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fffg.spad
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)if false
\documentclass{article}
\usepackage{axiom, amsthm, amsmath, amssymb}
\newtheorem{ToDo}{ToDo}[section]
\begin{document}
\title{fffg.spad}
\author{Martin Rubey}
\maketitle
\begin{abstract}
The packages defined in this file provide fast fraction free rational
interpolation algorithms.
\end{abstract}
\tableofcontents
\section{package FAMR2 FiniteAbelianMonoidRingFunctions2}
)endif
)abbrev package FAMR2 FiniteAbelianMonoidRingFunctions2
++ Author: Martin Rubey
++ Description:
++ This package provides a mapping function for \spadtype{FiniteAbelianMonoidRing}
FiniteAbelianMonoidRingFunctions2(E : OrderedAbelianMonoid,
R1 : Ring,
A1 : FiniteAbelianMonoidRing(R1, E),
R2 : Ring,
A2 : FiniteAbelianMonoidRing(R2, E)) _
: Exports == Implementation where
Exports == with
map : (R1 -> R2, A1) -> A2
++ \spad{map}(f, a) applies the map f to each coefficient in a. It is
++ assumed that f maps 0 to 0
Implementation == add
map(f : R1 -> R2, a : A1) : A2 ==
if zero? a then 0$A2
else
monomial(f leadingCoefficient a, degree a)$A2 + map(f, reductum a)
)if false
\section{package FFFG FractionFreeFastGaussian}
)endif
)abbrev package FFFG FractionFreeFastGaussian
++ Author: Martin Rubey
++ Description:
++ This package implements the interpolation algorithm proposed in Beckermann,
++ Bernhard and Labahn, George, Fraction-free computation of matrix rational
++ interpolants and matrix GCDs, SIAM Journal on Matrix Analysis and
++ Applications 22.
FractionFreeFastGaussian(D, V) : Exports == Implementation where
D : Join(IntegralDomain, GcdDomain)
V : AbelianMonoidRing(D, NonNegativeInteger) -- for example, SUP D
SUP ==> SparseUnivariatePolynomial
cFunction ==> (NonNegativeInteger, Vector SUP D) -> D
CoeffAction ==> (NonNegativeInteger, NonNegativeInteger, V) -> D
Exports == with
fffg : (List D, cFunction, Vector Integer, NonNegativeInteger) -> Matrix SUP D
++ \spad{fffg(C, c, vd, K)} is the general algorithm as proposed
++ by Beckermann and Labahn.
++
++ The first argument is the list of c_{i, i}. These are the only values
++ of C explicitly needed in \spad{fffg}.
++
++ The second argument c, computes c_k(M), i.e., c_k(.) is the dual basis
++ of the vector space V, but also knows about the special multiplication
++ rule as described in Equation (2). Note that the information about f
++ is therefore encoded in c.
++
++ vd is modified by the routine, on input it is the vector of degree
++ bounds n, as introduced in Definition 2.1. On output it is
++ vector of defects (degree bound minus degree of solution).
++
++ K is requested order of solution.
fffg : (List D, cFunction, List NonNegativeInteger) -> Matrix SUP D
++ \spad{fffg(C, c, eta)} is version of fffg which uses sum
++ of eta as order
interpolate : (List D, List D, NonNegativeInteger) -> Fraction SUP D
++ \spad{interpolate(xlist, ylist, deg)} returns the rational
++ function with
++ numerator degree at most \spad{deg} and denominator degree at most
++ \spad{#xlist-deg-1} that interpolates the given points using
++ fraction free arithmetic. Note that rational interpolation does not
++ guarantee that all given points are interpolated correctly:
++ unattainable points may make this impossible.
)if false
\begin{ToDo}
The following function could be moved to [[FFFGF]], parallel to
[[generalInterpolation]]. However, the reason for moving
[[generalInterpolation]] for fractions to a separate package was the need of
a generic signature, hence the extra argument [[VF]] to [[FFFGF]]. In the
special case of rational interpolation, this extra argument is not necessary,
since we are always returning a fraction of [[SUP]]s, and ignore [[V]]. In
fact, [[V]] is not needed for [[fffg]] itself, only if we want to specify a
[[CoeffAction]].
Thus, maybe it would be better to move [[fffg]] to a separate package?
