DTGSY2 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 DLACON.


 DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL


 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


 DTGSYL. See DTGSYL for details.

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. (DGECON 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION


          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 DOUBLE PRECISION


          On entry, the sum of squares of computed contributions to


          the Dif-estimate under computation by DTGSYL, 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 DTGSY2 is called by DTGSYL.

RDSCAL

          RDSCAL is DOUBLE PRECISION


          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 DTGSY2 is called by


                DTGSYL.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

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