SRC/ctgsy2.f(3)

✨ NAME ✨

SRC/ctgsy2.f — bestie, welcome to the LAPACK Library Functions Manual page for this file 💅

🌸 SYNOPSIS 🌸

🧚 Functions/Subroutines 🧚

• subroutine ctgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info) — CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

🦋 Function/Subroutine Documentation 🦋

subroutine ctgsy2 (character trans, integer ijob, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( ldd, * ) d, integer ldd, complex, dimension( lde, * ) e, integer lde, complex, dimension( ldf, * ) f, integer ldf, real scale, real rdsum, real rdscal, integer info)

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

💖 Purpose 💖

Okay so CTGSY2 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. 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 solving equation (1) corresponds 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) ],

Ik is the identity matrix of size k and X**H is the 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 is used to compute an estimate of

Dif[(A, D), (B, E)] =
= sigma_min(Z)

using reverse communication with CLACON.

CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL of an upper bound on the separation between two matrix pairs. Then the input (A, D), (B, E) are sub-pencils of two matrix pairs in CTGSYL.

🎀 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. (SGECON 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 COMPLEX array, dimension (LDA, M). On entry, A contains an upper triangular matrix.
• LDA — LDA is INTEGER. The leading dimension of the matrix A. LDA >= max(1, M).
• B — B is COMPLEX array, dimension (LDB, N). On entry, B contains an upper triangular matrix.
• LDB — LDB is INTEGER. The leading dimension of the matrix B. LDB >= max(1, N).
• C — C is COMPLEX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 REAL. 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 REAL. On entry, the sum of squares of computed contributions to the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by CTGSYL.
• RDSCAL — RDSCAL is REAL. 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 CTGSY2 is called by CTGSYL.
• INFO — INFO is INTEGER. On exit, if INFO is set to:
  • =0: Successful exit
  • <0: If INFO = -i, input argument number i is illegal.
  • >0: The matrix pairs (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.

🌟 Contributors: 🌟

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

Definition at line 256 of file ctgsy2.f.

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

---

xoxo, LAPACK Version 3.12.0 💖 — source/version footer points at SRC/ctgsy2.f(3). No separate date is shown on the footer, so I’m not making one up 😌🫶