tgsyl(3)

📦

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🌸 NAME 🌸

tgsyl — literally tgsyl: Sylvester equation. That's the whole name, bestie. The manual calls it tgsyl(3), and it's your go-to for the generalized Sylvester equation. 💖

✨ SYNOPSIS ✨

🫶 Functions 🫶

Okay so there are four precision flavors, and yes they all ate:

CTGSYL — the complex one, exact signature:

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

DTGSYL — the double precision one:

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

STGSYL — the single precision one:

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

ZTGSYL — the double complex one:

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

🦋 Detailed Description 🦋

The original page goes straight from here into the function docs — no extra lore, just pure linear algebra business. 💅

💖 Function Documentation 💖

🌟 `ctgsyl` — complex precision, zero chill 🌟

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)

✨ Purpose ✨

Okay so CTGSYL solves the generalized Sylvester equation, and it's giving major linear algebra main character energy:

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.

🎀 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.

🌈 `dtgsyl` — double precision era 🌈

DTGSYL

subroutine dtgsyl (character trans, integer ijob, integer m, integer n,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldb, * ) b, integer ldb, double precision, dimension( ldc, *
    ) c, integer ldc, double precision, dimension( ldd, * ) d, integer ldd,
    double precision, dimension( lde, * ) e, integer lde, double precision,
    dimension( ldf, * ) f, integer ldf, double precision scale, double
    precision dif, double precision, dimension( * ) work, integer lwork,
    integer, dimension( * ) iwork, integer info)

🎀 Purpose 🎀

DTGSYL solves the generalized Sylvester equation, no cap:

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

where 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. (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. 💅

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow. Iconic behavior honestly.

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. Yes bestie, that -kron is doing main character math energy. 🦋

If TRANS = 'T', DTGSYL 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 DLACON.

If IJOB >= 1, DTGSYL 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.

This is a level 3 BLAS algorithm. Slay. 🔥

🎀 Parameters 🎀

💜 TRANS 💜

TRANS is CHARACTER*1
= 'N': solve the generalized Sylvester equation (1).
= 'T': solve the 'transposed' system (3).

= '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. (DGECON 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 = 'T', DIF is not touched.

🎀 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, C and F hold the solutions R and L, respectively, to the homogeneous system with C = F = 0. Normally, SCALE = 1.

💜 WORK 💜

WORK is DOUBLE PRECISION 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+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.

Watch the sign, bestie: negative INFO means an argument was illegal, and positive INFO means (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.

📚 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 296 of file dtgsyl.f.

   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 bestie, 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.

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**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.

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.

This is a level 3 BLAS algorithm. 💅

✨ 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: 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 — INTEGER — The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
N — INTEGER — The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
A — REAL array, dimension (LDA, M) — The upper quasi triangular matrix A.
LDA — INTEGER — The leading dimension of the array A. LDA >= max(1, M).
B — REAL array, dimension (LDB, N) — The upper quasi triangular matrix B.
LDB — INTEGER — The leading dimension of the array 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) 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 — INTEGER — The leading dimension of the array C. LDC >= max(1, M).
D — REAL array, dimension (LDD, M) — The upper triangular matrix D.
LDD — INTEGER — The leading dimension of the array D. LDD >= max(1, M).
E — REAL array, dimension (LDE, N) — The upper triangular matrix E.
LDE — INTEGER — The leading dimension of the array 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) 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 — INTEGER — The leading dimension of the array F. LDF >= max(1, M).
DIF — 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.
SCALE — 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 — REAL array, dimension (MAX(1,LWORK)) — On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — 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 — INTEGER array, dimension (M+N+6).
INFO — 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.

🦋 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 296 of file stgsyl.f.

💫 `ztgsyl` 💫

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

✨ Purpose ✨

okay so 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 — the switch that tells us which equation we’re solving:
- = 'N': solve the generalized sylvester equation (1).
- = 'C': solve the system (3).
IJOB is INTEGER — specifies what kind of functionality is 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. (ZGECON on sub-systems is used).

Not referenced if TRANS = 'C'.

M is INTEGER — the order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
N is INTEGER — the order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
A is COMPLEX*16 array, dimension (LDA, M) — the upper triangular matrix A.
LDA is INTEGER — the leading dimension of the array A. LDA >= max(1, M).
B is COMPLEX*16 array, dimension (LDB, N) — the upper triangular matrix B.
LDB is INTEGER — the leading dimension of the array B. LDB >= max(1, N).
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.
LDC is INTEGER — the leading dimension of the array C. LDC >= max(1, M).
D is COMPLEX*16 array, dimension (LDD, M) — the upper triangular matrix D.
LDD is INTEGER — the leading dimension of the array D. LDD >= max(1, M).
E is COMPLEX*16 array, dimension (LDE, N) — the upper triangular matrix E.
LDE is INTEGER — the leading dimension of the array E. LDE >= max(1, N).
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 is INTEGER — the leading dimension of the array F. LDF >= max(1, M).
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 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, respectively, 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 is COMPLEX*16 array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, WORK(1) returns the optimal 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).

IWORK is INTEGER array, dimension (M+N+2).
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

Definition at line 292 of file ztgsyl.f.

👑 Author (the auto-generated one)

Generated automatically by Doxygen for LAPACK from the source code.

okay that’s the whole page, bestie — shoutout to LAPACK Version 3.12.0 and the tgsyl(3) footer label, and we are so out 💅✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

tgsyl(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

🫶 Functions 🫶

🦋 Detailed Description 🦋

💖 Function Documentation 💖

🌟 ctgsyl — complex precision, zero chill 🌟

✨ Purpose ✨

🎀 Parameters 🎀

👑 Author 👑

🌸 Contributors 🌸

📚 References 📚

🌈 dtgsyl — double precision era 🌈

🎀 Purpose 🎀

🎀 Parameters 🎀

💜 TRANS 💜

💖 IJOB 💖

🌸 M 🌸

🌼 N 🌼

🦋 A 🦋

🎀 LDA 🎀

💜 B 💜

💖 LDB 💖

🌸 C 🌸

🌼 LDC 🌼

🦋 D 🦋

🎀 LDD 🎀

💜 E 💜

💖 LDE 💖

🌸 F 🌸

🌼 LDF 🌼

🦋 DIF 🦋

🎀 SCALE 🎀

💜 WORK 💜

💖 LWORK 💖

🌸 IWORK 🌸

🌼 INFO 🌼

🌈 Author 🌈

🫶 Contributors: 🫶

📚 References: 📚

✨ Parameters ✨

🌸 Author 🌸

🎀 Contributors 🎀

🦋 References 🦋

💫 ztgsyl 💫

✨ Purpose ✨

🎀 Parameters 🎀

👑 Author

🌟 Contributors:

📚 References:

🧾 Definition at line

👑 Author (the auto-generated one)

🌟 `ctgsyl` — complex precision, zero chill 🌟

🌈 `dtgsyl` — double precision era 🌈

💫 `ztgsyl` 💫