# sglgesylv - Man Page

## Name

sglgesylv — Single Precision

— Single Precision routines for standard Sylvester equations.

## Synopsis

### Functions

subroutine sla_gesylv (facta, factb, transa, transb, sgn, m, n, a, lda, b, ldb, qa, ldqa, qb, ldqb, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Standard Sylvester Equations.
subroutine sla_gesylv2 (facta, factb, transa, transb, sgn, m, n, a, lda, b, ldb, qa, ldqa, qb, ldqb, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Standard Sylvester Equations.
subroutine sla_gesylv2_refine (transa, transb, guess, sgn, m, n, a, lda, b, ldb, x, ldx, y, ldy, as, ldas, bs, ldbs, q, ldq, u, ldu, maxit, tau, convlog, work, ldwork, info)
Iterative Refinement for the standard Sylvester Equations.
subroutine sla_gesylv_refine (transa, transb, guess, sgn, m, n, a, lda, b, ldb, x, ldx, y, ldy, as, ldas, bs, ldbs, q, ldq, u, ldu, maxit, tau, convlog, work, ldwork, info)
Iterative Refinement for the standard Sylvester Equations.

## Detailed Description

Single Precision routines for standard Sylvester equations.

This subsection contains the solvers for standard Sylvester equations with general coefficient matrices in single precision arithmetic. The Schur decompositions are computed in single precision with the help of LAPACK.

## Function Documentation

### subroutine sla_gesylv (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda,*) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldqa, *) qa, integer ldqa, real, dimension(ldqb, *) qb, integer ldqb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Standard Sylvester Equations.

Purpose:

 SLA_GESYLV solves a Sylvester equation of the following forms

op1(A) * X  +  X * op2(B) = SCALE * Y                              (1)

or

op1(A) * X  -  X * op2(B) = SCALE * Y                              (2)

where A is a M-by-M matrix and B is a N-by-N matrix. The right hand
side Y and the solution X are M-by-N matrices. The matrices A and B can be
either a general unreduced matrix or a (quasi-) upper triangular factor.
In the later case QA and QB provide the Schur-vectors of the matrices A
and B.
Parameters

FACTA

          FACTA is CHARACTER
Specifies how the matrix A is given.
== 'N':  The matrix A is given as a general matrix and its Schur decomposition
A = QA*S*QA**T will be computed.
== 'F':  The matrix A is given as its Schur decomposition in terms of S and QA
form A = QA*S*QA**T

FACTB

          FACTB is CHARACTER
Specifies how the matrix B is given.
== 'N':  The matrix B is given as a general matrix and its Schur decomposition
B = QB*R*QB**T will be computed.
== 'F':  The matrix A is given as its Schur decomposition in terms of R and QB
form A = QB*R*QB**T

TRANSA

          TRANSA is CHARACTER
Specifies the form of the system of equations with respect to A:
== 'N':  op1(A) = A
== 'T':  op1(A) = A**T

TRANSB

          TRANSB is CHARACTER
Specifies the form of the system of equations with respect to B:
== 'N':  op2(B) = B,
== 'T':  op2(B) = B**T

SGN

          SGN is REAL, allowed values: +/-1
Specifies the sign between the two parts of the Sylvester equation.
= 1 :  Solve Equation (1)
== -1:  Solve Equation (2)

M

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

N

          N is INTEGER
The order of the matrices B and D.  N >= 0.

A

          A is REAL 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.

LDA

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

B

          B is REAL array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with its
schur decomposition R.
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix R being the
Schur decomposition of B.

LDB

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

QA

          QA is REAL array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
Schur vectors of A on output.
If FACT == 'F', the matrix QA contains the Schur vectors of A.

LDQA

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

QB

          QB is REAL array, dimension (LDQB,N)
If FACT == 'N', the matrix QB is an empty N-by-N matrix on input and contains the
Schur vectors of B on output.
If FACT == 'F', the matrix QB contains the Schur vectors of B.

LDQB

          LDQB is INTEGER
The leading dimension of the array QB.  LDQB >= max(1,N).

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 M-by-N 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:  SGEES failed
= 2:  SLA_SORT_EV failed
= 3:  SLA_TRLYAP_DAG failed
< 0:  if INFO = -i, the i-th argument had an illegal value

SLA_TRSYLV_L3

SLA_TRSYLV_L3_2S

SLA_TRSYLV_DAG

SLA_TRSYLV_L2_UNOPT

SLA_TRSYLV_L2

SLA_TRSYLV_L2_REORDER

SLA_TRSYLV_L2_LOCAL_COPY

SLA_TRSYLV_L2_LOCAL_COPY_32

SLA_TRSYLV_L2_LOCAL_COPY_64

SLA_TRSYLV_L2_LOCAL_COPY_96

SLA_TRSYLV_L2_LOCAL_COPY_128

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 238 of file sla_gesylv.f90.

