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fdalg.spad
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fdalg.spad
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-- Free Fields in FriCAS
-- Created: Mon 2016-03-21 13:07
-- Changed: Mon 2017-06-12 13:07
-- Changed: Die 2018-07-03 19:25
-- Changed: Don 2018-08-30 09:09
-- Changed: Fre 2018-09-07 14:24
)abbrev domain FDALG FreeDivisionAlgebra
++ Author: Konrad Schrempf <schrempf@math.tugraz.at>
++ Date Created: Mit 2016-08-17 09:27
++ Date Changed: Fre 2018-09-07 14:24
++ Basic Functions:
++ Related Constructors: LinearMultivariateMatrixPencil, XDistributedPolynomial, FreeAssociative Algebra
++ Also See: XDistributedPolynomialFunctions
++ AMS Classifications:
++ Keywords: admissible linear system, minimal linear representation, noncommutative rational function, linearization, realization
++ References: Cohn and Reutenauer 1999, Schrempf 2017/2018
++ Description:
++ The elements of the Free Field are represented by Admissible
++ Linear Systems (ALS) in standard form ...
FreeDivisionAlgebra(VAR, F) : Exports == Implementation where
F : Field
VAR : OrderedSet
FLGDBG ==> true
ALTOUT ==> true
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
OF ==> OutputForm
LMMP ==> LinearMultivariateMatrixPencil(F)
G ==> Polynomial(F)
EQG ==> Equation(G)
FG ==> Fraction(G)
EQFG ==> Equation(FG)
M1 ==> FreeMonoid(VAR)
XDP ==> XDistributedPolynomial(VAR, F)
TERM ==> Record(k:M1, c:F)
IDXMIN ==> 1$NNI
IDXMAX ==> 2::NNI
IDXSZE ==> 3::NNI
IDXREF ==> 4::NNI
Exports == Join(DivisionRing(), Algebra(F)) with
-- Declaration (internal)
-- Changed: Fre 2018-09-07 14:27
interval : (NNI, NNI) -> List(NNI)
++ \spad{interval(i, j)} creates list [i, i+1, ..., j]
-- Declaration (basics)
-- Changed: Mit 2018-08-15 20:23
-- Changed: Sam 2018-09-01 23:07
qnew : (NNI) -> %
++ \spad{qnew(n)} creates an empty ALS of dimension n.
qnew : (NNI, List(M1)) -> %
++ \spad{qnew(n, lst)} creates an empty ALS of dimension n.
new : (LMMP, List(M1)) -> %
++ \spad{new(lp, lst)} creates an element by the linear
++ multivariate matrix pencil lp and the list of monomials lst.
new : (c:F) -> %
++ \spad{new(c)} creates a constant element.
new : (m:M1, c:F) -> %
++ \spad{new(m,c)} creates a monomial element with coefficient c.
copy : (%) -> %
++ \spad{copy(f)} gives a copy of the element f.
copy : (%, F) -> %
++ \spad{copy(f, alpha)} gives a copy of element f
++ multiplied by alpha.
coerce : (m:M1) -> %
++ \spad{coerce(m)} converts the monoid m into an element of
++ the free field represented by an ALS in minimal refined form.
coerce : (c:F) -> %
++ \spad{coerce(c)} converts the constant c into an element of
++ the free field represented by an ALS in minimal refined form.
enableDebugOutput : (f:%) -> %
++ \spad{enableDebugOutput(f)} enable displaying the ALS.
disableDebugOutput : (f:%) -> %
++ \spad{disableDebugOutput(f)} disable displaying the ALS.
toggleDebugOutput : (f:%) -> %
++ \spad{toggleDebugOutput(f)} enable/disable ALS.
enableAlternativeOutput : (f:%) -> %
++ \spad{enableAlternativeOutput(f)} enable output as
++ rational expression.
disableAlternativeOutput : (f:%) -> %
++ \spad{disableAlternativeOutput(f)} disable output as
++ rational expression.
toggleAlternativeOutput : (f:%) -> %
++ \spad{toggleAlternativeOutput(f)} enable/disable output
++ as rational expression.
minimal? : (%) -> Boolean
++ \spad{minimal?(f)} is f represented by a minimal ALS?
mutable? : (%) -> Boolean
++ \spad{mutable?(f)} is the underlying ALS of f mutable?
qregular? : (%, NNI, NNI) -> Boolean
++ \spad{qregular?(f, i_min, i_max)} does the specified
++ diagonal block define a regular element?
scalar? : (%, NNI, NNI) -> Boolean
++ \spad{scalar?(f, i, j)} is A(i,j) scalar?
zero? : (%, NNI, NNI) -> Boolean
++ \spad{zero?(f, i, j)} is A(i,j) zero?
scalar? : (%) -> Boolean
++ \spad{scalar?(f)} is f scalar?
regular? : (%) -> Boolean
++ \spad{regular?(f)} is f a regular element?
polynomial? : (%) -> Boolean
++ \spad{polynomial?(f)} is the ALS in polynomial form?
dimension : (%) -> NNI
++ \spad{dimension(f)} returns the dimension of the ALS.
variables : (%) -> List(M1)
++ \spad{variables(f)} returns a list of the variables.
elt : (%, NNI) -> F
++ \spad{elt(f, i)} returns v(i) from the ALS of f.
qelt : (%, NNI, NNI) -> XDP
++ \spad{qelt(f, i, j)} returns A(i,j) from the ALS of f.
elt : (%, NNI, NNI) -> XDP
++ \spad{elt(f, i, j)} returns A(i,j) from the ALS of f.
setelt! : (%, NNI, F) -> F
++ \spad{setelt!(f, i, alpha)} sets v(i) = alpha in the ALS of f.
setelt! : (%, NNI, NNI, XDP) -> XDP
++ \spad{setelt!(f, i, j, p)} sets A(i,j) = p in the ALS of f
++ (where p has degree less equal one) if the system is in
++ polynomial form and j>i.
vector : (%) -> Matrix(F)
++ \spad{vector(f)} returns v from the ALS of f.
vector : (%) -> Matrix(XDP)
++ \spad{vector(f)} returns v from the ALS of f.
matrix : (%, M1) -> Matrix(F)
++ \spad{matrix(f,m)} returns the coefficient matrix for
++ the monomial m of the ALS of f.
matrix : (%) -> Matrix(XDP)
++ \spad{matrix(f)} returns the matrix of the ALS of f.
appendSupport! : (%, List(M1)) -> %
++ \spad{appendSupport! (f, lst)} appends variables not in the support.
