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SUBROUTINE CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
$ IHIZ, Z, LDZ, INFO )
*
* -- LAPACK auxiliary routine (instrumented to count operations) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
LOGICAL WANTT, WANTZ
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* ..
* .. Array Arguments ..
COMPLEX H( LDH, * ), W( * ), Z( LDZ, * )
* ..
* Common block to return operation count.
* .. Common blocks ..
COMMON / LATIME / OPS, ITCNT
* ..
* .. Scalars in Common ..
REAL ITCNT, OPS
* ..
*
* Purpose
* =======
*
* CLAHQR is an auxiliary routine called by CHSEQR to update the
* eigenvalues and Schur decomposition already computed by CHSEQR, by
* dealing with the Hessenberg submatrix in rows and columns ILO to IHI.
*
* 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).
* CLAHQR 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 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, with any 2-by-2 diagonal blocks in
* standard form. 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 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 array, dimension (LDZ,N)
* If WANTZ is .TRUE., on entry Z must contain the current
* matrix Z of transformations accumulated by CHSEQR, 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, CLAHQR 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.
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ZERO, ONE
PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ),
$ ONE = ( 1.0E+0, 0.0E+0 ) )
REAL RZERO, HALF
PARAMETER ( RZERO = 0.0E+0, HALF = 0.5E+0 )
REAL DAT1
PARAMETER ( DAT1 = 0.75E+0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, I2, ITN, ITS, J, K, L, M, NH, NZ
REAL H10, H21, OPST, RTEMP, S, SMLNUM, T2, TST1, ULP
COMPLEX CDUM, H11, H11S, H22, SUM, T, T1, TEMP, U, V2,
$ X, Y
* ..
* .. Local Arrays ..
REAL RWORK( 1 )
COMPLEX V( 2 )
* ..
* .. External Functions ..
REAL CLANHS, SLAMCH
COMPLEX CLADIV
EXTERNAL CLANHS, SLAMCH, CLADIV
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CLARFG, CSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CONJG, MAX, MIN, REAL, SQRT
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
INFO = 0
***
* Initialize
OPST = 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.
*
ULP = SLAMCH( 'Precision' )
SMLNUM = SLAMCH( 'Safe minimum' ) / 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. 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
IF( I.LT.ILO )
$ GO TO 130
*
* Perform QR iterations on rows and columns ILO to I until a
* submatrix of order 1 splits off at the bottom because a
* subdiagonal element has become negligible.
*
L = ILO
DO 110 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 ) THEN
TST1 = CLANHS( '1', I-L+1, H( L, L ), LDH, RWORK )
***
* Increment op count
OPS = OPS + 5*( I-L+1 )*( I-L ) / 2
***
END IF
IF( ABS( REAL( H( K, K-1 ) ) ).LE.MAX( ULP*TST1, SMLNUM ) )
$ GO TO 30
20 CONTINUE
30 CONTINUE
L = K
***
* Increment op count
OPST = OPST + 5*( I-L+1 )
***
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 has split off.
*
IF( L.GE.I )
$ GO TO 120
*
* 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 = DAT1*ABS( REAL( H( I, I-1 ) ) )
T = S + H( I, I )
***
* Increment op count
OPST = OPST + 1
***
ELSE
*
* Wilkinson's shift.
