*> \brief \b CLAHQR 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
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*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
* IHIZ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* COMPLEX H( LDH, * ), W( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \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 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.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is COMPLEX array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO is zero and if WANTT is .TRUE., then H
*> is upper triangular in rows and columns ILO:IHI. If INFO
*> is zero and if WANTT is .FALSE., then the contents of H
*> are unspecified on exit. The output state of H in case
*> INF is positive is 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] W
*> \verbatim
*> W is 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).
*> \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 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.
*> \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
*> .GT. 0: if INFO = i, CLAHQR failed to compute all the
*> eigenvalues ILO to IHI in a total of 30 iterations
*> per eigenvalue; elements i+1:ihi of W contain
*> those eigenvalues which have been successfully
*> computed.
*>
*> If INFO .GT. 0 and WANTT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the
*> eigenvalues of the upper Hessenberg matrix
*> rows and columns ILO thorugh INFO of the final,
*> output value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*> (*) (initial value of H)*U = U*(final value of H)
*> where U is an orthognal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 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 complexOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> \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 CLAHQR 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 CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, 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 ..
COMPLEX H( LDH, * ), W( * ), Z( LDZ, * )
* ..
*
* =========================================================
*
* .. Parameters ..
COMPLEX ZERO, ONE
PARAMETER ( ZERO = ( 0.0e0, 0.0e0 ),
$ ONE = ( 1.0e0, 0.0e0 ) )
REAL RZERO, RONE, HALF
PARAMETER ( RZERO = 0.0e0, RONE = 1.0e0, HALF = 0.5e0 )
REAL DAT1
PARAMETER ( DAT1 = 3.0e0 / 4.0e0 )
* ..
* .. Local Scalars ..
COMPLEX CDUM, H11, H11S, H22, SC, SUM, T, T1, TEMP, U,
$ V2, X, Y
REAL AA, AB, BA, BB, H10, H21, RTEMP, S, SAFMAX,
$ SAFMIN, SMLNUM, SX, T2, TST, ULP
INTEGER I, I1, I2, ITS, ITMAX, J, JHI, JLO, K, L, M,
$ NH, NZ
* ..
* .. Local Arrays ..
COMPLEX V( 2 )
* ..
* .. External Functions ..
COMPLEX CLADIV
REAL SLAMCH
EXTERNAL CLADIV, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CLARFG, CSCAL, SLABAD
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CONJG, MAX, MIN, REAL, SQRT
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( 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
*
* ==== 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
* ==== ensure that subdiagonal entries are real ====
IF( WANTT ) THEN
JLO = 1
JHI = N
ELSE
JLO = ILO
JHI = IHI
END IF
DO 20 I = ILO + 1, IHI
IF( AIMAG( H( I, I-1 ) ).NE.RZERO ) THEN
* ==== The following redundant normalization
* . avoids problems with both gradual and
* . sudden underflow in ABS(H(I,I-1)) ====
SC = H( I, I-1 ) / CABS1( H( I, I-1 ) )
SC = CONJG( SC ) / ABS( SC )
H( I, I-1 ) = ABS( H( I, I-1 ) )
CALL CSCAL( JHI-I+1, SC, H( I, I ), LDH )
CALL CSCAL( MIN( JHI, I+1 )-JLO+1, CONJG( SC ), H( JLO, I ),
$ 1 )
IF( WANTZ )
$ CALL CSCAL( IHIZ-ILOZ+1, CONJG( SC ), Z( ILOZ, I ), 1 )
END IF
20 CONTINUE
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
* Set machine-dependent constants for the stopping criterion.
*
SAFMIN = SLAMCH( 'SAFE MINIMUM' )
SAFMAX = RONE / SAFMIN
CALL SLABAD( SAFMIN, SAFMAX )
ULP = SLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( REAL( 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. 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
30 CONTINUE
IF( I.LT.ILO )
$ GO TO 150
*
* 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 130 ITS = 0, ITMAX
*
* Look for a single small subdiagonal element.