-- FIXME: For testing and debugging only (should return a copy).
pencil : (%) -> LMMP
++ \spad{pencil(f)} returns a pointer to the underlying pencil.
swapRowsColumns! : (%, NNI, NNI) -> %
++ \spad{swapRowsColumns!(f, i, j)} exchanges rows i and j and
++ columns j and i in the ALS of f.
swapRows! : (%, NNI, NNI) -> %
++ \spad{swapRows!(f, i, j)} exchanges rows i and j in the ALS of f.
qswapRows! : (%, NNI, NNI) -> %
++ \spad{swapRows!(f, i, j)} exchanges rows i and j in the ALS of f.
addRowsColumns! : (%, NNI, NNI, F) -> %
++ \spad{addRowsColumns!(f, i, j, alpha)} adds alpha*row(i) to
++ row(j) and subtracts alpha*column(j) from column(i) in the ALS of f.
addColumnsRows! : (%, NNI, NNI, F) -> %
++ \spad{addColumnsRows!(f, i, j, alpha)} adds alpha*column(i) to
++ column(j) and subtracts alpha*row(j) from row(i) in the ALS of f.
addRows! : (%, NNI, NNI, F) -> %
++ \spad{addRows!(f, i, j, alpha)} adds alpha*row(i) to row(j) in
++ the ALS of f.
qaddRows! : (%, NNI, NNI, F) -> %
++ \spad{addRows!(f, i, j, alpha)} adds alpha*row(i) to row(j) in
++ the ALS of f.
multiplyRow! : (%, NNI, F) -> %
++ \spad{multiplyRow!(f, i, alpha)} multiplies row(i) by alpha
++ in the ALS of f.
multiplyColumn! : (%, NNI, F) -> %
++ \spad{multiplyColumn!(f, i, alpha)} multiplies column(i) by alpha
++ in the ALS of f.
qswapColumns! : (%, NNI, NNI) -> %
++ \spad{swapColumns!(f, i, j)} exchanges columns i and j in
++ the ALS of f.
swapColumns! : (%, NNI, NNI) -> %
++ \spad{swapColumns!(f, i, j)} exchanges columns i and j in
++ the ALS of f.
qaddColumns! : (%, NNI, NNI, F) -> %
++ \spad{addColumns!(f, i, j, alpha)} adds alpha*column(i) to
++ column(j) in A and subtracts row(j) from row(i) in s (in the
++ ALS of f), i.e. (A*U)(U^-1*s) = v.
addColumns! : (%, NNI, NNI, F) -> %
++ \spad{addColumns!(f, i, j, alpha)} adds alpha*column(i) to
++ column(j) in A and subtracts row(j) from row(i) in s (in the
++ ALS of f), i.e. (A*U)(U^-1*s) = v.
removeRowsColumns : (%, List(NNI), List(NNI)) -> %
++ \spad{removeRowsColumns(f, lst_row, lst_col)} returns
++ a new system with the specified rows and columns removed.
++ The number of rows and columns have to be the same!
insertRowsColumns : (%, List(NNI), List(NNI)) -> %
++ \spad{insertRowsColumns(f, lst_row, lst_col)} returns
++ a new system with rows and columns inserted. An index
++ k means a new row/column between k and k+1. The number
++ of rows and columns have to be the same!
qzero? : (%, NNI, NNI, NNI, NNI) -> Boolean
++ \spad{qzero?(f, i_min, i_max, j_min, j_max)} is the
++ spezified block zero (in the system matrix)?
qzero? : (%, NNI, NNI, NNI, NNI, NNI) -> Boolean
++ \spad{qzero?(f, i_min, i_max, j_min, j_max, l)} is the
++ spezified block zero in matrix l?
qzero? : (%, NNI, NNI, NNI, NNI, M1) -> Boolean
++ \spad{qzero?(f, i_min, i_max, j_min, j_max, m)} is the
++ spezified block zero in the (system) matrix corresponding
++ to the monomial m?
blockStructure : (%) -> Matrix(NNI)
++ \spad{blockStructure(f)} analyzes the structure of the ALS
++ of f and detects blocks with respect to an upper triangular
++ structure. Entry (i,1) contains the first row, (i,2) the
++ last row, (i,3) the size and (i,4) if block i is refined.
refined? : (%) -> Boolean
++ \spad{refined?(f)} is f represented by a refined ALS?
blockElimination : (%, List(NNI), List(NNI), Boolean, _
List(NNI), List(NNI), Boolean) -> List(Matrix(F))
++ \spad{blockElimination(f, rsrc, rdst, flg_u, csrc, cdst, flg_v)}
++ returns transformation matrices if it is possible to eliminate
++ all entries in rdst x cdst (including columns in u if flg_u = true,
++ including rows in v if flg_v = true) by using rows in rsrc and
++ columns in csrc. Otherwise an empty list.
