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SUBROUTINE <a name="ZTGSY2.1"></a><a href="ztgsy2.f.html#ZTGSY2.1">ZTGSY2</a>( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
$ INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
DOUBLE PRECISION RDSCAL, RDSUM, SCALE
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="ZTGSY2.22"></a><a href="ztgsy2.f.html#ZTGSY2.1">ZTGSY2</a> solves the generalized Sylvester equation
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A * R - L * B = scale * C (1)
</span><span class="comment">*</span><span class="comment"> D * R - L * E = scale * F
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,
</span><span class="comment">*</span><span class="comment"> (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
</span><span class="comment">*</span><span class="comment"> N-by-N and M-by-N, respectively. A, B, D and E are upper triangular
</span><span class="comment">*</span><span class="comment"> (i.e., (A,D) and (B,E) in generalized Schur form).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
</span><span class="comment">*</span><span class="comment"> scaling factor chosen to avoid overflow.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> In matrix notation solving equation (1) corresponds to solve
</span><span class="comment">*</span><span class="comment"> Zx = scale * b, where Z is defined as
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z = [ kron(In, A) -kron(B', Im) ] (2)
</span><span class="comment">*</span><span class="comment"> [ kron(In, D) -kron(E', Im) ],
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Ik is the identity matrix of size k and X' is the transpose of X.
</span><span class="comment">*</span><span class="comment"> kron(X, Y) is the Kronecker product between the matrices X and Y.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If TRANS = 'C', y in the conjugate transposed system Z'y = scale*b
</span><span class="comment">*</span><span class="comment"> is solved for, which is equivalent to solve for R and L in
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A' * R + D' * L = scale * C (3)
</span><span class="comment">*</span><span class="comment"> R * B' + L * E' = scale * -F
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This case is used to compute an estimate of Dif[(A, D), (B, E)] =
</span><span class="comment">*</span><span class="comment"> = sigma_min(Z) using reverse communicaton with <a name="ZLACON.51"></a><a href="zlacon.f.html#ZLACON.1">ZLACON</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="ZTGSY2.53"></a><a href="ztgsy2.f.html#ZTGSY2.1">ZTGSY2</a> also (IJOB >= 1) contributes to the computation in <a name="ZTGSYL.53"></a><a href="ztgsyl.f.html#ZTGSYL.1">ZTGSYL</a>
</span><span class="comment">*</span><span class="comment"> of an upper bound on the separation between to matrix pairs. Then
</span><span class="comment">*</span><span class="comment"> the input (A, D), (B, E) are sub-pencils of two matrix pairs in
</span><span class="comment">*</span><span class="comment"> <a name="ZTGSYL.56"></a><a href="ztgsyl.f.html#ZTGSYL.1">ZTGSYL</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TRANS (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N', solve the generalized Sylvester equation (1).
</span><span class="comment">*</span><span class="comment"> = 'T': solve the 'transposed' system (3).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IJOB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Specifies what kind of functionality to be performed.
</span><span class="comment">*</span><span class="comment"> =0: solve (1) only.
</span><span class="comment">*</span><span class="comment"> =1: A contribution from this subsystem to a Frobenius
</span><span class="comment">*</span><span class="comment"> norm-based estimate of the separation between two matrix
</span><span class="comment">*</span><span class="comment"> pairs is computed. (look ahead strategy is used).
</span><span class="comment">*</span><span class="comment"> =2: A contribution from this subsystem to a Frobenius
</span><span class="comment">*</span><span class="comment"> norm-based estimate of the separation between two matrix
</span><span class="comment">*</span><span class="comment"> pairs is computed. (<a name="DGECON.73"></a><a href="dgecon.f.html#DGECON.1">DGECON</a> on sub-systems is used.)
</span><span class="comment">*</span><span class="comment"> Not referenced if TRANS = 'T'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> M (input) INTEGER
</span><span class="comment">*</span><span class="comment"> On entry, M specifies the order of A and D, and the row
</span><span class="comment">*</span><span class="comment"> dimension of C, F, R and L.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> On entry, N specifies the order of B and E, and the column
</span><span class="comment">*</span><span class="comment"> dimension of C, F, R and L.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input) COMPLEX*16 array, dimension (LDA, M)
</span><span class="comment">*</span><span class="comment"> On entry, A contains an upper triangular matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix A. LDA >= max(1, M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input) COMPLEX*16 array, dimension (LDB, N)
</span><span class="comment">*</span><span class="comment"> On entry, B contains an upper triangular matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix B. LDB >= max(1, N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> C (input/output) COMPLEX*16 array, dimension (LDC, N)
</span><span class="comment">*</span><span class="comment"> On entry, C contains the right-hand-side of the first matrix
</span><span class="comment">*</span><span class="comment"> equation in (1).
