654 lines
23 KiB
FortranFixed
654 lines
23 KiB
FortranFixed
SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
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$ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF,
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$ WORK, LWORK, IWORK, LIWORK, INFO )
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*
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* -- LAPACK routine (version 3.1.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* January 2007
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*
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* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
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*
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* .. Scalar Arguments ..
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LOGICAL WANTQ, WANTZ
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INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
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$ M, N
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DOUBLE PRECISION PL, PR
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* ..
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* .. Array Arguments ..
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LOGICAL SELECT( * )
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INTEGER IWORK( * )
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DOUBLE PRECISION DIF( * )
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COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
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$ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
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* ..
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*
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* Purpose
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* =======
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*
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* ZTGSEN reorders the generalized Schur decomposition of a complex
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* matrix pair (A, B) (in terms of an unitary equivalence trans-
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* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
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* appears in the leading diagonal blocks of the pair (A,B). The leading
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* columns of Q and Z form unitary bases of the corresponding left and
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* right eigenspaces (deflating subspaces). (A, B) must be in
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* generalized Schur canonical form, that is, A and B are both upper
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* triangular.
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*
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* ZTGSEN also computes the generalized eigenvalues
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*
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* w(j)= ALPHA(j) / BETA(j)
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*
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* of the reordered matrix pair (A, B).
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*
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* Optionally, the routine computes estimates of reciprocal condition
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* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
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* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
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* between the matrix pairs (A11, B11) and (A22,B22) that correspond to
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* the selected cluster and the eigenvalues outside the cluster, resp.,
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* and norms of "projections" onto left and right eigenspaces w.r.t.
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* the selected cluster in the (1,1)-block.
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*
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*
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* Arguments
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* =========
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*
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* IJOB (input) integer
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* Specifies whether condition numbers are required for the
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* cluster of eigenvalues (PL and PR) or the deflating subspaces
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* (Difu and Difl):
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* =0: Only reorder w.r.t. SELECT. No extras.
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* =1: Reciprocal of norms of "projections" onto left and right
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* eigenspaces w.r.t. the selected cluster (PL and PR).
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* =2: Upper bounds on Difu and Difl. F-norm-based estimate
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* (DIF(1:2)).
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* =3: Estimate of Difu and Difl. 1-norm-based estimate
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* (DIF(1:2)).
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* About 5 times as expensive as IJOB = 2.
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* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
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* version to get it all.
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* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
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*
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* WANTQ (input) LOGICAL
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* .TRUE. : update the left transformation matrix Q;
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* .FALSE.: do not update Q.
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*
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* WANTZ (input) LOGICAL
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* .TRUE. : update the right transformation matrix Z;
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* .FALSE.: do not update Z.
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*
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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 an eigenvalue w(j), SELECT(j) must be set to
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* .TRUE..
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*
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* N (input) INTEGER
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* The order of the matrices A and B. N >= 0.
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*
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* A (input/output) COMPLEX*16 array, dimension(LDA,N)
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* On entry, the upper triangular matrix A, in generalized
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* Schur canonical form.
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* On exit, A is overwritten by the reordered matrix A.
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,N).
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*
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* B (input/output) COMPLEX*16 array, dimension(LDB,N)
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* On entry, the upper triangular matrix B, in generalized
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* Schur canonical form.
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* On exit, B is overwritten by the reordered matrix B.
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*
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* LDB (input) INTEGER
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* The leading dimension of the array B. LDB >= max(1,N).
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*
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* ALPHA (output) COMPLEX*16 array, dimension (N)
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* BETA (output) COMPLEX*16 array, dimension (N)
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* The diagonal elements of A and B, respectively,
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* when the pair (A,B) has been reduced to generalized Schur
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* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized
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* eigenvalues.
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*
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* Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
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* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
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* On exit, Q has been postmultiplied by the left unitary
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* transformation matrix which reorder (A, B); The leading M
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* columns of Q form orthonormal bases for the specified pair of
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* left eigenspaces (deflating subspaces).
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* If WANTQ = .FALSE., Q is not referenced.
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*
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* LDQ (input) INTEGER
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* The leading dimension of the array Q. LDQ >= 1.
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* If WANTQ = .TRUE., LDQ >= N.
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*
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* Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
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* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
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* On exit, Z has been postmultiplied by the left unitary
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* transformation matrix which reorder (A, B); The leading M
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* columns of Z form orthonormal bases for the specified pair of
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* left eigenspaces (deflating subspaces).
