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dlahqr.f
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*> \brief \b DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAHQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAHQR is an auxiliary routine called by DHSEQR to update the
*> eigenvalues and Schur decomposition already computed by DHSEQR, by
*> dealing with the Hessenberg submatrix in rows and columns ILO to
*> IHI.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper quasi-triangular in
*> rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
*> ILO = 1). DLAHQR 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.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO is zero and if WANTT is .TRUE., H is upper
*> quasi-triangular in rows and columns ILO:IHI, with any
*> 2-by-2 diagonal blocks in standard form. If INFO is zero
*> and WANTT is .FALSE., the contents of H are unspecified on
*> exit. The output state of H if INFO is nonzero is given
*> below under the description of INFO.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is DOUBLE PRECISION array, dimension (N)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues ILO to IHI are stored in the corresponding
*> elements of WR and WI. If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
*> eigenvalues are stored in the same order as on the diagonal
*> of the Schur form returned in H, with WR(i) = H(i,i), and, if
*> H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
*> WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> If WANTZ is .TRUE., on entry Z must contain the current
*> matrix Z of transformations accumulated by DHSEQR, 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.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: If INFO = i, DLAHQR failed to compute all the
*> eigenvalues ILO to IHI in a total of 30 iterations
*> per eigenvalue; elements i+1:ihi of WR and WI
*> contain those eigenvalues which have been
*> successfully computed.
*>
*> If INFO > 0 and WANTT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the
*> eigenvalues of the upper Hessenberg matrix rows
*> and columns ILO through INFO of the final, output
*> value of H.
*>
*> If INFO > 0 and WANTT is .TRUE., then on exit
*> (*) (initial value of H)*U = U*(final value of H)
*> where U is an orthogonal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO > 0 and WANTZ is .TRUE., then on exit
*> (final value of Z) = (initial value of Z)*U
*> where U is the orthogonal matrix in (*)
*> (regardless of the value of WANTT.)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 02-96 Based on modifications by
*> David Day, Sandia National Laboratory, USA
*>
*> 12-04 Further modifications by
*> Ralph Byers, University of Kansas, USA
*> This is a modified version of DLAHQR from LAPACK version 3.0.
*> It is (1) more robust against overflow and underflow and
*> (2) adopts the more conservative Ahues & Tisseur stopping
*> criterion (LAWN 122, 1997).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
* =========================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0, TWO = 2.0d0 )
DOUBLE PRECISION DAT1, DAT2
PARAMETER ( DAT1 = 3.0d0 / 4.0d0, DAT2 = -0.4375d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S,
$ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX,
$ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST,
$ ULP, V2, V3
INTEGER I, I1, I2, ITS, ITMAX, J, K, L, M, NH, NR, NZ
* ..
* .. Local Arrays ..
DOUBLE PRECISION V( 3 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLABAD, DLANV2, DLARFG, DROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== clear out the trash ====
DO 10 J = ILO, IHI - 3
H( J+2, J ) = ZERO
H( J+3, J ) = ZERO
10 CONTINUE
IF( ILO.LE.IHI-2 )
$ H( IHI, IHI-2 ) = ZERO
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
* Set machine-dependent constants for the stopping criterion.
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( 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
*
* ITMAX is the total number of QR iterations allowed.
*
ITMAX = 30 * MAX( 10, 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
20 CONTINUE
L = ILO
IF( I.LT.ILO )
$ GO TO 160
*
* 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 140 ITS = 0, ITMAX
*
* Look for a single small subdiagonal element.
*
DO 30 K = I, L + 1, -1
IF( ABS( H( K, K-1 ) ).LE.SMLNUM )
$ GO TO 40
TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
IF( TST.EQ.ZERO ) THEN
IF( K-2.GE.ILO )
$ TST = TST + ABS( H( K-1, K-2 ) )
IF( K+1.LE.IHI )
$ TST = TST + ABS( H( K+1, K ) )
END IF
* ==== The following is a conservative small subdiagonal
* . deflation criterion due to Ahues & Tisseur (LAWN 122,
* . 1997). It has better mathematical foundation and
* . improves accuracy in some cases. ====
IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN
AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
AA = MAX( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
BB = MIN( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
S = AA + AB
IF( BA*( AB / S ).LE.MAX( SMLNUM,
$ ULP*( BB*( AA / S ) ) ) )GO TO 40
END IF
30 CONTINUE
40 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 150
*
* 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 ) THEN
*
* Exceptional shift.