*
T = H( I, I )
U = H( I-1, I )*REAL( H( I, I-1 ) )
***
* Increment op count
OPST = OPST + 2
***
IF( U.NE.ZERO ) THEN
X = HALF*( H( I-1, I-1 )-T )
Y = SQRT( X*X+U )
IF( REAL( X )*REAL( Y )+AIMAG( X )*AIMAG( Y ).LT.RZERO )
$ Y = -Y
T = T - CLADIV( U, ( X+Y ) )
***
* Increment op count
OPST = OPST + 20
***
END IF
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 40 M = I - 1, L + 1, -1
*
* Determine the effect of starting the single-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 )
H11S = H11 - T
H21 = H( M+1, M )
S = CABS1( H11S ) + ABS( H21 )
H11S = H11S / S
H21 = H21 / S
V( 1 ) = H11S
V( 2 ) = H21
H10 = H( M, M-1 )
TST1 = CABS1( H11S )*( CABS1( H11 )+CABS1( H22 ) )
IF( ABS( H10*H21 ).LE.ULP*TST1 )
$ GO TO 50
40 CONTINUE
H11 = H( L, L )
H22 = H( L+1, L+1 )
H11S = H11 - T
H21 = H( L+1, L )
S = CABS1( H11S ) + ABS( H21 )
H11S = H11S / S
H21 = H21 / S
V( 1 ) = H11S
V( 2 ) = H21
50 CONTINUE
***
* Increment op count
OPST = OPST + 14*( I-M )
***
*
* Single-shift QR step
*
DO 100 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.
*
* V(2) is always real before the call to CLARFG, and hence
* after the call T2 ( = T1*V(2) ) is also real.
*
IF( K.GT.M )
$ CALL CCOPY( 2, H( K, K-1 ), 1, V, 1 )
CALL CLARFG( 2, V( 1 ), V( 2 ), 1, T1 )
***
* Increment op count
OPST = OPST + 38
***
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
END IF
V2 = V( 2 )
T2 = REAL( T1*V2 )
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 60 J = K, I2
SUM = CONJG( T1 )*H( K, J ) + T2*H( K+1, J )
H( K, J ) = H( K, J ) - SUM
H( K+1, J ) = H( K+1, J ) - SUM*V2
60 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+2,I).
*
DO 70 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*CONJG( V2 )
70 CONTINUE
***
* Increment op count
OPS = OPS + 20*( I2-I1+2+MIN( 2, I-K ) )
***
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 80 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*CONJG( V2 )
80 CONTINUE
***
* Increment op count
OPS = OPS + 20*NZ
***
END IF
*
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 extra
* scaling must be performed to ensure that H(M,M-1) remains
* real.
*
TEMP = ONE - T1
TEMP = TEMP / ABS( TEMP )
H( M+1, M ) = H( M+1, M )*CONJG( TEMP )
IF( M+2.LE.I )
$ H( M+2, M+1 ) = H( M+2, M+1 )*TEMP
DO 90 J = M, I
IF( J.NE.M+1 ) THEN
IF( I2.GT.J )
$ CALL CSCAL( I2-J, TEMP, H( J, J+1 ), LDH )
CALL CSCAL( J-I1, CONJG( TEMP ), H( I1, J ), 1 )
***
* Increment op count
OPST = OPST + 6*( I2-I1+3 )
***
IF( WANTZ ) THEN
CALL CSCAL( NZ, CONJG( TEMP ), Z( ILOZ, J ), 1 )
***
* Increment op count
OPST = OPST + 6*NZ
***
END IF
END IF
90 CONTINUE
END IF
100 CONTINUE
*
* Ensure that H(I,I-1) is real.
*
TEMP = H( I, I-1 )
IF( AIMAG( TEMP ).NE.RZERO ) THEN
RTEMP = ABS( TEMP )
H( I, I-1 ) = RTEMP
TEMP = TEMP / RTEMP
IF( I2.GT.I )
$ CALL CSCAL( I2-I, CONJG( TEMP ), H( I, I+1 ), LDH )
CALL CSCAL( I-I1, TEMP, H( I1, I ), 1 )
***
* Increment op count
OPST = OPST + 6*( I2-I1+1 )
***
IF( WANTZ ) THEN
CALL CSCAL( NZ, TEMP, Z( ILOZ, I ), 1 )
***
* Increment op count
OPST = OPST + 6*NZ
***
END IF
END IF
*
110 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
120 CONTINUE
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
W( I ) = H( I, I )
*
* 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
*
130 CONTINUE
***
* Compute final op count
OPS = OPS + OPST
***
RETURN
*
* End of CLAHQR
*
END
|