blockElimination : (%, List(NNI), List(NNI), List(NNI), List(NNI)) -> List(Matrix(F))
++ \spad{blockElimination(f, rsrc, rdst, csrc, cdst)}
++ flg_u = true, flg_v = true
display : (%, OutputForm) -> OutputForm
++ \spad{display(f,sol)} prints the element f as A*sol = v.
display : (%, List(Symbol)) -> OutputForm
++ \spad{display(f,[s])} prints the element f as A*(s_1,s_2,...,s_n)' = v.
admissibleLinearSystem : (%) -> OutputForm
++ \spad{admissibleLinearSystem(f)} output as ALS.
linearization : (%) -> Matrix(XDP)
++ \spad{linearization(f)} returns the element f as linearization.
linearization : (%) -> Matrix(%)
++ \spad{linearization(f)} returns the element f as linearization
++ with entries represented by admissible linear systems in minimal
++ refined form.
representation : (%) -> List(Matrix(XDP))
++ \spad{representation(f)} returns the element f as linear
++ representation (u,A,v).
_* : (f:%, U:Matrix(F)) -> %
++ \spad{f * U} column transformation
_* : (T:Matrix(F), f:%) -> %
++ \spad{T * f} row transformation
transformationMatrix : (%) -> Matrix(F)
++ \spad{transformationMatrix(f)} returns the identity matrix
++ to be modified and used within transformRows! or transformColumns!
transformRows! : (%, Matrix(F)) -> %
++ \spad{transformRows!(f, T)} transforms the ALS of f from the left.
transformColumns! : (%, Matrix(F)) -> %
++ \spad{transformColumns!(f, U)} transforms the ALS of f from the right.
-- Declaration (calculation)
-- Changed: Son 2018-08-12 11:09
-- Changed: Sam 2018-09-01 23:31
normalizeRHS! : (%) -> %
++ \spad{normalizeRHS!(f)} eliminates non-zero entries in the
++ right hand side of ALS with respect of the non-zero entry
++ with the highest index.
normalizePLS! : (%) -> %
++ \spad{normalizePLS!(f)} scales and rearranges rows and
++ columns of the system matrix such that the constant part
++ of the system matrix is the identity matrix (possibly of
++ smaller size).
normalizeDIAG! : (%) -> %
++ \spad{normalizeDIAG!(f)} rescales the rows such that the
++ first nonzero entry of the coefficient matrix in the diagonal
++ is one.
normalize! : (%) -> %
++ \spad{normalize!(f)} rescales the rows such that the first
++ nonzero entry of the coefficient matrix in the diagonal is
++ one. The right hand side is normalized by normalizeRHS!
++ and the non-zero entry is in the last row of the corresponding
++ block.
polynomial : (%) -> XDP
++ \spad{polynomial(f)} returns f as XDPOLY (if possible)
solutionVector : (%) -> Matrix(XDP)
++ \spad{solutionVector(f)} computes the solution vector s
++ of As=v if f is polynomial.
solutionVector : (%, NNI) -> Matrix(XDP)
++ \spad{solutionVector(f, k)} computes the approximated
++ solution vector up to powers M^k v' where PAs=Pv=v' with
++ PA = I-M.
columnSpan : (%) -> Stream(Matrix(XDP))
++ \spad{columnSpan(f)} computes the column span for a regular
++ element, that is (v', Mv', M^2v', ...) where PAs=Pv=v' with
++ PA = I-M.
rowSpan : (%) -> Stream(Matrix(XDP))
++ \spad{rowSpan(f)} computes the row span for a regular element,
++ that is (u; uM; uM^2; ...) where PAs=Pv with PA = I-M.
scaleALS : (%, F) -> %
++ \spad{scaleALS(f, alpha)} computes alpha*f by scaling
++ the right hand side of the ALS for f.
addALS : (%, %) -> %
++ \spad{addALS(f,g)} computes f+g in terms of the admissible
++ linear systems for f and g (without minimization).
multiplyALS : (%, %) -> %
++ \spad{multiplyALS(f,g)} computes f*g in terms of the admissible
++ linear systems for f and g (without minimization).
invertALS : (%) -> %
++ \spad{invertALS(f)} computes f^-1 in terms of the ALS for f.
++ There is no check if f is invertible!
invertSTD : (%) -> %
++ \spad{invertSTD(f)} computes the standard inverse of f in
++ terems of the admissible linear system There is no check if
++ f is invertible!
-- Declaration (factorization)
-- Changed: Sam 2018-09-01 23:45
factors : (%) -> List(%)
++ \spad{factors(f)} analysis the block structure of the
++ system matrix of the ALS of f to split f into factors.
summands : (%) -> List(%)
++ \spad{summands(f)} analysis the block structure of the
++ system matrix of the ALS of f to split f into summands.
leftFamily : (%) -> List(OutputForm)
++ \spad{leftFamily(f)} prints the left family s=A^-1*v.
rightFamily : (%) -> List(OutputForm)
++ \spad{rightFamily(f)} prints the right family t=u*A^-1.
ratexprInverse : (%, Boolean) -> OutputForm
++ \spad{ratexprInverse(f, flg)} returns f in output form if
++ it is a polynomial, (f)^-1 if f^-1 is a polynomial, "r<rank>"
++ if the system is minimal and "d<dim>" in general.
ratexpr : (%) -> OutputForm
++ \spad{ratexpr(f)} analysis the block structure of the
++ admissible linear system to write f as rational expression.
coerce : (%) -> OutputForm
++ \spad{coerce(f)} prints the ALS of f if the debug flag is
++ set and a rational expression if the alternative output
++ flag is set.
leftFactor : (%, NNI) -> %
++ \spad{leftFactor(f,k)} returns the left factor of rank k
++ of a polynomial f or 1 if it's not possible by linear
++ techniques.
rightFactor : (%, NNI) -> %
++ \spad{rightFactor(f,k)} returns the right factor of rank k
++ of a polynomial f or 1 if it's not possible by linear
++ techniques.
