SUBROUTINE PDLAED2( ICTXT, K, N, N1, NB, D, DROW, DCOL, Q, LDQ,
$ RHO, Z, W, DLAMDA, Q2, LDQ2, QBUF, CTOT, PSM,
$ NPCOL, INDX, INDXC, INDXP, INDCOL, COLTYP, NN,
$ NN1, NN2, IB1, IB2 )
*
* -- ScaLAPACK auxiliary routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* December 31, 1998
*
* .. Scalar Arguments ..
INTEGER DCOL, DROW, IB1, IB2, ICTXT, K, LDQ, LDQ2, N,
$ N1, NB, NN, NN1, NN2, NPCOL
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
INTEGER COLTYP( * ), CTOT( 0: NPCOL-1, 4 ),
$ INDCOL( N ), INDX( * ), INDXC( * ), INDXP( * ),
$ PSM( 0: NPCOL-1, 4 )
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ),
$ Q2( LDQ2, * ), QBUF( * ), W( * ), Z( * )
* ..
*
* Purpose
* =======
*
* PDLAED2 sorts the two sets of eigenvalues together into a single
* sorted set. Then it tries to deflate the size of the problem.
* There are two ways in which deflation can occur: when two or more
* eigenvalues are close together or if there is a tiny entry in the
* Z vector. For each such occurrence the order of the related secular
* equation problem is reduced by one.
*
* Arguments
* =========
*
* ICTXT (global input) INTEGER
* The BLACS context handle, indicating the global context of
* the operation on the matrix. The context itself is global.
*
* K (output) INTEGER
* The number of non-deflated eigenvalues, and the order of the
* related secular equation. 0 <= K <=N.
*
* N (input) INTEGER
* The dimension of the symmetric tridiagonal matrix. N >= 0.
*
* N1 (input) INTEGER
* The location of the last eigenvalue in the leading sub-matrix.
* min(1,N) < N1 < N.
*
* NB (global input) INTEGER
* The blocking factor used to distribute the columns of the
* matrix. NB >= 1.
*
* D (input/output) DOUBLE PRECISION array, dimension (N)
* On entry, D contains the eigenvalues of the two submatrices to
* be combined.
* On exit, D contains the trailing (N-K) updated eigenvalues
* (those which were deflated) sorted into increasing order.
*
* DROW (global input) INTEGER
* The process row over which the first row of the matrix D is
* distributed. 0 <= DROW < NPROW.
*
* DCOL (global input) INTEGER
* The process column over which the first column of the
* matrix D is distributed. 0 <= DCOL < NPCOL.
*
* Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
* On entry, Q contains the eigenvectors of two submatrices in
* the two square blocks with corners at (1,1), (N1,N1)
* and (N1+1, N1+1), (N,N).
* On exit, Q contains the trailing (N-K) updated eigenvectors
* (those which were deflated) in its last N-K columns.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= max(1,NQ).
*
* RHO (global input/output) DOUBLE PRECISION
* On entry, the off-diagonal element associated with the rank-1
* cut which originally split the two submatrices which are now
* being recombined.
* On exit, RHO has been modified to the value required by
* PDLAED3.
*
* Z (global input) DOUBLE PRECISION array, dimension (N)
* On entry, Z contains the updating vector (the last
* row of the first sub-eigenvector matrix and the first row of
* the second sub-eigenvector matrix).
* On exit, the contents of Z have been destroyed by the updating
* process.
*
* DLAMDA (global output) DOUBLE PRECISION array, dimension (N)
* A copy of the first K eigenvalues which will be used by
* SLAED3 to form the secular equation.
*
* W (global output) DOUBLE PRECISION array, dimension (N)
* The first k values of the final deflation-altered z-vector
* which will be passed to SLAED3.
*
* Q2 (output) DOUBLE PRECISION array, dimension (LDQ2, NQ)
* A copy of the first K eigenvectors which will be used by
*
* LDQ2 (input) INTEGER
* The leading dimension of the array Q2.
*
* QBUF (workspace) DOUBLE PRECISION array, dimension 3*N
*
* CTOT (workspace) INTEGER array, dimension( NPCOL, 4)
*
* PSM (workspace) INTEGER array, dimension( NPCOL, 4)
*
* NPCOL (global input) INTEGER
* The total number of columns over which the distributed
* submatrix is distributed.
*
* INDX (workspace) INTEGER array, dimension (N)
* The permutation used to sort the contents of DLAMDA into
* ascending order.
*
* INDXC (output) INTEGER array, dimension (N)
* The permutation used to arrange the columns of the deflated
* Q matrix into three groups: the first group contains non-zero
* elements only at and above N1, the second contains
* non-zero elements only below N1, and the third is dense.
*
* INDXP (workspace) INTEGER array, dimension (N)
* The permutation used to place deflated values of D at the end
* of the array. INDXP(1:K) points to the nondeflated D-values
* and INDXP(K+1:N) points to the deflated eigenvalues.
