DTGSYL 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', DTGSYL 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 DLACON.


 If IJOB >= 1, DTGSYL 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.

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.


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


          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 DOUBLE PRECISION


          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 DOUBLE PRECISION 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

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

  [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.