SRC/stgsyl.f(3)

📦

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✨ SRC/stgsyl.f — from the LAPACK Library Functions Manual ✨

okay so bestie, buckle up, because this one is a whole level-3 BLAS moment 👑

🌸 NAME 🌸

SRC/stgsyl.f — like, the file itself, that's the name, that's the vibe 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

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

STGSYL — say it with me, icon behavior 💜

🎀 Function/Subroutine Documentation 🎀

subroutine stgsyl (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 dif, real,
    dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
    integer info)

STGSYL 🫶

💖 Purpose 💖

okay so here's the tea, straight from the !> block, no cap:

STGSYL 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 real entries. (A, D) and (B, E) must be in generalized (real) Schur canonical form, i.e. A, B are upper quasi triangular and D, E are upper triangular. literally the glow-up requirements ✨

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow. 🚨 (yes, overflow is the enemy, respect the SCALE)

In matrix notation (1) is equivalent to solve Zx = scale b, where Z is defined as

            Z = [ kron(In, A)  -kron(B**T, Im) ]         (2)
                [ kron(In, D)  -kron(E**T, Im) ].

Here 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. (kron is giving main character energy fr fr 🌈)

If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b, 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 (TRANS = 'T') 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 SLACON. 💫

If IJOB >= 1, STGSYL 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. See [1-2] for more information. iykyk 📚

This is a level 3 BLAS algorithm. 🔥 absolute queen

👑 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: 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 IJOB = 1 is used).
- = 4: Only an estimate of Dif[(A,D), (B,E)] is computed. (SGECON on sub-systems is used).
- Not referenced if TRANS = 'T'. 🚩
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 REAL array, dimension (LDA, M). The upper quasi triangular matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1, M).
B — B is REAL array, dimension (LDB, N). The upper quasi triangular matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1, N).
C — C is REAL 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. 💅
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1, M).
D — D is REAL 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 REAL 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 REAL 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 REAL. 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 = 'T', DIF is not touched. 🙈 (hands off, bestie)
SCALE — SCALE is REAL. 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, C and F hold the solutions R and L, respectively, to the homogeneous system with C = F = 0. Normally, SCALE = 1.
WORK — WORK is REAL 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).
- 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. (the -1 trick, classic 👀)
IWORK — IWORK is INTEGER array, dimension (M+N+6).
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 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, honestly

📚 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.

citation game is unmatched, she read all three, she's not kidding 📖✨

⭐ Definition ⭐

Definition at line 296 of file stgsyl.f.

💜 Author (the meta one) 💜

Generated automatically by Doxygen for LAPACK from the source code. (not a human, just vibes 🤖)

and that's the whole page bestie, no notes 💅 slay — this cutie came to you courtesy of LAPACK, Version 3.12.0, and the file formally known as SRC/stgsyl.f(3) 🎀🦋✨

📦 where this page comes from

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