# dblgelyap - Man Page

## Name

dblgelyap — Double Precision

— Double precision solvers for standard Lyapunov and Stein equations with general coefficient matrices.

## Synopsis

### Functions

subroutine dla_gelyap (fact, trans, m, a, lda, q, ldq, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Standard Lyapunov Equations.
subroutine dla_gestein (fact, trans, m, a, lda, q, ldq, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Standard Stein Equations.
subroutine dla_gelyap_refine (trans, guess, m, a, lda, x, ldx, y, ldy, as, ldas, q, ldq, maxit, tau, convlog, work, ldwork, info)
Iterative Refinement for the Standard Lyapunov Equation.
subroutine dla_gestein_refine (trans, guess, m, a, lda, x, ldx, y, ldy, as, ldas, q, ldq, maxit, tau, convlog, work, ldwork, info)
Iterative Refinement for the Standard Stein Equations.

## Detailed Description

Double precision solvers for standard Lyapunov and Stein equations with general coefficient matrices.

## Function Documentation

### subroutine dla_gelyap (character, dimension(1) fact, character, dimension(1) trans, integer m, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldq, *) q, integer ldq, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Standard Lyapunov Equations.

Purpose:

 DLA_GELYAP solves a Lyapunov equation of the following forms

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

or

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

where A is a M-by-M general matrix or a matrix in upper Hessenberg form.
The right hand side Y and the solution X are M-by-M matrices.
The general matrix A can be supplied factorized in terms of its
Schur decomposition.
Parameters

FACT

          FACT is CHARACTER
Specifies how the matrix A is given.
== 'N':  The matrix A is given as a general matrix and its Schur decomposition
A = Q*S*Q**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 A is given in upper Hessenberg form and its Schur decomposition
A = Q*S*Q**T will be computed

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 matrix A.  M >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with its
schur decomposition S.
If FACT == 'H', the matrix A is an upper Hessenberg matrix and it is overwritten
with its schur decomposition S.
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S being the
Schur decomposition of A.

LDA

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

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,M)
If FACT == 'N', the matrix Q is an empty M-by-M matrix on input and contains the
Schur vectors of A on output.
If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the
Schur vectors of A on output.
If FACT == 'F', the matrix Q contains the Schur vectors of A.

LDQ

          LDQ is INTEGER
The leading dimension of the array Q.  LDQ >= 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)
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.  LDB >= 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 (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 again
without performing any computations.

INFO

          INFO is INTEGER
== 0:  successful exit
= 1:  DHGEES failed
= 2:  DLA_SORT_EV failed
= 3:  Internal solver failed
< 0:  if INFO = -i, the i-th argument had an illegal value

DLA_TRLYAP_L3

DLA_TRLYAP_L3_2S

DLA_TRLYAP_DAG

DLA_TRLYAP_L2

DLA_TRLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 175 of file dla_gelyap.f90.

### subroutine dla_gelyap_refine (character, dimension(1) trans, character, dimension(1) guess, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension( ldx, *) x, integer ldx, double precision, dimension(ldy, *) y, integer ldy, double precision, dimension(ldas, *) as, integer ldas, double precision, dimension(ldq, *) q, integer ldq, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Standard Lyapunov Equation.

Purpose:

 DLA_GELYAP_REFINE solves a standard Lyapunov equation of the following forms

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

or

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

where A is a M-by-M matrix using iterative refinement.
The right hand side Y and the solution X are M-by-M matrices.
The matrix A needs to be provided as the original data
as well as in 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 matrix A.  M >= 0.

A

          A is DOUBLE PRECISION 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).

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 Schur decomposition of A.

LDAS

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

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,M)
The array Q contains the Schur vectors for A as returned by DGEES.

LDQ

          LDQ is INTEGER
The leading dimension of the array Q.  LDQ >= 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 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 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.

DLA_TRLYAP_L3

DLA_TRLYAP_L2

DLA_TRLYAP_L3_2S

DLA_TRLYAP_DAG

DLA_TRLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 201 of file dla_gelyap_refine.f90.

### subroutine dla_gestein (character, dimension(1) fact, character, dimension(1) trans, integer m, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldq, *) q, integer ldq, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Standard Stein Equations.

Purpose:

 DLA_GESTEIN solves a standard Stein equation of the following forms

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

or

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

where A is a M-by-M general matrix or a matrix in upper Hessenberg form.
The right hand side Y and the solution X are M-by-M matrices.
The general matrix A can be supplied factorized in terms of its
Schur decomposition.
Parameters

FACT

          FACT is CHARACTER
Specifies how the matrix A is given.
== 'N':  The matrix A is given as a general matrix and its Schur decomposition
A = Q*S*Q**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 A is given in upper Hessenberg form and its Schur decomposition
A = Q*S*Q**T will be computed

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 and C.  M >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix A is a general matrix and it is overwritten with its
schur decomposition S.
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix S being the
Schur decomposition of A.
If FACT == 'H', the matrix A is an upper Hessenberg matrix and it is overwritten
with its schur decomposition S.

LDA

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

Q

          Q is DOUBLE PRECISION array, dimension (LDA,M)
If FACT == 'N', the matrix Q is an empty M-by-M matrix on input and contains the
Schur vectors of A on output.
If FACT == 'F', the matrix Q contains the Schur vectors of A.
If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the
Schur vectors of A on output.

LDQ

          LDQ is INTEGER
The leading dimension of the array Q.  LDQ >= 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)
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.  LDB >= 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 (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 again
without performing any computations.

INFO

          INFO is INTEGER
== 0:  successful exit
= 1:  DHGEES failed
= 2:  DLA_SORT_EV failed
= 3:  Inner solver failed
< 0:  if INFO = -i, the i-th argument had an illegal value

DLA_TRSTEIN_L3

DLA_TRSTEIN_L3_2S

DLA_TRSTEIN_DAG

DLA_TRSTEIN_L2

DLA_TRSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 175 of file dla_gestein.f90.

### subroutine dla_gestein_refine (character, dimension(1) trans, character, dimension(1) guess, integer m, double precision, dimension(lda, *) a, integer lda, double precision, dimension( ldx, *) x, integer ldx, double precision, dimension(ldy, *) y, integer ldy, double precision, dimension(ldas, *) as, integer ldas, double precision, dimension(ldq, *) q, integer ldq, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the Standard Stein Equations.

Purpose:

 DLA_GESTEIN_REFINE solves a standard Stein equation of the following forms

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

or

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

where A is a M-by-M matrix using iterative refinement.
The right hand side Y and the solution X are M-by-M matrices.
The matrix A needs to be provided as the original data
as well as in 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 eqaution.

LDA

          LDA is INTEGER
The leading dimension of the array A.  LDA >= 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 Schur decomposition of A.

LDAS

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

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,M)
The array Q contains the Schur vectors for A as returned by DGEES.

LDQ

          LDQ is INTEGER
The leading dimension of the array Q.  LDQ >= 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 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 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.

DLA_TRSTEIN_L3

DLA_TRSTEIN_L2

DLA_TRSTEIN_L3_2S

DLA_TRSTEIN_DAG

DLA_TRSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 202 of file dla_gestein_refine.f90.

## Author

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## Info

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