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
- See also
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.
- See also
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
- See also
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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