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---
:name: dtgsy2
:md5sum: aac9343f26eb26fda333d93013fb1815
:category: :subroutine
:arguments:
- trans:
:type: char
:intent: input
- ijob:
:type: integer
:intent: input
- m:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- a:
:type: doublereal
:intent: input
:dims:
- lda
- m
- lda:
:type: integer
:intent: input
- b:
:type: doublereal
:intent: input
:dims:
- ldb
- n
- ldb:
:type: integer
:intent: input
- c:
:type: doublereal
:intent: input/output
:dims:
- ldc
- n
- ldc:
:type: integer
:intent: input
- d:
:type: doublereal
:intent: input
:dims:
- ldd
- m
- ldd:
:type: integer
:intent: input
- e:
:type: doublereal
:intent: input
:dims:
- lde
- n
- lde:
:type: integer
:intent: input
- f:
:type: doublereal
:intent: input/output
:dims:
- ldf
- n
- ldf:
:type: integer
:intent: input
- scale:
:type: doublereal
:intent: output
- rdsum:
:type: doublereal
:intent: input/output
- rdscal:
:type: doublereal
:intent: input/output
- iwork:
:type: integer
:intent: workspace
:dims:
- m+n+2
- pq:
:type: integer
:intent: output
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DTGSY2 solves the generalized Sylvester equation:\n\
*\n\
* A * R - L * B = scale * C (1)\n\
* D * R - L * E = scale * F,\n\
*\n\
* using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,\n\
* (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,\n\
* N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)\n\
* must be in generalized Schur canonical form, i.e. A, B are upper\n\
* quasi triangular and D, E are upper triangular. The solution (R, L)\n\
* overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor\n\
* chosen to avoid overflow.\n\
*\n\
* In matrix notation solving equation (1) corresponds to solve\n\
* Z*x = scale*b, where Z is defined as\n\
*\n\
* Z = [ kron(In, A) -kron(B', Im) ] (2)\n\
* [ kron(In, D) -kron(E', Im) ],\n\
*\n\
* Ik is the identity matrix of size k and X' is the transpose of X.\n\
* kron(X, Y) is the Kronecker product between the matrices X and Y.\n\
* In the process of solving (1), we solve a number of such systems\n\
* where Dim(In), Dim(In) = 1 or 2.\n\
*\n\
* If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,\n\
* which is equivalent to solve for R and L in\n\
*\n\
* A' * R + D' * L = scale * C (3)\n\
* R * B' + L * E' = scale * -F\n\
*\n\
* This case is used to compute an estimate of Dif[(A, D), (B, E)] =\n\
* sigma_min(Z) using reverse communicaton with DLACON.\n\
*\n\
* DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL\n\
* of an upper bound on the separation between to matrix pairs. Then\n\
* the input (A, D), (B, E) are sub-pencils of the matrix pair in\n\
* DTGSYL. See DTGSYL for details.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* TRANS (input) CHARACTER*1\n\
* = 'N', solve the generalized Sylvester equation (1).\n\
* = 'T': solve the 'transposed' system (3).\n\
*\n\
* IJOB (input) INTEGER\n\
* Specifies what kind of functionality to be performed.\n\
* = 0: solve (1) only.\n\
* = 1: A contribution from this subsystem to a Frobenius\n\
* norm-based estimate of the separation between two matrix\n\
* pairs is computed. (look ahead strategy is used).\n\
* = 2: A contribution from this subsystem to a Frobenius\n\
* norm-based estimate of the separation between two matrix\n\
* pairs is computed. (DGECON on sub-systems is used.)\n\
* Not referenced if TRANS = 'T'.\n\
*\n\
* M (input) INTEGER\n\
* On entry, M specifies the order of A and D, and the row\n\
* dimension of C, F, R and L.\n\
*\n\
* N (input) INTEGER\n\