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* If WANTZ = .FALSE., Z is not referenced.
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*
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* LDZ (input) INTEGER
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* The leading dimension of the array Z. LDZ >= 1.
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* If WANTZ = .TRUE., LDZ >= N.
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*
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* M (output) INTEGER
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* The dimension of the specified pair of left and right
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* eigenspaces, (deflating subspaces) 0 <= M <= N.
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*
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* PL (output) DOUBLE PRECISION
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* PR (output) DOUBLE PRECISION
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* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
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* reciprocal of the norm of "projections" onto left and right
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* eigenspace with respect to the selected cluster.
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* 0 < PL, PR <= 1.
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* If M = 0 or M = N, PL = PR = 1.
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* If IJOB = 0, 2 or 3 PL, PR are not referenced.
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*
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* DIF (output) DOUBLE PRECISION array, dimension (2).
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* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
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* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
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* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
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* estimates of Difu and Difl, computed using reversed
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* communication with ZLACN2.
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* If M = 0 or N, DIF(1:2) = F-norm([A, B]).
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* If IJOB = 0 or 1, DIF is not referenced.
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*
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
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* IF IJOB = 0, WORK is not referenced. Otherwise,
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* on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*
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* LWORK (input) INTEGER
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* The dimension of the array WORK. LWORK >= 1
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* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M)
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* If IJOB = 3 or 5, LWORK >= 4*M*(N-M)
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*
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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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*
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* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
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* IF IJOB = 0, IWORK is not referenced. Otherwise,
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* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
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*
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* LIWORK (input) INTEGER
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* The dimension of the array IWORK. LIWORK >= 1.
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* If IJOB = 1, 2 or 4, LIWORK >= N+2;
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* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
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*
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* If LIWORK = -1, then a workspace query is assumed; the
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* routine only calculates the optimal size of the IWORK array,
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* returns this value as the first entry of the IWORK array, and
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* no error message related to LIWORK is issued by XERBLA.
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*
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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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* =1: Reordering of (A, B) failed because the transformed
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* matrix pair (A, B) would be too far from generalized
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* Schur form; the problem is very ill-conditioned.
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* (A, B) may have been partially reordered.
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* If requested, 0 is returned in DIF(*), PL and PR.
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*
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*
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* Further Details
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* ===============
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*
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* ZTGSEN first collects the selected eigenvalues by computing unitary
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* U and W that move them to the top left corner of (A, B). In other
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* words, the selected eigenvalues are the eigenvalues of (A11, B11) in
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*
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* U'*(A, B)*W = (A11 A12) (B11 B12) n1
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* ( 0 A22),( 0 B22) n2
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* n1 n2 n1 n2
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*
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* where N = n1+n2 and U' means the conjugate transpose of U. The first
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* n1 columns of U and W span the specified pair of left and right
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* eigenspaces (deflating subspaces) of (A, B).
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*
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* If (A, B) has been obtained from the generalized real Schur
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* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
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* reordered generalized Schur form of (C, D) is given by
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*
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* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
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*
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* and the first n1 columns of Q*U and Z*W span the corresponding
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* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
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*
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* Note that if the selected eigenvalue is sufficiently ill-conditioned,
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* then its value may differ significantly from its value before
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* reordering.
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*
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* The reciprocal condition numbers of the left and right eigenspaces
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* spanned by the first n1 columns of U and W (or Q*U and Z*W) may
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* be returned in DIF(1:2), corresponding to Difu and Difl, resp.
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*
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* The Difu and Difl are defined as:
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*
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* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
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* and
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* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
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*
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* where sigma-min(Zu) is the smallest singular value of the
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* (2*n1*n2)-by-(2*n1*n2) matrix
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*
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* Zu = [ kron(In2, A11) -kron(A22', In1) ]
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* [ kron(In2, B11) -kron(B22', In1) ].
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*
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* Here, Inx is the identity matrix of size nx and A22' is the
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* transpose of A22. kron(X, Y) is the Kronecker product between
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* the matrices X and Y.
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*
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* When DIF(2) is small, small changes in (A, B) can cause large changes
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* in the deflating subspace. An approximate (asymptotic) bound on the
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* maximum angular error in the computed deflating subspaces is
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*
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* EPS * norm((A, B)) / DIF(2),
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*
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* where EPS is the machine precision.