*
S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) )
H11 = DAT1*S + H( L, L )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE IF( ITS.EQ.20 ) THEN
*
* Exceptional shift.
*
S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
H11 = DAT1*S + H( I, I )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE
*
* Prepare to use Francis' double shift
* (i.e. 2nd degree generalized Rayleigh quotient)
*
H11 = H( I-1, I-1 )
H21 = H( I, I-1 )
H12 = H( I-1, I )
H22 = H( I, I )
END IF
S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 )
IF( S.EQ.ZERO ) THEN
RT1R = ZERO
RT1I = ZERO
RT2R = ZERO
RT2I = ZERO
ELSE
H11 = H11 / S
H21 = H21 / S
H12 = H12 / S
H22 = H22 / S
TR = ( H11+H22 ) / TWO
DET = ( H11-TR )*( H22-TR ) - H12*H21
RTDISC = SQRT( ABS( DET ) )
IF( DET.GE.ZERO ) THEN
*
* ==== complex conjugate shifts ====
*
RT1R = TR*S
RT2R = RT1R
RT1I = RTDISC*S
RT2I = -RT1I
ELSE
*
* ==== real shifts (use only one of them) ====
*
RT1R = TR + RTDISC
RT2R = TR - RTDISC
IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN
RT1R = RT1R*S
RT2R = RT1R
ELSE
RT2R = RT2R*S
RT1R = RT2R
END IF
RT1I = ZERO
RT2I = ZERO
END IF
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 50 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. (The following uses scaling to avoid
* overflows and most underflows.)
*
H21S = H( M+1, M )
S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S )
H21S = H( M+1, M ) / S
V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )*
$ ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S )
V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R )
V( 3 ) = H21S*H( M+2, M+1 )
S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) )
V( 1 ) = V( 1 ) / S
V( 2 ) = V( 2 ) / S
V( 3 ) = V( 3 ) / S
IF( M.EQ.L )
$ GO TO 60
IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE.
$ ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M,
$ M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60
50 CONTINUE
60 CONTINUE
*
* Double-shift QR step
*
DO 130 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 DCOPY( NR, H( K, K-1 ), 1, V, 1 )
CALL DLARFG( 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
* ==== Use the following instead of
* . H( K, K-1 ) = -H( K, K-1 ) to
* . avoid a bug when v(2) and v(3)
* . underflow. ====
H( K, K-1 ) = H( K, K-1 )*( ONE-T1 )
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 70 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
H( K+2, J ) = H( K+2, J ) - SUM*T3
70 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 80 J = I1, MIN( K+3, I )
SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
H( J, K+2 ) = H( J, K+2 ) - SUM*T3
80 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 90 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3
90 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 100 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
100 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 110 J = I1, I
SUM = H( J, K ) + V2*H( J, K+1 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
110 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 120 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
120 CONTINUE
END IF
END IF
130 CONTINUE
*
140 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
150 CONTINUE
*
IF( L.EQ.I ) THEN
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
WR( I ) = H( I, I )
WI( I ) = ZERO
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 DLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
$ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ),
$ CS, SN )
*
IF( WANTT ) THEN
*
* Apply the transformation to the rest of H.
*
IF( I2.GT.I )
$ CALL DROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
$ CS, SN )
CALL DROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN )
END IF
IF( WANTZ ) THEN
*
* Apply the transformation to Z.
*
CALL DROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN )
END IF
END IF
*
* return to start of the main loop with new value of I.
*
I = L - 1
GO TO 20
*
160 CONTINUE
RETURN
*
* End of DLAHQR
*
END