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