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*
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* The reciprocal norm of the projectors on the left and right
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* eigenspaces associated with (A11, B11) may be returned in PL and PR.
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* They are computed as follows. First we compute L and R so that
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* P*(A, B)*Q is block diagonal, where
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*
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* P = ( I -L ) n1 Q = ( I R ) n1
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* ( 0 I ) n2 and ( 0 I ) n2
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* n1 n2 n1 n2
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*
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* and (L, R) is the solution to the generalized Sylvester equation
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*
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* A11*R - L*A22 = -A12
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* B11*R - L*B22 = -B12
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*
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* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
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* An approximate (asymptotic) bound on the average absolute error of
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* the selected eigenvalues is
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*
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* EPS * norm((A, B)) / PL.
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*
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* There are also global error bounds which valid for perturbations up
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* to a certain restriction: A lower bound (x) on the smallest
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* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
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* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
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* (i.e. (A + E, B + F), is
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*
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* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
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*
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* An approximate bound on x can be computed from DIF(1:2), PL and PR.
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*
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* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
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* (L', R') and unperturbed (L, R) left and right deflating subspaces
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* associated with the selected cluster in the (1,1)-blocks can be
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* bounded as
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*
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* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
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* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
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*
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* See LAPACK User's Guide section 4.11 or the following references
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* for more information.
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*
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* Note that if the default method for computing the Frobenius-norm-
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* based estimate DIF is not wanted (see ZLATDF), then the parameter
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* IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF
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* (IJOB = 2 will be used)). See ZTGSYL for more details.
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*
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* Based on contributions by
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* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
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* Umea University, S-901 87 Umea, Sweden.
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*
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* References
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* ==========
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*
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* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
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* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
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* M.S. Moonen et al (eds), Linear Algebra for Large Scale and
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* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
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*
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* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
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* Eigenvalues of a Regular Matrix Pair (A, B) and Condition
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* Estimation: Theory, Algorithms and Software, Report
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* UMINF - 94.04, Department of Computing Science, Umea University,
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* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
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* To appear in Numerical Algorithms, 1996.
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*
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* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
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* for Solving the Generalized Sylvester Equation and Estimating the
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* Separation between Regular Matrix Pairs, Report UMINF - 93.23,
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* Department of Computing Science, Umea University, S-901 87 Umea,
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* Sweden, December 1993, Revised April 1994, Also as LAPACK working
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* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
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* 1996.
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*
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* =====================================================================
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*
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* .. Parameters ..
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INTEGER IDIFJB
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PARAMETER ( IDIFJB = 3 )
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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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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, SWAP, WANTD, WANTD1, WANTD2, WANTP
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INTEGER I, IERR, IJB, K, KASE, KS, LIWMIN, LWMIN, MN2,
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$ N1, N2
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DOUBLE PRECISION DSCALE, DSUM, RDSCAL, SAFMIN
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COMPLEX*16 TEMP1, TEMP2
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* ..
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* .. Local Arrays ..
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INTEGER ISAVE( 3 )
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZLACN2, ZLACPY, ZLASSQ, ZSCAL, ZTGEXC,
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$ ZTGSYL
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DCMPLX, DCONJG, MAX, SQRT
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH
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EXTERNAL DLAMCH
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* ..
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* .. Executable Statements ..
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*
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* Decode and test the input parameters
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*
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INFO = 0
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LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
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*
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IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -5
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -9
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ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
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INFO = -13
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ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
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INFO = -15
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZTGSEN', -INFO )
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RETURN
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END IF
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*
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IERR = 0
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*
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WANTP = IJOB.EQ.1 .OR. IJOB.GE.4
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WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4
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WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5
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WANTD = WANTD1 .OR. WANTD2
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*
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* Set M to the dimension of the specified pair of deflating
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* subspaces.
