364 lines
12 KiB
FortranFixed
364 lines
12 KiB
FortranFixed
SUBROUTINE ZLAED8( K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA,
|
|
$ Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR,
|
|
$ GIVCOL, GIVNUM, INFO )
|
|
*
|
|
* -- LAPACK routine (version 3.1) --
|
|
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
|
|
* November 2006
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
|
|
DOUBLE PRECISION RHO
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
|
|
$ INDXQ( * ), PERM( * )
|
|
DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ),
|
|
$ Z( * )
|
|
COMPLEX*16 Q( LDQ, * ), Q2( LDQ2, * )
|
|
* ..
|
|
*
|
|
* Purpose
|
|
* =======
|
|
*
|
|
* ZLAED8 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
|
|
* =========
|
|
*
|
|
* K (output) INTEGER
|
|
* Contains the number of non-deflated eigenvalues.
|
|
* This is 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 unitary matrix used to reduce
|
|
* the dense or band matrix to tridiagonal form.
|
|
* QSIZ >= N if ICOMPQ = 1.
|
|
*
|
|
* Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
|
|
* 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 ).
|
|
*
|
|
* 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.
|
|
*
|
|
* RHO (input/output) DOUBLE PRECISION
|
|
* Contains the off diagonal element associated with the rank-1
|
|
* cut which originally split the two submatrices which are now
|
|
* being recombined. RHO is modified during the computation to
|
|
* the value required by DLAED3.
|
|
*
|
|
* CUTPNT (input) INTEGER
|
|
* Contains the location of the last eigenvalue in the leading
|
|
* sub-matrix. MIN(1,N) <= CUTPNT <= N.
|
|
*
|
|
* Z (input) DOUBLE PRECISION array, dimension (N)
|
|
* On input this vector contains the updating vector (the last
|
|
* row of the first sub-eigenvector matrix and the first row of
|
|
* the second sub-eigenvector matrix). The contents of Z are
|
|
* destroyed during the updating process.
|
|
*
|
|
* DLAMDA (output) DOUBLE PRECISION array, dimension (N)
|
|
* Contains a copy of the first K eigenvalues which will be used
|
|
* by DLAED3 to form the secular equation.
|
|
*
|
|
* Q2 (output) COMPLEX*16 array, dimension (LDQ2,N)
|
|
* If ICOMPQ = 0, Q2 is not referenced. Otherwise,
|
|
* Contains a copy of the first K eigenvectors which will be used
|
|
* by DLAED7 in a matrix multiply (DGEMM) to update the new
|
|
* eigenvectors.
|
|
*
|
|
* LDQ2 (input) INTEGER
|
|
* The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
|
|
*
|
|
* W (output) DOUBLE PRECISION array, dimension (N)
|
|
* This will hold the first k values of the final
|
|
* deflation-altered z-vector and will be passed to DLAED3.
|
|
*
|
|
* INDXP (workspace) INTEGER array, dimension (N)
|
|
* This will contain the permutation used to place deflated
|
|
* values of D at the end of the array. On output 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)
|
|
* This will contain the permutation used to sort the contents of
|
|
* D into ascending order.
|
|
*
|
|
* INDXQ (input) INTEGER array, dimension (N)
|
|
* This contains 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.
|
|
*
|
|
* PERM (output) INTEGER array, dimension (N)
|
|
* Contains the permutations (from deflation and sorting) to be
|
|
* applied to each eigenblock.
|
|
*
|
|
* GIVPTR (output) INTEGER
|
|
* Contains 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) DOUBLE PRECISION array, dimension (2, N)
|
|
* Each number indicates the S value to be used in the
|
|
* corresponding Givens rotation.
|
|
*
|
|
* INFO (output) INTEGER
|
|
* = 0: successful exit.
|
|
* < 0: if INFO = -i, the i-th argument had an illegal value.
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. 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 I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
|
|
DOUBLE PRECISION C, EPS, S, T, TAU, TOL
|
|
* ..
|
|
* .. External Functions ..
|
|
INTEGER IDAMAX
|
|
DOUBLE PRECISION DLAMCH, DLAPY2
|
|
EXTERNAL IDAMAX, DLAMCH, DLAPY2
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DCOPY, DLAMRG, DSCAL, XERBLA, ZCOPY, ZDROT,
|
|
$ ZLACPY
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC ABS, MAX, MIN, SQRT
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input parameters.
|
|
*
|
|
INFO = 0
|
|
*
|
|
IF( N.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( QSIZ.LT.N ) THEN
|
|
INFO = -3
|
|
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
|
|
INFO = -5
|
|
ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN
|
|
INFO = -8
|
|
ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN
|
|
INFO = -12
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZLAED8', -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 DSCAL( 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 DSCAL( 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 DLAMRG( 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 tolerance
|
|
*
|
|
IMAX = IDAMAX( N, Z, 1 )
|
|
JMAX = IDAMAX( N, D, 1 )
|
|
EPS = DLAMCH( '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
|
|
DO 50 J = 1, N
|
|
PERM( J ) = INDXQ( INDX( J ) )
|
|
CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
|
|
50 CONTINUE
|
|
CALL ZLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), LDQ )
|
|
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 60 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 100
|
|
ELSE
|
|
JLAM = J
|
|
GO TO 70
|
|
END IF
|
|
60 CONTINUE
|
|
70 CONTINUE
|
|
J = J + 1
|
|
IF( J.GT.N )
|
|
$ GO TO 90
|
|
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 = DLAPY2( 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
|
|
CALL ZDROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1,
|
|
$ Q( 1, INDXQ( INDX( J ) ) ), 1, C, S )
|
|
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
|
|
80 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 80
|
|
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 70
|
|
90 CONTINUE
|
|
*
|
|
* Record the last eigenvalue.
|
|
*
|
|
K = K + 1
|
|
W( K ) = Z( JLAM )
|
|
DLAMDA( K ) = D( JLAM )
|
|
INDXP( K ) = JLAM
|
|
*
|
|
100 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.
|
|
*
|
|
DO 110 J = 1, N
|
|
JP = INDXP( J )
|
|
DLAMDA( J ) = D( JP )
|
|
PERM( J ) = INDXQ( INDX( JP ) )
|
|
CALL ZCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 )
|
|
110 CONTINUE
|
|
*
|
|
* 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
|
|
CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 )
|
|
CALL ZLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, Q( 1, K+1 ),
|
|
$ LDQ )
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZLAED8
|
|
*
|
|
END
|