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RedHat 9 (Linux i386) - man page for stgsen (redhat section l)

STGSEN(l)					)					STGSEN(l)

NAME
       STGSEN - reorder the generalized real Schur decomposition of a real matrix pair (A, B) (in
       terms of an orthonormal equivalence trans- formation Q' * (A, B) * Z), so that a  selected
       cluster	of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangu-
       lar matrix A and the upper triangular B

SYNOPSIS
       SUBROUTINE STGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q,
			  LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )

	   LOGICAL	  WANTQ, WANTZ

	   INTEGER	  IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, M, N

	   REAL 	  PL, PR

	   LOGICAL	  SELECT( * )

	   INTEGER	  IWORK( * )

	   REAL 	  A(  LDA,  * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), DIF( *
			  ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )

PURPOSE
       STGSEN reorders the generalized real Schur decomposition of a real matrix pair (A, B)  (in
       terms  of an orthonormal equivalence trans- formation Q' * (A, B) * Z), so that a selected
       cluster of eigenvalues appears in the leading diagonal blocks of the upper  quasi-triangu-
       lar  matrix  A and the upper triangular B. The leading columns of Q and Z form orthonormal
       bases of the corresponding left and right eigen- spaces (deflating subspaces). (A, B) must
       be  in generalized real Schur canonical form (as returned by SGGES), i.e. A is block upper
       triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

       STGSEN also computes the generalized eigenvalues

		   w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

       of the reordered matrix pair (A, B).

       Optionally, STGSEN computes the estimates of reciprocal condition numbers for  eigenvalues
       and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e.
       the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the
       selected  cluster  and  the  eigenvalues outside the cluster, resp., and norms of "projec-
       tions" onto left and right eigenspaces w.r.t.  the selected cluster in the (1,1)-block.

ARGUMENTS
       IJOB    (input) INTEGER
	       Specifies whether condition numbers are required for the  cluster  of  eigenvalues
	       (PL and PR) or the deflating subspaces (Difu and Difl):
	       =0: Only reorder w.r.t. SELECT. No extras.
	       =1:  Reciprocal	of  norms of "projections" onto left and right eigenspaces w.r.t.
	       the selected cluster (PL and PR).  =2: Upper bounds on Difu and Difl. F-norm-based
	       estimate
	       (DIF(1:2)).
	       =3: Estimate of Difu and Difl. 1-norm-based estimate
	       (DIF(1:2)).   About  5 times as expensive as IJOB = 2.  =4: Compute PL, PR and DIF
	       (i.e. 0, 1 and 2 above): Economic version to get it all.  =5: Compute PL,  PR  and
	       DIF (i.e. 0, 1 and 3 above)

       WANTQ   (input) LOGICAL

       WANTZ   (input) LOGICAL

       SELECT  (input) LOGICAL array, dimension (N)
	       SELECT specifies the eigenvalues in the selected cluster.  To select a real eigen-
	       value w(j), SELECT(j) must be set to 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 either both
	       included in the cluster or both excluded.

       N       (input) INTEGER
	       The order of the matrices A and B. N >= 0.

       A       (input/output) REAL array, dimension(LDA,N)
	       On entry, the upper quasi-triangular matrix A, with (A,	B)  in	generalized  real
	       Schur canonical form.  On exit, A is overwritten by the reordered matrix A.

       LDA     (input) INTEGER
	       The leading dimension of the array A. LDA >= max(1,N).

       B       (input/output) REAL array, dimension(LDB,N)
	       On  entry,  the	upper  triangular matrix B, with (A, B) in generalized real Schur
	       canonical form.	On exit, B is overwritten by the reordered matrix B.

       LDB     (input) INTEGER
	       The leading dimension of the array B. LDB >= max(1,N).

