<Section Label="semiechelon">
<Heading>Semi echelonised bases and cleaning</Heading>
A semi echelonised basis is a record with the following components:
<C>vectors</C> is a list of vectors of equal length, optionally
(and optimally) they can be wrapped to a matrix. <C>pivots</C>
is a list of integers. The number <M>i</M> in position <M>j</M> indicates
that the <M>j</M>th vector in <C>vectors</C> has a one in column number
<M>i</M> and all vectors with a number bigger than <M>i</M> in <C>vectors</C>
have a zero in column number <M>i</M>.
<P/>
Note that for technical reasons another component <C>helper</C> is bound
containing a <C>cvec</C> of length <M>1</M> over the same field.
<P/>
Note further that the output of the &GAP; library operation
<Ref Oper="SemiEchelonMat" BookName="Ref"/> is not compatible to this
setup, because it contains a component <C>heads</C> that contains at
position <M>i</M>, if nonzero, the number of the row
for that the pivot element is in column <M>i</M>.
<ManSection>
<Meth Name="EmptySemiEchelonBasis" Arg="v"/>
<Returns> a new mutable empty semi echelonised basis </Returns>
<Description>
The argument <A>v</A> must be a sample vector or matrix.
If it is a matrix, then one of its rows is taken as sample vector.
</Description>
</ManSection>
<ManSection>
<Oper Name="Vectors" Arg="b"/>
<Returns> a matrix </Returns>
<Description>
The argument <A>b</A> must be a semi echelonised basis. This
operation returns a (mutable) matrix whose rows are the basis
vectors.
</Description>
</ManSection>
<ManSection>
<Oper Name="Length" Arg="b" Label="for a semi echelonised basis"/>
<Returns> an integer </Returns>
<Description>
The argument <A>b</A> must be a semi echelonised basis. This
operation returns the number of vectors in the basis.
</Description>
</ManSection>
<ManSection>
<Meth Name="CleanRow" Arg="b, v, extend, dec"/>
<Returns> <C>true</C> or <C>false</C> </Returns>
<Description>
This is the basic operation for cleaning with a semi echelonised basis
<A>b</A>. The vector <A>v</A> is cleaned with the vectors in the
semi echelonised basis. The function returns <C>true</C> if <A>v</A>
lies in the span of <A>b</A> and false otherwise.
<P/>
The boolean
value <A>extend</A> indicates, whether the basis should be extended
in the latter case by the newly cleaned vector.
<P/>
The argument
<A>dec</A> is either <C>fail</C> in which case it is not used or
a vector over the same field as <A>v</A> that is at least as long
as the number <M>n</M> of vectors in <A>b</A> (plus one if <A>extend</A> is
<C>true</C>). In this case, the first <M>n</M> components of <A>dec</A>
are changed such that
<M>\sum_{{i=1}}^{n} <A>dec</A>_i <A>b</A>_i</M>. If <A>extend</A> is
<C>true</C> and <A>v</A> is not contained in the span of the vectors
in <A>b</A> and <A>dec</A> is a vector, then the first <M>n+1</M>
components of <A>dec</A> are changed such that
<M><A>v</A> = \sum_{{i=1}}^{n+1} <A>dec</A>_i <A>b</A>_i</M>.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutable" Arg="b" Label="for a semi echelonised basis"/>
<Returns> a semi echelonised basis </Returns>
<Description>
Turns the output <A>b</A> of <Ref Oper="SemiEchelonMat" BookName="Ref"/>
into a valid semi echelonised basis in the sense of the
<Package>cvec</Package> package. This means that the component
<C>pivots</C> is calculated from the component <C>heads</C>.
Use this function only if absolutely necessary. Instead, directly
invoke <Ref Meth="SemiEchelonBasisMutable"/> on the original matrix.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutable" Arg="m"/>
<Returns> a semi echelonised basis </Returns>
<Description>
The argument <A>m</A> must be a <C>cmat</C>. This method calculates
a semi echelonised basis for the row space of <A>m</A>.
<P/>
There are a
number of similar operations the names of which are derived from
<Ref Meth="SemiEchelonBasisMutable"/> by appending single letters: The letters
<Q>P</Q>, <Q>T</Q> and <Q>X</Q> are modifiers and there are operations
for most of the <M>8</M> combinations of those letters being present or not
respectively. Always give the present letters in alphabetical order.
<P/>
The <Q>X</Q> indicates, that the input matrix may be
destroyed, that is, the rows of <A>m</A> will be changed and the
very same objects be used in the semi echelonised result basis.