\end{ToDo}
)endif
interpolate : (List Fraction D, List Fraction D, NonNegativeInteger)
-> Fraction SUP D
++ \spad{interpolate(xlist, ylist, deg)} returns the rational
++ function with
++ numerator degree \spad{deg} that interpolates the given points using
++ fraction free arithmetic.
generalInterpolation : (List D, CoeffAction,
Vector V, List NonNegativeInteger) -> Matrix SUP D
++ \spad{generalInterpolation(C, CA, f, eta)} performs Hermite-Pade
++ approximation using the given action CA of polynomials on the elements
++ of f. The result is guaranteed to be correct up to order
++ |eta|-1. Given that eta is a "normal" point, the degrees on the
++ diagonal are given by eta. The degrees of column i are in this case
++ eta + e.i - [1, 1, ..., 1], where the degree of zero is -1.
++
++ The first argument C is the list of coefficients c_{k, k} in the
++ expansion <x^k> z g(x) = sum_{i=0}^k c_{k, i} <x^i> g(x).
++
++ The second argument, CA(k, l, f), should return the coefficient of x^k
++ in z^l f(x).
generalInterpolation : (List D, CoeffAction, Vector V,
Vector Integer, NonNegativeInteger) -> Matrix SUP D
++ \spad{generalInterpolation(C, CA, f, vd, K)} is like
++ \spad{generalInterpolation(C, CA, f, eta)} but solves up to
++ order K and modifies vd to return defects of solutions
generalCoefficient : (CoeffAction, Vector V,
NonNegativeInteger, Vector SUP D) -> D
++ \spad{generalCoefficient(action, f, k, p)} gives the coefficient of
++ x^k in p(z)\dot f(x), where the action of z^l on a polynomial in x is
++ given by action, i.e., action(k, l, f) should return the coefficient
++ of x^k in z^l f(x).
ShiftAction : (NonNegativeInteger, NonNegativeInteger, V) -> D
++ \spad{ShiftAction(k, l, g)} gives the coefficient of x^k in z^l g(x),
++ where \spad{z*(a+b*x+c*x^2+d*x^3+...) = (b*x+2*c*x^2+3*d*x^3+...)}. In
++ terms of sequences, z*u(n)=n*u(n).
ShiftC : NonNegativeInteger -> List D
++ \spad{ShiftC} gives the coefficients c_{k, k} in the expansion <x^k> z
++ g(x) = sum_{i=0}^k c_{k, i} <x^i> g(x), where z acts on g(x) by
++ shifting. In fact, the result is [0, 1, 2, ...]
DiffAction : (NonNegativeInteger, NonNegativeInteger, V) -> D
++ \spad{DiffAction(k, l, g)} gives the coefficient of x^k in z^l g(x),
++ where z*(a+b*x+c*x^2+d*x^3+...) = (a*x+b*x^2+c*x^3+...), i.e.,
++ multiplication with x.
DiffC : NonNegativeInteger -> List D
++ \spad{DiffC} gives the coefficients c_{k, k} in the expansion <x^k> z
++ g(x) = sum_{i=0}^k c_{k, i} <x^i> g(x), where z acts on g(x) by
++ shifting. In fact, the result is [0, 0, 0, ...]
qShiftAction : (D, NonNegativeInteger, NonNegativeInteger, V) -> D
++ \spad{qShiftAction(q, k, l, g)} gives the coefficient of x^k in z^l
++ g(x), where z*(a+b*x+c*x^2+d*x^3+...) =
++ (a+q*b*x+q^2*c*x^2+q^3*d*x^3+...). In terms of sequences,
++ z*u(n)=q^n*u(n).
qShiftC : (D, NonNegativeInteger) -> List D
++ \spad{qShiftC} gives the coefficients c_{k, k} in the expansion <x^k> z
++ g(x) = sum_{i=0}^k c_{k, i} <x^i> g(x), where z acts on g(x) by
++ shifting. In fact, the result is [1, q, q^2, ...]
genVectorStream : (NonNegativeInteger, NonNegativeInteger, _
NonNegativeInteger) -> Stream List NonNegativeInteger
++ \spad{genVectorStream(sumEta, maxEta, k)} generates stream
++ of all possible non-increasing lists \spad{eta}
++ with maximal entry \spad{maxEta} and sum of entries at most
++ \spad{sumEta}.
genVectorStream2 : (NonNegativeInteger, NonNegativeInteger, _
NonNegativeInteger) -> Stream List NonNegativeInteger
++ genVectorStream2 is like genVectorStream, but skips every second
++ vector.
