tgex2(3)

📦

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🎀 tgex2 — Library Functions Manual 🎀

okay so bestie, welcome to tgex2 — the page that's all about reordering the generalized Schur form. it's giving "linear algebra but make it a glow-up" ✨

💖 NAME 💖

tgex2 — reorder generalized Schur form

🌸 SYNOPSIS 🌸

💫 Functions 💫

subroutine ctgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1,
    info)

CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

subroutine dtgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1,
    n1, n2, work, lwork, info)

DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

subroutine stgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1,
    n1, n2, work, lwork, info)

STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

subroutine ztgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1,
    info)

ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

🌈 Detailed Description 🌈

(the source page keeps this one super quiet — no prose here at all. all the actual tea is down in the Function Documentation section, so scroll with me 💅)

🔥 Function Documentation 🔥

💜 subroutine ctgex2

subroutine ctgex2 (logical wantq, logical wantz, integer n, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,
    integer ldb, complex, dimension( ldq, * ) q, integer ldq, complex,
    dimension( ldz, * ) z, integer ldz, integer j1, integer info)

CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

✨ Purpose

!> !> CTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) !> in an upper triangular matrix pair (A, B) by an unitary equivalence !> transformation. !> !> (A, B) must be in generalized Schur canonical form, that is, A and !> B are both upper triangular. !> !> Optionally, the matrices Q and Z of generalized Schur vectors are !> updated. !> !> Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H !> Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H !> !>

🍓 Parameters

WANTQ — WANTQ is LOGICAL
- .TRUE. : update the left transformation matrix Q;
- .FALSE.: do not update Q.
WANTZ — WANTZ is LOGICAL
- .TRUE. : update the right transformation matrix Z;
- .FALSE.: do not update Z.
N — N is INTEGER. The order of the matrices A and B. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX array, dimension (LDB,N). On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Q — Q is COMPLEX array, dimension (LDQ,N). If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N.
Z — Z is COMPLEX array, dimension (LDZ,N). If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.
J1 — J1 is INTEGER. The index to the first block (A11, B11).
INFO — INFO is INTEGER
- =0: Successful exit.
- =1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill-conditioned.

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🦋 Further Details

In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.

💐 Contributors

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

📚 References

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

Definition at line 188 of file ctgex2.f.

💜 subroutine dtgex2

subroutine dtgex2 (logical wantq, logical wantz, integer n, double
    precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, *
    ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz,
    integer j1, integer n1, integer n2, double precision, dimension( * )
    work, integer lwork, integer info)

DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

✨ Purpose

!> !> DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) !> of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair !> (A, B) by an orthogonal equivalence transformation. !> !> (A, B) must be in generalized real Schur canonical form (as returned !> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 !> diagonal blocks. B is upper triangular. !> !> Optionally, the matrices Q and Z of generalized Schur vectors are !> updated. !> !> Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T !> Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T !> !>

🍓 Parameters

WANTQ — WANTQ is LOGICAL
- .TRUE. : update the left transformation matrix Q;
- .FALSE.: do not update Q.
WANTZ — WANTZ is LOGICAL
- .TRUE. : update the right transformation matrix Z;
- .FALSE.: do not update Z.
N — N is INTEGER. The order of the matrices A and B. N >= 0.
A — A is DOUBLE PRECISION array, dimensions (LDA,N). On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is DOUBLE PRECISION array, dimensions (LDB,N). On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Q — Q is DOUBLE PRECISION array, dimension (LDQ,N). On entry, if WANTQ = .TRUE., the orthogonal matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N.
Z — Z is DOUBLE PRECISION array, dimension (LDZ,N). On entry, if WANTZ =.TRUE., the orthogonal matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.
J1 — J1 is INTEGER. The index to the first block (A11, B11). 1 <= J1 <= N.
N1 — N1 is INTEGER. The order of the first block (A11, B11). N1 = 0, 1 or 2.
N2 — N2 is INTEGER. The order of the second block (A22, B22). N2 = 0, 1 or 2.
WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
INFO — INFO is INTEGER
- =0: Successful exit
- >0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too ill-conditioned.
- <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🦋 Further Details

💐 Contributors

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

📚 References

(and that's exactly where part 1 wraps up — refs are incoming in the next part, bestie 🫶)

okay so the rest of dtgex2's reference list is literally this, and we are not skipping the receipts:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

And the source location for that one is Definition at line 219 of file dtgex2.f..

💜 `stgex2` — the real-valued adjacent-block swapper

   subroutine stgex2 (logical wantq, logical wantz, integer n, real, dimension(
       lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
       dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer
       ldz, integer j1, integer n1, integer n2, real, dimension( * ) work,
       integer lwork, integer info)
       STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular
       matrix pair by an orthogonal equivalence transformation.

so STGEX2 is the real-valued one that swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. iconic, honestly.

