411 lines
13 KiB
Fortran
411 lines
13 KiB
Fortran
SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
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$ ILOZ, IHIZ, Z, LDZ, INFO )
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*
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* -- LAPACK auxiliary routine (version 2.0) --
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* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
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* Courant Institute, Argonne National Lab, and Rice University
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* October 31, 1992
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*
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* .. Scalar Arguments ..
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LOGICAL WANTT, WANTZ
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INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
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* ..
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*
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* Purpose
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* =======
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*
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* DLAHQR is an auxiliary routine called by DHSEQR to update the
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* eigenvalues and Schur decomposition already computed by DHSEQR, by
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* dealing with the Hessenberg submatrix in rows and columns ILO to IHI.
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*
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* Arguments
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* =========
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*
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* WANTT (input) LOGICAL
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* = .TRUE. : the full Schur form T is required;
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* = .FALSE.: only eigenvalues are required.
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*
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* WANTZ (input) LOGICAL
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* = .TRUE. : the matrix of Schur vectors Z is required;
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* = .FALSE.: Schur vectors are not required.
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*
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* N (input) INTEGER
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* The order of the matrix H. N >= 0.
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*
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* ILO (input) INTEGER
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* IHI (input) INTEGER
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* It is assumed that H is already upper quasi-triangular in
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* rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
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* ILO = 1). DLAHQR works primarily with the Hessenberg
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* submatrix in rows and columns ILO to IHI, but applies
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* transformations to all of H if WANTT is .TRUE..
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* 1 <= ILO <= max(1,IHI); IHI <= N.
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*
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* H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
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* On entry, the upper Hessenberg matrix H.
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* On exit, if WANTT is .TRUE., H is upper quasi-triangular in
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* rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in
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* standard form. If WANTT is .FALSE., the contents of H are
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* unspecified on exit.
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*
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* LDH (input) INTEGER
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* The leading dimension of the array H. LDH >= max(1,N).
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*
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* WR (output) DOUBLE PRECISION array, dimension (N)
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* WI (output) DOUBLE PRECISION array, dimension (N)
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* The real and imaginary parts, respectively, of the computed
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* eigenvalues ILO to IHI are stored in the corresponding
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* elements of WR and WI. If two eigenvalues are computed as a
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* complex conjugate pair, they are stored in consecutive
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* elements of WR and WI, say the i-th and (i+1)th, with
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* WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
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* eigenvalues are stored in the same order as on the diagonal
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* of the Schur form returned in H, with WR(i) = H(i,i), and, if
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* H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
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* WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
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*
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* ILOZ (input) INTEGER
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* IHIZ (input) INTEGER
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* Specify the rows of Z to which transformations must be
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* applied if WANTZ is .TRUE..
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* 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
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*
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* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
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* If WANTZ is .TRUE., on entry Z must contain the current
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* matrix Z of transformations accumulated by DHSEQR, and on
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* exit Z has been updated; transformations are applied only to
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* the submatrix Z(ILOZ:IHIZ,ILO:IHI).
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* If WANTZ is .FALSE., Z is not referenced.
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*
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* LDZ (input) INTEGER
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* The leading dimension of the array Z. LDZ >= max(1,N).
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* > 0: DLAHQR failed to compute all the eigenvalues ILO to IHI
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* in a total of 30*(IHI-ILO+1) iterations; if INFO = i,
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* elements i+1:ihi of WR and WI contain those eigenvalues
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* which have been successfully computed.
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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DOUBLE PRECISION DAT1, DAT2
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PARAMETER ( DAT1 = 0.75D+0, DAT2 = -0.4375D+0 )
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* ..
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* .. Local Scalars ..
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INTEGER I, I1, I2, ITN, ITS, J, K, L, M, NH, NR, NZ
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DOUBLE PRECISION CS, H00, H10, H11, H12, H21, H22, H33, H33S,
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$ H43H34, H44, H44S, OVFL, S, SMLNUM, SN, SUM,
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$ T1, T2, T3, TST1, ULP, UNFL, V1, V2, V3
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* ..
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* .. Local Arrays ..
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DOUBLE PRECISION V( 3 ), WORK( 1 )
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH, DLANHS
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EXTERNAL DLAMCH, DLANHS
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* ..
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* .. External Subroutines ..