</span><span class="comment">*</span><span class="comment"> On exit, if IJOB = 0, C has been overwritten by the solution
</span><span class="comment">*</span><span class="comment"> R.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDC (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix C. LDC >= max(1, M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (input) COMPLEX*16 array, dimension (LDD, M)
</span><span class="comment">*</span><span class="comment"> On entry, D contains an upper triangular matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDD (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix D. LDD >= max(1, M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> E (input) COMPLEX*16 array, dimension (LDE, N)
</span><span class="comment">*</span><span class="comment"> On entry, E contains an upper triangular matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDE (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix E. LDE >= max(1, N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> F (input/output) COMPLEX*16 array, dimension (LDF, N)
</span><span class="comment">*</span><span class="comment"> On entry, F contains the right-hand-side of the second matrix
</span><span class="comment">*</span><span class="comment"> equation in (1).
</span><span class="comment">*</span><span class="comment"> On exit, if IJOB = 0, F has been overwritten by the solution
</span><span class="comment">*</span><span class="comment"> L.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDF (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix F. LDF >= max(1, M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SCALE (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
</span><span class="comment">*</span><span class="comment"> R and L (C and F on entry) will hold the solutions to a
</span><span class="comment">*</span><span class="comment"> slightly perturbed system but the input matrices A, B, D and
</span><span class="comment">*</span><span class="comment"> E have not been changed. If SCALE = 0, R and L will hold the
</span><span class="comment">*</span><span class="comment"> solutions to the homogeneous system with C = F = 0.
</span><span class="comment">*</span><span class="comment"> Normally, SCALE = 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RDSUM (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> On entry, the sum of squares of computed contributions to
</span><span class="comment">*</span><span class="comment"> the Dif-estimate under computation by <a name="ZTGSYL.136"></a><a href="ztgsyl.f.html#ZTGSYL.1">ZTGSYL</a>, where the
</span><span class="comment">*</span><span class="comment"> scaling factor RDSCAL (see below) has been factored out.
</span><span class="comment">*</span><span class="comment"> On exit, the corresponding sum of squares updated with the
</span><span class="comment">*</span><span class="comment"> contributions from the current sub-system.
</span><span class="comment">*</span><span class="comment"> If TRANS = 'T' RDSUM is not touched.
</span><span class="comment">*</span><span class="comment"> NOTE: RDSUM only makes sense when <a name="ZTGSY2.141"></a><a href="ztgsy2.f.html#ZTGSY2.1">ZTGSY2</a> is called by
</span><span class="comment">*</span><span class="comment"> <a name="ZTGSYL.142"></a><a href="ztgsyl.f.html#ZTGSYL.1">ZTGSYL</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RDSCAL (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> On entry, scaling factor used to prevent overflow in RDSUM.
</span><span class="comment">*</span><span class="comment"> On exit, RDSCAL is updated w.r.t. the current contributions
</span><span class="comment">*</span><span class="comment"> in RDSUM.
</span><span class="comment">*</span><span class="comment"> If TRANS = 'T', RDSCAL is not touched.
</span><span class="comment">*</span><span class="comment"> NOTE: RDSCAL only makes sense when <a name="ZTGSY2.149"></a><a href="ztgsy2.f.html#ZTGSY2.1">ZTGSY2</a> is called by
</span><span class="comment">*</span><span class="comment"> <a name="ZTGSYL.150"></a><a href="ztgsyl.f.html#ZTGSYL.1">ZTGSYL</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> On exit, if INFO is set to
</span><span class="comment">*</span><span class="comment"> =0: Successful exit
</span><span class="comment">*</span><span class="comment"> <0: If INFO = -i, input argument number i is illegal.
</span><span class="comment">*</span><span class="comment"> >0: The matrix pairs (A, D) and (B, E) have common or very
</span><span class="comment">*</span><span class="comment"> close eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Based on contributions by
</span><span class="comment">*</span><span class="comment"> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
</span><span class="comment">*</span><span class="comment"> Umea University, S-901 87 Umea, Sweden.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION ZERO, ONE
INTEGER LDZ
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, LDZ = 2 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> LOGICAL NOTRAN
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