       ALPHAR  (output) REAL array, dimension (N)
	       ALPHAI  (output) REAL array, dimension (N) BETA	  (output) REAL array,	dimension
	       (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized
	       eigenvalues.  ALPHAR(j) + ALPHAI(j)*i and BETA(j),j=1,...,N  are the diagonals  of
	       the  complex  Schur  form (S,T) that would result if the 2-by-2 diagonal blocks of
	       the real generalized Schur form of (A,B) were further reduced to  triangular  form
	       using complex unitary transformations.  If ALPHAI(j) is zero, then the j-th eigen-
	       value is real; if positive, then the j-th and (j+1)-st eigenvalues are  a  complex
	       conjugate pair, with ALPHAI(j+1) negative.

       Q       (input/output) REAL array, dimension (LDQ,N)
	       On  entry, if WANTQ = .TRUE., Q is an N-by-N matrix.  On exit, Q has been postmul-
	       tiplied by the left orthogonal transformation matrix which  reorder  (A,  B);  The
	       leading	M  columns  of	Q  form  orthonormal bases for the specified pair of left
	       eigenspaces (deflating subspaces).  If WANTQ = .FALSE., Q is not referenced.

       LDQ     (input) INTEGER
	       The leading dimension of the array Q.  LDQ >= 1; and if WANTQ = .TRUE., LDQ >= N.

       Z       (input/output) REAL array, dimension (LDZ,N)
	       On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.  On exit, Z has been  postmul-
	       tiplied	by  the  left  orthogonal transformation matrix which reorder (A, B); The
	       leading M columns of Z form orthonormal bases  for  the	specified  pair  of  left
	       eigenspaces (deflating subspaces).  If WANTZ = .FALSE., Z is not referenced.

       LDZ     (input) INTEGER
	       The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.

       M       (output) INTEGER
	       The  dimension  of  the	specified pair of left and right eigen- spaces (deflating
	       subspaces). 0 <= M <= N.

	       PL, PR  (output) REAL If IJOB = 1, 4 or 5, PL, PR are lower bounds on the recipro-
	       cal  of	the norm of "projections" onto left and right eigenspaces with respect to
	       the selected cluster.  0 < PL, PR <= 1.	If M = 0 or M = N, PL = PR  = 1.  If IJOB
	       = 0, 2 or 3, PL and PR are not referenced.

       DIF     (output) REAL array, dimension (2).
	       If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
	       If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
	       Difu  and  Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based estimates of Difu and
	       Difl.  If M = 0 or N, DIF(1:2) = F-norm([A, B]).  If IJOB = 0 or  1,  DIF  is  not
	       referenced.

       WORK    (workspace/output) REAL array, dimension (LWORK)
	       IF  IJOB  =  0,	WORK is not referenced.  Otherwise, on exit, if INFO = 0, WORK(1)
	       returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK. LWORK >=  4*N+16.  If IJOB = 1, 2 or 4, LWORK  >=
	       MAX(4*N+16, 2*M*(N-M)).	If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*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/output) INTEGER array, dimension (LIWORK)
	       IF  IJOB  = 0, IWORK is not referenced.	Otherwise, on exit, if INFO = 0, IWORK(1)
	       returns the optimal LIWORK.

       LIWORK  (input) INTEGER
	       The dimension of the array IWORK. LIWORK >= 1.  If IJOB = 1, 2  or  4,  LIWORK  >=
	       N+6.  If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

	       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 (A, B) failed because the transformed matrix pair (A, B) would
	       be too far from generalized Schur form; the problem is very ill-conditioned.   (A,
	       B)  may	have been partially reordered.	If requested, 0 is returned in DIF(*), PL
	       and PR.

FURTHER DETAILS
       STGSEN first collects the selected eigenvalues by computing orthogonal 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:

		     U'*(A, B)*W = (A11 A12) (B11 B12) n1
				   ( 0	A22),( 0  B22) n2
				     n1  n2    n1  n2

       where N = n1+n2 and U' means the transpose of U. The first n1 columns of U and W span  the
       specified pair of left and right eigenspaces (deflating subspaces) of (A, B).

