sglgglyap(3)

📦

version 1.1.2 · 3 versions

1.1.2 · fedora 43 (latest)
1.1.1 · fedora 43
1.1.2 · fedora 44

sglgglyap(3) 💅

MEPACK · section 3 · single precision, but make it fashion ✨

okay so bestie, buckle up 🫶 this page is all about the single precision frontends that solve generalized Lyapunov and Stein equations when your coefficient matrices are just... general. no cute structure to exploit, no shortcuts, just vibes, Schur decompositions and a lot of workspace. and ngl, this page is long, so we're doing it in parts. part 1 starts now 💖

NAME 🎀

<pink>sglgglyap</pink> — Single Precision

Single precision solvers for generalized Lyapunov and Stein equations with general coefficient matrices.

SYNOPSIS 🦋

Functions

sla_gglyap — frontend for the solution of Generalized Lyapunov Equations 👑

subroutine sla_gglyap (fact, trans, m, a, lda, b, ldb, q, ldq, z, ldz,
    x, ldx, scale, work, ldwork, info)

sla_ggstein — frontend for the solution of Generalized Stein Equations ✨

subroutine sla_ggstein (fact, trans, m, a, lda, b, ldb, q, ldq, z, ldz,
    x, ldx, scale, work, ldwork, info)

sla_gglyap_refine — Iterative Refinement for the Generalized Lyapunov Equations 💅

subroutine sla_gglyap_refine (trans, guess, m, a, lda, b, ldb, x, ldx,
    y, ldy, as, ldas, bs, ldbs, q, ldq, z, ldz, maxit, tau, convlog,
    work, ldwork, info)

Detailed Description 💜

Single precision solvers for generalized Lyapunov and Stein equations with general coefficient matrices.

Function Documentation 🔥

`subroutine sla_gglyap`

subroutine sla_gglyap (character, dimension(1) fact, character, dimension(1)
    trans, integer m, real, dimension(lda,*) a, integer lda, real,
    dimension(ldb,*) b, integer ldb, real, dimension(ldq, *) q, integer ldq,
    real, dimension(ldz,*) z, integer ldz, real, dimension(ldx, *) x,
    integer ldx, real scale, real, dimension(*) work, integer ldwork,
    integer info)

Frontend for the solution of Generalized Lyapunov Equations.

Purpose ✨

SLA_GGLYAP solves a generalized Lyapunov equation of the following forms:

A * X * B^T + B * X * A^T = SCALE * Y                                              (1)

A^T * X * B + B^T * X * A =  SCALE * Y                                             (2)

where (A,B) is a M-by-M matrix pencil. The right hand side Y and the solution X are M-by-M matrices. The matrix pencil (A,B) is either in general form, in generalized Hessenberg form, or in generalized Schur form where Q and Z also need to be provided. 🫶

