SRC/ztgsy2.f(3)

💖 SRC/ztgsy2.f(3) — Library Functions Manual 💖

✨ NAME ✨

SRC/ztgsy2.f — okay bestie, that’s literally the whole name, and it’s giving “the file is the moment” 💅

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🌈 SYNOPSIS 🌈

Functions/Subroutines

subroutine ztgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e,
    lde, f, ldf, scale, rdsum, rdscal, info)

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm). Slay, math icon behavior 🔥

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🦋 Function/Subroutine Documentation 🦋

subroutine ztgsy2 (character trans, integer ijob, integer m, integer n,
    complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
    ldb, * ) b, integer ldb, complex*16, dimension( ldc, * ) c, integer ldc,
    complex*16, dimension( ldd, * ) d, integer ldd, complex*16, dimension(
    lde, * ) e, integer lde, complex*16, dimension( ldf, * ) f, integer ldf,
    double precision scale, double precision rdsum, double precision rdscal,
    integer info)

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm). Again, because repetition is mother, and this subroutine is the main character 👑

🎀 Purpose 🎀

Okay so here’s the tea: ZTGSY2 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). Ngl, this is giving “I know my BLAS” energy 😭✨

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow. She said “no spillage” 💅

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 conjugate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. Honestly? Iconic notation behavior 🫶

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 ZLACON. Yes bestie, that’s a double equals in the source lore, we don’t fix, we observe 😌

ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are sub-pencils of two matrix pairs in ZTGSYL. The word “to” is right there in the original, I’m not gonna explain her choices 💀

💅 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 COMPLEX*16 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*16 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*16 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*16 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*16 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*16 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). Yep, M, exactly as written — not me editing, bestie 😭
• 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 ZTGSYL, 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 ZTGSY2 is called by ZTGSYL.
• 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 ZTGSY2 is called by ZTGSYL.
• 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 ztgsy2.f.

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📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code.

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💖 That’s the whole page, bestie! Source: SRC/ztgsy2.f(3), LAPACK Version 3.12.0. Mwah 💋✨