factorize : (%, NNI) -> List(%)
++ \spad{factorize(f,k)} factorizes f in f=g*h with rank(g)=k
++ if possible (if necessary by using non-linear techniques).
factor : (%) -> List(%)
++ \spad{factor(f)} factorizes f in f=f_1*f_2*...*f_k with
++ atoms (irreducible elements) f_i. Notice that this factorization
++ is unique only with respect to similiarity.
factorizationTransformations : (%, NNI, NNI, List(EQG)) -> List(Matrix(F))
++ \spad{factorizationTransformations(f,k_rows,k_cols,sol)}
++ for debugging purposes (interface LINPEN)
factorizationEquations : (%, NNI, NNI) -> List(G)
++ \spad{factorizationEquations(f,k_rows,k_cols)}
++ for debugging purposes (interface LINPEN)
factorizationGroebner : (%, NNI, NNI) -> List(G)
++ \spad{factorizationGroebner(f,k_rows,k_cols)}
++ for debugging purposes (interface LINPEN)
factorizationSolve : (%, NNI, NNI) -> List(List(EQG))
++ \spad{factorizationSolve(f,k_rows,k_cols)} returns a
++ (possible empty) list of solutions for an admissible
++ transformation to create an upper right block of zeros
++ of size k_rows times k_cols.
refineUR! : (%, NNI) -> %
++ \spad{refineUR!(f,k)} uses linear techniques to create an
++ upper right block of zeros with k rows (if possible).
refineUR! : (%) -> %
++ \spad{refineUR!(f)} uses linear techniques to create upper
++ right blocks of zeros in staircase form (as far as possible).
-- Declaration (minimization)
-- Changed: Fre 2018-09-07 15:55
extendedALS : (%) -> %
++ \spad{extendedALS(f)} returns an extended ALS for f, that is,
++ 1*f (with a scalar first row).
normalALS : (%) -> %
++ \spad{normalALS(f)} removes a scalar first row of an (extended) ALS.
leftMinimization : (%, NNI, NNI) -> %
++ \spad{leftMinimization(f, i_min, i_max)} tries to apply a
++ left minimization step with respect to the pivot block with
++ the rows/columns (i_min, ..., i_max).
rightMinimization : (%, NNI, NNI) -> %
++ \spad{rightMinimization(f, i_min, i_max)} tries to apply a
++ right minimization step with respect to the pivot block with
++ the rows/columns (i_min, ..., i_max).
minimize : (%) -> %
++ \spad{minimize(f)} minimizes the unterlying ALS by applying
++ left and right block minimization steps. Minimality is only
++ guaranteed if the remaining blocks are refined.
addMIN : (%, %) -> %
++ \spad{addMIN(f,g)} uses addALS(f,g) with minimization.
multiplyMIN : (%, %) -> %
++ \spad{multiplyMIN(f,g)} uses multiplyALS(f,g) with minimization.
invertMIN : (%) -> %
++ \spad{invertMIN(f)} uses invertSTD(f) and minimization to
++ construct a minimal system for f^-1. Linear techniques are
++ used to get a fine pivot block structure.
inverse : (%) -> %
++ \spad{inverse(f)} f^-1 using invertMIN.
_+ : (%, F) -> %
++ \spad{f + alpha} adds the scalar alpha to f.
_- : (%, F) -> %
++ \spad{f - alpha} subtracts the scalar alpha from f.
_+ : (F, %) -> %
++ \spad{alpha + f} adds the scalar alpha to f.
_- : (F, %) -> %
++ \spad{alpha - f} adds the scalar alpha to -f.
_/ : (%, F) -> %
++ \spad{f / alpha} computes f / alpha for nonzero alpha.
_/ : (F, %) -> %
++ \spad{alpha / f} computes alpha / f for nonzero f.
_/ : (%, %) -> %
++ \spad{f / g} computes f * g^-1 for nonzero element g.
_^ : (%, PositiveInteger) -> %
++ \spad{f^n} returns f^n.
_^ : (%, Integer) -> %
++ \spad{f^n} returns f^n.
setRefined! : (%, NNI) -> %
++ \spad{setRefined!(f,max_sze)} sets the internal flag if the
++ system is refined (over the ground field) up to the specified
++ block size.
refine! : (%, Boolean) -> %
++ \spad{refine!(f, flg)} refines the underlying admissible
++ admissible linear system using simple and linear techniques
++ and if flg=true also non-linear techniques (Groebner basis).
refine! : (%) -> %
++ \spad{refine!(f)} refines f using non-linear techniques.
refinementTransformations : (%, NNI, NNI, List(EQG)) -> List(Matrix(F))
++ \spad{refinementTransformations(f,i_min,i_max,sol)}
++ for debugging purposes (interface LINPEN)
refinementEquations : (%, NNI, NNI, NNI, Boolean, Boolean) -> List(G)
++ \spad{refinementEquations(f,i_min,i_max,k_rows,flg_u,flg_r)}
++ for debugging purposes (interface LINPEN)
refinementGroebner : (%, NNI, NNI, NNI, Boolean, Boolean) -> List(G)
++ \spad{refinementGroebner(f,i_min,i_max,k_rows,flg_u,flg_r)}
++ computes a Groebner basis for the ideal generated by the
++ equations for creating a zero block with k rows within the
++ pivot block i_min..i_max and conditions to guarantee invertible
++ transformations.
refinementSolve : (%, NNI, NNI, NNI, Boolean, Boolean) -> List(List(EQG))
++ \spad{refinementSolve(f,i_min,i_max,k_rows,flg_u,flg_r)}
++ Computes a list of solutions (for the entries in transformation
++ matrices) to create a lower left block of zeros with k rows in
++ the pivot block i_min..i_max.
rank : (%) -> NNI
++ \spad{rank(f)} returns the rank of the element f, that is,
++ the dimension of a minimal admissible linear system (for f).
coerce : (%) -> XDP
++ \spad{coerce(f)} converts the element to XDPOLY (if possible).
coerce : (XDP) -> %
++ \spad{coerce(p)} converts the polynomial p to an element in the
++ free field represented by an minimal admissible linear system.