Implementation ==> add
-------------------------------------------------------------------------------
-- Shift Operator
-------------------------------------------------------------------------------
-- ShiftAction(k, l, f) is the CoeffAction appropriate for the shift operator.
ShiftAction(k : NonNegativeInteger, l : NonNegativeInteger, f : V) : D ==
k^l*coefficient(f, k)
ShiftC(total : NonNegativeInteger) : List D ==
[i::D for i in 0..total-1]
-------------------------------------------------------------------------------
-- q-Shift Operator
-------------------------------------------------------------------------------
-- q-ShiftAction(k, l, f) is the CoeffAction appropriate for the q-shift operator.
qShiftAction(q : D, k : NonNegativeInteger, l : NonNegativeInteger, f : V) : D ==
q^(k*l)*coefficient(f, k)
qShiftC(q : D, total : NonNegativeInteger) : List D ==
[q^i for i in 0..total-1]
-------------------------------------------------------------------------------
-- Differentiation Operator
-------------------------------------------------------------------------------
-- DiffAction(k, l, f) is the CoeffAction appropriate for the differentiation
-- operator.
DiffAction(k : NonNegativeInteger, l : NonNegativeInteger, f : V) : D ==
if k < l then 0 else coefficient(f, (k-l)::NonNegativeInteger)
DiffC(total : NonNegativeInteger) : List D ==
[0 for i in 1..total]
-------------------------------------------------------------------------------
-- general - suitable for functions f
-------------------------------------------------------------------------------
-- get the coefficient of z^k in the scalar product of p and f, the action
-- being defined by coeffAction
generalCoefficient(coeffAction : CoeffAction, f : Vector V,
k : NonNegativeInteger, p : Vector SUP D) : D ==
res : D := 0
for i in 1..#f repeat
-- Defining a and b and summing only over those coefficients that might be
-- nonzero makes a huge difference in speed
a := f.i
b := p.i
for l in minimumDegree b..degree b repeat
if not zero? coefficient(b, l)
then res := res + coefficient(b, l) * coeffAction(k, l, a)
res
generalInterpolation(C : List D, coeffAction : CoeffAction,
f : Vector V,
eta : List NonNegativeInteger) : Matrix SUP D ==
c : cFunction := (x, y) +-> generalCoefficient(coeffAction, f,
(x - 1)::NonNegativeInteger, y)
fffg(C, c, eta)
generalInterpolation(C : List D, coeffAction : CoeffAction,
f : Vector V, vd : Vector Integer,
K : NonNegativeInteger) : Matrix SUP D ==
c : cFunction := (x, y) +-> generalCoefficient(coeffAction, f,
(x - 1)::NonNegativeInteger, y)
fffg(C, c, vd, K)
-------------------------------------------------------------------------------
-- general - suitable for functions f - trying all possible degree combinations
-------------------------------------------------------------------------------
-- The following function returns the lexicographically next vector with
-- non-negative components smaller than [[p]] with the same sum as [[v]].
nextVector!(p : NonNegativeInteger, v : List NonNegativeInteger)
: Union("failed", List NonNegativeInteger) ==
n := #v
pos := position(x +-> x < p, v)
zero? pos => return "failed"
if pos = 1 then
sum : Integer := v.1
for i in 2..n repeat
if v.i < p and sum > 0 then
v.i := v.i + 1
sum := sum - 1
for j in 1..i-1 repeat
if sum > p then
v.j := p
sum := sum - p
else
v.j := sum::NonNegativeInteger
sum := 0
return v
else sum := sum + v.i
return "failed"
else
v.pos := v.pos + 1
v.(pos-1) := (v.(pos-1) - 1)::NonNegativeInteger
v
)if false
The following function returns the stream of all possible degree vectors,
beginning with [[v]], where the degree vectors are sorted in reverse
lexicographic order. Furthermore, the entries are all less or equal to [[p]]
and their sum equals the sum of the entries of [[v]]. We assume that the
entries of [[v]] are also all less or equal to [[p]].