🎯 Purpose

STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation.

(A, B) must be in generalized real Schur canonical form (as returned by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are updated.

Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T

🧾 Parameters

WANTQ — WANTQ is LOGICAL. .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
WANTZ — WANTZ is LOGICAL. .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
N — N is INTEGER. The order of the matrices A and B. N >= 0.
A — A is REAL array, dimension (LDA,N). On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is REAL array, dimension (LDB,N). On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Q — Q is REAL array, dimension (LDQ,N). On entry, if WANTQ = .TRUE., the orthogonal matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N.
Z — Z is REAL array, dimension (LDZ,N). On entry, if WANTZ = .TRUE., the orthogonal matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.
J1 — J1 is INTEGER. The index to the first block (A11, B11). 1 <= J1 <= N.
N1 — N1 is INTEGER. The order of the first block (A11, B11). N1 = 0, 1 or 2.
N2 — N2 is INTEGER. The order of the second block (A22, B22). N2 = 0, 1 or 2.
WORK — WORK is REAL array, dimension (MAX(1,LWORK)).
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 ).
INFO — INFO is INTEGER.
- =0: Successful exit.
- >0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too ill-conditioned.
- <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔍 Further Details:

🫶 Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

📚 References:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

Definition at line 219 of file stgex2.f.

🦋 `ztgex2` — the `COMPLEX*16` adjacent-block swapper

   subroutine ztgex2 (logical wantq, logical wantz, integer n, complex*16,
       dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
       integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16,
       dimension( ldz, * ) z, integer ldz, integer j1, integer info)
       ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular
       matrix pair by an unitary equivalence transformation.

ZTGEX2 is the complex*16 version, and it swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation. <sparkle>slay</sparkle>.

🎯 Purpose

ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) in an upper triangular matrix pair (A, B) by an unitary equivalence transformation.

(A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are updated.

Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H

🧾 Parameters

WANTQ — WANTQ is LOGICAL. .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
WANTZ — WANTZ is LOGICAL. .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
N — N is INTEGER. The order of the matrices A and B. N >= 0.
A — A is COMPLEX*16 array, dimensions (LDA,N). On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX*16 array, dimensions (LDB,N). On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Q — Q is COMPLEX*16 array, dimension (LDQ,N). If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N.
Z — Z is COMPLEX*16 array, dimension (LDZ,N). If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.
J1 — J1 is INTEGER. The index to the first block (A11, B11).
INFO — INFO is INTEGER.
- =0: Successful exit.
- =1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill-conditioned.

👑 Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔍 Further Details:

🫶 Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

📚 References

okay so if you want the full lore — like, the actual math-girlie receipts behind all of this — here's where it lives, bestie 👇

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

ngl these are the citations you cite when someone asks "wait, why does reordering eigenvalues in a generalized real Schur form work like that" — like, the vibes are academic but the receipts are real 💅

Also, tiny housekeeping detail so you know exactly where the actual code lives: Definition at line 188 of file ztgex2.f.

✨ Author ✨

Generated automatically by Doxygen for LAPACK from the source code.

(it wasn't a human who sat down and wrote this — it's literally the computer doing its little documentation glow-up 🫶 the source code is the main character here, Doxygen just took the notes.)

and that's the whole page, bestie!! 🎀💖✨

Cute lil sign-off before you close the tab: this gorgeousness comes to you from LAPACK, Version 3.12.0, living on the tgex2(3) page — that's the footer energy right there, version stamp and all. Go forth and reorder those eigenvalues, you absolute icon 👑💜

📦 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

tgex2(3)

🎀 tgex2 — Library Functions Manual 🎀

💖 NAME 💖

🌸 SYNOPSIS 🌸

💫 Functions 💫

🌈 Detailed Description 🌈

🔥 Function Documentation 🔥

💜 subroutine ctgex2

✨ Purpose

🍓 Parameters

👑 Author

🦋 Further Details

💐 Contributors

📚 References

💜 subroutine dtgex2

✨ Purpose

🍓 Parameters

👑 Author

🦋 Further Details

💐 Contributors

📚 References

💜 stgex2 — the real-valued adjacent-block swapper

🎯 Purpose

🧾 Parameters

👑 Author

🔍 Further Details:

🫶 Contributors:

📚 References:

🦋 ztgex2 — the COMPLEX*16 adjacent-block swapper

🎯 Purpose

🧾 Parameters

👑 Author

🔍 Further Details:

🫶 Contributors:

📚 References

✨ Author ✨

💜 `stgex2` — the real-valued adjacent-block swapper

🦋 `ztgex2` — the `COMPLEX*16` adjacent-block swapper