SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
$ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* January 3, 2001
*
* .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
DOUBLE PRECISION S, SEP
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
$ WR( * )
* ..
*
* Purpose
* =======
*
* DTRSEN reorders the real Schur factorization of a real matrix
* A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
* the leading diagonal blocks of the upper quasi-triangular matrix T,
* and the leading columns of Q form an orthonormal basis of the
* corresponding right invariant subspace.
*
* Optionally the routine computes the reciprocal condition numbers of
* the cluster of eigenvalues and/or the invariant subspace.
*
* T must be in Schur canonical form (as returned by DHSEQR), that is,
* block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
* 2-by-2 diagonal block has its diagonal elemnts equal and its
* off-diagonal elements of opposite sign.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies whether condition numbers are required for the
* cluster of eigenvalues (S) or the invariant subspace (SEP):
* = 'N': none;
* = 'E': for eigenvalues only (S);
* = 'V': for invariant subspace only (SEP);
* = 'B': for both eigenvalues and invariant subspace (S and
* SEP).
*
* COMPQ (input) CHARACTER*1
* = 'V': update the matrix Q of Schur vectors;
* = 'N': do not update Q.
*
* SELECT (input) LOGICAL array, dimension (N)
* SELECT specifies the eigenvalues in the selected cluster. To
* select a real eigenvalue w(j), SELECT(j) must be set to
* .TRUE.. To select a complex conjugate pair of eigenvalues
* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
* either SELECT(j) or SELECT(j+1) or both must be set to
* .TRUE.; a complex conjugate pair of eigenvalues must be
* either both included in the cluster or both excluded.
*
* N (input) INTEGER
* The order of the matrix T. N >= 0.
*
* T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
* On entry, the upper quasi-triangular matrix T, in Schur
* canonical form.
* On exit, T is overwritten by the reordered matrix T, again in
* Schur canonical form, with the selected eigenvalues in the
* leading diagonal blocks.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= max(1,N).
*
* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
* On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
* On exit, if COMPQ = 'V', Q has been postmultiplied by the
* orthogonal transformation matrix which reorders T; the
* leading M columns of Q form an orthonormal basis for the
* specified invariant subspace.
* If COMPQ = 'N', Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q.
* LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
*
* WR (output) DOUBLE PRECISION array, dimension (N)
* WI (output) DOUBLE PRECISION array, dimension (N)
* The real and imaginary parts, respectively, of the reordered
* eigenvalues of T. The eigenvalues are stored in the same
* order as on the diagonal of T, with WR(i) = T(i,i) and, if
* T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
* WI(i+1) = -WI(i). Note that if a complex eigenvalue is
* sufficiently ill-conditioned, then its value may differ
* significantly from its value before reordering.
*
* M (output) INTEGER
* The dimension of the specified invariant subspace.
* 0 < = M <= N.
*
* S (output) DOUBLE PRECISION
* If JOB = 'E' or 'B', S is a lower bound on the reciprocal
* condition number for the selected cluster of eigenvalues.
* S cannot underestimate the true reciprocal condition number
* by more than a factor of sqrt(N). If M = 0 or N, S = 1.
* If JOB = 'N' or 'V', S is not referenced.
*
* SEP (output) DOUBLE PRECISION
* If JOB = 'V' or 'B', SEP is the estimated reciprocal
* condition number of the specified invariant subspace. If
* M = 0 or N, SEP = norm(T).
* If JOB = 'N' or 'E', SEP is not referenced.
*
* 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.
* If JOB = 'N', LWORK >= max(1,N);
* if JOB = 'E', LWORK >= max(1,M*(N-M));
* if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
*
* 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.
*
* IWORK (workspace) INTEGER array, dimension (LIWORK)
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK.
* If JOB = 'N' or 'E', LIWORK >= 1;
* if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
*
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal size of the IWORK array,
* returns this value as the first entry of the IWORK array, and
* no error message related to LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* = 1: reordering of T failed because some eigenvalues are too
* close to separate (the problem is very ill-conditioned);
* T may have been partially reordered, and WR and WI
* contain the eigenvalues in the same order as in T; S and
* SEP (if requested) are set to zero.
*
* Further Details
* ===============
*
* DTRSEN first collects the selected eigenvalues by computing an
* orthogonal transformation Z to move them to the top left corner of T.
* In other words, the selected eigenvalues are the eigenvalues of T11
* in:
*
* Z'*T*Z = ( T11 T12 ) n1
* ( 0 T22 ) n2
* n1 n2
*
* where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
* of Z span the specified invariant subspace of T.
*
* If T has been obtained from the real Schur factorization of a matrix
* A = Q*T*Q', then the reordered real Schur factorization of A is given
* by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
* the corresponding invariant subspace of A.
