version 1.5, 2010/08/07 13:18:09
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version 1.9, 2011/07/22 07:38:21
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$ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, |
$ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, |
$ WORK, LWORK, IWORK, LIWORK, INFO ) |
$ WORK, LWORK, IWORK, LIWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.2) -- |
* -- LAPACK routine (version 3.3.1) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* January 2007 |
* -- April 2011 -- |
* |
* |
* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. |
* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. |
* |
* |
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* |
* |
* ZTGSEN reorders the generalized Schur decomposition of a complex |
* ZTGSEN reorders the generalized Schur decomposition of a complex |
* matrix pair (A, B) (in terms of an unitary equivalence trans- |
* matrix pair (A, B) (in terms of an unitary equivalence trans- |
* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues |
* formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues |
* appears in the leading diagonal blocks of the pair (A,B). The leading |
* appears in the leading diagonal blocks of the pair (A,B). The leading |
* columns of Q and Z form unitary bases of the corresponding left and |
* columns of Q and Z form unitary bases of the corresponding left and |
* right eigenspaces (deflating subspaces). (A, B) must be in |
* right eigenspaces (deflating subspaces). (A, B) must be in |
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* U and W that move them to the top left corner of (A, B). In other |
* U and W that move them to the top left corner of (A, B). In other |
* words, the selected eigenvalues are the eigenvalues of (A11, B11) in |
* words, the selected eigenvalues are the eigenvalues of (A11, B11) in |
* |
* |
* U'*(A, B)*W = (A11 A12) (B11 B12) n1 |
* U**H*(A, B)*W = (A11 A12) (B11 B12) n1 |
* ( 0 A22),( 0 B22) n2 |
* ( 0 A22),( 0 B22) n2 |
* n1 n2 n1 n2 |
* n1 n2 n1 n2 |
* |
* |
* where N = n1+n2 and U' means the conjugate transpose of U. The first |
* where N = n1+n2 and U**H means the conjugate transpose of U. The first |
* n1 columns of U and W span the specified pair of left and right |
* n1 columns of U and W span the specified pair of left and right |
* eigenspaces (deflating subspaces) of (A, B). |
* eigenspaces (deflating subspaces) of (A, B). |
* |
* |
* If (A, B) has been obtained from the generalized real Schur |
* If (A, B) has been obtained from the generalized real Schur |
* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the |
* decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the |
* reordered generalized Schur form of (C, D) is given by |
* reordered generalized Schur form of (C, D) is given by |
* |
* |
* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', |
* (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H, |
* |
* |
* and the first n1 columns of Q*U and Z*W span the corresponding |
* and the first n1 columns of Q*U and Z*W span the corresponding |
* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). |
* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). |
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* where sigma-min(Zu) is the smallest singular value of the |
* where sigma-min(Zu) is the smallest singular value of the |
* (2*n1*n2)-by-(2*n1*n2) matrix |
* (2*n1*n2)-by-(2*n1*n2) matrix |
* |
* |
* Zu = [ kron(In2, A11) -kron(A22', In1) ] |
* Zu = [ kron(In2, A11) -kron(A22**H, In1) ] |
* [ kron(In2, B11) -kron(B22', In1) ]. |
* [ kron(In2, B11) -kron(B22**H, In1) ]. |
* |
* |
* Here, Inx is the identity matrix of size nx and A22' is the |
* Here, Inx is the identity matrix of size nx and A22**H is the |
* transpose of A22. kron(X, Y) is the Kronecker product between |
* conjugate transpose of A22. kron(X, Y) is the Kronecker product between |
* the matrices X and Y. |
* the matrices X and Y. |
* |
* |
* When DIF(2) is small, small changes in (A, B) can cause large changes |
* When DIF(2) is small, small changes in (A, B) can cause large changes |