DTGSY2(3)

💅 DTGSY2(3) ✨

okay so this is from the Library Functions Manual, and it's all about DTGSY2(3) — the unblocked generalized Sylvester equation solver. It's giving serious LAPACK main character energy, so let's get into it bestie.

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🎀 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dtgsy2.f

✨ SYNOPSIS

🫶 Functions/Subroutines

subroutine DTGSY2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e,
    lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info)

DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

💖 Function/Subroutine Documentation

subroutine DTGSY2 (character trans, integer ijob, integer m, integer n,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldb, * ) b, integer ldb, double precision, dimension( ldc, *
    ) c, integer ldc, double precision, dimension( ldd, * ) d, integer ldd,
    double precision, dimension( lde, * ) e, integer lde, double precision,
    dimension( ldf, * ) f, integer ldf, double precision scale, double
    precision rdsum, double precision rdscal, integer, dimension( * ) iwork,
    integer pq, integer info)

DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

🎯 Purpose

okay bestie, here's the whole point: 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 communication 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.

🌸 Parameters

• 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. so watch out for that >0, bestie — common or very close eigenvalues are the vibe killer here.

👑 Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🌸 Contributors

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

📍 Definition

Definition at line 271 of file dtgsy2.f.

🪄 Author

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🫶 final sign-off: this page is from LAPACK Version 3.12.0, footer path /home/abui...RC/dtgsy2.f(3). The page doesn't give a separate date, so I'm not inventing one — periodt. 💅✨