*
* The reciprocal condition number of the average of the eigenvalues of
* T11 may be returned in S. S lies between 0 (very badly conditioned)
* and 1 (very well conditioned). It is computed as follows. First we
* compute R so that
*
* P = ( I R ) n1
* ( 0 0 ) n2
* n1 n2
*
* is the projector on the invariant subspace associated with T11.
* R is the solution of the Sylvester equation:
*
* T11*R - R*T22 = T12.
*
* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
* the two-norm of M. Then S is computed as the lower bound
*
* (1 + F-norm(R)**2)**(-1/2)
*
* on the reciprocal of 2-norm(P), the true reciprocal condition number.
* S cannot underestimate 1 / 2-norm(P) by more than a factor of
* sqrt(N).
*
* An approximate error bound for the computed average of the
* eigenvalues of T11 is
*
* EPS * norm(T) / S
*
* where EPS is the machine precision.
*
* The reciprocal condition number of the right invariant subspace
* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
* SEP is defined as the separation of T11 and T22:
*
* sep( T11, T22 ) = sigma-min( C )
*
* where sigma-min(C) is the smallest singular value of the
* n1*n2-by-n1*n2 matrix
*
* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
*
* I(m) is an m by m identity matrix, and kprod denotes the Kronecker
* product. We estimate sigma-min(C) by the reciprocal of an estimate of
* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
*
* When SEP is small, small changes in T can cause large changes in
* the invariant subspace. An approximate bound on the maximum angular
* error in the computed right invariant subspace is
*
* EPS * norm(T) / SEP
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
$ WANTSP
INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
$ NN
DOUBLE PRECISION EST, RNORM, SCALE
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLANGE
EXTERNAL LSAME, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DLACON, DLACPY, DTREXC, DTRSYL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
WANTQ = LSAME( COMPQ, 'V' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
$ THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -8
ELSE
*
* Set M to the dimension of the specified invariant subspace,
* and test LWORK and LIWORK.
*
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( T( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
*
N1 = M
N2 = N - M
NN = N1*N2
*
IF( WANTSP ) THEN
LWMIN = MAX( 1, 2*NN )
LIWMIN = MAX( 1, NN )
ELSE IF( LSAME( JOB, 'N' ) ) THEN
LWMIN = MAX( 1, N )
LIWMIN = 1
ELSE IF( LSAME( JOB, 'E' ) ) THEN
LWMIN = MAX( 1, NN )
LIWMIN = 1
END IF
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTRSEN', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.N .OR. M.EQ.0 ) THEN
IF( WANTS )
$ S = ONE
IF( WANTSP )
$ SEP = DLANGE( '1', N, N, T, LDT, WORK )
GO TO 40
END IF
*
* Collect the selected blocks at the top-left corner of T.
*
KS = 0
PAIR = .FALSE.
DO 20 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
SWAP = SELECT( K )
IF( K.LT.N ) THEN
IF( T( K+1, K ).NE.ZERO ) THEN
PAIR = .TRUE.
SWAP = SWAP .OR. SELECT( K+1 )
END IF
END IF
IF( SWAP ) THEN
KS = KS + 1
*
* Swap the K-th block to position KS.
*
IERR = 0
KK = K
IF( K.NE.KS )
$ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
$ IERR )
IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
*
* Blocks too close to swap: exit.
*
INFO = 1
IF( WANTS )
$ S = ZERO
IF( WANTSP )
$ SEP = ZERO
GO TO 40
END IF
IF( PAIR )
$ KS = KS + 1
END IF
END IF
20 CONTINUE
*
IF( WANTS ) THEN
*
* Solve Sylvester equation for R:
*
* T11*R - R*T22 = scale*T12
*
CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
$ LDT, WORK, N1, SCALE, IERR )
*
* Estimate the reciprocal of the condition number of the cluster
* of eigenvalues.
*
RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
IF( RNORM.EQ.ZERO ) THEN
S = ONE
ELSE
S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
$ SQRT( RNORM ) )
END IF
END IF
*
IF( WANTSP ) THEN
*
* Estimate sep(T11,T22).
*
EST = ZERO
KASE = 0
30 CONTINUE
CALL DLACON( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve T11*R - R*T22 = scale*X.
*
CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
$ IERR )
ELSE
*
* Solve T11'*R - R*T22' = scale*X.
*
CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
$ IERR )
END IF
GO TO 30
END IF
*
SEP = SCALE / EST
END IF
*
40 CONTINUE
*
* Store the output eigenvalues in WR and WI.
*
DO 50 K = 1, N
WR( K ) = T( K, K )
WI( K ) = ZERO
50 CONTINUE
DO 60 K = 1, N - 1
IF( T( K+1, K ).NE.ZERO ) THEN
WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
$ SQRT( ABS( T( K+1, K ) ) )
WI( K+1 ) = -WI( K )
END IF
60 CONTINUE
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DTRSEN
*
END