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*
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M = 0
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DO 10 K = 1, N
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ALPHA( K ) = A( K, K )
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BETA( K ) = B( K, K )
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IF( K.LT.N ) THEN
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IF( SELECT( K ) )
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$ M = M + 1
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ELSE
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IF( SELECT( N ) )
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$ M = M + 1
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END IF
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10 CONTINUE
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*
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IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
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LWMIN = MAX( 1, 2*M*( N-M ) )
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LIWMIN = MAX( 1, N+2 )
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ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN
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LWMIN = MAX( 1, 4*M*( N-M ) )
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LIWMIN = MAX( 1, 2*M*( N-M ), N+2 )
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ELSE
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LWMIN = 1
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LIWMIN = 1
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END IF
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*
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WORK( 1 ) = LWMIN
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IWORK( 1 ) = LIWMIN
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*
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IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -21
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ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -23
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZTGSEN', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
|
|
* Quick return if possible.
|
|
*
|
|
IF( M.EQ.N .OR. M.EQ.0 ) THEN
|
|
IF( WANTP ) THEN
|
|
PL = ONE
|
|
PR = ONE
|
|
END IF
|
|
IF( WANTD ) THEN
|
|
DSCALE = ZERO
|
|
DSUM = ONE
|
|
DO 20 I = 1, N
|
|
CALL ZLASSQ( N, A( 1, I ), 1, DSCALE, DSUM )
|
|
CALL ZLASSQ( N, B( 1, I ), 1, DSCALE, DSUM )
|
|
20 CONTINUE
|
|
DIF( 1 ) = DSCALE*SQRT( DSUM )
|
|
DIF( 2 ) = DIF( 1 )
|
|
END IF
|
|
GO TO 70
|
|
END IF
|
|
*
|
|
* Get machine constant
|
|
*
|
|
SAFMIN = DLAMCH( 'S' )
|
|
*
|
|
* Collect the selected blocks at the top-left corner of (A, B).
|
|
*
|
|
KS = 0
|
|
DO 30 K = 1, N
|
|
SWAP = SELECT( K )
|
|
IF( SWAP ) THEN
|
|
KS = KS + 1
|
|
*
|
|
* Swap the K-th block to position KS. Compute unitary Q
|
|
* and Z that will swap adjacent diagonal blocks in (A, B).
|
|
*
|
|
IF( K.NE.KS )
|
|
$ CALL ZTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
|
|
$ LDZ, K, KS, IERR )
|
|
*
|
|
IF( IERR.GT.0 ) THEN
|
|
*
|
|
* Swap is rejected: exit.
|
|
*
|
|
INFO = 1
|
|
IF( WANTP ) THEN
|
|
PL = ZERO
|
|
PR = ZERO
|
|
END IF
|
|
IF( WANTD ) THEN
|
|
DIF( 1 ) = ZERO
|
|
DIF( 2 ) = ZERO
|
|
END IF
|
|
GO TO 70
|
|
END IF
|
|
END IF
|
|
30 CONTINUE
|
|
IF( WANTP ) THEN
|
|
*
|
|
* Solve generalized Sylvester equation for R and L:
|
|
* A11 * R - L * A22 = A12
|
|
* B11 * R - L * B22 = B12
|
|
*
|
|
N1 = M
|
|
N2 = N - M
|
|
I = N1 + 1
|
|
CALL ZLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 )
|
|
CALL ZLACPY( 'Full', N1, N2, B( 1, I ), LDB, WORK( N1*N2+1 ),
|
|
$ N1 )
|
|
IJB = 0
|
|
CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
|
|
$ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1,
|
|
$ DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
|
|
$ LWORK-2*N1*N2, IWORK, IERR )
|
|
*
|
|
* Estimate the reciprocal of norms of "projections" onto
|
|
* left and right eigenspaces
|
|
*
|
|
RDSCAL = ZERO
|
|
DSUM = ONE
|
|
CALL ZLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM )
|
|
PL = RDSCAL*SQRT( DSUM )
|
|
IF( PL.EQ.ZERO ) THEN
|
|
PL = ONE
|
|
ELSE
|
|
PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) )
|
|
END IF
|
|
RDSCAL = ZERO
|
|
DSUM = ONE
|
|
CALL ZLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM )
|
|
PR = RDSCAL*SQRT( DSUM )
|
|
IF( PR.EQ.ZERO ) THEN
|
|
PR = ONE
|
|
ELSE
|
|
PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) )
|
|
END IF
|
|
END IF
|
|
IF( WANTD ) THEN
|
|
*
|
|
* Compute estimates Difu and Difl.
|
|
*
|
|
IF( WANTD1 ) THEN
|
|
N1 = M
|
|
N2 = N - M
|
|
I = N1 + 1
|
|
IJB = IDIFJB
|
|
*
|
|
* Frobenius norm-based Difu estimate.
