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ztgsyl.f(3) LAPACK ztgsyl.f(3)

ztgsyl.f -


subroutine ztgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
 
ZTGSYL

ZTGSYL
Purpose:
 ZTGSYL 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 complex entries. A, B, D and E are upper triangular (i.e., (A,D) and (B,E) in generalized Schur form).
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**H, Im) ] (2) [ kron(In, D) -kron(E**H, Im) ],
Here Ix is the identity matrix of size x and X**H is the conjugate transpose of X. Kron(X, Y) is the Kronecker product between the matrices X and Y.
If TRANS = 'C', y in the conjugate transposed system Z**H *y = scale*b is solved for, which is equivalent to solve for R and L in
A**H * R + D**H * L = scale * C (3) R * B**H + L * E**H = scale * -F
This case (TRANS = 'C') 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 ZLACON.
If IJOB >= 1, ZTGSYL 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.
This is a level-3 BLAS algorithm.
Parameters:
TRANS
          TRANS is CHARACTER*1
          = 'N': solve the generalized sylvester equation (1).
          = 'C': solve the "conjugate 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 is used).
          =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
              (ZGECON on sub-systems is used).
          Not referenced if TRANS = 'C'.
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 COMPLEX*16 array, dimension (LDA, M)
          The upper triangular matrix A.
LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1, M).
B
          B is COMPLEX*16 array, dimension (LDB, N)
          The upper triangular matrix B.
LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1, N).
C
          C is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 = 'C', DIF is not referenced.
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, R and L will
          hold the solutions to the homogenious system with C = F = 0.
WORK
          WORK is COMPLEX*16 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+2)
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 very 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 294 of file ztgsyl.f.

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