SRC/ctgsyl.f(3)

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🌸 SRC/ctgsyl.f(3) — Library Functions Manual 🌸

okay bestie, this is the manual page for SRC/ctgsyl.f(3) from the Library Functions Manual, and it is lowkey iconic: one subroutine, a whole generalized Sylvester equation situation, and level-3 BLAS energy. 💅✨

✨ NAME ✨

SRC/ctgsyl.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

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

🎀 Function/Subroutine Documentation 🎀

🌟 subroutine ctgsyl 🌟

subroutine ctgsyl (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 dif, complex, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer info)
    CTGSYL

🫶 Purpose

okay so CTGSYL 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 CLACON.

If IJOB >= 1, CTGSYL 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. periodt.

🌸 Parameters

💜 TRANS

TRANS is CHARACTER*1

  • = 'N': solve the generalized sylvester equation (1).
  • = 'C': solve the 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 is used).
  • =4: Only an estimate of Dif[(A,D), (B,E)] is computed. (CGECON 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 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 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 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 COMPLEX 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 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 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 = 'C', DIF is not referenced.

💜 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, R and L will hold the solutions to the homogeneous system with C = F = 0.

💜 WORK

WORK is COMPLEX 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.

💜 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.

📚 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.

Definition at line 292 of file ctgsyl.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

okay bestie, that’s the whole page. the footer wraps it with LAPACK, Version 3.12.0, and SRC/ctgsyl.f(3) — that’s the source/version sign-off, and there’s no date listed. now go solve that generalized Sylvester equation and slay. 💖✨

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