### subroutine sla_gesylv2 (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, real sgn, integer m, integer n, real, dimension(lda,*) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldqa, *) qa, integer ldqa, real, dimension(ldqb, *) qb, integer ldqb, real, dimension(ldx, *) x, integer ldx, real scale, real, dimension(*) work, integer ldwork, integer info)

Frontend for the solution of Standard Sylvester Equations.

Purpose:

 SLA_GESYLV2 solves a Sylvester equation of the following forms

op1(A) * X * op2(B) +  X  = SCALE * Y                              (1)

or

op1(A) * X * op2(B) -  X  = SCALE * Y                              (2)

where A is a M-by-M matrix and B is a N-by-N matrix. The right hand
side Y and the solution X are M-by-N matrices. The matrices A and B can be
either a general unreduced matrix or a (quasi-) upper triangular factor.
In the later case QA and QB provide the Schur-vectors of the matrices A
and B.
Parameters

FACTA

          FACTA is CHARACTER
Specifies how the matrix A is given.
== 'N':  The matrix A is given as a general matrix and its Schur decomposition
A = QA*S*QA**T will be computed.
== 'F':  The matrix A is given as its Schur decomposition in terms of S and QA
form A = QA*S*QA**T

FACTB

          FACTB is CHARACTER
Specifies how the matrix B is given.
== 'N':  The matrix B is given as a general matrix and its Schur decomposition
B = QB*R*QB**T will be computed.
== 'F':  The matrix A is given as its Schur decomposition in terms of R and QB
form A = QB*R*QB**T

TRANSA

          TRANSA is CHARACTER
Specifies the form of the system of equations with respect to A:
== 'N':  op1(A) = A
== 'T':  op1(A) = A**T

TRANSB

          TRANSB is CHARACTER
Specifies the form of the system of equations with respect to B:
== 'N':  op2(B) = B,
== 'T':  op2(B) = B**T

SGN

          SGN is REAL, allowed values: +/-1
Specifies the sign between the two parts of the Sylvester equation.
= 1 :  Solve Equation (1)
== -1:  Solve Equation (2)

M

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

N

          N is INTEGER
The order of the matrices B and D.  N >= 0.

A

          A is REAL 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.

LDA

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

B

          B is REAL array, dimension (LDB,N)
If FACT == 'N', the matrix B is a general matrix and it is overwritten with its
schur decomposition R.
If FACT == 'F', the matrix A contains its (quasi-) upper triangular matrix R being the
Schur decomposition of B.

LDB

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

QA

          QA is REAL array, dimension (LDQA,M)
If FACT == 'N', the matrix QA is an empty M-by-M matrix on input and contains the
Schur vectors of A on output.
If FACT == 'F', the matrix QA contains the Schur vectors of A.

LDQA

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

QB

          QB is REAL array, dimension (LDQB,N)
If FACT == 'N', the matrix QB is an empty N-by-N matrix on input and contains the
Schur vectors of B on output.
If FACT == 'F', the matrix QB contains the Schur vectors of B.

LDQB

          LDQB is INTEGER
The leading dimension of the array QB.  LDQB >= max(1,N).

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 M-by-N 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:  SGEES failed
= 2:  SLA_SORT_EV failed
= 3:  SLA_TRLYAP_DAG failed
< 0:  if INFO = -i, the i-th argument had an illegal value

SLA_TRSYLV2_L3

SLA_TRSYLV2_L3_2S

SLA_TRSYLV2_DAG

SLA_TRSYLV2_L2_UNOPT

SLA_TRSYLV2_L2

SLA_TRSYLV2_L2_REORDER

SLA_TRSYLV2_L2_LOCAL_COPY

SLA_TRSYLV2_L2_LOCAL_COPY_32

SLA_TRSYLV2_L2_LOCAL_COPY_64

SLA_TRSYLV2_L2_LOCAL_COPY_96

SLA_TRSYLV2_L2_LOCAL_COPY_128

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 238 of file sla_gesylv2.f90.

### subroutine sla_gesylv2_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, real sgn, integer m, integer n, 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(ldu, *) u, integer ldu, integer maxit, real tau, real, dimension(*) convlog, real, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the standard Sylvester Equations.

Purpose:

 SLA_GESYLV2_REFINE solves a Sylvester equation of the following forms

op1(A) * X * op2(B)  +  X = Y                              (1)

or

op1(A) * X * op2(B)  -  X = Y                              (2)

where A is a M-by-M matrix and B is a N-by-N matrix using iterative refinement.
The right hand side Y and the solution X are M-by-N matrices.
The matrix A and B need to be given in the original form as well
as in their Schur decomposition since both are required in the
iterative refinement procedure.
Parameters

TRANSA

          TRANSA is CHARACTER
Specifies the form of the system of equations with respect to A:
== 'N':  op1(A) = A
== 'T':  op1(A) = A**T

TRANSB

          TRANSB is CHARACTER
Specifies the form of the system of equations with respect to B:
== 'N':  op2(B) = B,
== 'T':  op2(B) = B**T

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.

