S = WORK( INDS+I ) - SIGMA
50 CONTINUE
*
IF( .NOT.( S.GT.ZERO .OR. S.LT.ONE ) ) THEN
*
* Run a slower version of the above loop if a NaN is detected
*
SAWNAN = .TRUE.
J = B1 + 1
60 CONTINUE
IF( WORK( INDS+J ).GT.ZERO .OR. WORK( INDS+J ).LT.ONE ) THEN
J = J + 1
GO TO 60
END IF
WORK( INDS+J ) = LLD( J )
S = WORK( INDS+J ) - SIGMA
DO 70 I = J + 1, R2 - 1
OPS = OPS + REAL( 3 )
DPLUS = D( I ) + S
WORK( I ) = LD( I ) / DPLUS
IF( WORK( I ).EQ.ZERO ) THEN
WORK( INDS+I ) = LLD( I )
ELSE
OPS = OPS + REAL( 2 )
WORK( INDS+I ) = S*WORK( I )*L( I )
END IF
S = WORK( INDS+I ) - SIGMA
70 CONTINUE
END IF
OPS = OPS + REAL( 1 )
WORK( INDP+BN-1 ) = D( BN ) - SIGMA
DO 80 I = BN - 1, R1, -1
OPS = OPS + REAL( 5 )
DMINUS = LLD( I ) + WORK( INDP+I )
TMP = D( I ) / DMINUS
WORK( INDUMN+I ) = L( I )*TMP
WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - SIGMA
80 CONTINUE
TMP = WORK( INDP+R1-1 )
IF( .NOT.( TMP.GT.ZERO .OR. TMP.LT.ONE ) ) THEN
*
* Run a slower version of the above loop if a NaN is detected
*
SAWNAN = .TRUE.
J = BN - 3
90 CONTINUE
IF( WORK( INDP+J ).GT.ZERO .OR. WORK( INDP+J ).LT.ONE ) THEN
J = J - 1
GO TO 90
END IF
OPS = OPS + REAL( 1 )
WORK( INDP+J ) = D( J+1 ) - SIGMA
DO 100 I = J, R1, -1
OPS = OPS + REAL( 3 )
DMINUS = LLD( I ) + WORK( INDP+I )
TMP = D( I ) / DMINUS
WORK( INDUMN+I ) = L( I )*TMP
IF( TMP.EQ.ZERO ) THEN
OPS = OPS + REAL( 1 )
WORK( INDP+I-1 ) = D( I ) - SIGMA
ELSE
OPS = OPS + REAL( 2 )
WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - SIGMA
END IF
100 CONTINUE
END IF
*
* Find the index (from R1 to R2) of the largest (in magnitude)
* diagonal element of the inverse
*
MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 )
IF( MINGMA.EQ.ZERO )
$ MINGMA = EPS*WORK( INDS+R1-1 )
R = R1
DO 110 I = R1, R2 - 1
OPS = OPS + REAL( 1 )
TMP = WORK( INDS+I ) + WORK( INDP+I )
IF( TMP.EQ.ZERO ) THEN
OPS = OPS + REAL( 1 )
TMP = EPS*WORK( INDS+I )
END IF
IF( ABS( TMP ).LT.ABS( MINGMA ) ) THEN
MINGMA = TMP
R = I + 1
END IF
110 CONTINUE
*
* Compute the (scaled) r-th column of the inverse
*
ISUPPZ( 1 ) = B1
ISUPPZ( 2 ) = BN
Z( R ) = CONE
ZTZ = ONE
IF( .NOT.SAWNAN ) THEN
FROM = R - 1
TO = MAX( R-BLKSIZ, B1 )
120 CONTINUE
IF( FROM.GE.B1 ) THEN
DO 130 I = FROM, TO, -1
OPS = OPS + REAL( 9 )
Z( I ) = -( WORK( I )*Z( I+1 ) )
ZTZ = ZTZ + REAL( Z( I )*Z( I ) )
130 CONTINUE
IF( ABS( Z( TO ) ).LE.EPS .AND. ABS( Z( TO+1 ) ).LE.EPS )
$ THEN
ISUPPZ( 1 ) = TO + 2
ELSE
FROM = TO - 1
TO = MAX( TO-BLKSIZ, B1 )
GO TO 120
END IF
END IF
FROM = R + 1
TO = MIN( R+BLKSIZ, BN )
140 CONTINUE
IF( FROM.LE.BN ) THEN
DO 150 I = FROM, TO
OPS = OPS + REAL( 9 )
Z( I ) = -( WORK( INDUMN+I-1 )*Z( I-1 ) )
ZTZ = ZTZ + REAL( Z( I )*Z( I ) )
150 CONTINUE
IF( ABS( Z( TO ) ).LE.EPS .AND. ABS( Z( TO-1 ) ).LE.EPS )
$ THEN
ISUPPZ( 2 ) = TO - 2
ELSE
FROM = TO + 1
TO = MIN( TO+BLKSIZ, BN )
GO TO 140
END IF
END IF
ELSE
DO 160 I = R - 1, B1, -1
IF( Z( I+1 ).EQ.ZERO ) THEN
OPS = OPS + REAL( 3 )
Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 )
ELSE IF( ABS( Z( I+1 ) ).LE.EPS .AND. ABS( Z( I+2 ) ).LE.
$ EPS ) THEN
ISUPPZ( 1 ) = I + 3
GO TO 170
ELSE
OPS = OPS + REAL( 2 )
Z( I ) = -( WORK( I )*Z( I+1 ) )
END IF
OPS = OPS + REAL( 7 )
ZTZ = ZTZ + REAL( Z( I )*Z( I ) )
160 CONTINUE
170 CONTINUE
DO 180 I = R, BN - 1
IF( Z( I ).EQ.ZERO ) THEN
OPS = OPS + REAL( 3 )
Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 )
ELSE IF( ABS( Z( I ) ).LE.EPS .AND. ABS( Z( I-1 ) ).LE.EPS )
$ THEN
ISUPPZ( 2 ) = I - 2
GO TO 190
ELSE
OPS = OPS + REAL( 2 )
Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) )
END IF
OPS = OPS + REAL( 7 )
ZTZ = ZTZ + REAL( Z( I+1 )*Z( I+1 ) )
180 CONTINUE
190 CONTINUE
END IF
DO 200 I = B1, ISUPPZ( 1 ) - 3
Z( I ) = ZERO
200 CONTINUE
DO 210 I = ISUPPZ( 2 ) + 3, BN
Z( I ) = ZERO
210 CONTINUE
*
RETURN
*
* End of CLAR1V
*
END
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