sglgelyap - Man Page

subroutine sla_gelyap (fact, trans, m, a, lda, q, ldq, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Standard Lyapunov Equations.
subroutine sla_gestein (fact, trans, m, a, lda, q, ldq, x, ldx, scale, work, ldwork, info)
Frontend for the solution of Standard Stein Equations.
subroutine sla_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 sla_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.

Frontend for the solution of Standard Lyapunov Equations.

Purpose:

 SLA_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 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 == '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 REAL 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 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 symmetric M-by-M matrices.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDB >= 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 again
          without performing any computations.

INFO

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

See also

SLA_TRLYAP_L3

SLA_TRLYAP_L3_2S

SLA_TRLYAP_DAG

SLA_TRLYAP_L2

SLA_TRLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 175 of file sla_gelyap.f90.

Iterative Refinement for the Standard Lyapunov Equation.

Purpose:

 SLA_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 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).

X

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

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 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_TRLYAP_L3

SLA_TRLYAP_L2

SLA_TRLYAP_L3_2S

SLA_TRLYAP_DAG

SLA_TRLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 201 of file sla_gelyap_refine.f90.

Frontend for the solution of Standard Stein Equations.

Purpose:

 SLA_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 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.
          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 REAL 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 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 symmetric M-by-M matrices.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDB >= 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 again
          without performing any computations.

INFO

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

See also

SLA_TRSTEIN_L3

SLA_TRSTEIN_L3_2S

SLA_TRSTEIN_DAG

SLA_TRSTEIN_L2

SLA_TRSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 175 of file sla_gestein.f90.

Iterative Refinement for the Standard Stein Equations.

Purpose:

 SLA_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 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).

X

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

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 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_TRSTEIN_L3

SLA_TRSTEIN_L2

SLA_TRSTEIN_L3_2S

SLA_TRSTEIN_DAG

SLA_TRSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 202 of file sla_gestein_refine.f90.