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SUBROUTINE ZLAHQR2( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
$ IHIZ, Z, LDZ, INFO )
*
* -- ScaLAPACK routine (version 1.7) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 22, 2000
*
* .. Scalar Arguments ..
LOGICAL WANTT, WANTZ
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* ..
* .. Array Arguments ..
COMPLEX*16 H( LDH, * ), W( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* ZLAHQR2 is an auxiliary routine called by ZHSEQR to update the
* eigenvalues and Schur decomposition already computed by ZHSEQR, by
* dealing with the Hessenberg submatrix in rows and columns ILO to IHI.
* This version of ZLAHQR (not the standard LAPACK version) uses a
* double-shift algorithm (like LAPACK's DLAHQR).
* Unlike the standard LAPACK convention, this does not assume the
* subdiagonal is real, nor does it work to preserve this quality if
* given.
*
* Arguments
* =========
*
* WANTT (input) LOGICAL
* = .TRUE. : the full Schur form T is required;
* = .FALSE.: only eigenvalues are required.
*
* WANTZ (input) LOGICAL
* = .TRUE. : the matrix of Schur vectors Z is required;
* = .FALSE.: Schur vectors are not required.
*
* N (input) INTEGER
* The order of the matrix H. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that H is already upper triangular in rows and
* columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
* ZLAHQR works primarily with the Hessenberg submatrix in rows
* and columns ILO to IHI, but applies transformations to all of
* H if WANTT is .TRUE..
* 1 <= ILO <= max(1,IHI); IHI <= N.
*
* H (input/output) COMPLEX*16 array, dimension (LDH,N)
* On entry, the upper Hessenberg matrix H.
* On exit, if WANTT is .TRUE., H is upper triangular in rows
* and columns ILO:IHI. If WANTT is .FALSE., the contents of H
* are unspecified on exit.
*
* LDH (input) INTEGER
* The leading dimension of the array H. LDH >= max(1,N).
*
* W (output) COMPLEX*16 array, dimension (N)
* The computed eigenvalues ILO to IHI are stored in the
* corresponding elements of W. If WANTT is .TRUE., the
* eigenvalues are stored in the same order as on the diagonal
* of the Schur form returned in H, with W(i) = H(i,i).
*
* ILOZ (input) INTEGER
* IHIZ (input) INTEGER
* Specify the rows of Z to which transformations must be
* applied if WANTZ is .TRUE..
* 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*
* Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
* If WANTZ is .TRUE., on entry Z must contain the current
* matrix Z of transformations, and on exit Z has been updated;
* transformations are applied only to the submatrix
* Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not
* referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* > 0: if INFO = i, ZLAHQR failed to compute all the
* eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1)
* iterations; elements i+1:ihi of W contain those
* eigenvalues which have been successfully computed.
*
* Further Details
* ===============
*
* Modified by Mark R. Fahey, June, 2000
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 ZERO
PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) )
DOUBLE PRECISION RZERO, RONE
PARAMETER ( RZERO = 0.0D+0, RONE = 1.0D+0 )
DOUBLE PRECISION DAT1, DAT2
PARAMETER ( DAT1 = 0.75D+0, DAT2 = -0.4375D+0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, I2, ITN, ITS, J, K, L, M, NH, NR, NZ
DOUBLE PRECISION CS, OVFL, S, SMLNUM, TST1, ULP, UNFL
COMPLEX*16 CDUM, H00, H10, H11, H12, H21, H22, H33, H33S,
$ H43H34, H44, H44S, SN, SUM, T1, T2, T3, V1, V2,
$ V3
* ..
* .. Local Arrays ..
DOUBLE PRECISION RWORK( 1 )
COMPLEX*16 V( 3 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, ZLANHS
EXTERNAL DLAMCH, ZLANHS
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, ZCOPY, ZLANV2, ZLARFG, ZROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( ILO.EQ.IHI ) THEN
W( ILO ) = H( ILO, ILO )
RETURN
END IF
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
* Set machine-dependent constants for the stopping criterion.
* If norm(H) <= sqrt(OVFL), overflow should not occur.
*
UNFL = DLAMCH( 'Safe minimum' )
OVFL = RONE / UNFL
CALL DLABAD( UNFL, OVFL )
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( NH / ULP )
*
* I1 and I2 are the indices of the first row and last column of H
* to which transformations must be applied. If eigenvalues only are
* being computed, I1 and I2 are set inside the main loop.