--<-<SPAD:fdalg dcl05>>
Implementation == add
-- f = u * A^-1 * v, A full, u = [1,0,...,0]
Rep := Record(size:NNI, _
supp:List(M1), _
lmmp:LMMP, _
flg_minimal:Boolean, _
flg_refined:Boolean, _
flg_mutable:Boolean, _
flg_debug:Boolean, _
flg_output:Boolean)
-- Implementation (internal)
-- Changed: Fre 2018-09-07 14:28
-- The admissible linear system \pi = (u,A,v) of dimension n
-- for an element f=u*A^-1*v in the free field is represented
-- by a matrix pencil of size n+1: [0, u; v, A]. To simplify
-- the access to the system matrix A and the (scalar) vectors
-- u and v we use the following functions ...
POS_1 ==> 1$NNI -- The position of the coefficient matrix w.r.t. the empty word
NUM_R ==> 1$NNI -- Number of rows of the matrix u in u*A^-1*v
NUM_C ==> 1$NNI -- Number of columns of the matrix v in u*A^-1*v
characteristic() == characteristic()$F
-- Warning: Notice that NUM_R = NUM_C > 1 is a matrix algebra
-- (with zero divisors) over the free field and NOT a field
-- anymore. Everything except NUM_R = NUM_C = 1 has not been
-- tested. The notion of minimality is not clear for NUM_R > 1
-- or NUM_C > 1.
qeltA (p:LMMP, i:NNI, j:NNI, l:NNI) : F ==
qelt(p, NUM_R+i, NUM_C+j, l)
qeltv (p:LMMP, i:NNI, l:NNI) : F ==
qelt(p, NUM_R+i, 1, l)
qeltu (p:LMMP, j:NNI, l:NNI) : F ==
qelt(p, 1, NUM_C+j, l)
qlstA (p:LMMP, i:NNI, j:NNI) : List(F) ==
qelt(p, NUM_R+i, NUM_C+j)
qsetu! (p:LMMP, i:NNI, l:NNI, alpha:F) : F ==
qsetelt!(p, 1, NUM_C+i, l, alpha)
qsetA! (p:LMMP, i:NNI, j:NNI, l:NNI, alpha:F) : F ==
qsetelt!(p, NUM_R+i, NUM_C+j, l, alpha)
qsetv! (p:LMMP, i:NNI, l:NNI, alpha:F) : F ==
qsetelt!(p, NUM_R+i, 1, l, alpha)
qmtxu (p:LMMP, j1:NNI, j2:NNI, l:NNI) : Matrix(F) ==
subMatrix(p, 1, NUM_R, NUM_C+j1, NUM_C+j2, l)
qmtxA (p:LMMP, i1:NNI, i2:NNI, j1:NNI, j2:NNI, l:NNI) : Matrix(F) ==
subMatrix(p, NUM_R+i1, NUM_R+i2, NUM_C+j1, NUM_C+j2, l)
qmtxv (p:LMMP, i1:NNI, i2:NNI, l:NNI) : Matrix(F) ==
subMatrix(p, NUM_R+i1, NUM_R+i2, 1, NUM_C, l)
qscalev!(p:LMMP, i1:NNI, i2:NNI, l:NNI, alpha:F) : LMMP ==
qscaleBlock!(p, NUM_R+i1, NUM_R+i2, 1, NUM_C, l, alpha)
qscale!(f:%, alpha:F) : % ==
qscaleBlock!(f.lmmp, NUM_R+1, NUM_R+f.size, 1, NUM_C, POS_1, alpha)
f
qsetmtxu! (p:LMMP, j:NNI, l:NNI, a:Matrix(F)) : Matrix(F) ==
setsubMatrix!(p, 1, NUM_C+j, l, a)
qsetmtxA! (p:LMMP, i:NNI, j:NNI, l:NNI, a:Matrix(F)) : Matrix(F) ==
setsubMatrix!(p, NUM_R+i, NUM_C+j, l, a)
qsetmtxv! (p:LMMP, i:NNI, l:NNI, a:Matrix(F)) : Matrix(F) ==
setsubMatrix!(p, NUM_R+i, 1, l, a)
qzeroA? (p:LMMP, i:NNI, j:NNI) : Boolean ==
qzero?(p, NUM_R+i, NUM_C+j)
qzeromtxA? (p:LMMP, i1:NNI, i2:NNI, j1:NNI, j2:NNI) : Boolean ==
qzero?(p, NUM_R+i1, NUM_R+i2, NUM_C+j1, NUM_C+j2)
qsemizeroA? (p:LMMP, i1:NNI, i2:NNI, j1:NNI, j2:NNI, l:NNI) : Boolean ==
qsemizero?(p, NUM_R+i1, NUM_R+i2, NUM_C+j1, NUM_C+j2, l)
qzerov? (p:LMMP, i1:NNI, i2:NNI) : Boolean ==
qzero?(p, NUM_R+i1, NUM_R+i2, 1, NUM_C)
interval (i_min:NNI, i_max:NNI) : List(NNI) ==
lst_wrk : List(NNI) := []
for k in i_max .. i_min by -1 repeat
lst_wrk := cons(k, lst_wrk)
lst_wrk
-- Implementation (basics)
-- Changed: Mit 2018-08-22 08:52
-- Changed: Sam 2018-09-01 23:27
qnew (n:NNI) : % ==
lst_new := [1$M1]$List(M1)
lp := qnew(n+1, n+1, #lst_new)$LMMP
qsetelt!(lp, 1, 2, POS_1, 1$F)
flg_min := false
flg_ref := false
[n, lst_new, lp, flg_min, flg_ref, true, FLGDBG, ALTOUT]
qnew (n:NNI, lst:List(M1)) : % ==
lst_new := cons(1$M1, remove(1$M1, lst))
lp := qnew(n+1, n+1, #lst_new)$LMMP
qsetelt!(lp, 1, 2, POS_1, 1$F)
flg_min := false
flg_ref := false
[n, lst_new, lp, flg_min, flg_ref, true, FLGDBG, ALTOUT]
-- FIXME: Check entries in pencil (fullness of system matrix, etc.)