<P/>
The <Q>T</Q> indicates, that the transformation is calculated, that is,
the resulting record <C>r</C> will have a component <C>coeffs</C>, such
that <C>r.coeffs * <A>m</A></C> is equal to <C>r.vectors</C> component
of the semi echelonised result. Further, with given letter <Q>T</Q>
there will be a component <C>relations</C> which is a basis for the
(left) nullspace of <A>m</A>.
<P/>
The <Q>P</Q> indicates, that a component <C>r.p</C> is calculated such
that <C>r.p * r.vectors</C> is the original matrix <A>m</A>.
<P/>
Currently the variants with <Q>P</Q> and <Q>T</Q> both present are not
implement because they will probably not be very useful.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutableX" Arg="m"/>
<Returns> a semi echelonised basis </Returns>
<Description>
Same as <Ref Meth="SemiEchelonBasisMutable"/> but destructive in <A>m</A>.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutableT" Arg="m"/>
<Returns> a semi echelonised basis </Returns>
<Description>
Same as <Ref Meth="SemiEchelonBasisMutable"/> but in addition stores the
transformation, see <Ref Meth="SemiEchelonBasisMutable"/>.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutableTX" Arg="m"/>
<Returns> a semi echelonised basis </Returns>
<Description>
Same as <Ref Meth="SemiEchelonBasisMutableT"/> but destructive in <A>m</A>.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutableP" Arg="m"/>
<Returns> a semi echelonised basis </Returns>
<Description>
Same as <Ref Meth="SemiEchelonBasisMutable"/> but in addition stores the
inverse transformation, see <Ref Meth="SemiEchelonBasisMutable"/>.
</Description>
</ManSection>
<ManSection>
<Meth Name="SemiEchelonBasisMutablePX" Arg="m"/>
<Returns> a semi echelonised basis </Returns>
<Description>
Same as <Ref Meth="SemiEchelonBasisMutableP"/> but destructive in <A>m</A>.
</Description>
</ManSection>
<ManSection>
<Meth Name="MutableNullspaceMat" Arg="m"/>
<Returns> a <C>cmat</C> </Returns>
<Description>
Returns a <C>cmat</C> the rows of which are a basis of the (left) nullspace
of the <C>cmat</C> <A>m</A>. Internally, <Ref Meth="SemiEchelonBasisMutableT"/>
is used and the component <C>relations</C> of the result is returned.
The result is mutable, which is the reason for the introduction of
a new operation besides <Ref Oper="NullspaceMat" BookName="Ref"/>.
</Description>
</ManSection>
<ManSection>
<Meth Name="MutableNullspaceMatX" Arg="m"/>
<Returns> a <C>cmat</C> </Returns>
<Description>
Same as <Ref Meth="MutableNullspaceMat"/> but destructive in <A>m</A>.
</Description>
</ManSection>
<ManSection>
<Meth Name="NullspaceMat" Arg="m"/>
<Returns> an immutable <C>cmat</C> </Returns>
<Description>
Behaves exactly as expected. <A>m</A> must be a <C>cmat</C>.
</Description>
</ManSection>
<ManSection>
<Meth Name="NullspaceMatDestructive" Arg="m"/>
<Returns> an immutable <C>cmat</C> </Returns>
<Description>
Behaves exactly as expected. <A>m</A> must be a <C>cmat</C>.
</Description>
</ManSection>
</Section>
<Section Label="polynomials">
<Heading>Characteristic and minimal polynomial</Heading>
<ManSection>
<Meth Name="CharacteristicPolynomialOfMatrix" Arg="m[, indetnr]"/>
<Returns> a record </Returns>
<Description>
Calculates the characteristic polynomial of the <C>cmat</C> <A>m</A>
over its base field. Because during the calculations the polynomial
automatically comes as a product of some not necessarily irreducible
factors, the result is returned in a record with two components.
The <C>poly</C> component contains the full characteristic polynomial.
The <C>factors</C> component contains a list of not necessarily
irreducibly factors the product of which is the characteristic polynomial.
If an indeterminate number is given as second argument, the polynomials
are written in that indeterminate, otherwise in the first indeterminate
over the base field.
