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stgsyl.f(3)				      LAPACK				      stgsyl.f(3)

NAME
       stgsyl.f -

SYNOPSIS
   Functions/Subroutines
       subroutine stgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF,
	   SCALE, DIF, WORK, LWORK, IWORK, INFO)
	   STGSYL

Function/Subroutine Documentation
   subroutine stgsyl (characterTRANS, integerIJOB, integerM, integerN, real, dimension( lda, *
       )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( ldc, * )C,
       integerLDC, real, dimension( ldd, * )D, integerLDD, real, dimension( lde, * )E,
       integerLDE, real, dimension( ldf, * )F, integerLDF, realSCALE, realDIF, real, dimension( *
       )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
       STGSYL

       Purpose:

	    STGSYL solves the generalized Sylvester equation:

			A * R - L * B = scale * C		  (1)
			D * R - L * E = scale * F

	    where R and L are unknown m-by-n matrices, (A, D), (B, E) and
	    (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
	    respectively, with real entries. (A, D) and (B, E) must be in
	    generalized (real) Schur canonical form, i.e. A, B are upper quasi
	    triangular and D, E are upper triangular.

	    The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
	    scaling factor chosen to avoid overflow.

	    In matrix notation (1) is equivalent to solve  Zx = scale b, where
	    Z is defined as

		       Z = [ kron(In, A)  -kron(B**T, Im) ]	    (2)
			   [ kron(In, D)  -kron(E**T, Im) ].

	    Here Ik is the identity matrix of size k and X**T is the transpose of
	    X. kron(X, Y) is the Kronecker product between the matrices X and Y.

	    If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b,
	    which is equivalent to solve for R and L in

			A**T * R + D**T * L = scale * C 	  (3)
			R * B**T + L * E**T = scale * -F

	    This case (TRANS = 'T') is used to compute an one-norm-based estimate
	    of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
	    and (B,E), using SLACON.

	    If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate
	    of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
	    reciprocal of the smallest singular value of Z. See [1-2] for more
	    information.

	    This is a level 3 BLAS algorithm.

       Parameters:
	   TRANS

		     TRANS is CHARACTER*1
		     = 'N', solve the generalized Sylvester equation (1).
		     = 'T', solve the 'transposed' system (3).

	   IJOB

		     IJOB is INTEGER
		     Specifies what kind of functionality to be performed.
		      =0: solve (1) only.
		      =1: The functionality of 0 and 3.
		      =2: The functionality of 0 and 4.
		      =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
			  (look ahead strategy IJOB  = 1 is used).
		      =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
			  ( SGECON on sub-systems is used ).
		     Not referenced if TRANS = 'T'.

	   M

		     M is INTEGER
		     The order of the matrices A and D, and the row dimension of
		     the matrices C, F, R and L.

	   N

		     N is INTEGER
		     The order of the matrices B and E, and the column dimension
		     of the matrices C, F, R and L.

	   A

		     A is REAL array, dimension (LDA, M)
		     The upper quasi triangular matrix A.

	   LDA

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

	   B

		     B is REAL array, dimension (LDB, N)
		     The upper quasi triangular matrix B.

	   LDB

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

	   C

		     C is REAL array, dimension (LDC, N)
		     On entry, C contains the right-hand-side of the first matrix
		     equation in (1) or (3).
		     On exit, if IJOB = 0, 1 or 2, C has been overwritten by
		     the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
		     the solution achieved during the computation of the
		     Dif-estimate.

	   LDC

		     LDC is INTEGER
		     The leading dimension of the array C. LDC >= max(1, M).

	   D

		     D is REAL array, dimension (LDD, M)
		     The upper triangular matrix D.

	   LDD

		     LDD is INTEGER
		     The leading dimension of the array D. LDD >= max(1, M).

	   E

		     E is REAL array, dimension (LDE, N)
		     The upper triangular matrix E.

	   LDE

		     LDE is INTEGER
		     The leading dimension of the array E. LDE >= max(1, N).

	   F

		     F is REAL array, dimension (LDF, N)
		     On entry, F contains the right-hand-side of the second matrix
		     equation in (1) or (3).
		     On exit, if IJOB = 0, 1 or 2, F has been overwritten by
		     the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
		     the solution achieved during the computation of the
		     Dif-estimate.

	   LDF

		     LDF is INTEGER
		     The leading dimension of the array F. LDF >= max(1, M).

	   DIF

		     DIF is REAL
		     On exit DIF is the reciprocal of a lower bound of the
		     reciprocal of the Dif-function, i.e. DIF is an upper bound of
		     Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
		     IF IJOB = 0 or TRANS = 'T', DIF is not touched.

	   SCALE

		     SCALE is REAL
		     On exit SCALE is the scaling factor in (1) or (3).
		     If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
		     to a slightly perturbed system but the input matrices A, B, D
		     and E have not been changed. If SCALE = 0, C and F hold the
		     solutions R and L, respectively, to the homogeneous system
		     with C = F = 0. Normally, SCALE = 1.

	   WORK

		     WORK is REAL array, dimension (MAX(1,LWORK))
		     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

	   LWORK

		     LWORK is INTEGER
		     The dimension of the array WORK. LWORK > = 1.
		     If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

		     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

		     IWORK is INTEGER array, dimension (M+N+6)

	   INFO

		     INFO is INTEGER
		       =0: successful exit
		       <0: If INFO = -i, the i-th argument had an illegal value.
		       >0: (A, D) and (B, E) have common or close eigenvalues.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Contributors:
	   Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
	   87 Umea, Sweden.

       References:

	     [1] 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.

	     [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
		 Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
		 Appl., 15(4):1045-1060, 1994

	     [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
		 Condition Estimators for Solving the Generalized Sylvester
		 Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
		 July 1989, pp 745-751.

       Definition at line 298 of file stgsyl.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      stgsyl.f(3)
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