new (lp:LMMP, lst:List(M1)) : % ==
for mon in lst repeat
length(mon) > 1 =>
error "FDALG: new(lp, lst) - monomials not linear."
nrows(lp) ~= ncols(lp) =>
error "FDALG: new(lp,lst) - Linear Pencil is not square."
#lst ~= nelem(lp) =>
error "FDALG: new(lp,lst) - lengths do not agree."
n := (nrows(lp) - 1)::NNI
flg_min := false
flg_ref := false
[n, lst, lp, flg_min, flg_ref, true, FLGDBG, ALTOUT]
getSupport (var:List(VAR)) : List(M1) ==
lst_mon := new(1+#var, 1$M1)$List(M1)
for k in 1 .. #var repeat
lst_mon(k+1) := var(k)::M1
lst_mon
new (c:F) : % ==
if zero?(c) then
return(copy(0))
als := qnew(1)
qsetA!(als.lmmp, 1, 1, POS_1, 1$F)
qsetv!(als.lmmp, 1, POS_1, c)
als.flg_minimal := true
als.flg_refined := true
als
new (m:M1, c:F) : % ==
if zero? c then
als := qnew(1)
qsetA!(als.lmmp, 1, 1, POS_1, 1$F)
else
n := length(m) + 1
als := qnew(n, getSupport(varList(m)))
l := 0$NNI
-- e.g. for m = xyyx
-- factors(m) = [[gen=x,exp=1], [gen=y,exp=2], [gen=x,exp=1]]
for fct in factors(m) repeat
pos := position((fct.gen)::M1, als.supp)::NNI
for i in 1 .. fct.exp repeat
qsetA!(als.lmmp, 1+l, 1+l+1, pos, -1$F)
l := l + 1
for k in 1 .. n repeat
qsetA!(als.lmmp, k, k, POS_1, 1$F)
qsetv!(als.lmmp, n, POS_1, c)
als.flg_minimal := true
als.flg_refined := true
als
-- Info: Matrix size 0 causes problems, therefore a matrix of dimension 1 is used.
0 : % ==
als := qnew(1)
qsetA!(als.lmmp, 1, 1, POS_1, 1$F)
als.flg_minimal := true
als.flg_refined := true
als.flg_mutable := false
als.flg_debug := false
als.flg_output := false
als
1 : % ==
als := qnew(1)
qsetA!(als.lmmp, 1, 1, POS_1, 1$F)
qsetv!(als.lmmp, 1, POS_1, 1$F)
als.flg_minimal := true
als.flg_refined := true
als.flg_mutable := false
als.flg_debug := false
als.flg_output := false
als
copy (f:%) : % ==
lmmp_new := copy(f.lmmp)
supp_new := copy(f.supp)
[f.size, supp_new, lmmp_new, f.flg_minimal, f.flg_refined, _
true, f.flg_debug, f.flg_output]
copy (f:%, alpha:F) : % ==
if zero?(alpha) then
return(copy(0))
lmmp_new := copy(f.lmmp)
supp_new := copy(f.supp)
qscalev!(lmmp_new, 1, f.size, POS_1, alpha)
[f.size, supp_new, lmmp_new, f.flg_minimal, f.flg_refined, _
true, f.flg_debug, f.flg_output]
coerce (m:M1) : % ==
new(m, 1$F)
coerce (c:F) : % ==
copy(1, c)
enableDebugOutput (f:%) : % ==
f.flg_debug := true
f
disableDebugOutput (f:%) : % ==
f.flg_debug := false
f
toggleDebugOutput (f:%) : % ==
f.flg_debug := not(f.flg_debug)
f
enableAlternativeOutput (f:%) : % ==
f.flg_output := true
f
disableAlternativeOutput (f:%) : % ==
f.flg_output := false
f
toggleAlternativeOutput (f:%) : % ==
f.flg_output := not(f.flg_output)
f
minimal? (f:%) : Boolean ==
f.flg_minimal
mutable? (f:%) : Boolean ==
f.flg_mutable
qregular? (f:%, i_min:NNI, i_max:NNI) : Boolean ==
if not(qdiagonal?(f.lmmp, NUM_R+i_min, NUM_R+i_max, POS_1)) then
return(false)
lst_ele := qdiagonal(f.lmmp, NUM_R+i_min, NUM_R+i_max, POS_1)
for i in 1 .. #lst_ele repeat
if not(lst_ele(i) = 1$F) then
return(false)
true
-- FIXME: Index check?