Parameters

FACT — it's a CHARACTER, and it decides how the matrix A shows up.
- == 'N': the matrix pencil (A,B) is given as a general matrix pencil, and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**T will be computed.
- == 'F': the matrix A is given as its Schur decomposition in terms of S and Q form A = Q*S*Q**T
- == 'H': the matrix pencil (A,B) is given in generalized Hessenberg form and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**T will be computed.
TRANS — also a CHARACTER, and it picks the shape of the system of equations with respect to A:
- == 'N': Equation (1) is solved.
- == 'T': Equation (2) is solved.
M — INTEGER. the order of the matrices A, B, Y and X. M >= 0.
A — REAL array, dimension (LDA,M).
- If FACT == , the matrix A is a general matrix and it is overwritten with the quasi upper triangular matrix S of the generalized schur decomposition.
- If FACT == , the matrix A contains its (quasi-) upper triangular matrix S of the generalized Schur decomposition of (A,B).
- If FACT == , the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,B) and it is overwritten with the quasi upper triangular matrix S of the generalized Schur decomposition.
LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,M).
B — REAL array, dimension (LDB,M).
- If FACT == , the matrix B a general matrix and it is overwritten with the upper triangular matrix of the generalized Schur decomposition.
- If FACT == , the matrix B contains its upper triangular matrix R of the generalized schur Schur decomposition of (A,B).
- If FACT == , the matrix B is the upper triangular matrix of the generalized Hessenberg form (A,B) and it is overwritten with the upper triangular matrix of the generalized Schur decomposition.
LDB — INTEGER. the leading dimension of the array B. LDB >= max(1,M).
Q — REAL array, dimension (LDQ,M).
- If FACT == , the matrix Q is an empty M-by-M matrix on input and contains the left Schur vectors of (A,B) on output.
- If FACT == , the matrix Q contains the left Schur vectors of (A,B).
- If FACT == , the matrix Q is an empty M-by-M matrix on input and contains the left Schur vectors of (A,B) on output.
LDQ — INTEGER. the leading dimension of the array Q. LDQ >= max(1,M).
Z — REAL array, dimension (LDZ,M).
- If FACT == , the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output.
- If FACT == , the matrix Z contains the right Schur vectors of (A,B).
- If FACT == , the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output.
LDZ — INTEGER. the leading dimension of the array Z. LDZ >= max(1,M).
X — REAL array, dimension (LDX,N).
- On input, the matrix X contains the right hand side Y.
- On output, the matrix X contains the solution of Equation (1) or (2)
- Right hand side Y and the solution X are symmetric M-by-M matrices.
LDX — INTEGER. the leading dimension of the array X. LDX >= max(1,M).
SCALE — REAL. SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true.
WORK — REAL array, dimension (MAX(1,LDWORK)). Workspace for the algorithm. The optmimal workspace is given either by \ref mepack_memory_frontend or a previous call to the this routine with LDWORK === -1.
LDWORK — INTEGER. Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend . Alternatively, if LDWORK == -1 on input, the subroutine will return the required size of the workspace in LDWORK without performing any computations.
INFO — INTEGER.
- == 0: successful exit 🎉
- = 1: SGGES failed
- = 2: SLA_SORT_GEV failed
- = 3: Inner solver failed
- < 0: if INFO = -i, the i-th argument had an illegal value

Author

Martin Koehler, MPI Magdeburg

Date

October 2025

Definition

Defined at line 207 of file sla_gglyap.f90.

`subroutine sla_gglyap_refine`

subroutine sla_gglyap_refine (character, dimension(1) trans, character,
    dimension(1) guess, integer m, real, dimension(lda, *) a, integer lda,
    real, dimension(ldb, *) b, integer ldb, real, dimension( ldx, *) x,
    integer ldx, real, dimension(ldy, *) y, integer ldy, real,
    dimension(ldas, *) as, integer ldas, real, dimension(ldbs,*) bs, integer
    ldbs, real, dimension(ldq, *) q, integer ldq, real, dimension(ldz, *) z,
    integer ldz, integer maxit, real tau, real, dimension(*) convlog, real,
    dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Generalized Lyapunov Equations.

Purpose 💫

SLA_GGLYAP_REFINE solves a generalized Lyapunov equation of the following forms:

A * X * B^T + B * X * A^T = SCALE * Y                                              (1)

A^T * X * B + B^T * X * A =  SCALE * Y                                             (2)

where (A,B) is a M-by-M matrix pencil using iterative refinement. The right hand side Y and the solution X are M-by-M matrices. The matrix pencil (A,B) needs to provide as the original data as well as in generalized Schur decomposition since both are required in the iterative refinement process. 🧚