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EXTERNAL DCOPY, DLABAD, DLANV2, DLARFG, DROT
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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IF( ILO.EQ.IHI ) THEN
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WR( ILO ) = H( ILO, ILO )
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WI( ILO ) = ZERO
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RETURN
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END IF
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*
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NH = IHI - ILO + 1
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NZ = IHIZ - ILOZ + 1
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*
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* Set machine-dependent constants for the stopping criterion.
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* If norm(H) <= sqrt(OVFL), overflow should not occur.
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*
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UNFL = DLAMCH( 'Safe minimum' )
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OVFL = ONE / UNFL
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CALL DLABAD( UNFL, OVFL )
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ULP = DLAMCH( 'Precision' )
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SMLNUM = UNFL*( NH / ULP )
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*
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* I1 and I2 are the indices of the first row and last column of H
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* to which transformations must be applied. If eigenvalues only are
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* being computed, I1 and I2 are set inside the main loop.
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*
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IF( WANTT ) THEN
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I1 = 1
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I2 = N
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END IF
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*
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* ITN is the total number of QR iterations allowed.
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*
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ITN = 30*NH
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*
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* The main loop begins here. I is the loop index and decreases from
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* IHI to ILO in steps of 1 or 2. Each iteration of the loop works
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* with the active submatrix in rows and columns L to I.
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* Eigenvalues I+1 to IHI have already converged. Either L = ILO or
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* H(L,L-1) is negligible so that the matrix splits.
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*
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I = IHI
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10 CONTINUE
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L = ILO
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IF( I.LT.ILO )
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$ GO TO 150
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*
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* Perform QR iterations on rows and columns ILO to I until a
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* submatrix of order 1 or 2 splits off at the bottom because a
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* subdiagonal element has become negligible.
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*
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DO 130 ITS = 0, ITN
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*
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* Look for a single small subdiagonal element.
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*
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DO 20 K = I, L + 1, -1
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TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
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IF( TST1.EQ.ZERO )
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$ TST1 = DLANHS( '1', I-L+1, H( L, L ), LDH, WORK )
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IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) )
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$ GO TO 30
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20 CONTINUE
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30 CONTINUE
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L = K
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IF( L.GT.ILO ) THEN
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*
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* H(L,L-1) is negligible
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*
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H( L, L-1 ) = ZERO
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END IF
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*
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* Exit from loop if a submatrix of order 1 or 2 has split off.
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*
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IF( L.GE.I-1 )
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$ GO TO 140
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*
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* Now the active submatrix is in rows and columns L to I. If
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* eigenvalues only are being computed, only the active submatrix
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* need be transformed.
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*
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IF( .NOT.WANTT ) THEN
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I1 = L
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I2 = I
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END IF
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*
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IF( ITS.EQ.10 .OR. ITS.EQ.20 ) THEN
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*
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* Exceptional shift.
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*
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S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
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H44 = DAT1*S
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H33 = H44
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H43H34 = DAT2*S*S
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ELSE
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*
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* Prepare to use Wilkinson's double shift
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*
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H44 = H( I, I )
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H33 = H( I-1, I-1 )
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H43H34 = H( I, I-1 )*H( I-1, I )
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END IF
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*
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* Look for two consecutive small subdiagonal elements.
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*
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DO 40 M = I - 2, L, -1
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*
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* Determine the effect of starting the double-shift QR
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* iteration at row M, and see if this would make H(M,M-1)
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* negligible.
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*
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H11 = H( M, M )
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H22 = H( M+1, M+1 )
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H21 = H( M+1, M )
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H12 = H( M, M+1 )
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H44S = H44 - H11
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H33S = H33 - H11
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V1 = ( H33S*H44S-H43H34 ) / H21 + H12
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V2 = H22 - H11 - H33S - H44S
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V3 = H( M+2, M+1 )
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S = ABS( V1 ) + ABS( V2 ) + ABS( V3 )
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V1 = V1 / S
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V2 = V2 / S
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V3 = V3 / S
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V( 1 ) = V1
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V( 2 ) = V2
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V( 3 ) = V3
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IF( M.EQ.L )
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$ GO TO 50
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H00 = H( M-1, M-1 )
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H10 = H( M, M-1 )
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TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) )
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IF( ABS( H10 )*( ABS( V2 )+ABS( V3 ) ).LE.ULP*TST1 )
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$ GO TO 50
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40 CONTINUE
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50 CONTINUE
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*
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* Double-shift QR step
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*
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DO 120 K = M, I - 1
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*
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* The first iteration of this loop determines a reflection G
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* from the vector V and applies it from left and right to H,
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* thus creating a nonzero bulge below the subdiagonal.