* On entry, N specifies the order of B and E, and the column\n\
* dimension of C, F, R and L.\n\
*\n\
* A (input) DOUBLE PRECISION array, dimension (LDA, M)\n\
* On entry, A contains an upper quasi triangular matrix.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the matrix A. LDA >= max(1, M).\n\
*\n\
* B (input) DOUBLE PRECISION array, dimension (LDB, N)\n\
* On entry, B contains an upper quasi triangular matrix.\n\
*\n\
* LDB (input) INTEGER\n\
* The leading dimension of the matrix B. LDB >= max(1, N).\n\
*\n\
* C (input/output) DOUBLE PRECISION array, dimension (LDC, N)\n\
* On entry, C contains the right-hand-side of the first matrix\n\
* equation in (1).\n\
* On exit, if IJOB = 0, C has been overwritten by the\n\
* solution R.\n\
*\n\
* LDC (input) INTEGER\n\
* The leading dimension of the matrix C. LDC >= max(1, M).\n\
*\n\
* D (input) DOUBLE PRECISION array, dimension (LDD, M)\n\
* On entry, D contains an upper triangular matrix.\n\
*\n\
* LDD (input) INTEGER\n\
* The leading dimension of the matrix D. LDD >= max(1, M).\n\
*\n\
* E (input) DOUBLE PRECISION array, dimension (LDE, N)\n\
* On entry, E contains an upper triangular matrix.\n\
*\n\
* LDE (input) INTEGER\n\
* The leading dimension of the matrix E. LDE >= max(1, N).\n\
*\n\
* F (input/output) DOUBLE PRECISION array, dimension (LDF, N)\n\
* On entry, F contains the right-hand-side of the second matrix\n\
* equation in (1).\n\
* On exit, if IJOB = 0, F has been overwritten by the\n\
* solution L.\n\
*\n\
* LDF (input) INTEGER\n\
* The leading dimension of the matrix F. LDF >= max(1, M).\n\
*\n\
* SCALE (output) DOUBLE PRECISION\n\
* On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions\n\
* R and L (C and F on entry) will hold the solutions to a\n\
* slightly perturbed system but the input matrices A, B, D and\n\
* E have not been changed. If SCALE = 0, R and L will hold the\n\
* solutions to the homogeneous system with C = F = 0. Normally,\n\
* SCALE = 1.\n\
*\n\
* RDSUM (input/output) DOUBLE PRECISION\n\
* On entry, the sum of squares of computed contributions to\n\
* the Dif-estimate under computation by DTGSYL, where the\n\
* scaling factor RDSCAL (see below) has been factored out.\n\
* On exit, the corresponding sum of squares updated with the\n\
* contributions from the current sub-system.\n\
* If TRANS = 'T' RDSUM is not touched.\n\
* NOTE: RDSUM only makes sense when DTGSY2 is called by DTGSYL.\n\
*\n\
* RDSCAL (input/output) DOUBLE PRECISION\n\
* On entry, scaling factor used to prevent overflow in RDSUM.\n\
* On exit, RDSCAL is updated w.r.t. the current contributions\n\
* in RDSUM.\n\
* If TRANS = 'T', RDSCAL is not touched.\n\
* NOTE: RDSCAL only makes sense when DTGSY2 is called by\n\
* DTGSYL.\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (M+N+2)\n\
*\n\
* PQ (output) INTEGER\n\
* On exit, the number of subsystems (of size 2-by-2, 4-by-4 and\n\
* 8-by-8) solved by this routine.\n\
*\n\
* INFO (output) INTEGER\n\
* On exit, if INFO is set to\n\
* =0: Successful exit\n\
* <0: If INFO = -i, the i-th argument had an illegal value.\n\
* >0: The matrix pairs (A, D) and (B, E) have common or very\n\
* close eigenvalues.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* Based on contributions by\n\
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,\n\
* Umea University, S-901 87 Umea, Sweden.\n\
*\n\
* =====================================================================\n\
* Replaced various illegal calls to DCOPY by calls to DLASET.\n\
* Sven Hammarling, 27/5/02.\n\
*\n"
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