# dbltglyap - Man Page

## Name

dbltglyap — Double Precision

— Double precision solvers for generalized Lyapunov and Stein equations with triangular coefficient matrices.

## Synopsis

### Functions

subroutine dla_tglyap_dag (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation with DAG scheduling.
subroutine dla_tgstein_dag (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-3 Bartels-Stewart Algorithm for the generalized Stein equation with DAG scheduling.
subroutine dla_tglyap_l2 (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation.
subroutine dla_tglyap_l2_unopt (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation (Unoptimized)
subroutine dla_tgstein_l2 (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-2 Bartels-Stewart Algorithm for the generalized Stein equation.
subroutine dla_tglyap_l3 (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation.
subroutine dla_tglyap_l3_2s (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation two stage.
subroutine dla_tgstein_l3 (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-3 Bartels-Stewart Algorithm for the generalized Stein equation.
subroutine dla_tgstein_l3_2s (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Level-3 Bartels-Stewart Algorithm for the generalized Stein equation (Two Stage)
recursive subroutine dla_tglyap_recursive (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Recursive Blocking Algorithm for the generalized Lyapunov equation.
recursive subroutine dla_tgstein_recursive (trans, m, a, lda, b, ldb, x, ldx, scale, work, info)
Recursive Blocking Algorithm for the generalized Stein equation.
subroutine dla_ggstein_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

Double precision solvers for generalized Lyapunov and Stein equations with triangular coefficient matrices.

## Function Documentation

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

Iterative Refinement for the Generalized Lyapunov Equations.

Purpose:

``` DLA_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 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 DOUBLE PRECISION array, dimension (LDA,M)
The array A contains the original matrix A defining the equation.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The array B contains the original matrix B defining the equation.```

LDB

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

X

```          X is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDQ,M)
The array Q contains the left generalized Schur vectors for (A,B) as returned by DGGES.```

LDQ

```          LDQ is INTEGER
The leading dimension of the array Q.  LDQ >= max(1,M).```

Z

```          Z is DOUBLE PRECISION array, dimension (LDZ,M)
The array Z contains the right generalized Schur vectors for (A,B) as returned by DGGES.```

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, MAXIT <= 100
On exit, MAXIT contains the number of iteration steps taken by the algorithm.```

TAU

```          TAU is DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LDWORK))
Workspace for the algorithm. The optimal 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.```

DLA_TGSTEIN_L3

DLA_TGSTEIN_L2

DLA_TGSTEIN_L3_2S

DLA_TGSTEIN_DAG

DLA_TGSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 243 of file dla_ggstein_refine.f90.

### subroutine dla_tglyap_dag (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb,*) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation with DAG scheduling.

Purpose:

``` DLA_TGLYAP_DAG solves a generalized Lyapunov  equation of the following forms

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

or

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

where A is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 3 operations and DAG .

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 154 of file dla_tglyap_dag.f90.

### subroutine dla_tglyap_l2 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation.

Purpose:

``` DLA_TGLYAP_L2 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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 2 operations.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANS.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 155 of file dla_tglyap_l2.f90.

### subroutine dla_tglyap_l2_unopt (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the generalized Lyapunov equation (Unoptimized)

Purpose:

``` DLA_TGLYAP_L2 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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 2 operations but works with inefficient DSYR2 calls in the case TRANS='T'.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANS.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 156 of file dla_tglyap_l2_unopt.f90.

### subroutine dla_tglyap_l3 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation.

Purpose:

``` DLA_TGLYAP_L3 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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 3 operations.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 154 of file dla_tglyap_l3.f90.

### subroutine dla_tglyap_l3_2s (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Lyapunov equation two stage.

Purpose:

``` DLA_TGLYAP_L3_2S 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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 3 operations with two stage blocking (DAG inner solves)

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 153 of file dla_tglyap_l3_2stage.f90.

### recursive subroutine dla_tglyap_recursive (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb,*) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Recursive Blocking Algorithm for the generalized Lyapunov equation.

Purpose:

``` DLA_TGLYAP_RECURSIVE 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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using recursive blocking.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension M*M
Workspace for the algorithm.```

INFO

```          INFO is INTEGER
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 140 of file dla_tglyap_recursive.f90.

### subroutine dla_tgstein_dag (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb,*) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation with DAG scheduling.

Purpose:

``` DLA_TGSTEIN_DAG solves a generalized Lyapunov  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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 3 operations and DAG .

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 152 of file dla_tgstein_dag.f90.

### subroutine dla_tgstein_l2 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-2 Bartels-Stewart Algorithm for the generalized Stein equation.

Purpose:

``` DLA_TGSTEIN_L2 solves a generalized Lyapunov  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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 2 operations.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 154 of file dla_tgstein_l2.f90.

### subroutine dla_tgstein_l3 (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation.

Purpose:

``` DLA_TGSTEIN_L3 solves a generalized Lyapunov  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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 3 operations.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 153 of file dla_tgstein_l3.f90.

### subroutine dla_tgstein_l3_2s (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Level-3 Bartels-Stewart Algorithm for the generalized Stein equation (Two Stage)

Purpose:

``` DLA_TGSTEIN_L3_2S solves a generalized Lyapunov  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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using BLAS level 3 operations with two level blocking.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension LWORK
Workspace for the algorithm.
The workspace needs to queried before the running the computation.
The query is performed by calling the subroutine with INFO == -1 on input.
The required workspace is then returned in INFO.```

INFO

```          INFO is INTEGER

On input:
== -1 : Perform a workspace query
<> -1: normal operation

On exit, workspace query:
< 0 :  if INFO = -i, the i-th argument had an illegal value
>= 0:  The value of INFO is the required number of elements in the workspace.

On exit, normal operation:
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 153 of file dla_tgstein_l3_2stage.f90.

### recursive subroutine dla_tgstein_recursive (character, dimension(1) trans, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer info)

Recursive Blocking Algorithm for the generalized Stein equation.

Purpose:

``` DLA_TGSTEIN_RECURSIVE solves a generalized Lyapunov  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 is a M-by-M quasi upper triangular matrix, B is a M-by-M upper triangular,
and X and Y are symmetric  M-by-M matrices.
Typically the matrix pencil (A,B) is created by DGGES from LAPACK.```
Attention

The algorithm is implemented using recursive blocking.

Parameters

TRANS

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

M

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
The matrix A must be (quasi-) upper triangular.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,M)
The matrix B must be upper triangular.```

LDB

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

X

```          X is DOUBLE PRECISION 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)
as selected by TRANSA, TRANSB, and SGN.
Right hand side Y and the solution X are M-by-N matrices.```

LDX

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

SCALE

```          SCALE is DOUBLE PRECISION
SCALE is a scaling factor to prevent the overflow in the result.
If INFO == 0 then SCALE is 1.0D0 otherwise if one of the inner systems
could not be solved correctly, 0 < SCALE <= 1 holds true.```

WORK

```          WORK is DOUBLE PRECISION array, dimension M*M
Workspace for the algorithm.```

INFO

```          INFO is INTEGER
== 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  The equation is not solved correctly. One of the arising inner
system got singular.```
Author

Martin Koehler, MPI Magdeburg

Date

October 2023

Definition at line 140 of file dla_tgstein_recursive.f90.

## Author

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## Info

Wed Oct 18 2023 00:00:00 Version 1.1.0 MEPACK