*
DO 40 K = I, L + 1, -1
IF( CABS1( H( K, K-1 ) ).LE.SMLNUM )
$ GO TO 50
TST = CABS1( H( K-1, K-1 ) ) + CABS1( H( K, K ) )
IF( TST.EQ.ZERO ) THEN
IF( K-2.GE.ILO )
$ TST = TST + ABS( REAL( H( K-1, K-2 ) ) )
IF( K+1.LE.IHI )
$ TST = TST + ABS( REAL( 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 examples. ====
IF( ABS( REAL( H( K, K-1 ) ) ).LE.ULP*TST ) THEN
AB = MAX( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) )
BA = MIN( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) )
AA = MAX( CABS1( H( K, K ) ),
$ CABS1( H( K-1, K-1 )-H( K, K ) ) )
BB = MIN( CABS1( H( K, K ) ),
$ CABS1( H( K-1, K-1 )-H( K, K ) ) )
S = AA + AB
IF( BA*( AB / S ).LE.MAX( SMLNUM,
$ ULP*( BB*( AA / S ) ) ) )GO TO 50
END IF
40 CONTINUE
50 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 has split off.
*
IF( L.GE.I )
$ 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 ) THEN
*
* Exceptional shift.
*
S = DAT1*ABS( REAL( H( L+1, L ) ) )
T = S + H( L, L )
ELSE IF( ITS.EQ.20 ) THEN
*
* Exceptional shift.
*
S = DAT1*ABS( REAL( H( I, I-1 ) ) )
T = S + H( I, I )
ELSE
*
* Wilkinson's shift.
*
T = H( I, I )
U = SQRT( H( I-1, I ) )*SQRT( H( I, I-1 ) )
S = CABS1( U )
IF( S.NE.RZERO ) THEN
X = HALF*( H( I-1, I-1 )-T )
SX = CABS1( X )
S = MAX( S, CABS1( X ) )
Y = S*SQRT( ( X / S )**2+( U / S )**2 )
IF( SX.GT.RZERO ) THEN
IF( REAL( X / SX )*REAL( Y )+AIMAG( X / SX )*
$ AIMAG( Y ).LT.RZERO )Y = -Y
END IF
T = T - U*CLADIV( U, ( X+Y ) )
END IF
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 60 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 = REAL( H( M+1, M ) )
S = CABS1( H11S ) + ABS( H21 )
H11S = H11S / S
H21 = H21 / S
V( 1 ) = H11S
V( 2 ) = H21
H10 = REAL( H( M, M-1 ) )
IF( ABS( H10 )*ABS( H21 ).LE.ULP*
$ ( CABS1( H11S )*( CABS1( H11 )+CABS1( H22 ) ) ) )
$ GO TO 70
60 CONTINUE
H11 = H( L, L )
H22 = H( L+1, L+1 )
H11S = H11 - T
H21 = REAL( H( L+1, L ) )
S = CABS1( H11S ) + ABS( H21 )
H11S = H11S / S
H21 = H21 / S
V( 1 ) = H11S
V( 2 ) = H21
70 CONTINUE
*
* Single-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.
*
* 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 )
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 80 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
80 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+2,I).
*
DO 90 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 )
90 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 100 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 )
100 CONTINUE
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 110 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 )
IF( WANTZ ) THEN
CALL CSCAL( NZ, CONJG( TEMP ), Z( ILOZ, J ), 1 )
END IF
END IF
110 CONTINUE
END IF
120 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 )
IF( WANTZ ) THEN
CALL CSCAL( NZ, TEMP, Z( ILOZ, I ), 1 )
END IF
END IF
*
130 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
140 CONTINUE
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
W( I ) = H( I, I )
*
* return to start of the main loop with new value of I.
*
I = L - 1
GO TO 30
*
150 CONTINUE
RETURN
*
* End of CLAHQR
*
END