sglgesylv - Man Page

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.

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 an upper Hessenberg form
 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
          == 'H':  The matrix A is given as an upper Hessenberg form and its Schur
                  decomposition A = QA*S*QA**T will be computed

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 B is given as its Schur decomposition in terms of R and QB
                  form B = QB*R*QB**T
          == 'H':  The matrix B is given as an upper Hessenberg form and its Schur
                  decomposition B = QB*R*QB**T will be computed

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

B

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

LDB

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

QA

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

LDQA

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

QB

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

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

See also

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

January 2024

Definition at line 250 of file sla_gesylv.f90.

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 an upper Hessenberg form
 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
          == 'H':  The matrix A is given as an upper Hessenberg form and its Schur
                  decomposition A = QA*S*QA**T will be computed

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 B is given as its Schur decomposition in terms of R and QB
                  form B = QB*R*QB**T
          == 'H':  The matrix B is given as an upper Hessenberg form and its Schur
                  decomposition B = QB*R*QB**T will be computed

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

B

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

LDB

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

QA

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

LDQA

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

QB

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

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

See also

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

January 2024

Definition at line 250 of file sla_gesylv2.f90.

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.

See also

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

January 2024

Definition at line 271 of file sla_gesylv2_refine.f90.

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.

See also

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

January 2024

Definition at line 272 of file sla_gesylv_refine.f90.