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SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
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$ SEP, WORK, LWORK, INFO )
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* -- LAPACK routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
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* .. Scalar Arguments ..
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INTEGER INFO, LDQ, LDT, LWORK, M, N
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DOUBLE PRECISION S, SEP
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* .. Array Arguments ..
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COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
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* ZTRSEN reorders the Schur factorization of a complex matrix
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* A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
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* the leading positions on the diagonal of the upper triangular matrix
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* T, and the leading columns of Q form an orthonormal basis of the
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* corresponding right invariant subspace.
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* Optionally the routine computes the reciprocal condition numbers of
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* the cluster of eigenvalues and/or the invariant subspace.
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* JOB (input) CHARACTER*1
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* Specifies whether condition numbers are required for the
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* cluster of eigenvalues (S) or the invariant subspace (SEP):
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* = 'E': for eigenvalues only (S);
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* = 'V': for invariant subspace only (SEP);
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* = 'B': for both eigenvalues and invariant subspace (S and
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* COMPQ (input) CHARACTER*1
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* = 'V': update the matrix Q of Schur vectors;
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* = 'N': do not update Q.
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* SELECT (input) LOGICAL array, dimension (N)
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* SELECT specifies the eigenvalues in the selected cluster. To
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* select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
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* The order of the matrix T. N >= 0.
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* T (input/output) COMPLEX*16 array, dimension (LDT,N)
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* On entry, the upper triangular matrix T.
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* On exit, T is overwritten by the reordered matrix T, with the
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* selected eigenvalues as the leading diagonal elements.
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* The leading dimension of the array T. LDT >= max(1,N).
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* Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
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* On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
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* On exit, if COMPQ = 'V', Q has been postmultiplied by the
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* unitary transformation matrix which reorders T; the leading M
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* columns of Q form an orthonormal basis for the specified
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* If COMPQ = 'N', Q is not referenced.
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* The leading dimension of the array Q.
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* LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
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* W (output) COMPLEX*16 array, dimension (N)
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* The reordered eigenvalues of T, in the same order as they
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* appear on the diagonal of T.
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* The dimension of the specified invariant subspace.
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* S (output) DOUBLE PRECISION
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* If JOB = 'E' or 'B', S is a lower bound on the reciprocal
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* condition number for the selected cluster of eigenvalues.
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* S cannot underestimate the true reciprocal condition number
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* by more than a factor of sqrt(N). If M = 0 or N, S = 1.
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* If JOB = 'N' or 'V', S is not referenced.
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* SEP (output) DOUBLE PRECISION
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* If JOB = 'V' or 'B', SEP is the estimated reciprocal
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* condition number of the specified invariant subspace. If
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* M = 0 or N, SEP = norm(T).
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* If JOB = 'N' or 'E', SEP is not referenced.
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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* LWORK (input) INTEGER
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* The dimension of the array WORK.
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* If JOB = 'N', LWORK >= 1;
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* if JOB = 'E', LWORK = max(1,M*(N-M));
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* if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
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* If LWORK = -1, then a workspace query is assumed; the routine
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* only calculates the optimal size of the WORK array, returns
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* this value as the first entry of the WORK array, and no error
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* message related to LWORK is issued by XERBLA.
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -i, the i-th argument had an illegal value
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* ZTRSEN first collects the selected eigenvalues by computing a unitary
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* transformation Z to move them to the top left corner of T. In other
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* words, the selected eigenvalues are the eigenvalues of T11 in:
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* Z'*T*Z = ( T11 T12 ) n1
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* where N = n1+n2 and Z' means the conjugate transpose of Z. The first
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* n1 columns of Z span the specified invariant subspace of T.
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* If T has been obtained from the Schur factorization of a matrix
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* A = Q*T*Q', then the reordered Schur factorization of A is given by
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* A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the
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* corresponding invariant subspace of A.