scalar? (f:%, i:NNI, j:NNI) : Boolean ==
a_lst := qlstA(f.lmmp, i, j)
flg_wrk := true
for k in 2 .. #(f.supp) repeat
if not(zero?(a_lst(k))) then
flg_wrk := false
break
flg_wrk
zero? (f:%, i:NNI, j:NNI) : Boolean ==
flg_wrk := scalar?(f, i, j) and zero?(qeltA(f.lmmp, i, j, POS_1))
flg_wrk
scalar? (f:%) : Boolean ==
if f.size > 1 then
return(false)
if not(qregular?(f, 1, f.size)) then
return(false)
for k in 2 .. #(f.supp) repeat
if not(qnilpotent?(f.lmmp, NUM_R+1, NUM_R+f.size, k)) then
return(false)
true
regular? (f:%) : Boolean ==
n := f.size
flg_wrk := false
if rank(qmtxA(f.lmmp, 1, n, 1, n, POS_1)) = n then
flg_wrk := true
flg_wrk
polynomialForm? (f:%) : Boolean ==
if not(quppertriangular?(f.lmmp, NUM_R+1, NUM_R+f.size, POS_1)) then
return(false)
for k in 2 .. #(f.supp) repeat
if not(qnilpotent?(f.lmmp, NUM_R+1, NUM_R+f.size, k)) then
return(false)
lst_ele := qdiagonal(f.lmmp, NUM_R+1, NUM_R+f.size, POS_1)
for i in 1 .. #lst_ele repeat
if not(lst_ele(i) = 1$F) then
return(false)
true
polynomial? (f:%) : Boolean ==
polynomialForm?(f)
dimension (f:%) : NNI ==
f.size
variables (f:%) : List(M1) ==
copy(f.supp)
elt (f:%, i:NNI) : F ==
(i < 1) or (i > f.size) =>
error "FDALG: elt(%,i) - index out of range."
qeltv(f.lmmp, i, POS_1)
qelt (f:%, i:NNI, j:NNI) : XDP ==
a_tmp := 0$XDP
for k in 1 .. #(f.supp) repeat
a_tmp := a_tmp + qeltA(f.lmmp, i, j, k)*elt(f.supp, k)::XDP
a_tmp
elt (f:%, i:NNI, j:NNI) : XDP ==
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: elt(%,i,j) - index out of range."
qelt(f, i, j)
setelt! (f:%, i:NNI, alpha:F) : F ==
not(f.flg_mutable) =>
error "FDALG: setelt!(%,i,alpha) - element is not mutable."
(i < 1) or (i > f.size) =>
error "FDALG: setelt!(%,i,alpha) - index out of range."
v_old := qeltv(f.lmmp, i, POS_1)
if not(v_old = alpha) then
qsetv!(f.lmmp, i, POS_1, alpha)
f.flg_minimal := false
f.flg_refined := false
alpha
setelt! (f:%, i:NNI, j:NNI, p:XDP) : XDP ==
not(f.flg_mutable) =>
error "FDALG: setelt!(%,i,j,p) - element is not mutable."
not(polynomialForm?(f)) =>
error "FDALG: setelt!(%,i,j,p) - system not in polynomial form."
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: setelt!(%,i,j,p) - index out of range."
i > j =>
error "FDALG: setelt!(%,i,j,p) - entry not above diagonal."
if not(zero?(p)) then
degree(p) > 1 =>
error "FDALG: setelt!(%,i,j,p) - degree higher than one."
p_old := qlstA(f.lmmp, i, j)
for k in 1 .. #(f.supp) repeat
qsetA!(f.lmmp, i, j, k, 0$F)
for mon in getSupport(varList(p)) repeat
pos := position(mon, f.supp)::NNI
pos = 0 =>
error "FDALG: setelt!(%,i,j,p) - monomial not available."
qsetA!(f.lmmp, i, j, pos, coefficient(p, mon))
p_new := qlstA(f.lmmp, i, j)
if not(p_old = p_new) then
f.flg_minimal := false
p
vector (f:%) : Matrix(F) ==
qmtxv(f.lmmp, 1, f.size, POS_1)
vector (f:%) : Matrix(XDP) ==
v_tmp := qmtxv(f.lmmp, 1, f.size, POS_1)
v_wrk := new(nrows(v_tmp), 1, 0$XDP)$Matrix(XDP)
for i in 1 .. nrows(v_tmp) repeat
qsetelt!(v_wrk, i, 1, qelt(v_tmp, i, 1)::XDP)
v_wrk
matrix (f:%, m:M1) : Matrix(F) ==
pos := position(m, f.supp)::NNI
pos = 0 =>
error "FDALG: matrix(%,m) - monoid not available."
qmtxA(f.lmmp, 1, f.size, 1, f.size, pos)
matrix (f:%) : Matrix(XDP) ==
n := f.size
A_wrk := new(n,n,0$XDP)$Matrix(XDP)
for k in 1 .. #(f.supp) repeat
A_tmp := qmtxA(f.lmmp, 1, f.size, 1, f.size, k)
for i in 1 .. n repeat
for j in 1 .. n repeat
qsetelt!(A_wrk, i, j, qelt(A_wrk, i, j) _
+ qelt(A_tmp, i, j)*elt(f.supp, k)::XDP)
A_wrk
appendSupport! (f:%, lst:List(M1)) : % ==
lst_new : List(M1) := []
for mon in lst repeat
pos := position(mon, f.supp)::NNI
if zero?(pos) then
lst_new := cons(mon, lst_new)
if #lst_new > 0 then
append!(f.lmmp, #lst_new)
f.supp := append(f.supp, lst_new)
f
pencil (f:%) : LMMP ==
f.lmmp
qswapRows! (f:%, i:NNI, j:NNI) : % ==
if not(i = j) then
qswapRows!(f.lmmp, NUM_R+i, NUM_R+j)
f
-- FIXME: Swapping rows within a block should not change refinement flag.
swapRows! (f:%, i:NNI, j:NNI) : % ==
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: swapRows!(%,i,j) - index/indices out of range."
if not(i = j) then
qswapRows!(f.lmmp, NUM_R+i, NUM_R+j)
f.flg_refined := false
f
qaddRows! (f:%, i:NNI, j:NNI, alpha:F) : % ==
if not(i=j) then
qaddRows!(f.lmmp, NUM_R+i, NUM_R+j, alpha)
f
addRowsColumns! (f:%, i:NNI, j:NNI, alpha:F) : % ==
i = j =>
error "FDALG: addRowsColumns!(%,i,j,alpha) - indices must be different."
