dblgglyap - Man Page

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.

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, in generalized
 Hessenberg form, or in generalized Schur form where Q and Z also need 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
          == 'H': The matrix pencil (A,B) is given in generalized Hessenberg form and its Schur decomposition
                  A = Q*S*Z**T, B = Q*R*Z**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, 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 the
          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).
          If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
          Hessenberg form (A,B) and it is overwritten with the quasi upper triangular matrix S
          of the generalized Schur decomposition.

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).
          If FACT == 'H', the matrix B is the upper triangular matrix of the generalized Hessenberg form
          (A,B) and it is overwritten with the upper triangular matrix of the generalized Schur decomposition.

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).
          If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the
          left Schur vectors of (A,B) on output.

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).
          If FACT == 'H', the matrix Z is an empty M-by-M matrix on input and contains the
          right Schur vectors of (A,B) on output.

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

See also

DLA_TGLYAP_L3

DLA_TGLYAP_L3_2S

DLA_TGLYAP_DAG

DLA_TGLYAP_L2

DLA_TGLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 207 of file dla_gglyap.f90.

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.

See also

DLA_TGLYAP_L3

DLA_TGLYAP_L2

DLA_TGLYAP_L3_2S

DLA_TGLYAP_DAG

DLA_TGLYAP_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 244 of file dla_gglyap_refine.f90.

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, in generalized
 Hessenberg form, or in generalized Schur form where Q and Z also need 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
          == 'H': The matrix pencil (A,B) is given in generalized Hessenberg form and its Schur decomposition
                  A = Q*S*Z**T, B = Q*R*Z**T will be computed.

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 the
          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).
          If FACT == 'H', the matrix A is an upper Hessenberg matrix of the generalized
          Hessenberg form (A,B) and it is overwritten with the quasi upper triangular matrix S
          of the generalized Schur decomposition.

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).
          If FACT == 'H', the matrix B is the upper triangular matrix of the generalized Hessenberg form
          (A,B) and it is overwritten with the upper triangular matrix of the generalized Schur decomposition.

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).
          If FACT == 'H', the matrix Q is an empty M-by-M matrix on input and contains the
          left Schur vectors of (A,B) on output.

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).
          If FACT == 'H', the matrix Z is an empty M-by-M matrix on input and contains the
          right Schur vectors of (A,B) on output.

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

See also

DLA_TGSTEIN_L3

DLA_TGSTEIN_L3_2S

DLA_TGSTEIN_DAG

DLA_TGSTEIN_L2

DLA_TGSTEIN_RECURSIVE

Author

Martin Koehler, MPI Magdeburg

Date

January 2024

Definition at line 208 of file dla_ggstein.f90.