)abbrev package VHPSOLV VectorHermitePadeSolver
VectorHermitePadeSolver : Exports == Implementation where
  F ==> Expression(Integer)
  SUP ==> SparseUnivariatePolynomial(F)
  Exports ==> with
      hp_solve : (List Vector(SUP), List(NonNegativeInteger),
                  NonNegativeInteger) -> Matrix SUP
        ++ hp_solve(lv, eta, K) solves Hermite-Pade problem with degree
        ++ bound eta up to order K.
  Implementation ==> add

    FFFG ==> FractionFreeFastGaussian(F, SUP)

    power_action(m : NonNegativeInteger
                ) : ((NonNegativeInteger, NonNegativeInteger, SUP) -> F) ==
        (k, l, g) +-> DiffAction(k, m*l, g)$FFFG

    hp_solve(lv, eta, K) ==
        m := #first(lv)
        lpp : List(SUP) := []
        for v in lv repeat
            #v ~= m =>
                error "hp_solve: vectors must be of the same length"
            pp : SUP := 0
            for i in 1..m repeat
                pp1 := multiplyExponents(v(i), m)
                pp := pp + monomial(1, (i-1)::NonNegativeInteger)$SUP*pp1
            lpp := cons(pp, lpp)
        lpp := reverse!(lpp)
        vd : Vector Integer := vector([ei::Integer for ei in eta])
        C : List(F) := [0 for i in 1..K]
        res1 := generalInterpolation(C, power_action(m), vector(lpp),
                                     vd, K)$FFFG
        n := #vd
        n1 : NonNegativeInteger := 0
        for i in 1..n | vd(i) >= 0 repeat
            n1 := n1 + 1
        res := new(n, n1, 0)$Matrix(SUP)
        i1 : NonNegativeInteger := 0
        for i in 1..n | vd(i) >= 0 repeat
            i1 := i1 + 1
            for j in 1..n repeat
                res(j, i1) := res1(j, i)
        res

)abbrev package LODOF2 LinearOrdinaryDifferentialOperatorFactorizer2
)boot $tryRecompileArguments := nil
++ Description:
++ \spadtype{LinearOrdinaryDifferentialOperatorFactorizer2} provides a
++ factorizer for linear ordinary differential operators whose coefficients
++ are rational functions.
++ Keywords: differential equation, ODE, LODO, factoring
LinearOrdinaryDifferentialOperatorFactorizer2(var, cen) : Exports == Impl where
  var : Symbol
  cen : Expression Integer

  F ==> Expression Integer
  K ==> Kernel F
  P ==> SparseMultivariatePolynomial(Integer, K)
  UP ==> UnivariatePolynomial(var, F)
  RF ==> Fraction UP
  SUP ==> SparseUnivariatePolynomial(F)
  L  ==> LinearOrdinaryDifferentialOperator1 RF
  L3 ==> LinearOrdinaryDifferentialOperator3(F, UP, RF)
  UTS ==> UnivariateTaylorSeries(F, var, cen)
  US ==> UnivariateLaurentSeries(F, var, cen)
  LL ==> LinearOrdinaryDifferentialOperator3(F, UTS, US)
  PO ==> Record(x : Integer, y : Integer)
  FZ ==> Fraction Integer
  FA ==> Record(factor : UP, exponent : NonNegativeInteger)
  F2 ==> Record(Qt : LL, Rt : LL)
  NL ==> Record(main : F2, nf : F2, error : LL)
  LW ==> Record(lr : NL, n_lifts : PositiveInteger)
  TERM ==> Record(k : Integer, c : F)
  LC ==> Record(llc : List Stream TERM, rlc : List Stream TERM)
  LAU ==> Record(laurl : List US, laurr : List US)
  SP ==> List(Record(op : LL, ram : UP, expart : US))
  LA ==> Record(ope : LL, dext : PositiveInteger, alpha : F, rami : UP)
  RS ==> List Union(LL, LA)
  FRI ==> Union(SP, RS)
  SG ==> Record(point : Union(F, "infinity"), lpf : LL, dxt : PositiveInteger)
  CB ==> List SG
  FG ==> List Record(sing : SG, arf : RS)
  TF ==> Union(List L, "failed")
  T2 ==> Union(List SUP, "failed")
  PL ==> Record(apr : List Polynomial UTS, ac : Integer)
  GE ==> List Record(ecs : List US, ecr : UP, ect : RF)
  POLYCATQ ==> PolynomialCategoryQuotientFunctions(IndexedExponents(K),
                                                   K, Integer, P, F)
  GEM ==> List Record(singularity : SG, fos : SP, mge : List US)

