📄 dhgeqz.f
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SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
$ LWORK, INFO )
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* DHGEQZ implements a single-/double-shift version of the QZ method for
* finding the generalized eigenvalues
*
* w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation
*
* det( A - w(i) B ) = 0
*
* In addition, the pair A,B may be reduced to generalized Schur form:
* B is upper triangular, and A is block upper triangular, where the
* diagonal blocks are either 1-by-1 or 2-by-2, the 2-by-2 blocks having
* complex generalized eigenvalues (see the description of the argument
* JOB.)
*
* If JOB='S', then the pair (A,B) is simultaneously reduced to Schur
* form by applying one orthogonal transformation (usually called Q) on
* the left and another (usually called Z) on the right. The 2-by-2
* upper-triangular diagonal blocks of B corresponding to 2-by-2 blocks
* of A will be reduced to positive diagonal matrices. (I.e.,
* if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and
* B(j+1,j+1) will be positive.)
*
* If JOB='E', then at each iteration, the same transformations
* are computed, but they are only applied to those parts of A and B
* which are needed to compute ALPHAR, ALPHAI, and BETAR.
*
* If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
* transformations used to reduce (A,B) are accumulated into the arrays
* Q and Z s.t.:
*
* Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
* Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
*
* Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
* Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
* pp. 241--256.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* = 'E': compute only ALPHAR, ALPHAI, and BETA. A and B will
* not necessarily be put into generalized Schur form.
* = 'S': put A and B into generalized Schur form, as well
* as computing ALPHAR, ALPHAI, and BETA.
*
* COMPQ (input) CHARACTER*1
* = 'N': do not modify Q.
* = 'V': multiply the array Q on the right by the transpose of
* the orthogonal transformation that is applied to the
* left side of A and B to reduce them to Schur form.
* = 'I': like COMPQ='V', except that Q will be initialized to
* the identity first.
*
* COMPZ (input) CHARACTER*1
* = 'N': do not modify Z.
* = 'V': multiply the array Z on the right by the orthogonal
* transformation that is applied to the right side of
* A and B to reduce them to Schur form.
* = 'I': like COMPZ='V', except that Z will be initialized to
* the identity first.
*
* N (input) INTEGER
* The order of the matrices A, B, Q, and Z. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that A is already upper triangular in rows and
* columns 1:ILO-1 and IHI+1:N.
* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
* On entry, the N-by-N upper Hessenberg matrix A. Elements
* below the subdiagonal must be zero.
* If JOB='S', then on exit A and B will have been
* simultaneously reduced to generalized Schur form.
* If JOB='E', then on exit A will have been destroyed.
* The diagonal blocks will be correct, but the off-diagonal
* portion will be meaningless.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max( 1, N ).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, the N-by-N upper triangular matrix B. Elements
* below the diagonal must be zero. 2-by-2 blocks in B
* corresponding to 2-by-2 blocks in A will be reduced to
* positive diagonal form. (I.e., if A(j+1,j) is non-zero,
* then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be
* positive.)
* If JOB='S', then on exit A and B will have been
* simultaneously reduced to Schur form.
* If JOB='E', then on exit B will have been destroyed.
* Elements corresponding to diagonal blocks of A will be
* correct, but the off-diagonal portion will be meaningless.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max( 1, N ).
*
* ALPHAR (output) DOUBLE PRECISION array, dimension (N)
* ALPHAR(1:N) will be set to real parts of the diagonal
* elements of A that would result from reducing A and B to
* Schur form and then further reducing them both to triangular
* form using unitary transformations s.t. the diagonal of B
* was non-negative real. Thus, if A(j,j) is in a 1-by-1 block
* (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=A(j,j).
* Note that the (real or complex) values
* (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the
* generalized eigenvalues of the matrix pencil A - wB.
*
* ALPHAI (output) DOUBLE PRECISION array, dimension (N)
* ALPHAI(1:N) will be set to imaginary parts of the diagonal
* elements of A that would result from reducing A and B to
* Schur form and then further reducing them both to triangular
* form using unitary transformations s.t. the diagonal of B
* was non-negative real. Thus, if A(j,j) is in a 1-by-1 block
* (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0.
* Note that the (real or complex) values
* (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the
* generalized eigenvalues of the matrix pencil A - wB.
*
* BETA (output) DOUBLE PRECISION array, dimension (N)
* BETA(1:N) will be set to the (real) diagonal elements of B
* that would result from reducing A and B to Schur form and
* then further reducing them both to triangular form using
* unitary transformations s.t. the diagonal of B was
* non-negative real. Thus, if A(j,j) is in a 1-by-1 block
* (i.e., A(j+1,j)=A(j,j+1)=0), then BETA(j)=B(j,j).
* Note that the (real or complex) values
* (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the
* generalized eigenvalues of the matrix pencil A - wB.
* (Note that BETA(1:N) will always be non-negative, and no
* BETAI is necessary.)
*
* Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
* If COMPQ='N', then Q will not be referenced.
* If COMPQ='V' or 'I', then the transpose of the orthogonal
* transformations which are applied to A and B on the left
* will be applied to the array Q on the right.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1.