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* The reciprocal condition number of the average of the eigenvalues of
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* T11 may be returned in S. S lies between 0 (very badly conditioned)
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* and 1 (very well conditioned). It is computed as follows. First we
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* is the projector on the invariant subspace associated with T11.
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* R is the solution of the Sylvester equation:
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* T11*R - R*T22 = T12.
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* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
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* the two-norm of M. Then S is computed as the lower bound
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* (1 + F-norm(R)**2)**(-1/2)
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* on the reciprocal of 2-norm(P), the true reciprocal condition number.
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* S cannot underestimate 1 / 2-norm(P) by more than a factor of
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* An approximate error bound for the computed average of the
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* eigenvalues of T11 is
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* where EPS is the machine precision.
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* The reciprocal condition number of the right invariant subspace
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* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
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* SEP is defined as the separation of T11 and T22:
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* sep( T11, T22 ) = sigma-min( C )
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* where sigma-min(C) is the smallest singular value of the
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* n1*n2-by-n1*n2 matrix
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* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
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* I(m) is an m by m identity matrix, and kprod denotes the Kronecker
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* product. We estimate sigma-min(C) by the reciprocal of an estimate of
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* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
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* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
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* When SEP is small, small changes in T can cause large changes in
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* the invariant subspace. An approximate bound on the maximum angular
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* error in the computed right invariant subspace is
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* EPS * norm(T) / SEP
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* =====================================================================
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* .. Local Scalars ..
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LOGICAL LQUERY, WANTBH, WANTQ, WANTS, WANTSP
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INTEGER IERR, K, KASE, KS, LWMIN, N1, N2, NN
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DOUBLE PRECISION EST, RNORM, SCALE
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DOUBLE PRECISION RWORK( 1 )
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* .. External Functions ..
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DOUBLE PRECISION ZLANGE
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EXTERNAL LSAME, ZLANGE
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZLACN2, ZLACPY, ZTREXC, ZTRSYL
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* .. Intrinsic Functions ..
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* .. Executable Statements ..
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* Decode and test the input parameters.
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WANTBH = LSAME( JOB, 'B' )
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WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
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WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
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WANTQ = LSAME( COMPQ, 'V' )
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* Set M to the number of selected eigenvalues.
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LQUERY = ( LWORK.EQ.-1 )
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LWMIN = MAX( 1, 2*NN )
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ELSE IF( LSAME( JOB, 'N' ) ) THEN
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ELSE IF( LSAME( JOB, 'E' ) ) THEN
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IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
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ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
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ELSE IF( N.LT.0 ) THEN
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ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
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ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
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ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
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CALL XERBLA( 'ZTRSEN', -INFO )
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ELSE IF( LQUERY ) THEN
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* Quick return if possible
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IF( M.EQ.N .OR. M.EQ.0 ) THEN
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$ SEP = ZLANGE( '1', N, N, T, LDT, RWORK )
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* Collect the selected eigenvalues at the top left corner of T.
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IF( SELECT( K ) ) THEN
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* Swap the K-th eigenvalue to position KS.
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$ CALL ZTREXC( COMPQ, N, T, LDT, Q, LDQ, K, KS, IERR )
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* Solve the Sylvester equation for R:
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* T11*R - R*T22 = scale*T12
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CALL ZLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
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CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
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$ LDT, WORK, N1, SCALE, IERR )
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* Estimate the reciprocal of the condition number of the cluster
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RNORM = ZLANGE( 'F', N1, N2, WORK, N1, RWORK )
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IF( RNORM.EQ.ZERO ) THEN
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S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
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* Estimate sep(T11,T22).
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CALL ZLACN2( NN, WORK( NN+1 ), WORK, EST, KASE, ISAVE )
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* Solve T11*R - R*T22 = scale*X.
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CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
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$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
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* Solve T11'*R - R*T22' = scale*X.
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CALL ZTRSYL( 'C', 'C', -1, N1, N2, T, LDT,
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$ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
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* Copy reordered eigenvalues to W.
added
removed
libcruft/lapack/ztrsen.f