</Description>
</ManSection>
<ManSection>
<Meth Name="FactorsOfCharacteristicPolynomial" Arg="m[, indetnr]"/>
<Returns> a list </Returns>
<Description>
Calculates the characteristic polynomial of the <C>cmat</C> <A>m</A>
over its base field. The result is a list of irreducible factors of
the characteristic polynomial of <A>m</A>, the product of which is
the characteristic polynomial. Because during the calculations the
polynomial
automatically comes as a product of some not necessarily irreducible
factors, this is more efficient than just calculating the characteristic
polynomial and then factoring it.
If an indeterminate number is given as second argument, the polynomials
are written in that indeterminate, otherwise in the first indeterminate
over the base field.
</Description>
</ManSection>
<ManSection>
<Meth Name="MinimalPolynomialOfMatrix" Arg="m[, indetnr]"/>
<Returns> a polynomial over the base field of <A>m</A> </Returns>
<Description>
Calculates the minimal polynomial of the <C>cmat</C> <A>m</A>
over its base field. The polynomial is returned.
If an indeterminate number is given as second argument, the polynomials
are written in that indeterminate, otherwise in the first indeterminate
over the base field.
</Description>
</ManSection>
<ManSection>
<Meth Name="CharAndMinimalPolynomialOfMatrix" Arg="m[, indetnr]"/>
<Returns> a record </Returns>
<Description>
Calculates the characteristic and minimal polynomial of the
<C>cmat</C> <A>m</A>
over its base field. Because during the calculation of the minimal
polynomial the characteristic polynomial is calculated anyway, the
result is returned in a record with five components:
The <C>charpoly</C> component contains the full characteristic polynomial.
The <C>irreds</C> component contains the set of irreducible factors
of the characteristic polynomial as a list. The <C>mult</C> component
contains a corresponding list of multiplicities, that is in position
<M>i</M> is the multiplicity of the irreducible factor number <M>i</M>
in the characteristic polynomial. The component <C>minpoly</C> contains
the minimal polynomial and the component <C>multmin</C> the corresponding
multiplicities of the irreducible factors of the characteristic polynomial
in the minimal polynomial.
If an indeterminate number is given as second argument, the polynomials
are written in that indeterminate, otherwise in the first indeterminate
over the base field.
</Description>
</ManSection>
<ManSection>
<Oper Name="MinimalPolynomialOfMatrixMC" Arg="m, prob[, indetnr]"/>
<Returns> a record </Returns>
<Description>
This method calculates the characteristic and minimal polynomials of the
row list matrix <A>m</A> over its base domain.
It uses the Monte Carlo algorithm by Neunhoeffer
and Praeger. The second argument <A>prob</A> is an upper bound for the
error probability, it can be <M>0</M> in which case a deterministic
verification is done. The optional argument <A>indetnr</A> is the number
of the indeterminate over the base domain to be used to express polynomials.
<P/>
The result is a record with the following components bound: The component
<C>charpoly</C> is the characteristic polynomial which is guaranteed to
be correct. The component <C>minpoly</C> is always a divisor of the minimal
polynomial and usually is equal to it.
See below for details. The component <C>irreds</C> is a sorted
list of the irreducible factors of the characteristic polynomial. The
component <C>mult</C> is a corresponding list of the same length containing
the multiplicities of these irreducible factors in the characteristic
polynomial. The component <C>minmult</C> is a corresponding list of the same
length containing the multiplicities of these irreducible factors in the
polynomial given in <C>minpoly</C>. The component <C>proof</C> is set to
<C>true</C> if the result is proved to be correct, which can happen even
if <A>prob</A> was non-zero (for example in the case of a cyclic matrix).
The component <C>iscyclic</C> is set to <C>true</C> if and only if the
minimal polynomial is equal to the characteristic polynomial. The component
<C>prob</C> is set to the probability given in <A>prob</A>, if that was
zero then it is set to <M>1/10000</M> but <C>proof</C> will be <C>true</C>.
The components <C>A</C>, <C>B</C> and <C>S</C> describe a base change
for <A>m</A> to a sparse form which is obtained as a byproduct.
<C>S</C> is a semi echelonised basis which was obtained by a spin-up
computation with <A>m</A> and if <M>Y</M> is the sparse basis
then <M>Y = A \cdot S</M> and <M>B=A^{-1}</M>.
<P/>The given result for the minimal polynomial
could be a proper divisor of the real minimal polynomial
if <A>prob</A> was non-zero, however, the probability for this outcome is
guaranteed to be less than or equal to <A>prob</A>. Note that it is
always guaranteed that all irreducible factors of the minimal polynomial are
actually present, it can only happen that the multiplicities are too small.
</Description>
</ManSection>
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