)endif
vectorStream(p : NonNegativeInteger, v : List NonNegativeInteger)
: Stream List NonNegativeInteger == delay
next := nextVector!(p, copy v)
(next case "failed") => empty()$Stream(List NonNegativeInteger)
cons(next, vectorStream(p, next))
-- [[vectorStream2]] skips every second entry of [[vectorStream]].
vectorStream2(p : NonNegativeInteger, v : List NonNegativeInteger)
: Stream List NonNegativeInteger == delay
next := nextVector!(p, copy v)
(next case "failed") => empty()$Stream(List NonNegativeInteger)
next2 := nextVector!(p, copy next)
(next2 case "failed") => cons(next, empty())
cons(next2, vectorStream2(p, next2))
)if false
If [[f]] consists of two elements only, we can skip every second degree vector:
note that [[fffg]], and thus also [[generalInterpolation]], returns a matrix
with [[#f]] columns, each corresponding to a solution of the interpolation
problem. More precisely, the $i$\textsuperscript{th} column is a solution with
degrees [[eta]]$-(1, 1, \dots, 1)+e_i$. Thus, in the case of $2\times 2$ matrices,
[[vectorStream]] would produce solutions corresponding to $(d, 0), (d-1, 1);
(d-1, 1), (d-2, 2); (d-2, 2), (d-3, 3)\dots$, i.e., every second matrix is
redundant.
Although some redundancy exists also for higher dimensional [[f]], the scheme
becomes much more complicated, thus we did not implement it.
We need to generate an initial degree vector, being the minimal element in
reverse lexicographic order, i.e., $m, m, \dots, m, k, 0, 0, \dots$, where $m$
is [[maxEta]] and $k$ is the remainder of [[sumEta]] divided by
[[maxEta]]. This is done by the following code:
)endif
initialVector(sum : NonNegativeInteger, maxEta : NonNegativeInteger,
n : NonNegativeInteger)_
: List NonNegativeInteger ==
entry : Integer
[(if sum < maxEta _
then (entry := sum; sum := 0) _
else (entry := maxEta; sum := (sum - maxEta)::NonNegativeInteger); _
entry::NonNegativeInteger) for i in 1..n]
genVectorStream(sum : NonNegativeInteger, max : NonNegativeInteger,
k : NonNegativeInteger) : Stream List NonNegativeInteger ==
eta := initialVector(sum, max, k)
cons(eta, vectorStream(max, eta))
genVectorStream2(sum : NonNegativeInteger, max : NonNegativeInteger,
k : NonNegativeInteger) : Stream List NonNegativeInteger ==
eta := initialVector(sum, max, k)
cons(eta, vectorStream2(max, eta))
-------------------------------------------------------------------------------
-- rational interpolation
-------------------------------------------------------------------------------
interpolate(x : List Fraction D, y : List Fraction D, d : NonNegativeInteger)
: Fraction SUP D ==
gx := splitDenominator(x)$InnerCommonDenominator(D, Fraction D, _
List D, _
List Fraction D)
gy := splitDenominator(y)$InnerCommonDenominator(D, Fraction D, _
List D, _
List Fraction D)
r := interpolate(gx.num, gy.num, d)
elt(numer r, monomial(gx.den, 1))/(gy.den*elt(denom r, monomial(gx.den, 1)))
interpolate(x : List D, y : List D, d : NonNegativeInteger) : Fraction SUP D ==
-- compute interpolants of grade d and N-d-1
if (N := #x) ~= #y then
error "interpolate: number of points and values must match"
if N <= d then
error "interpolate: numerator degree must be smaller than number of data points"
c : cFunction := (s, u) +-> y.s * elt(u.2, x.s) - elt(u.1, x.s)
eta : List NonNegativeInteger := [d, (N-d)::NonNegativeInteger]
M := fffg(x, c, eta)
if zero?(M.(2, 1)) then M.(1, 2)/M.(2, 2)
else M.(1, 1)/M.(2, 1)
)if false
Because of Lemma~5.3, [[M.1.(2, 1)]] and [[M.1.(2, 2)]] cannot both vanish,
since [[eta_sigma]] is always $\sigma$-normal by Theorem~7.2 and therefore also
para-normal, see Definition~4.2.
Because of Lemma~5.1 we have that [[M.1.(*, 2)]] is a solution of the
interpolation problem, if [[M.1.(2, 1)]] vanishes.
