SRC/stgsy2.f(3)

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💅 SRC/stgsy2.f — Library Functions Manual 💅

🌟 NAME 🌟

SRC/stgsy2.f — bestie, that’s the whole name section, no cap 💖

💖 SYNOPSIS 💖

🫶 Functions/Subroutines 🫶

subroutine stgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e,
    lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info)
  • stgsy2 — STGSY2 solves the generalized Sylvester equation (unblocked algorithm). ✨

🧚 Function/Subroutine Documentation 🧚

🦋 subroutine stgsy2

subroutine stgsy2 (character trans, integer ijob, integer m, integer n,
       real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,
       integer ldb, real, dimension( ldc, * ) c, integer ldc, real, dimension(
       ldd, * ) d, integer ldd, real, dimension( lde, * ) e, integer lde, real,
       dimension( ldf, * ) f, integer ldf, real scale, real rdsum, real rdscal,
       integer, dimension( * ) iwork, integer pq, integer info)

STGSY2 solves the generalized Sylvester equation (unblocked algorithm). 💅

🎯 Purpose 🎯

okay so STGSY2 solves the generalized Sylvester equation:

A * R - L * B = scale * C                (1)
D * R - L * E = scale * F,

It uses Level 1 and 2 BLAS. Here R and L are unknown M-by-N matrices, and (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. The pairs (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). Also, 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow. 🫶

In matrix notation, solving equation (1) corresponds to solving 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 solving 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 SLACON. 🔥

STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL 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 STGSYL. See STGSYL for details. 👑

🎀 Parameters 🎀

  • TRANS — CHARACTER*1. = 'N': solve the generalized Sylvester equation (1). = 'T': solve the 'transposed' system (3). 💫
  • IJOB — 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 — INTEGER. On entry, M specifies the order of A and D, and the row dimension of C, F, R and L. 🌸
  • N — INTEGER. On entry, N specifies the order of B and E, and the column dimension of C, F, R and L. 🌸
  • A — REAL array, dimension (LDA, M). On entry, A contains an upper quasi triangular matrix. 💜
  • LDA — INTEGER. The leading dimension of the matrix A. LDA >= max(1, M). ✨
  • B — REAL array, dimension (LDB, N). On entry, B contains an upper quasi triangular matrix. 💜
  • LDB — INTEGER. The leading dimension of the matrix B. LDB >= max(1, N). ✨
  • C — REAL 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 — INTEGER. The leading dimension of the matrix C. LDC >= max(1, M). ✨
  • D — REAL array, dimension (LDD, M). On entry, D contains an upper triangular matrix. 💜
  • LDD — INTEGER. The leading dimension of the matrix D. LDD >= max(1, M). ✨
  • E — REAL array, dimension (LDE, N). On entry, E contains an upper triangular matrix. 💜
  • LDE — INTEGER. The leading dimension of the matrix E. LDE >= max(1, N). ✨
  • F — REAL 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 — INTEGER. The leading dimension of the matrix F. LDF >= max(1, M). ✨
  • SCALE — 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 — REAL. On entry, the sum of squares of computed contributions to the Dif-estimate under computation by STGSYL, 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 STGSY2 is called by STGSYL. 🍓
  • RDSCAL — 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 STGSY2 is called by STGSYL. 🍓
  • IWORK — INTEGER array, dimension (M+N+2). 🧚
  • PQ — INTEGER. On exit, the number of subsystems (of size 2-by-2, 4-by-4 and 8-by-8) solved by this routine. 👑
  • INFO — 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. 😭

🌸 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 271 of file stgsy2.f. 📌

🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. ✨

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