       If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair
       (C, D) = Q*(A, B)*Z', then the reordered generalized real Schur form of (C, D) is given by

		(C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',

       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.).

       Note  that  if the selected eigenvalue is sufficiently ill-conditioned, then its value may
       differ significantly from its value before reordering.

       The reciprocal condition numbers of the left and right eigenspaces spanned by the first n1
       columns of U and W (or Q*U and Z*W) may be returned in DIF(1:2), corresponding to Difu and
       Difl, resp.

       The Difu and Difl are defined as:

	    Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
       and
	    Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

       where sigma-min(Zu) is the smallest singular value of the (2*n1*n2)-by-(2*n1*n2) matrix

	    Zu = [ kron(In2, A11)  -kron(A22', In1) ]
		 [ kron(In2, B11)  -kron(B22', In1) ].

       Here, Inx is the identity matrix of size nx and A22' is the transpose of A22.  kron(X,  Y)
       is the Kronecker product between the matrices X and Y.

       When  DIF(2)  is  small,  small changes in (A, B) can cause large changes in the deflating
       subspace. An approximate (asymptotic) bound on the maximum angular error in  the  computed
       deflating subspaces is

	    EPS * norm((A, B)) / DIF(2),

       where EPS is the machine precision.

       The  reciprocal	norm  of the projectors on the left and right eigenspaces associated with
       (A11, B11) may be returned in PL and PR.  They are computed as follows. First we compute L
       and R so that P*(A, B)*Q is block diagonal, where

	    P = ( I -L ) n1	      Q = ( I R ) n1
		( 0  I ) n2    and	  ( 0 I ) n2
		  n1 n2 		   n1 n2

       and (L, R) is the solution to the generalized Sylvester equation

	    A11*R - L*A22 = -A12
	    B11*R - L*B22 = -B12

       Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).  An approximate (as-
       ymptotic) bound on the average absolute error of the selected eigenvalues is

	    EPS * norm((A, B)) / PL.

       There are also global error bounds which valid for perturbations up to a certain  restric-
       tion:  A lower bound (x) on the smallest F-norm(E,F) for which an eigenvalue of (A11, B11)
       may move and coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), (i.e.  (A
       + E, B + F), is

	x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

       An approximate bound on x can be computed from DIF(1:2), PL and PR.

       If  y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L', R') and unperturbed
       (L, R) left and right deflating subspaces associated with  the  selected  cluster  in  the
       (1,1)-blocks can be bounded as

	max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
	max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

       See LAPACK User's Guide section 4.11 or the following references for more information.

       Note  that  if  the default method for computing the Frobenius-norm- based estimate DIF is
       not wanted (see SLATDF), then the parameter IDIFJB (see below) should be changed from 3 to
       4 (routine SLATDF (IJOB = 2 will be used)). See STGSYL for more details.

       Based on contributions by
	  Bo Kagstrom and Peter Poromaa, Department of Computing Science,
	  Umea University, S-901 87 Umea, Sweden.

       References
       ==========

       [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
	   Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
	   M.S. Moonen et al (eds), Linear Algebra for Large Scale and
	   Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

       [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
	   Eigenvalues of a Regular Matrix Pair (A, B) and Condition
	   Estimation: Theory, Algorithms and Software,
	   Report UMINF - 94.04, Department of Computing Science, Umea
	   University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
	   Note 87. To appear in Numerical Algorithms, 1996.

       [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
	   for Solving the Generalized Sylvester Equation and Estimating the
	   Separation between Regular Matrix Pairs, Report UMINF - 93.23,
	   Department of Computing Science, Umea University, S-901 87 Umea,
	   Sweden, December 1993, Revised April 1994, Also as LAPACK Working
	   Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
	   1996.

LAPACK version 3.0			   15 June 2000 				STGSEN(l)


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