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*
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* Each subsequent iteration determines a reflection G to
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* restore the Hessenberg form in the (K-1)th column, and thus
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* chases the bulge one step toward the bottom of the active
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* submatrix. NR is the order of G.
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*
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NR = MIN( 3, I-K+1 )
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IF( K.GT.M )
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$ CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 )
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CALL DLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
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IF( K.GT.M ) THEN
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H( K, K-1 ) = V( 1 )
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H( K+1, K-1 ) = ZERO
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IF( K.LT.I-1 )
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$ H( K+2, K-1 ) = ZERO
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ELSE IF( M.GT.L ) THEN
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H( K, K-1 ) = -H( K, K-1 )
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END IF
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V2 = V( 2 )
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T2 = T1*V2
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IF( NR.EQ.3 ) THEN
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V3 = V( 3 )
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T3 = T1*V3
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*
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* Apply G from the left to transform the rows of the matrix
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* in columns K to I2.
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*
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DO 60 J = K, I2
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SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
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H( K, J ) = H( K, J ) - SUM*T1
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H( K+1, J ) = H( K+1, J ) - SUM*T2
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H( K+2, J ) = H( K+2, J ) - SUM*T3
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60 CONTINUE
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*
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* Apply G from the right to transform the columns of the
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* matrix in rows I1 to min(K+3,I).
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*
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DO 70 J = I1, MIN( K+3, I )
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SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
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H( J, K ) = H( J, K ) - SUM*T1
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H( J, K+1 ) = H( J, K+1 ) - SUM*T2
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H( J, K+2 ) = H( J, K+2 ) - SUM*T3
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70 CONTINUE
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*
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IF( WANTZ ) THEN
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*
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* Accumulate transformations in the matrix Z
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*
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DO 80 J = ILOZ, IHIZ
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SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 )
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Z( J, K ) = Z( J, K ) - SUM*T1
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Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
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Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3
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80 CONTINUE
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END IF
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ELSE IF( NR.EQ.2 ) THEN
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*
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* Apply G from the left to transform the rows of the matrix
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* in columns K to I2.
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*
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DO 90 J = K, I2
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SUM = H( K, J ) + V2*H( K+1, J )
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H( K, J ) = H( K, J ) - SUM*T1
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H( K+1, J ) = H( K+1, J ) - SUM*T2
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90 CONTINUE
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*
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* Apply G from the right to transform the columns of the
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* matrix in rows I1 to min(K+3,I).
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*
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DO 100 J = I1, I
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SUM = H( J, K ) + V2*H( J, K+1 )
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H( J, K ) = H( J, K ) - SUM*T1
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H( J, K+1 ) = H( J, K+1 ) - SUM*T2
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100 CONTINUE
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*
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IF( WANTZ ) THEN
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*
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* Accumulate transformations in the matrix Z
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*
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DO 110 J = ILOZ, IHIZ
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SUM = Z( J, K ) + V2*Z( J, K+1 )
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Z( J, K ) = Z( J, K ) - SUM*T1
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Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
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110 CONTINUE
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END IF
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END IF
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120 CONTINUE
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*
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130 CONTINUE
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*
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* Failure to converge in remaining number of iterations
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*
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INFO = I
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RETURN
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*
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140 CONTINUE
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*
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IF( L.EQ.I ) THEN
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*
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* H(I,I-1) is negligible: one eigenvalue has converged.
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*
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WR( I ) = H( I, I )
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WI( I ) = ZERO
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ELSE IF( L.EQ.I-1 ) THEN
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*
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* H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
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*
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* Transform the 2-by-2 submatrix to standard Schur form,
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* and compute and store the eigenvalues.
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*
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CALL DLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
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$ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ),
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$ CS, SN )
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*
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IF( WANTT ) THEN
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*
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* Apply the transformation to the rest of H.
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*
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IF( I2.GT.I )
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$ CALL DROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
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$ CS, SN )
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CALL DROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN )
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END IF
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IF( WANTZ ) THEN
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*
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* Apply the transformation to Z.
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*
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CALL DROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN )
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END IF
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END IF
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*
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* Decrement number of remaining iterations, and return to start of
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* the main loop with new value of I.
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*
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ITN = ITN - ITS
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I = L - 1
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GO TO 10
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*
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150 CONTINUE
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RETURN
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*
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* End of DLAHQR
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*
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END
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