|
|
*
|
|
CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
|
|
$ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ),
|
|
$ N1, DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
|
|
$ LWORK-2*N1*N2, IWORK, IERR )
|
|
*
|
|
* Frobenius norm-based Difl estimate.
|
|
*
|
|
CALL ZTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, WORK,
|
|
$ N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ),
|
|
$ N2, DSCALE, DIF( 2 ), WORK( N1*N2*2+1 ),
|
|
$ LWORK-2*N1*N2, IWORK, IERR )
|
|
ELSE
|
|
*
|
|
* Compute 1-norm-based estimates of Difu and Difl using
|
|
* reversed communication with ZLACN2. In each step a
|
|
* generalized Sylvester equation or a transposed variant
|
|
* is solved.
|
|
*
|
|
KASE = 0
|
|
N1 = M
|
|
N2 = N - M
|
|
I = N1 + 1
|
|
IJB = 0
|
|
MN2 = 2*N1*N2
|
|
*
|
|
* 1-norm-based estimate of Difu.
|
|
*
|
|
40 CONTINUE
|
|
CALL ZLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 1 ), KASE,
|
|
$ ISAVE )
|
|
IF( KASE.NE.0 ) THEN
|
|
IF( KASE.EQ.1 ) THEN
|
|
*
|
|
* Solve generalized Sylvester equation
|
|
*
|
|
CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA,
|
|
$ WORK, N1, B, LDB, B( I, I ), LDB,
|
|
$ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
|
|
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
|
|
$ IERR )
|
|
ELSE
|
|
*
|
|
* Solve the transposed variant.
|
|
*
|
|
CALL ZTGSYL( 'C', IJB, N1, N2, A, LDA, A( I, I ), LDA,
|
|
$ WORK, N1, B, LDB, B( I, I ), LDB,
|
|
$ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
|
|
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
|
|
$ IERR )
|
|
END IF
|
|
GO TO 40
|
|
END IF
|
|
DIF( 1 ) = DSCALE / DIF( 1 )
|
|
*
|
|
* 1-norm-based estimate of Difl.
|
|
*
|
|
50 CONTINUE
|
|
CALL ZLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 2 ), KASE,
|
|
$ ISAVE )
|
|
IF( KASE.NE.0 ) THEN
|
|
IF( KASE.EQ.1 ) THEN
|
|
*
|
|
* Solve generalized Sylvester equation
|
|
*
|
|
CALL ZTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA,
|
|
$ WORK, N2, B( I, I ), LDB, B, LDB,
|
|
$ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
|
|
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
|
|
$ IERR )
|
|
ELSE
|
|
*
|
|
* Solve the transposed variant.
|
|
*
|
|
CALL ZTGSYL( 'C', IJB, N2, N1, A( I, I ), LDA, A, LDA,
|
|
$ WORK, N2, B, LDB, B( I, I ), LDB,
|
|
$ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
|
|
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
|
|
$ IERR )
|
|
END IF
|
|
GO TO 50
|
|
END IF
|
|
DIF( 2 ) = DSCALE / DIF( 2 )
|
|
END IF
|
|
END IF
|
|
*
|
|
* If B(K,K) is complex, make it real and positive (normalization
|
|
* of the generalized Schur form) and Store the generalized
|
|
* eigenvalues of reordered pair (A, B)
|
|
*
|
|
DO 60 K = 1, N
|
|
DSCALE = ABS( B( K, K ) )
|
|
IF( DSCALE.GT.SAFMIN ) THEN
|
|
TEMP1 = DCONJG( B( K, K ) / DSCALE )
|
|
TEMP2 = B( K, K ) / DSCALE
|
|
B( K, K ) = DSCALE
|
|
CALL ZSCAL( N-K, TEMP1, B( K, K+1 ), LDB )
|
|
CALL ZSCAL( N-K+1, TEMP1, A( K, K ), LDA )
|
|
IF( WANTQ )
|
|
$ CALL ZSCAL( N, TEMP2, Q( 1, K ), 1 )
|
|
ELSE
|
|
B( K, K ) = DCMPLX( ZERO, ZERO )
|
|
END IF
|
|
*
|
|
ALPHA( K ) = A( K, K )
|
|
BETA( K ) = B( K, K )
|
|
*
|
|
60 CONTINUE
|
|
*
|
|
70 CONTINUE
|
|
*
|
|
WORK( 1 ) = LWMIN
|
|
IWORK( 1 ) = LIWMIN
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZTGSEN
|
|
*
|
|
END
|