Parameters

TRANS — CHARACTER. Specifies the form of the system of equations with respect to A :
- == 'N': Equation (1) is solved
- == 'T': Equation (2) is solved
GUESS — CHARACTER. Specifies whether X contains an initial guess or nor not.
- = 'I': X contains an initial guess
- = 'N': No initial guess, X is set to zero at the begin of the iteration.
M — INTEGER. The order of the matrices A and B. M >= 0.
A — REAL array, dimension (LDA,M). The array A contains the original matrix A defining the eqaution.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
B — REAL array, dimension (LDB,M). The array B contains the original matrix B defining the eqaution.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,M).
X — REAL array, dimension (LDX,M).
- On input, the array X contains the initial guess, if GUESS = 'I'.
- On output, the array X contains the solution X.
LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,M).
Y
Y — Y is a REAL array, dimension (LDY,M). On input, the array Y contains the right hand side Y. The array stays unchanged during the iteration. 💅
LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,M). ✨
AS — AS is a REAL array, dimension (LDAS,M). The array AS contains the generalized Schur decomposition of the A. 🦋
LDAS — LDAS is INTEGER. The leading dimension of the array AS. LDAS >= max(1,M). 💖
BS — BS is a REAL array, dimension (LDBS,M). The array AS contains the generalized Schur decomposition of the B. 🌸
LDBS — LDBS is INTEGER. The leading dimension of the array BS. LDBS >= max(1,M). 🎀
Q — Q is a REAL array, dimension (LDQ,M). The array Q contains the left generalized Schur vectors for (A,B) as returned by SGGES. 👑
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,M). ⭐
Z — Z is a REAL array, dimension (LDZ,M). The array Z contains the right generalized Schur vectors for (A,B) as returned by SGGES. 🌈
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= max(1,M). 💜
MAXIT — MAXIT is INTEGER. On input, MAXIT contains the maximum number of iteration that are performed, 2 <= MAXIT <= 100. On exit, MAXIT contains the number of iteration steps taken by the algorithm. 🔥
TAU — TAU is REAL. On input, TAU contains the additional security factor for the stopping criterion, typical values are 0.1. On exit, TAU contains the last relative residual when the stopping criterion got valid. ✨
CONVLOG — CONVLOG is a REAL array, dimension (MAXIT). The CONVLOG array contains the convergence history of the iterative refinement. CONVLOG(I) contains the maximum relative residual before it is solved for the I-th time. 🍓
WORK — WORK is a REAL array, dimension (MAX(1,LDWORK)). Workspace for the algorithm. The optmimal workspace is returned in LDWORK, if LDWORK == -1 on input. In this case no computations are performed. 💀
LDWORK — LDWORK is INTEGER. If LDWORK == -1 the subroutine will return the required size of the workspace in LDWORK on exit. No computations are performed and none of the arrays are referenced. 🫶
INFO — INFO is INTEGER.
- == 0: Success
- > 0: Iteration failed in step INFO
- < 0: if INFO = -i, the i-th argument had an illegal value
- = -50: Some of the internal settings like NB,... are incorrect. 😭

Author 👑

Martin Koehler, MPI Magdeburg

Date 📅

October 2025

Definition 📍

Definition at line 244 of file sla_gglyap_refine.f90.

`subroutine sla_ggstein` 🧚

subroutine sla_ggstein (character, dimension(1) fact, character,
    dimension(1) trans, integer m, real, dimension(lda,*) a, integer lda,
    real, dimension(ldb,*) b, integer ldb, real, dimension(ldq, *) q,
    integer ldq, real, dimension(ldz,*) z, integer ldz, real, dimension(ldx,
    *) x, integer ldx, real scale, real, dimension(*) work, integer ldwork,
    integer info)

Frontend for the solution of Generalized Stein Equations. 💖

Purpose 🌟

Okay so SLA_GGSTEIN solves a generalized Stein equation, and it's giving two possible forms, bestie:

A * X * A^T - B * X * B^T = SCALE * Y                                              (1)

A^T * X * A - B^T * X * B =  SCALE * Y                                             (2)

where (A,B) is a M-by-M matrix pencil. The right hand side Y and the solution X are M-by-M matrices. The matrix pencil (A,B) is either in general form, in generalized Hessenberg form, or in generalized Schur form where Q and Z also need to be provided. 💅