* If COMPQ='V' or 'I', then LDQ >= N.
*
* Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
* If COMPZ='N', then Z will not be referenced.
* If COMPZ='V' or 'I', then the orthogonal transformations
* which are applied to A and B on the right will be applied
* to the array Z on the right.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If COMPZ='V' or 'I', then LDZ >= N.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
* On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* = 1,...,N: the QZ iteration did not converge. (A,B) is not
* in Schur form, but ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO+1,...,N should be correct.
* = N+1,...,2*N: the shift calculation failed. (A,B) is not
* in Schur form, but ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO-N+1,...,N should be correct.
* > 2*N: various "impossible" errors.
*
* Further Details
* ===============
*
* Iteration counters:
*
* JITER -- counts iterations.
* IITER -- counts iterations run since ILAST was last
* changed. This is therefore reset only when a 1-by-1 or
* 2-by-2 block deflates off the bottom.
*
* =====================================================================
*
* .. Parameters ..
* $ SAFETY = 1.0E+0 )
DOUBLE PRECISION HALF, ZERO, ONE, SAFETY
PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
$ SAFETY = 1.0D+2 )
* ..
* .. Local Scalars ..
LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
$ LQUERY
INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
$ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
$ JR, MAXIT
DOUBLE PRECISION A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
$ AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
$ AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
$ B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
$ BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
$ CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
$ SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T,
$ TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L,
$ U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR,
$ WR2
* ..
* .. Local Arrays ..
DOUBLE PRECISION V( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3
EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3
* ..
* .. External Subroutines ..
EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Decode JOB, COMPQ, COMPZ
*
IF( LSAME( JOB, 'E' ) ) THEN
ILSCHR = .FALSE.
ISCHUR = 1
ELSE IF( LSAME( JOB, 'S' ) ) THEN
ILSCHR = .TRUE.
ISCHUR = 2
ELSE
ISCHUR = 0
END IF
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Check Argument Values
*
INFO = 0
WORK( 1 ) = MAX( 1, N )
LQUERY = ( LWORK.EQ.-1 )
IF( ISCHUR.EQ.0 ) THEN
INFO = -1
ELSE IF( ICOMPQ.EQ.0 ) THEN
INFO = -2
ELSE IF( ICOMPZ.EQ.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( ILO.LT.1 ) THEN
INFO = -5
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -6
ELSE IF( LDA.LT.N ) THEN
INFO = -8
ELSE IF( LDB.LT.N ) THEN
INFO = -10
ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
INFO = -15
ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
INFO = -17
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DHGEQZ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = DBLE( 1 )
RETURN
END IF
*
* Initialize Q and Z
*
IF( ICOMPQ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
* Machine Constants
*
IN = IHI + 1 - ILO
SAFMIN = DLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
ANORM = DLANHS( 'F', IN, A( ILO, ILO ), LDA, WORK )
BNORM = DLANHS( 'F', IN, B( ILO, ILO ), LDB, WORK )
ATOL = MAX( SAFMIN, ULP*ANORM )
BTOL = MAX( SAFMIN, ULP*BNORM )
ASCALE = ONE / MAX( SAFMIN, ANORM )
BSCALE = ONE / MAX( SAFMIN, BNORM )
*
* Set Eigenvalues IHI+1:N
*
DO 30 J = IHI + 1, N
IF( B( J, J ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 10 JR = 1, J
A( JR, J ) = -A( JR, J )
B( JR, J ) = -B( JR, J )
10 CONTINUE
ELSE
A( J, J ) = -A( J, J )
B( J, J ) = -B( J, J )
END IF
IF( ILZ ) THEN
DO 20 JR = 1, N
Z( JR, J ) = -Z( JR, J )
20 CONTINUE
END IF
END IF
ALPHAR( J ) = A( J, J )
ALPHAI( J ) = ZERO
BETA( J ) = B( J, J )
30 CONTINUE
*
* If IHI < ILO, skip QZ steps
*
IF( IHI.LT.ILO )
$ GO TO 380
*
* MAIN QZ ITERATION LOOP
*
* Initialize dynamic indices
*
* Eigenvalues ILAST+1:N have been found.
* Column operations modify rows IFRSTM:whatever.
* Row operations modify columns whatever:ILASTM.
*
* If only eigenvalues are being computed, then
* IFRSTM is the row of the last splitting row above row ILAST;
* this is always at least ILO.
* IITER counts iterations since the last eigenvalue was found,
* to tell when to use an extraordinary shift.
* MAXIT is the maximum number of QZ sweeps allowed.
*
ILAST = IHI
IF( ILSCHR ) THEN
IFRSTM = 1
ILASTM = N
ELSE
IFRSTM = ILO
ILASTM = IHI
END IF
IITER = 0
ESHIFT = ZERO
MAXIT = 30*( IHI-ILO+1 )
*
DO 360 JITER = 1, MAXIT
*
* Split the matrix if possible.
*
* Two tests:
* 1: A(j,j-1)=0 or j=ILO
* 2: B(j,j)=0
*
IF( ILAST.EQ.ILO ) THEN
*
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