-------------------------------------------------------------------------------
-- fffg
-------------------------------------------------------------------------------
[[recurrence]] computes the new matrix $M$, according to the following formulas
(cf. Table~2 in Beckermann and Labahn):
\begin{align*}
&\text{Increase order}\\
&\quad\quad\text{for $\ell=1\dots m$, $\ell\neq\pi$}\\
&\quad\quad\quad\quad\mathbf M_{\sigma+1}^{(., \ell)} :=
\left(\mathbf M_{\sigma}^{(., \ell)}r^{(\pi)}
- \mathbf M_{\sigma}^{(., \pi)}r^{(\ell)}\right)/d_\sigma\\
&\text{Increase order in column $\pi$}\\
&\quad\quad\mathbf M_{\sigma+1}^{(., \pi)} :=
\left(z-c_{\sigma, \sigma}\right)\mathbf M_{\sigma}^{(., \pi)}\\
&\text{Adjust degree constraints : }\\
&\quad\quad\mathbf M_{\sigma+1}^{(., \pi)} :=
\left(\mathbf M_{\sigma+1}^{(., \pi)}r^{(\pi)}
- \sum_{\ell\neq\pi}\mathbf M_{\sigma+1}^{(., \ell)}p^{(\ell)}
\right)/d_\sigma
\end{align*}
Since we do not need the matrix $\mathbf M_{\sigma}$ anymore, we drop the index
and update the matrix destructively. In the following, we write [[Ck]] for
$c_{\sigma, \sigma}$.
)endif
-- a major part of the time is spent here
recurrence(M : Matrix SUP D, pi : NonNegativeInteger,
m : NonNegativeInteger, r : Vector D, d : D, z : SUP D,
Ck : D, p : Vector D, vd : Vector Integer) : Matrix SUP D ==
rPi : D := qelt(r, pi)
polyf : SUP D := rPi * (z - Ck::SUP D)
for i in 1..m repeat
MiPi : SUP D := qelt(M, i, pi)
newMiPi : SUP D := polyf * MiPi
-- update columns ~= pi and calculate their sum
for l in 1..m | l ~= pi and vd(l) >= 0 repeat
rl : D := qelt(r, l)
-- I need the coercion to SUP D, since exquo returns an element of
-- Union("failed", SUP D)...
Mil : SUP D := ((qelt(M, i, l) * rPi - MiPi * rl) exquo d)::SUP D
qsetelt!(M, i, l, Mil)
pl : D := qelt(p, l)
newMiPi := newMiPi - pl * Mil
-- update column pi
qsetelt!(M, i, pi, (newMiPi exquo d)::SUP D)
M
fffg(C : List D, c : cFunction, vd : Vector Integer,
K : NonNegativeInteger) : Matrix SUP D ==
z : SUP D := monomial(1, 1)
m : NonNegativeInteger := #vd
M : Matrix SUP D := scalarMatrix(m, 1)
d : D := 1
etak : Vector NonNegativeInteger := zero(m)
r : Vector D := zero(m)
p : Vector D := zero(m)
lambdaMax : Integer
lambda : NonNegativeInteger
for k in 1..K repeat
-- k = sigma+1
lambda := 0
lambdaMax := -1
for l in 1..m repeat
(vdl := vd(l)) < 0 => iterate
r(l) := c(k, column(M, l))
r(l) = 0 => iterate
if vdl > lambdaMax then
lambdaMax := vdl
lambda := l
-- if Lambda is empty, then M, d and etak remain unchanged. Otherwise, we look
-- for the next closest para-normal point.
lambda = 0 => iterate
-- Calculate leading coefficients
for l in 1..m | l ~= lambda repeat
if etak(l) > 0 and vd(l) >= 0 then
p(l) := coefficient(M(l, lambda),
(etak(l) - 1)::NonNegativeInteger)
else
p(l) := 0
-- increase order and adjust degree constraints
M := recurrence(M, lambda, m, r, d, z, C.k, p, vd)
d := r.lambda
etak(lambda) := etak(lambda) + 1
vd(lambda) := vd(lambda) - 1
M
fffg(C : List D, c : cFunction, eta : List NonNegativeInteger
) : Matrix SUP D ==
vd : Vector Integer := vector([ei::Integer for ei in eta])
fffg(C, c, vd, reduce(_+, eta))
)if false
\section{package FFFGF FractionFreeFastGaussianFractions}
)endif
)abbrev package FFFGF FractionFreeFastGaussianFractions
++ Author: Martin Rubey
++ Description:
++ This package lifts the interpolation functions from
++ \spadtype{FractionFreeFastGaussian} to fractions.
