dblgesylv - Man Page
Name
dblgesylv — Double Precision
— Double Precision routines for standard Sylvester equations.
Synopsis
Functions
subroutine dla_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 dla_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 dla_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 dla_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
Double Precision routines for standard Sylvester equations.
This subsection contains the solvers for standard Sylvester equations with general coefficient matrices in double precision arithmetic. The Schur decompositions are computed in double precision with the help of LAPACK.
Function Documentation
subroutine dla_gesylv (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, double precision sgn, integer m, integer n, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldqa, *) qa, integer ldqa, double precision, dimension(ldqb, *) qb, integer ldqb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)
Frontend for the solution of Standard Sylvester Equations.
Purpose:
DLA_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 DOUBLE PRECISION, 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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 M-by-N 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: DGEES failed = 2: DLA_SORT_EV failed = 3: DLA_TRLYAP_DAG failed < 0: if INFO = -i, the i-th argument had an illegal value
- See also
DLA_TRSYLV_L3
DLA_TRSYLV_L3_2S
DLA_TRSYLV_DAG
DLA_TRSYLV_L2_UNOPT
DLA_TRSYLV_L2
DLA_TRSYLV_L2_REORDER
DLA_TRSYLV_L2_LOCAL_COPY
DLA_TRSYLV_L2_LOCAL_COPY_32
DLA_TRSYLV_L2_LOCAL_COPY_64
DLA_TRSYLV_L2_LOCAL_COPY_96
DLA_TRSYLV_L2_LOCAL_COPY_128
- Author
Martin Koehler, MPI Magdeburg
- Date
Januar 2023
Definition at line 238 of file dla_gesylv.f90.
subroutine dla_gesylv2 (character, dimension(1) facta, character, dimension(1) factb, character, dimension(1) transa, character, dimension(1) transb, double precision sgn, integer m, integer n, double precision, dimension(lda,*) a, integer lda, double precision, dimension(ldb, *) b, integer ldb, double precision, dimension(ldqa, *) qa, integer ldqa, double precision, dimension(ldqb, *) qb, integer ldqb, double precision, dimension(ldx, *) x, integer ldx, double precision scale, double precision, dimension(*) work, integer ldwork, integer info)
Frontend for the solution of Standard Sylvester Equations.
Purpose:
DLA_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 DOUBLE PRECISION, 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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 M-by-N 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: DGEES failed = 2: DLA_SORT_EV failed = 3: DLA_TRLYAP_DAG failed < 0: if INFO = -i, the i-th argument had an illegal value
- See also
DLA_TRSYLV2_L3
DLA_TRSYLV2_L3_2S
DLA_TRSYLV2_DAG
DLA_TRSYLV2_L2_UNOPT
DLA_TRSYLV2_L2
DLA_TRSYLV2_L2_REORDER
DLA_TRSYLV2_L2_LOCAL_COPY
DLA_TRSYLV2_L2_LOCAL_COPY_32
DLA_TRSYLV2_L2_LOCAL_COPY_64
DLA_TRSYLV2_L2_LOCAL_COPY_96
DLA_TRSYLV2_L2_LOCAL_COPY_128
- Author
Martin Koehler, MPI Magdeburg
- Date
Januar 2023
Definition at line 238 of file dla_gesylv2.f90.
subroutine dla_gesylv2_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, double precision sgn, integer m, integer n, 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(ldu, *) u, integer ldu, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)
Iterative Refinement for the standard Sylvester Equations.
Purpose:
DLA_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 DOUBLE PRECISION, 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 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,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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors of A as returned by DGEES.
LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).
U
U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the Schur vectors of B as returned by DGEES.
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 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_TRSYLV2_DAG
DLA_TRSYLV2_LEVEL3
DLA_TRSYLV2_L3_2S
DLA_TRSYLV2_L2_UNOPT
DLA_TRSYLV2_L2
DLA_TRSYLV2_L2_REORDER
DLA_TRSYLV2_L2_LOCAL_COPY_32
DLA_TRSYLV2_L2_LOCAL_COPY_64
DLA_TRSYLV2_L2_LOCAL_COPY_96
DLA_TRSYLV2_L2_LOCAL_COPY_128
DLA_TRSYLV2_L2_LOCAL_COPY
- Author
Martin Koehler, MPI Magdeburg
- Date
Januar 2023
Definition at line 271 of file dla_gesylv2_refine.f90.
subroutine dla_gesylv_refine (character, dimension(1) transa, character, dimension(1) transb, character, dimension(1) guess, double precision sgn, integer m, integer n, 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(ldu, *) u, integer ldu, integer maxit, double precision tau, double precision, dimension(*) convlog, double precision, dimension(*) work, integer ldwork, integer info)
Iterative Refinement for the standard Sylvester Equations.
Purpose:
DLA_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 DOUBLE PRECISION, 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDQ,M) The array Q contains the Schur vectors of A as returned by DGEES.
LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).
U
U is DOUBLE PRECISION array, dimension (LDU,N) The array U contains the Schur vectors of B as returned by DGEES.
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 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 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
DLA_TRSYLV_DAG
DLA_TRSYLV_LEVEL3
DLA_TRSYLV_L3_2S
DLA_TRSYLV_L2_UNOPT
DLA_TRSYLV_L2
DLA_TRSYLV_L2_REORDER
DLA_TRSYLV_L2_LOCAL_COPY_32
DLA_TRSYLV_L2_LOCAL_COPY_64
DLA_TRSYLV_L2_LOCAL_COPY_96
DLA_TRSYLV_L2_LOCAL_COPY_128
DLA_TRSYLV_L2_LOCAL_COPY
- Author
Martin Koehler, MPI Magdeburg
- Date
Januar 2023
Definition at line 272 of file dla_gesylv_refine.f90.
Author
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