*
* INDCOL (workspace) INTEGER array, dimension (N)
*
* COLTYP (workspace/output) INTEGER array, dimension (N)
* During execution, a label which will indicate which of the
* following types a column in the Q2 matrix is:
* 1 : non-zero in the upper half only;
* 2 : dense;
* 3 : non-zero in the lower half only;
* 4 : deflated.
*
* NN (global output) INTEGER, the order of matrix U, (PDLAED1).
* NN1 (global output) INTEGER, the order of matrix Q1, (PDLAED1).
* NN2 (global output) INTEGER, the order of matrix Q2, (PDLAED1).
* IB1 (global output) INTEGER, pointeur on Q1, (PDLAED1).
* IB2 (global output) INTEGER, pointeur on Q2, (PDLAED1).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
$ TWO = 2.0D0, EIGHT = 8.0D0 )
* ..
* .. Local Scalars ..
INTEGER COL, CT, I, IAM, IE1, IE2, IMAX, INFO, J, JJQ2,
$ JJS, JMAX, JS, K2, MYCOL, MYROW, N1P1, N2, NJ,
$ NJCOL, NJJ, NP, NPROCS, NPROW, PJ, PJCOL, PJJ
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX, INDXG2L, INDXL2G, NUMROC
DOUBLE PRECISION DLAPY2, PDLAMCH
EXTERNAL IDAMAX, INDXG2L, INDXL2G, NUMROC, PDLAMCH,
$ DLAPY2
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, BLACS_PINFO, DCOPY, DGERV2D,
$ DGESD2D, DLAPST, DROT, DSCAL, INFOG1L
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, MOD, SQRT
* ..
* .. External Functions ..
* ..
* .. Local Arrays ..
INTEGER PTT( 4 )
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
CALL BLACS_PINFO( IAM, NPROCS )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
NP = NUMROC( N, NB, MYROW, DROW, NPROW )
*
N2 = N - N1
N1P1 = N1 + 1
*
IF( RHO.LT.ZERO ) THEN
CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
*
* Normalize z so that norm(z) = 1. Since z is the concatenation of
* two normalized vectors, norm2(z) = sqrt(2).
*
T = ONE / SQRT( TWO )
CALL DSCAL( N, T, Z, 1 )
*
* RHO = ABS( norm(z)**2 * RHO )
*
RHO = ABS( TWO*RHO )
*
* Calculate the allowable deflation tolerance
*
IMAX = IDAMAX( N, Z, 1 )
JMAX = IDAMAX( N, D, 1 )
EPS = PDLAMCH( ICTXT, 'Epsilon' )
TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
*
* If the rank-1 modifier is small enough, no more needs to be done
* except to reorganize Q so that its columns correspond with the
* elements in D.
*
IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
K = 0
GO TO 220
END IF
*
* If there are multiple eigenvalues then the problem deflates. Here
* the number of equal eigenvalues are found. As each equal
* eigenvalue is found, an elementary reflector is computed to rotate
* the corresponding eigensubspace so that the corresponding
* components of Z are zero in this new basis.
*
*
CALL DLAPST( 'I', N, D, INDX, INFO )
*
DO 10 I = 1, N1
COLTYP( I ) = 1
10 CONTINUE
DO 20 I = N1P1, N
COLTYP( I ) = 3
20 CONTINUE
COL = DCOL
DO 40 I = 1, N, NB
DO 30 J = 0, NB - 1
IF( I+J.LE.N )
$ INDCOL( I+J ) = COL
30 CONTINUE
COL = MOD( COL+1, NPCOL )
40 CONTINUE
*
K = 0
K2 = N + 1
DO 50 J = 1, N
NJ = INDX( J )
IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
COLTYP( NJ ) = 4
INDXP( K2 ) = NJ
IF( J.EQ.N )
$ GO TO 80
ELSE
PJ = NJ
GO TO 60
END IF
50 CONTINUE
60 CONTINUE
J = J + 1
NJ = INDX( J )
IF( J.GT.N )
$ GO TO 80
IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
COLTYP( NJ ) = 4
INDXP( K2 ) = NJ
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( PJ )
C = Z( NJ )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = DLAPY2( C, S )
T = D( NJ ) - D( PJ )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( NJ ) = TAU
Z( PJ ) = ZERO
IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
$ COLTYP( NJ ) = 2
COLTYP( PJ ) = 4
CALL INFOG1L( NJ, NB, NPCOL, MYCOL, DCOL, NJJ, NJCOL )
CALL INFOG1L( PJ, NB, NPCOL, MYCOL, DCOL, PJJ, PJCOL )
IF( INDCOL( PJ ).EQ.INDCOL( NJ ) .AND. MYCOL.EQ.NJCOL ) THEN
CALL DROT( NP, Q( 1, PJJ ), 1, Q( 1, NJJ ), 1, C, S )
ELSE IF( MYCOL.EQ.PJCOL ) THEN
CALL DGESD2D( ICTXT, NP, 1, Q( 1, PJJ ), NP, MYROW,
$ NJCOL )
CALL DGERV2D( ICTXT, NP, 1, QBUF, NP, MYROW, NJCOL )
CALL DROT( NP, Q( 1, PJJ ), 1, QBUF, 1, C, S )
ELSE IF( MYCOL.EQ.NJCOL ) THEN
CALL DGESD2D( ICTXT, NP, 1, Q( 1, NJJ ), NP, MYROW,
$ PJCOL )
CALL DGERV2D( ICTXT, NP, 1, QBUF, NP, MYROW, PJCOL )
CALL DROT( NP, QBUF, 1, Q( 1, NJJ ), 1, C, S )
END IF
T = D( PJ )*C**2 + D( NJ )*S**2
D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
D( PJ ) = T
K2 = K2 - 1
I = 1
70 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = PJ
I = I + 1
GO TO 70
ELSE
INDXP( K2+I-1 ) = PJ
END IF
ELSE
INDXP( K2+I-1 ) = PJ
END IF
PJ = NJ
ELSE
K = K + 1
DLAMDA( K ) = D( PJ )
W( K ) = Z( PJ )
INDXP( K ) = PJ
PJ = NJ
END IF
END IF
GO TO 60
80 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
DLAMDA( K ) = D( PJ )
W( K ) = Z( PJ )
INDXP( K ) = PJ
*
* Count up the total number of the various types of columns, then
* form a permutation which positions the four column types into
* four uniform groups (although one or more of these groups may be
* empty).
