# dblgglyap - Man Page

## Name

dblgglyap — Double Precision

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

## Synopsis

### Functions

subroutine dla_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 dla_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 dla_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

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

## Function Documentation

### subroutine dla_gglyap (character, dimension(1) fact, character, dimension(1) trans, integer m, 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, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Generalized Lyapunov Equations.

Purpose:

 DLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.  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 (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:  DGGES failed
= 2:  DLA_SORT_GEV failed
= 3:  Inner solver failed
< 0:  if INFO = -i, the i-th argument had an illegal value

DLA_TGLYAP_L3

DLA_TGLYAP_L3_2S

DLA_TGLYAP_DAG

DLA_TGLYAP_L2

DLA_TGLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 196 of file dla_gglyap.f90.

### subroutine dla_gglyap_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_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 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).

B

          B is DOUBLE PRECISION 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 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, 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_TGLYAP_L3

DLA_TGLYAP_L2

DLA_TGLYAP_L3_2S

DLA_TGLYAP_DAG

DLA_TGLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 244 of file dla_gglyap_refine.f90.

### subroutine dla_ggstein (character, dimension(1) fact, character, dimension(1) trans, integer m, 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, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Generalized Stein 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. 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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.  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 (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:  DGGES failed
= 2:  DLA_SORT_GEV failed
= 3:  Inner solver failed
< 0:  if INFO = -i, the i-th argument had an illegal value

DLA_TGSTEIN_L3

DLA_TGSTEIN_L3_2S

DLA_TGSTEIN_DAG

DLA_TGSTEIN_L2

DLA_TGSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 197 of file dla_ggstein.f90.

## Author

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## Info

Tue Mar 7 2023 Version 1.0.3 MEPACK