*
IF( WANTT ) THEN
I1 = 1
I2 = N
END IF
*
* ITN is the total number of QR iterations allowed.
*
ITN = 30*NH
*
* The main loop begins here. I is the loop index and decreases from
* IHI to ILO in steps of 1 or 2. Each iteration of the loop works
* with the active submatrix in rows and columns L to I.
* Eigenvalues I+1 to IHI have already converged. Either L = ILO, or
* H(L,L-1) is negligible so that the matrix splits.
*
I = IHI
10 CONTINUE
L = ILO
IF( I.LT.ILO )
$ GO TO 150
*
* Perform QR iterations on rows and columns ILO to I until a
* submatrix of order 1 or 2 splits off at the bottom because a
* subdiagonal element has become negligible.
*
DO 130 ITS = 0, ITN
*
* Look for a single small subdiagonal element.
*
DO 20 K = I, L + 1, -1
TST1 = CABS1( H( K-1, K-1 ) ) + CABS1( H( K, K ) )
IF( TST1.EQ.RZERO )
$ TST1 = ZLANHS( '1', I-L+1, H( L, L ), LDH, RWORK )
IF( CABS1( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) )
$ GO TO 30
20 CONTINUE
30 CONTINUE
L = K
IF( L.GT.ILO ) THEN
*
* H(L,L-1) is negligible
*
H( L, L-1 ) = ZERO
END IF
*
* Exit from loop if a submatrix of order 1 or 2 has split off.
*
IF( L.GE.I-1 )
$ GO TO 140
*
* Now the active submatrix is in rows and columns L to I. If
* eigenvalues only are being computed, only the active submatrix
* need be transformed.
*
IF( .NOT.WANTT ) THEN
I1 = L
I2 = I
END IF
*
IF( ITS.EQ.10 .OR. ITS.EQ.20 ) THEN
*
* Exceptional shift.
*
* S = ABS( DBLE( H( I,I-1 ) ) ) + ABS( DBLE( H( I-1,I-2 ) ) )
S = CABS1( H( I, I-1 ) ) + CABS1( H( I-1, I-2 ) )
H44 = DAT1*S
H33 = H44
H43H34 = DAT2*S*S
ELSE
*
* Prepare to use Wilkinson's shift.
*
H44 = H( I, I )
H33 = H( I-1, I-1 )
H43H34 = H( I, I-1 )*H( I-1, I )
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 40 M = I - 2, L, -1
*
* Determine the effect of starting the double-shift QR
* iteration at row M, and see if this would make H(M,M-1)
* negligible.
*
H11 = H( M, M )
H22 = H( M+1, M+1 )
H21 = H( M+1, M )
H12 = H( M, M+1 )
H44S = H44 - H11
H33S = H33 - H11
V1 = ( H33S*H44S-H43H34 ) / H21 + H12
V2 = H22 - H11 - H33S - H44S
V3 = H( M+2, M+1 )
S = CABS1( V1 ) + CABS1( V2 ) + ABS( V3 )
V1 = V1 / S
V2 = V2 / S
V3 = V3 / S
V( 1 ) = V1
V( 2 ) = V2
V( 3 ) = V3
IF( M.EQ.L )
$ GO TO 50
H00 = H( M-1, M-1 )
H10 = H( M, M-1 )
TST1 = CABS1( V1 )*( CABS1( H00 )+CABS1( H11 )+
$ CABS1( H22 ) )
IF( CABS1( H10 )*( CABS1( V2 )+CABS1( V3 ) ).LE.ULP*TST1 )
$ GO TO 50
40 CONTINUE
50 CONTINUE
*
* Double-shift QR step
*
DO 120 K = M, I - 1
*
* The first iteration of this loop determines a reflection G
* from the vector V and applies it from left and right to H,
* thus creating a nonzero bulge below the subdiagonal.
*
* Each subsequent iteration determines a reflection G to
* restore the Hessenberg form in the (K-1)th column, and thus
* chases the bulge one step toward the bottom of the active
* submatrix. NR is the order of G
*
NR = MIN( 3, I-K+1 )
IF( K.GT.M )
$ CALL ZCOPY( NR, H( K, K-1 ), 1, V, 1 )
CALL ZLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
IF( K.LT.I-1 )
$ H( K+2, K-1 ) = ZERO
ELSE IF( M.GT.L ) THEN
* The real double-shift code uses H( K, K-1 ) = -H( K, K-1 )
* instead of the following.
