SRC/ztgsyl.f(3)

📦
version 3.12.1 · 4 versions
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🦋 SRC/ztgsyl.f

okay bestie, gather round, because this is a LAPACK library function page for SRC/ztgsyl.f and it's genuinely iconic 💅 it's giving "level-3 BLAS algorithm that solves a generalized Sylvester equation" and honestly? slay.

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ztgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e,
    lde, f, ldf, scale, dif, work, lwork, iwork, info)

ZTGSYL 🎀

💖 Function/Subroutine Documentation 💖

subroutine ztgsyl (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 dif, complex*16, dimension( * )
    work, integer lwork, integer, dimension( * ) iwork, integer info)

ZTGSYL — yes that's the name twice, it's just her main character energy ✨

🌸 Purpose 🌸

okay so here's the whole vibe, bestie, quoted with love:

ZTGSYL solves the generalized Sylvester equation:
            A * R - L * B = scale * C            (1)
            D * R - L * E = scale * F
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 complex entries. 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 (1) is equivalent 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) ],
Here Ix is the identity matrix of size x and X**H is the conjugate 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 (TRANS = 'C') is used to compute an one-norm-based estimate of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and (B,E), using ZLACON.
If IJOB >= 1, ZTGSYL computes a Frobenius norm-based estimate of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the reciprocal of the smallest singular value of Z.
This is a level-3 BLAS algorithm. 🔥

👑 Parameters 👑

  • TRANS — TRANS is CHARACTER*1 💫
    • = 'N': solve the generalized sylvester equation (1).
    • = 'C': solve the system (3).
  • IJOB — IJOB is INTEGER, and she specifies what kind of functionality to be performed, no cap:
    • =0: solve (1) only.
    • =1: The functionality of 0 and 3.
    • =2: The functionality of 0 and 4.
    • =3: Only an estimate of Dif[(A,D), (B,E)] is computed. (look ahead strategy is used).
    • =4: Only an estimate of Dif[(A,D), (B,E)] is computed. (ZGECON on sub-systems is used).
    • Not referenced if TRANS = 'C'.
  • M — M is INTEGER. The order of the matrices A and D, and the row dimension of the matrices C, F, R and L. 🍓
  • N — N is INTEGER. The order of the matrices B and E, and the column dimension of the matrices C, F, R and L. 🍓
  • A — A is COMPLEX*16 array, dimension (LDA, M). The upper triangular matrix A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1, M).
  • B — B is COMPLEX*16 array, dimension (LDB, N). The upper triangular matrix B.
  • LDB — LDB is INTEGER. The leading dimension of the array 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) or (3). On exit, if IJOB = 0, 1 or 2, C has been overwritten by the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R, the solution achieved during the computation of the Dif-estimate. 💀 (she really does overwrite, pay attention)
  • LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1, M).
  • D — D is COMPLEX*16 array, dimension (LDD, M). The upper triangular matrix D.
  • LDD — LDD is INTEGER. The leading dimension of the array D. LDD >= max(1, M).
  • E — E is COMPLEX*16 array, dimension (LDE, N). The upper triangular matrix E.
  • LDE — LDE is INTEGER. The leading dimension of the array 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) or (3). On exit, if IJOB = 0, 1 or 2, F has been overwritten by the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L, the solution achieved during the computation of the Dif-estimate. 🙈
  • LDF — LDF is INTEGER. The leading dimension of the array F. LDF >= max(1, M).
  • DIF — DIF is DOUBLE PRECISION. On exit DIF is the reciprocal of a lower bound of the reciprocal of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2). IF IJOB = 0 or TRANS = 'C', DIF is not referenced.
  • SCALE — SCALE is DOUBLE PRECISION. On exit SCALE is the scaling factor in (1) or (3). If 0 < SCALE < 1, C and F hold the solutions R and L, resp., 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.
  • WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ✨
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK > = 1. If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N). Also: if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🫶
  • IWORK — IWORK is INTEGER array, dimension (M+N+2).
  • INFO — INFO is INTEGER:
    • =0: successful exit 🎉
    • <0: If INFO = -i, the i-th argument had an illegal value.
    • >0: (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. 🧚 (legends, fr fr)

⭐ References: ⭐

  1. B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
  2. B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.
  3. B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

oh and one last thing bestie — Definition at line 292 of file ztgsyl.f. 📍

mwah 💋 that's the whole page! it's LAPACK, Version 3.12.0, page SRC/ztgsyl.f(3), and this rewrite was generated automatically by Doxygen for LAPACK from the source code. go solve those generalized Sylvester equations, you absolute icon 👑✨

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