*> \brief \b STRSEN * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download STRSEN + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, * M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER COMPQ, JOB * INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N * REAL S, SEP * .. * .. Array Arguments .. * LOGICAL SELECT( * ) * INTEGER IWORK( * ) * REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), * \$ WR( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> STRSEN 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 SHSEQR), that is, *> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each *> 2-by-2 diagonal block has its diagonal elements equal and its *> off-diagonal elements of opposite sign. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOB *> \verbatim *> JOB is 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). *> \endverbatim *> *> \param[in] COMPQ *> \verbatim *> COMPQ is CHARACTER*1 *> = 'V': update the matrix Q of Schur vectors; *> = 'N': do not update Q. *> \endverbatim *> *> \param[in] SELECT *> \verbatim *> SELECT is 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. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix T. N >= 0. *> \endverbatim *> *> \param[in,out] T *> \verbatim *> T is REAL 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. *> \endverbatim *> *> \param[in] LDT *> \verbatim *> LDT is INTEGER *> The leading dimension of the array T. LDT >= max(1,N). *> \endverbatim *> *> \param[in,out] Q *> \verbatim *> Q is REAL 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. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. *> LDQ >= 1; and if COMPQ = 'V', LDQ >= N. *> \endverbatim *> *> \param[out] WR *> \verbatim *> WR is REAL array, dimension (N) *> \endverbatim *> *> \param[out] WI *> \verbatim *> WI is REAL 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. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The dimension of the specified invariant subspace. *> 0 < = M <= N. *> \endverbatim *> *> \param[out] S *> \verbatim *> S is REAL *> 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. *> \endverbatim *> *> \param[out] SEP *> \verbatim *> SEP is REAL *> 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. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is 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. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is 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. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date April 2012 * *> \ingroup realOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> STRSEN 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 * T * Z = ( T11 T12 ) n1 *> ( 0 T22 ) n2 *> n1 n2 *> *> where N = n1+n2 and Z**T 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**T, then the reordered real Schur factorization of A is given *> by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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 *> \endverbatim *> * ===================================================================== SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, \$ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) * * -- LAPACK computational 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..-- * April 2012 * * .. Scalar Arguments .. CHARACTER COMPQ, JOB INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N REAL S, SEP * .. * .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), \$ WR( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS, \$ WANTSP INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2, \$ NN REAL EST, RNORM, SCALE * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Functions .. LOGICAL LSAME REAL SLANGE EXTERNAL LSAME, SLANGE * .. * .. External Subroutines .. EXTERNAL SLACN2, SLACPY, STREXC, STRSYL, 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( 'STRSEN', -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 = SLANGE( '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 STREXC( 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 SLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) CALL STRSYL( '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 = SLANGE( '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 SLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Solve T11*R - R*T22 = scale*X. * CALL STRSYL( 'N', 'N', -1, N1, N2, T, LDT, \$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, \$ IERR ) ELSE * * Solve T11**T*R - R*T22**T = scale*X. * CALL STRSYL( '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 STRSEN * END