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: addRowsColumns!(%,i,j,alpha) - index/indices out of range."
j = 1 =>
error "FDALG: addRowsColumns!(%,i,j,alpha) - first column must not be used."
qaddRows!(f.lmmp, NUM_R+i, NUM_R+j, alpha)
qaddColumns!(f.lmmp, NUM_C+j, NUM_C+i, -alpha)
if i < j then
f.flg_refined := false
f
addColumnsRows! (f:%, i:NNI, j:NNI, alpha:F) : % ==
i = j =>
error "FDALG: addColumnsRows!(%,i,j,alpha) - indices must be different."
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: addColumnsRows!(%,i,j,alpha) - index/indices out of range."
i = 1 =>
error "FDALG: addColumnsRows!(%,i,j,alpha) - first column must not be used."
qaddColumns!(f.lmmp, NUM_C+i, NUM_C+j, alpha)
qaddRows!(f.lmmp, NUM_R+j, NUM_R+i, -alpha)
f
addRows! (f:%, i:NNI, j:NNI, alpha:F) : % ==
i = j =>
error "FDALG: addRows!(%,i,j,alpha) - indices must be different."
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: addRows!(%,i,j,alpha) - index/indices out of range."
addRows!(f.lmmp, NUM_R+i, NUM_R+j, alpha)
if i < j then
f.flg_refined := false
f
qmultiplyRow! (f:%, i:NNI, alpha:F) : % ==
qmultiplyRow!(f.lmmp, NUM_R+i, alpha)
f
multiplyRow! (f:%, i:NNI, alpha:F) : % ==
if alpha = 1$F then
return(f)
(i < 1) or (i > f.size) =>
error "FDALG: multiplyRow!(%,i,alpha) - index out of range."
zero?(alpha) =>
error "FDALG: multiplyRow!(%,i,alpha) - zero factor."
qmultiplyRow!(f.lmmp, NUM_R+i, alpha)
f
qmultiplyColumn! (f:%, i:NNI, alpha:F) : % ==
qmultiplyColumn!(f.lmmp, NUM_C+i, alpha)
f
multiplyColumn! (f:%, i:NNI, alpha:F) : % ==
if alpha = 1$F then
return(f)
(i < 1) or (i > f.size) =>
error "FDALG: multiplyColumn!(%,i,alpha) - index out of range."
zero?(alpha) =>
error "FDALG: multiplyColumn!(%,i,alpha) - zero factor."
qmultiplyColumn!(f.lmmp, NUM_C+i, alpha)
f
qswapColumns! (f:%, i:NNI, j:NNI) : % ==
if not(i = j) then
qswapColumns!(f.lmmp, NUM_C+i, NUM_C+j)
f
-- FIXME: Swapping columns within a block should not change refinement flag.
swapColumns! (f:%, i:NNI, j:NNI) : % ==
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: swapColumns!(%,i,j) - index/indices out of range."
i = 1 or j = 1 =>
error "FDALG: swapColumns!(%,i,j) - first column must not be changed."
if not(i = j) then
qswapColumns!(f.lmmp, NUM_C+i, NUM_C+j)
f.flg_refined := false
f
swapRowsColumns! (f:%, i:NNI, j:NNI) : % ==
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: swapRowsColumns!(%,i,j) - index/indices out of range."
i = 1 or j = 1 =>
error "FDALG: swapRowsColumns!(%,i,j) - first column must not be changed."
if not(i = j) then
qswapRows!(f.lmmp, NUM_R+i, NUM_R+j)
qswapColumns!(f.lmmp, NUM_C+i, NUM_C+j)
f.flg_refined := false
f
addColumns! (f:%, i:NNI, j:NNI, alpha:F) : % ==
i = j =>
error "FDALG: addColumns!(%,i,j,alpha) - indices must be different."
(i < 1) or (i > f.size) or (j < 1) or (j > f.size) =>
error "FDALG: addColumns!(%,i,j,alpha) - index/indices out of range."
i = 1 =>
error "FDALG: addColumns!(%,i,j,alpha) - first column must not be used."
addColumns!(f.lmmp, NUM_C+i, NUM_C+j, alpha)
if i > j then
f.flg_refined := false
f
qaddColumns! (f:%, i:NNI, j:NNI, alpha:F) : % ==
if not(i=j) then
addColumns!(f.lmmp, NUM_C+i, NUM_C+j, alpha)
f
-- FIXME: Check row and column indices?
removeRowsColumns (f:%, lst_row:List(NNI), lst_col:List(NNI)) : % ==
lst_r := map(a +-> a+1, removeDuplicates(lst_row))$List(NNI)
lst_c := map(a +-> a+1, removeDuplicates(lst_col))$List(NNI)
#lst_r ~= #lst_c =>
error "FDALG: removeRowsColumns(%,rows,cols) - number of rows and columns differ."
f_new := new(removeRowsColumns(f.lmmp, lst_r, lst_c), f.supp)
f_new
insertRowsColumns (f:%, lst_row:List(NNI), lst_col:List(NNI)) : % ==
lst_r := map(a +-> a+1, lst_row)$List(NNI)
lst_c := map(a +-> a+1, lst_col)$List(NNI)
#lst_r ~= #lst_c =>
error "FDALG: insertRowsColumns(%,rows,cols) - number of rows and columns differ."
f_new := new(insertRowsColumns(f.lmmp, lst_r, lst_c), f.supp)
for k in 1 .. #lst_r repeat
qsetA!(f_new.lmmp, k+lst_row(k), k+lst_col(k), POS_1, 1$F)
f_new