FractionFreeFastGaussianFractions(D, V, VF) : Exports == Implementation where
D : Join(IntegralDomain, GcdDomain)
V : FiniteAbelianMonoidRing(D, NonNegativeInteger)
VF : FiniteAbelianMonoidRing(Fraction D, NonNegativeInteger)
F ==> Fraction D
SUP ==> SparseUnivariatePolynomial
FFFG ==> FractionFreeFastGaussian
FAMR2 ==> FiniteAbelianMonoidRingFunctions2
cFunction ==> (NonNegativeInteger, Vector SUP D) -> D
CoeffAction ==> (NonNegativeInteger, NonNegativeInteger, V) -> D
-- coeffAction(k, l, f) is the coefficient of x^k in z^l f(x)
Exports == with
generalInterpolation : (List D, CoeffAction, Vector VF, List NonNegativeInteger)
-> Matrix SUP D
++ \spad{generalInterpolation(l, CA, f, eta)} performs Hermite-Pade
++ approximation using the given action CA of polynomials on the elements
++ of f. The result is guaranteed to be correct up to order
++ |eta|-1. Given that eta is a "normal" point, the degrees on the
++ diagonal are given by eta. The degrees of column i are in this case
++ eta + e.i - [1, 1, ..., 1], where the degree of zero is -1.
Implementation == add
multiplyRows!(v : Vector D, M : Matrix SUP D) : Matrix SUP D ==
n := #v
for i in 1..n repeat
for j in 1..n repeat
M.(i, j) := v.i*M.(i, j)
M
generalInterpolation(C : List D, coeffAction : CoeffAction,
f : Vector VF, eta : List NonNegativeInteger) : Matrix SUP D ==
n := #f
g : Vector V := new(n, 0)
den : Vector D := new(n, 0)
for i in 1..n repeat
c := coefficients(f.i)
den.i := commonDenominator(c)$CommonDenominator(D, F, List F)
g.i := map(x +-> retract(x*den.i)@D, f.i)
$FAMR2(NonNegativeInteger, Fraction D, VF, D, V)
M := generalInterpolation(C, coeffAction, g, eta)$FFFG(D, V)
-- The following is necessary since I'm multiplying each row with a factor, not
-- each column. Possibly I could factor out gcd den, but I'm not sure whether
-- this is efficient.
multiplyRows!(den, M)
)if false
\section{package NEWTON NewtonInterpolation}
)endif
)abbrev package NEWTON NewtonInterpolation
++ Description:
++ This package exports Newton interpolation for the special case where the
++ result is known to be in the original integral domain
NewtonInterpolation F : Exports == Implementation where
F : IntegralDomain
Exports == with
newton : List F -> SparseUnivariatePolynomial F
++ \spad{newton}(l) returns the interpolating polynomial for the values
++ l, where the x-coordinates are assumed to be [1, 2, 3, ..., n] and the
++ coefficients of the interpolating polynomial are known to be in the
++ domain F. I.e., it is a very streamlined version for a special case of
++ interpolation.
Implementation == add
differences(yl : List F) : List F ==
[y2-y1 for y1 in yl for y2 in rest yl]
z : SparseUnivariatePolynomial(F) := monomial(1, 1)
-- we assume x=[1, 2, 3, ..., n]
newtonAux(k : F, fact : F, yl : List F) : SparseUnivariatePolynomial(F) ==
if empty? rest yl
then ((yl.1) exquo fact)::F::SparseUnivariatePolynomial(F)
else ((yl.1) exquo fact)::F::SparseUnivariatePolynomial(F)
+ (z-k::SparseUnivariatePolynomial(F)) _
* newtonAux(k+1$F, fact*k, differences yl)
newton yl == newtonAux(1$F, 1$F, yl)
--Copyright (c) 2006-2007, Martin Rubey <Martin.Rubey@univie.ac.at>
--
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--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
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--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
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