# sglgglyap - Man Page

## Name

sglgglyap — Single Precision

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

## Synopsis

### Functions

subroutine sla_gglyap (fact, trans, m, a, lda, b, ldb, q, ldq, z, ldz, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Generalized Lyapunov Equations.
subroutine sla_ggstein (fact, trans, m, a, lda, b, ldb, q, ldq, z, ldz, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Generalized Stein 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)
Iterative Refinement for the Generalized Lyapunov Equations.

## Detailed Description

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

## Function Documentation

### 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)

or

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 or in generalized
Schur form where Q and Z also needs 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

TRANS

          TRANS is CHARACTER
Specifies the form of the system of equations with respect to A:
== '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 == 'N', the matrix A is a general matrix and it is overwritten with then
quasi upper triangular matrix S of the generalized schur decomposition.
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the
generalized  Schur decomposition of (A,B).

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 == 'N', the matrix B a general matrix and it is overwritten with the upper triangular
matrix of the generalized Schur decomposition.
If FACT == 'F', the matrix B contains its upper triangular matrix R of the generalized schur
Schur decomposition of (A,B).

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 == 'N', 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 == 'F', the matrix Q contains the left Schur vectors of (A,B).

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 == 'N', 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 == 'F', the matrix Z contains the right Schur vectors of (A,B).

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

SLA_TGLYAP_L3

SLA_TGLYAP_L3_2S

SLA_TGLYAP_DAG

SLA_TGLYAP_L2

SLA_TGLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 196 of file sla_gglyap.f90.

### 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)

or

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

          TRANS is CHARACTER
Specifies the form of the system of equations with respect to A :
== 'N':  Equation (1) is solved
== 'T':  Equation (2) is solved

GUESS

          GUESS is 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

          M is INTEGER
The order of the matrices A and B.  M >= 0.

A

          A is REAL array, dimension (LDA,M)
The array A contains the original matrix A defining the eqaution.

LDA

          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).

B

          B is REAL array, dimension (LDB,M)
The array B contains the original matrix B defining the eqaution.

LDB

          LDB is INTEGER
The leading dimension of the array B.  LDB >= max(1,M).

X

          X is 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

          LDX is INTEGER
The leading dimension of the array X.  LDX >= max(1,M).

Y

          Y is 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 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 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 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 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 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 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.

SLA_TGLYAP_L3

SLA_TGLYAP_L2

SLA_TGLYAP_L3_2S

SLA_TGLYAP_DAG

SLA_TGLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 244 of file sla_gglyap_refine.f90.

### 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:

 SLA_GGSTEIN solves a generalized Stein equation of the following forms

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

or

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
M-by-M matrices.  The matrix pencil (A,B) is either in general form or in generalized
Schur form where Q and Z also needs 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

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 == 'N', the matrix A is a general matrix and it is overwritten with then
quasi upper triangular matrix S of the generalized schur decomposition.
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S of the
generalized  Schur decomposition of (A,B).

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 == 'N', the matrix B a general matrix and it is overwritten with the upper triangular
matrix of the generalized Schur decomposition.
If FACT == 'F', the matrix B contains its upper triangular matrix R of the generalized schur
Schur decomposition of (A,B).

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 == 'N', 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 == 'F', the matrix Q contains the left Schur vectors of (A,B).

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 == 'N', 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 == 'F', the matrix Z contains the right Schur vectors of (A,B).

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

SLA_TGSTEIN_L3

SLA_TGSTEIN_L3_2S

SLA_TGSTEIN_DAG

SLA_TGSTEIN_L2

SLA_TGSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 197 of file sla_ggstein.f90.

## Author

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## Info

Tue Mar 7 2023 Version 1.0.3 MEPACK