SGN

          SGN is REAL, allowed values: +/-1
Specifies the sign between both terms.

M

          M is INTEGER
The order of the matrix A.  M >= 0.

N

          N is INTEGER
The order of the matrix B.  N >= 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,N)
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,N).

X

          X is REAL array, dimension (LDX,N)
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,N)
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 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,N)
The array BS contains the Schur decomposition of B.

LDBS

          LDBS is INTEGER
The leading dimension of the array BS.  LDBS >= max(1,N).

Q

          Q is REAL array, dimension (LDQ,M)
The array Q contains the Schur vectors of A as returned by SGEES.

LDQ

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

U

          U is REAL array, dimension (LDU,N)
The array U contains the Schur vectors of B as returned by SGEES.

LDU

          LDU is INTEGER
The leading dimension of the array U.  LDU >= max(1,N).

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_TRSYLV2_DAG

DLA_TRSYLV2_LEVEL3

SLA_TRSYLV2_L3_2S

SLA_TRSYLV2_L2_UNOPT

SLA_TRSYLV2_L2

SLA_TRSYLV2_L2_REORDER

SLA_TRSYLV2_L2_LOCAL_COPY_32

SLA_TRSYLV2_L2_LOCAL_COPY_64

SLA_TRSYLV2_L2_LOCAL_COPY_96

SLA_TRSYLV2_L2_LOCAL_COPY_128

SLA_TRSYLV2_L2_LOCAL_COPY

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 271 of file sla_gesylv2_refine.f90.

### subroutine sla_gesylv_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, real sgn, integer m, integer n, 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(ldu, *) u, integer ldu, integer maxit, real tau, real, dimension(*) convlog, real, dimension(*) work, integer ldwork, integer info)

Iterative Refinement for the standard Sylvester Equations.

Purpose:

 SLA_GESYLV_REFINE solves a Sylvester equation of the following forms

op1(A) * X  +  X * op2(B) = Y                              (1)

or

op1(A) * X  -  X * op2(B) = Y                              (2)

where A is a M-by-M matrix and B is a N-by-N matrix using iterative refinement.
The right hand side Y and the solution X are M-by-N matrices.
The matrix A and B need to be given in the original form as well
as in their Schur decomposition since both are required in the
iterative refinement procedure.
Parameters

TRANSA

          TRANSA is CHARACTER
Specifies the form of the system of equations with respect to A:
== 'N':  op1(A) = A
== 'T':  op1(A) = A**T

TRANSB

          TRANSB is CHARACTER
Specifies the form of the system of equations with respect to B:
== 'N':  op2(B) = B,
== 'T':  op2(B) = B**T

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.

SGN

          SGN is REAL, allowed values: +/-1
Specifies the sign between both terms.

M

          M is INTEGER
The order of the matrix A.  M >= 0.

N

          N is INTEGER
The order of the matrix B.  N >= 0.

A

          A is REAL 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 REAL array, dimension (LDB,N)
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,N).

X

          X is REAL array, dimension (LDX,N)
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,N)
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 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,N)
The array BS contains the Schur decomposition of B.

LDBS

          LDBS is INTEGER
The leading dimension of the array BS.  LDBS >= max(1,N).

Q

          Q is REAL array, dimension (LDQ,M)
The array Q contains the Schur vectors of A as returned by SGEES.

LDQ

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

U

          U is REAL array, dimension (LDU,N)
The array U contains the Schur vectors of B as returned by SGEES.

LDU

          LDU is INTEGER
The leading dimension of the array U.  LDU >= max(1,N).

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 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.

SLA_TRSYLV_DAG

DLA_TRSYLV_LEVEL3

SLA_TRSYLV_L3_2S

SLA_TRSYLV_L2_UNOPT

SLA_TRSYLV_L2

SLA_TRSYLV_L2_REORDER

SLA_TRSYLV_L2_LOCAL_COPY_32

SLA_TRSYLV_L2_LOCAL_COPY_64

SLA_TRSYLV_L2_LOCAL_COPY_96

SLA_TRSYLV_L2_LOCAL_COPY_128

SLA_TRSYLV_L2_LOCAL_COPY

Author

Martin Koehler, MPI Magdeburg

Date

Januar 2023

Definition at line 272 of file sla_gesylv_refine.f90.

## Author

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## Info

Tue Mar 7 2023 Version 1.0.3 MEPACK