H( K, K-1 ) = H( K, K-1 ) - DCONJG( T1 )*H( K, K-1 )
END IF
V2 = V( 2 )
T2 = T1*V2
IF( NR.EQ.3 ) THEN
V3 = V( 3 )
T3 = T1*V3
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 60 J = K, I2
SUM = DCONJG( T1 )*H( K, J ) +
$ DCONJG( T2 )*H( K+1, J ) +
$ DCONJG( T3 )*H( K+2, J )
H( K, J ) = H( K, J ) - SUM
H( K+1, J ) = H( K+1, J ) - SUM*V2
H( K+2, J ) = H( K+2, J ) - SUM*V3
60 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 70 J = I1, MIN( K+3, I )
SUM = T1*H( J, K ) + T2*H( J, K+1 ) + T3*H( J, K+2 )
H( J, K ) = H( J, K ) - SUM
H( J, K+1 ) = H( J, K+1 ) - SUM*DCONJG( V2 )
H( J, K+2 ) = H( J, K+2 ) - SUM*DCONJG( V3 )
70 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 80 J = ILOZ, IHIZ
SUM = T1*Z( J, K ) + T2*Z( J, K+1 ) +
$ T3*Z( J, K+2 )
Z( J, K ) = Z( J, K ) - SUM
Z( J, K+1 ) = Z( J, K+1 ) - SUM*DCONJG( V2 )
Z( J, K+2 ) = Z( J, K+2 ) - SUM*DCONJG( V3 )
80 CONTINUE
END IF
ELSE IF( NR.EQ.2 ) THEN
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 90 J = K, I2
SUM = DCONJG( T1 )*H( K, J ) +
$ DCONJG( T2 )*H( K+1, J )
H( K, J ) = H( K, J ) - SUM
H( K+1, J ) = H( K+1, J ) - SUM*V2
90 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+2,I).
*
DO 100 J = I1, MIN( K+2, I )
SUM = T1*H( J, K ) + T2*H( J, K+1 )
H( J, K ) = H( J, K ) - SUM
H( J, K+1 ) = H( J, K+1 ) - SUM*DCONJG( V2 )
100 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 110 J = ILOZ, IHIZ
SUM = T1*Z( J, K ) + T2*Z( J, K+1 )
Z( J, K ) = Z( J, K ) - SUM
Z( J, K+1 ) = Z( J, K+1 ) - SUM*DCONJG( V2 )
110 CONTINUE
END IF
END IF
*
* Since at the start of the QR step we have for M > L
* H( K, K-1 ) = H( K, K-1 ) - DCONJG( T1 )*H( K, K-1 )
* then we don't need to do the following
* IF( K.EQ.M .AND. M.GT.L ) THEN
* If the QR step was started at row M > L because two
* consecutive small subdiagonals were found, then H(M,M-1)
* must also be updated by a factor of (1-T1).
* TEMP = ONE - T1
* H( m, m-1 ) = H( m, m-1 )*DCONJG( TEMP )
* END IF
120 CONTINUE
*
* Ensure that H(I,I-1) is real.
*
130 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
140 CONTINUE
*
IF( L.EQ.I ) THEN
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
W( I ) = H( I, I )
*
ELSE IF( L.EQ.I-1 ) THEN
*
* H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
*
* Transform the 2-by-2 submatrix to standard Schur form,
* and compute and store the eigenvalues.
*
CALL ZLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
$ H( I, I ), W( I-1 ), W( I ), CS, SN )
*
IF( WANTT ) THEN
*
* Apply the transformation to the rest of H.
*
IF( I2.GT.I )
$ CALL ZROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
$ CS, SN )
CALL ZROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS,
$ DCONJG( SN ) )
END IF
IF( WANTZ ) THEN
*
* Apply the transformation to Z.
*
CALL ZROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS,
$ DCONJG( SN ) )
END IF
*
END IF
*
* Decrement number of remaining iterations, and return to start of
* the main loop with new value of I.
*
ITN = ITN - ITS
I = L - 1
GO TO 10
*
150 CONTINUE
RETURN
*
* End of ZLAHQR2
*
END
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