SUBROUTINE SLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO,
$ CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR,
$ GIVCOL, GIVNUM, INDXP, INDX, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
$ QSIZ
REAL RHO
* ..
* .. Array Arguments ..
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
$ INDXQ( * ), PERM( * )
REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ),
$ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
* ..
*
* Purpose
* =======
*
* SLAED8 merges 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 element in the
* Z vector. For each such occurrence the order of the related secular
* equation problem is reduced by one.
*
* Arguments
* =========
*
* ICOMPQ (input) INTEGER
* = 0: Compute eigenvalues only.
* = 1: Compute eigenvectors of original dense symmetric matrix
* also. On entry, Q contains the orthogonal matrix used
* to reduce the original matrix to tridiagonal form.
*
* K (output) INTEGER
* The number of non-deflated eigenvalues, and the order of the
* related secular equation.
*
* N (input) INTEGER
* The dimension of the symmetric tridiagonal matrix. N >= 0.
*
* QSIZ (input) INTEGER
* The dimension of the orthogonal matrix used to reduce
* the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
*
* D (input/output) REAL array, dimension (N)
* On entry, the eigenvalues of the two submatrices to be
* combined. On exit, the trailing (N-K) updated eigenvalues
* (those which were deflated) sorted into increasing order.
*
* Q (input/output) REAL array, dimension (LDQ,N)
* If ICOMPQ = 0, Q is not referenced. Otherwise,
* on entry, Q contains the eigenvectors of the partially solved
* system which has been previously updated in matrix
* multiplies with other partially solved eigensystems.
* 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,N).
*
* INDXQ (input) INTEGER array, dimension (N)
* The permutation which separately sorts the two sub-problems
* in D into ascending order. Note that elements in the second
* half of this permutation must first have CUTPNT added to
* their values in order to be accurate.
*
* RHO (input/output) REAL
* 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
* SLAED3.
*
* CUTPNT (input) INTEGER
* The location of the last eigenvalue in the leading
* sub-matrix. min(1,N) <= CUTPNT <= N.
*
* Z (input) REAL 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 are destroyed by the updating
* process.
*
* DLAMDA (output) REAL array, dimension (N)
* A copy of the first K eigenvalues which will be used by
* SLAED3 to form the secular equation.
*
* Q2 (output) REAL array, dimension (LDQ2,N)
* If ICOMPQ = 0, Q2 is not referenced. Otherwise,
* a copy of the first K eigenvectors which will be used by
* SLAED7 in a matrix multiply (SGEMM) to update the new
* eigenvectors.
*
* LDQ2 (input) INTEGER
* The leading dimension of the array Q2. LDQ2 >= max(1,N).
*
* W (output) REAL array, dimension (N)
* The first k values of the final deflation-altered z-vector and
* will be passed to SLAED3.
*
* PERM (output) INTEGER array, dimension (N)
* The permutations (from deflation and sorting) to be applied
* to each eigenblock.
*
* GIVPTR (output) INTEGER
* The number of Givens rotations which took place in this
* subproblem.
*
* GIVCOL (output) INTEGER array, dimension (2, N)
* Each pair of numbers indicates a pair of columns to take place
* in a Givens rotation.
*
* GIVNUM (output) REAL array, dimension (2, N)
* Each number indicates the S value to be used in the
* corresponding Givens rotation.
*
* 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.
*
* INDX (workspace) INTEGER array, dimension (N)
* The permutation used to sort the contents of D into ascending
* order.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* Based on contributions by
* Jeff Rutter, Computer Science Division, University of California
* at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL MONE, ZERO, ONE, TWO, EIGHT
PARAMETER ( MONE = -1.0E0, ZERO = 0.0E0, ONE = 1.0E0,
$ TWO = 2.0E0, EIGHT = 8.0E0 )
* ..
* .. Local Scalars ..
