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stgsy2.f(3)				      LAPACK				      stgsy2.f(3)

NAME
       stgsy2.f -

SYNOPSIS
   Functions/Subroutines
       subroutine stgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF,
	   SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO)
	   STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Function/Subroutine Documentation
   subroutine stgsy2 (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, realRDSUM, realRDSCAL,
       integer, dimension( * )IWORK, integerPQ, integerINFO)
       STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

       Purpose:

	    STGSY2 solves the generalized Sylvester equation:

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

	    using Level 1 and 2 BLAS. 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 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 solving equation (1) corresponds to solve
	    Z*x = scale*b, where Z is defined as

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

	    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.
	    In the process of solving (1), we solve a number of such systems
	    where Dim(In), Dim(In) = 1 or 2.

	    If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y,
	    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 is used to compute an estimate of Dif[(A, D), (B, E)] =
	    sigma_min(Z) using reverse communicaton with SLACON.

	    STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL
	    of an upper bound on the separation between to matrix pairs. Then
	    the input (A, D), (B, E) are sub-pencils of the matrix pair in
	    STGSYL. See STGSYL for details.

       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: A contribution from this subsystem to a Frobenius
			  norm-based estimate of the separation between two matrix
			  pairs is computed. (look ahead strategy is used).
		     = 2: A contribution from this subsystem to a Frobenius
			  norm-based estimate of the separation between two matrix
			  pairs is computed. (SGECON on sub-systems is used.)
		     Not referenced if TRANS = 'T'.

	   M

		     M is INTEGER
		     On entry, M specifies the order of A and D, and the row
		     dimension of C, F, R and L.

	   N

		     N is INTEGER
		     On entry, N specifies the order of B and E, and the column
		     dimension of C, F, R and L.

	   A

		     A is REAL array, dimension (LDA, M)
		     On entry, A contains an upper quasi triangular matrix.

	   LDA

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

	   B

		     B is REAL array, dimension (LDB, N)
		     On entry, B contains an upper quasi triangular matrix.

	   LDB

		     LDB is INTEGER
		     The leading dimension of the matrix 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).
		     On exit, if IJOB = 0, C has been overwritten by the
		     solution R.

	   LDC

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

	   D

		     D is REAL array, dimension (LDD, M)
		     On entry, D contains an upper triangular matrix.

	   LDD

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

	   E

		     E is REAL array, dimension (LDE, N)
		     On entry, E contains an upper triangular matrix.

	   LDE

		     LDE is INTEGER
		     The leading dimension of the matrix 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).
		     On exit, if IJOB = 0, F has been overwritten by the
		     solution L.

	   LDF

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

	   SCALE

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

	   RDSUM

		     RDSUM is REAL
		     On entry, the sum of squares of computed contributions to
		     the Dif-estimate under computation by STGSYL, where the
		     scaling factor RDSCAL (see below) has been factored out.
		     On exit, the corresponding sum of squares updated with the
		     contributions from the current sub-system.
		     If TRANS = 'T' RDSUM is not touched.
		     NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL.

	   RDSCAL

		     RDSCAL is REAL
		     On entry, scaling factor used to prevent overflow in RDSUM.
		     On exit, RDSCAL is updated w.r.t. the current contributions
		     in RDSUM.
		     If TRANS = 'T', RDSCAL is not touched.
		     NOTE: RDSCAL only makes sense when STGSY2 is called by
			   STGSYL.

	   IWORK

		     IWORK is INTEGER array, dimension (M+N+2)

	   PQ

		     PQ is INTEGER
		     On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
		     8-by-8) solved by this routine.

	   INFO

		     INFO is INTEGER
		     On exit, if INFO is set to
		       =0: Successful exit
		       <0: If INFO = -i, the i-th argument had an illegal value.
		       >0: The matrix pairs (A, D) and (B, E) have common or very
			   close eigenvalues.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   September 2012

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

       Definition at line 273 of file stgsy2.f.

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

Version 3.4.2				 Tue Sep 25 2012			      stgsy2.f(3)
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