*
DO 100 J = 1, 4
DO 90 I = 0, NPCOL - 1
CTOT( I, J ) = 0
90 CONTINUE
PTT( J ) = 0
100 CONTINUE
DO 110 J = 1, N
CT = COLTYP( J )
COL = INDCOL( J )
CTOT( COL, CT ) = CTOT( COL, CT ) + 1
110 CONTINUE
*
* PSM(*) = Position in SubMatrix (of types 1 through 4)
*
DO 120 COL = 0, NPCOL - 1
PSM( COL, 1 ) = 1
PSM( COL, 2 ) = 1 + CTOT( COL, 1 )
PSM( COL, 3 ) = PSM( COL, 2 ) + CTOT( COL, 2 )
PSM( COL, 4 ) = PSM( COL, 3 ) + CTOT( COL, 3 )
120 CONTINUE
PTT( 1 ) = 1
DO 140 I = 2, 4
CT = 0
DO 130 J = 0, NPCOL - 1
CT = CT + CTOT( J, I-1 )
130 CONTINUE
PTT( I ) = PTT( I-1 ) + CT
140 CONTINUE
*
* Fill out the INDXC array so that the permutation which it induces
* will place all type-1 columns first, all type-2 columns next,
* then all type-3's, and finally all type-4's.
*
DO 150 J = 1, N
JS = INDXP( J )
COL = INDCOL( JS )
CT = COLTYP( JS )
I = INDXL2G( PSM( COL, CT ), NB, COL, DCOL, NPCOL )
INDX( J ) = I
INDXC( PTT( CT ) ) = I
PSM( COL, CT ) = PSM( COL, CT ) + 1
PTT( CT ) = PTT( CT ) + 1
150 CONTINUE
*
*
DO 160 J = 1, N
JS = INDXP( J )
JJS = INDXG2L( JS, NB, J, J, NPCOL )
COL = INDCOL( JS )
IF( COL.EQ.MYCOL ) THEN
I = INDX( J )
JJQ2 = INDXG2L( I, NB, J, J, NPCOL )
CALL DCOPY( NP, Q( 1, JJS ), 1, Q2( 1, JJQ2 ), 1 )
END IF
160 CONTINUE
*
*
* The deflated eigenvalues and their corresponding vectors go back
* into the last N - K slots of D and Q respectively.
*
CALL DCOPY( N, D, 1, Z, 1 )
DO 170 J = K + 1, N
JS = INDXP( J )
I = INDX( J )
D( I ) = Z( JS )
170 CONTINUE
*
PTT( 1 ) = 1
DO 190 I = 2, 4
CT = 0
DO 180 J = 0, NPCOL - 1
CT = CT + CTOT( J, I-1 )
180 CONTINUE
PTT( I ) = PTT( I-1 ) + CT
190 CONTINUE
*
*
IB1 = INDXC( 1 )
IE1 = IB1
IB2 = INDXC( PTT( 2 ) )
IE2 = IB2
DO 200 I = 2, PTT( 3 ) - 1
IB1 = MIN( IB1, INDXC( I ) )
IE1 = MAX( IE1, INDXC( I ) )
200 CONTINUE
DO 210 I = PTT( 2 ), PTT( 4 ) - 1
IB2 = MIN( IB2, INDXC( I ) )
IE2 = MAX( IE2, INDXC( I ) )
210 CONTINUE
NN1 = IE1 - IB1 + 1
NN2 = IE2 - IB2 + 1
NN = MAX( IE1, IE2 ) - MIN( IB1, IB2 ) + 1
220 CONTINUE
RETURN
*
* End of PDLAED2
*
END