*
INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
REAL C, EPS, S, T, TAU, TOL
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SLAMCH, SLAPY2
EXTERNAL ISAMAX, SLAMCH, SLAPY2
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLACPY, SLAMRG, SROT, SSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
INFO = -10
ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLAED8', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
N1 = CUTPNT
N2 = N - N1
N1P1 = N1 + 1
*
IF( RHO.LT.ZERO ) THEN
CALL SSCAL( N2, MONE, Z( N1P1 ), 1 )
END IF
*
* Normalize z so that norm(z) = 1
*
T = ONE / SQRT( TWO )
DO 10 J = 1, N
INDX( J ) = J
10 CONTINUE
CALL SSCAL( N, T, Z, 1 )
RHO = ABS( TWO*RHO )
*
* Sort the eigenvalues into increasing order
*
DO 20 I = CUTPNT + 1, N
INDXQ( I ) = INDXQ( I ) + CUTPNT
20 CONTINUE
DO 30 I = 1, N
DLAMDA( I ) = D( INDXQ( I ) )
W( I ) = Z( INDXQ( I ) )
30 CONTINUE
I = 1
J = CUTPNT + 1
CALL SLAMRG( N1, N2, DLAMDA, 1, 1, INDX )
DO 40 I = 1, N
D( I ) = DLAMDA( INDX( I ) )
Z( I ) = W( INDX( I ) )
40 CONTINUE
*
* Calculate the allowable deflation tolerence
*
IMAX = ISAMAX( N, Z, 1 )
JMAX = ISAMAX( N, D, 1 )
EPS = SLAMCH( 'Epsilon' )
TOL = EIGHT*EPS*ABS( D( JMAX ) )
*
* 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
IF( ICOMPQ.EQ.0 ) THEN
DO 50 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
50 CONTINUE
ELSE
DO 60 J = 1, N
PERM( J ) = INDXQ( INDX( J ) )
CALL SCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
60 CONTINUE
CALL SLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ),
$ LDQ )
END IF
RETURN
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.
*
K = 0
GIVPTR = 0
K2 = N + 1
DO 70 J = 1, N
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
IF( J.EQ.N )
$ GO TO 110
ELSE
JLAM = J
GO TO 80
END IF
70 CONTINUE
80 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 100
IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
INDXP( K2 ) = J
ELSE
*
* Check if eigenvalues are close enough to allow deflation.
*
S = Z( JLAM )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = SLAPY2( C, S )
T = D( J ) - D( JLAM )
C = C / TAU
S = -S / TAU
IF( ABS( T*C*S ).LE.TOL ) THEN
*
* Deflation is possible.
*
Z( J ) = TAU
Z( JLAM ) = ZERO
*
* Record the appropriate Givens rotation
*
GIVPTR = GIVPTR + 1
GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) )
GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) )
GIVNUM( 1, GIVPTR ) = C
GIVNUM( 2, GIVPTR ) = S
IF( ICOMPQ.EQ.1 ) THEN
CALL SROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
END IF
T = D( JLAM )*C*C + D( J )*S*S
D( J ) = D( JLAM )*S*S + D( J )*C*C
D( JLAM ) = T
K2 = K2 - 1
I = 1
90 CONTINUE
IF( K2+I.LE.N ) THEN
IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN
INDXP( K2+I-1 ) = INDXP( K2+I )
INDXP( K2+I ) = JLAM
I = I + 1
GO TO 90
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
ELSE
INDXP( K2+I-1 ) = JLAM
END IF
JLAM = J
ELSE
K = K + 1
W( K ) = Z( JLAM )
DLAMDA( K ) = D( JLAM )
INDXP( K ) = JLAM
JLAM = J
END IF
END IF
GO TO 80
100 CONTINUE
*
* Record the last eigenvalue.
*
K = K + 1
W( K ) = Z( JLAM )
DLAMDA( K ) = D( JLAM )
INDXP( K ) = JLAM
*
110 CONTINUE
*
* Sort the eigenvalues and corresponding eigenvectors into DLAMDA
* and Q2 respectively. The eigenvalues/vectors which were not
* deflated go into the first K slots of DLAMDA and Q2 respectively,
* while those which were deflated go into the last N - K slots.
*
IF( ICOMPQ.EQ.0 ) THEN
DO 120 J = 1, N
JP = INDXP( J )
DLAMDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
120 CONTINUE
ELSE
DO 130 J = 1, N
JP = INDXP( J )
DLAMDA( J ) = D( JP )
PERM( J ) = INDXQ( INDX( JP ) )
CALL SCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
130 CONTINUE
END IF
*
* The deflated eigenvalues and their corresponding vectors go back
* into the last N - K slots of D and Q respectively.
*
IF( K.LT.N ) THEN
IF( ICOMPQ.EQ.0 ) THEN
CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
ELSE
CALL SCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
CALL SLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2,
$ Q( 1, K+1 ), LDQ )
END IF
END IF
*
RETURN
*
* End of SLAED8
*
END