Parameters 🎀

FACT — FACT is CHARACTER. Specifies how the matrix A is given.
- == 'N': The matrix pencil (A,B) is given as a general matrix pencil and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**T will be computed.
- == 'F': The matrix A is given as its Schur decomposition in terms of S and Q form A = Q*S*Q**T
- == 'H': The matrix pencil (A,B) is given in generalized Hessenberg form and its Schur decomposition A = Q*S*Z**T, B = Q*R*Z**T will be computed. 🦋
TRANS — TRANS is CHARACTER. Specifies the form of the system of equations with respect to (A,B) :
- == 'N': Equation (1) is solved.
- == 'T': Equation (2) is solved. ✨
M — M is INTEGER. The order of the matrices A, B, Y and X. M >= 0. 💖
A — A is REAL array, dimension (LDA,M). If FACT == , the matrix A is a general matrix and it is overwritten with the quasi upper triangular matrix S of the generalized schur decomposition. If FACT == , the matrix A contains its (quasi-) upper triangular matrix S of the generalized Schur decomposition of (A,B). If FACT == , the matrix A is an upper Hessenberg matrix of the generalized Hessenberg form (A,B) and it is overwritten with the quasi upper triangular matrix S of the generalized Schur decomposition. 🌸
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M). 👑
B — B is REAL array, dimension (LDB,M). If FACT == , the matrix B a general matrix and it is overwritten with the upper triangular matrix of the generalized Schur decomposition. If FACT == , the matrix B contains its upper triangular matrix R of the generalized schur Schur decomposition of (A,B). If FACT == , the matrix B is the upper triangular matrix of the generalized Hessenberg form (A,B) and it is overwritten with the upper triangular matrix of the generalized Schur decomposition. ⭐
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,M). 🌈
Q — Q is REAL array, dimension (LDQ,M). If FACT == , the matrix Q is an empty M-by-M matrix on input and contains the left Schur vectors of (A,B) on output. If FACT == , the matrix Q contains the left Schur vectors of (A,B). If FACT == , the matrix Q is an empty M-by-M matrix on input and contains the left Schur vectors of (A,B) on output. 💜
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,M). 🔥
Z — Z is REAL array, dimension (LDZ,M). If FACT == , the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output. If FACT == , the matrix Z contains the right Schur vectors of (A,B). If FACT == , the matrix Z is an empty M-by-M matrix on input and contains the right Schur vectors of (A,B) on output. 🍓
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= max(1,M). 💀
X — X is REAL array, dimension (LDX,N). On input, the matrix X contains the right hand side Y. On output, the matrix X contains the solution of Equation (1) or (2). Right hand side Y and the solution X are symmetric M-by-M matrices. 🫶
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,M). 😭
SCALE — SCALE is REAL. SCALE is a scaling factor to prevent the overflow in the result. If INFO == 0 then SCALE is 1.0 otherwise if one of the inner systems could not be solved correctly, 0 < SCALE <= 1 holds true. ✨
WORK — WORK is REAL array, dimension (MAX(1,LDWORK)). Workspace for the algorithm. The optmimal workspace is given either by \ref mepack_memory_frontend or a previous call to the this routine with LDWORK === -1. 💅
LDWORK — LDWORK is INTEGER. Size of the workspace for the algorithm. This can be determined by a call \ref mepack_memory_frontend. Alternatively, if LDWORK == -1 on input the subroutine will return the required size of the workspace in LDWORK without performing any computations. 🌟
INFO — INFO is INTEGER.
- == 0: successful exit
- = 1: SGGES failed
- = 2: SLA_SORT_GEV failed
- = 3: Inner solver failed
- < 0: if INFO = -i, the i-th argument had an illegal value. 🧚

Author 👑

Martin Koehler, MPI Magdeburg

Date 📅

October 2025

Definition 📍

Definition at line 208 of file sla_ggstein.f90.

This whole thing was generated automatically by Doxygen for MEPACK from the source code, version 1.1.2, dated Thu Oct 9 2025 00:00:00, and the man page is sglgglyap(3). That's the vibe, bestie, periodt. 💅✨

📦 where this page comes from

fedora 43: mepack-devel 1.1.2-1.fc43 · GPL-3.0-or-later and LGPL-2.1-or-later and BSD-3-Clause
fedora 43: mepack-devel 1.1.2-1.fc43 · GPL-3.0-or-later and LGPL-2.1-or-later and BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

sglgglyap(3)

sglgglyap(3) 💅

NAME 🎀

SYNOPSIS 🦋

Functions

Detailed Description 💜

Function Documentation 🔥

`subroutine sla_gglyap`

Purpose ✨

Parameters

See also

Author

Date

Definition

`subroutine sla_gglyap_refine`

Purpose 💫

Parameters

See also ✨

Author 👑

Date 📅

Definition 📍

`subroutine sla_ggstein` 🧚

Purpose 🌟

Parameters 🎀

See also ✨

Author 👑

Date 📅

Definition 📍

sglgglyap(3)

sglgglyap(3) 💅

NAME 🎀

SYNOPSIS 🦋

Functions

Detailed Description 💜

Function Documentation 🔥

subroutine sla_gglyap

Purpose ✨

Parameters

See also

Author

Date

Definition

subroutine sla_gglyap_refine

Purpose 💫

Parameters

See also ✨

Author 👑

Date 📅

Definition 📍

subroutine sla_ggstein 🧚

Purpose 🌟

Parameters 🎀

See also ✨

Author 👑

Date 📅

Definition 📍

`subroutine sla_gglyap`

`subroutine sla_gglyap_refine`

`subroutine sla_ggstein` 🧚