  Exports ==> with
    testnp : L3 -> List(Record(point : PO, slope : FZ, npoly : UP))
    ++ test function.
    testfn : (L3, SUP -> Factored(SUP), String) -> List LL
    ++ test function.
    testf2 : (L3, UP, FZ) -> F2
    ++ test function.
    testll : (FZ, L3, UP, UP, FZ, Integer) -> LAU
    ++ test function.
    testlc : (FZ, L3, UP, UP, FZ, Integer) -> LC
    ++ test function.
    testln : (FZ, L3, UP, UP, FZ, Integer) -> NL
    ++ test function.
    testlw : (FZ, L3, UP, UP, FZ, Integer) -> Stream F2
    ++ test function.
    testcp : (L3, FZ, Integer) -> LL
    ++ test function.
    testco : (L3, FZ, Integer) -> Boolean
    ++ test function.
    testcx : (L3, FZ) -> UP
    ++ test function.
    testpd : UP -> LL
    ++ test function.
    testfr : (L3, SUP -> Factored(SUP), String) -> FRI
    ++ test function.
    testsb : (Union(UP, US), L3) -> LL
    ++ test function.
    testro : (L3, F, PositiveInteger) -> LL
    ++ test function.
    testfo : (L3, SUP -> Factored(SUP), String) -> FRI
    ++ test function.
    testsc : (L3, L3, SUP -> Factored(SUP)) -> Boolean
    ++ test function.
    testfg : (L, SUP -> Factored(SUP)) -> List L
    ++ test function.
    testlp : (L, Union(F, "infinity")) -> LL
    ++ test function.
    factor : (L, (SUP, List F) -> Factored(SUP), List F) -> List L
    ++ to be documented later.
    testis : L -> Boolean
    ++ test function.
    testgt : (F, K) -> F
    ++ test function.
    gen_exp : (L, Union(F, "infinity"), (SUP, List F) -> Factored(SUP), List F) -> GE
    ++ to be documented later.
    ge_minimal : (L, SUP -> Factored(SUP), String) -> GEM
    ++ to be documented later.

  Impl ==> add
    import from IntegerRetractions(F)
    import from StreamFunctions2(LW, F2)
    import from StreamFunctions2(F2, LL)
    import from StreamFunctions3(LL, Integer, TERM)
    import from US
    import from Integer
    import from Fraction(Integer)
    import from Factored(UP)
    import from UP
    import from RS
    import from RF

    convertUPtoUTS : UP -> UTS
    convertUTStoUP : (UTS, Integer) -> UP
    convertL3toLL : L3 -> LL
    newtonpolygonPoints : LL -> List PO
    newtonpolygon : LL -> List(Record(point : PO, slope : FZ, npoly : UP))
    factor_newton : (LL, SUP -> Factored(SUP), String) -> List LL
    factorUP : (UP, SUP -> Factored(SUP)) -> Factored(UP)
    factor_newton2 : (LL, UP, Record(point : PO, slope : FZ, npoly : UP)) -> F2
    laurent_op : (List US, Integer) -> LL
    list_laurent : (LC, Integer, Integer) -> LAU
    list_laurentop : (List Stream TERM, Integer) -> List US
    listream_coefs : (Stream F2, Integer, Integer, FZ, FZ) -> LC
    listream_coefsop : (Stream LL, Integer, Integer, FZ) -> List Stream TERM
    lift_newton : (FZ, LL, UP, UP, FZ, Integer, F2, LL, PositiveInteger) -> NL
    ln_wrapper : (FZ, LL, UP, UP, FZ, Integer) -> Stream F2
    coefs_poly : (LL, FZ, Integer) -> UP
    coefs_operator : (UP, FZ, Integer) -> LL
    coeffx : (LL, FZ) -> UP
    plug_delta : UP -> LL
    factor_riccati : (LL, SUP -> Factored(SUP), String) -> FRI
    substitute : (Union(UP, US), Union(LL, LA)) -> Union(LL, LA)
    ramification_of : (LL, F, PositiveInteger) -> LL
    factor_op : (LL, SUP -> Factored(SUP), String) -> FRI
    make_rightfactor : (LL, LA) -> LL
    lift_rightfactor : (PositiveInteger, LA) -> LL
    same_charclass? : (LL, LL, SUP -> Factored(SUP)) -> Boolean
    try_factorization : (LL, Integer, List FZ, SG, L, Integer, Integer, String) -> TF
    try_factorization2 : (List List UTS, Integer, List FZ, Integer) -> T2
    flist : (LL, Integer) -> List List US
    xDn_modr : (Integer, LL) -> LL
    factor_global : (L, SUP -> Factored(SUP)) -> List L
    factor_minmult1 : (List FZ, SG, L, Integer, Integer, LL, SUP -> Factored(SUP)) -> List L
    l_p : (L, Union(F, "infinity")) -> LL
    subsup : (UP, RF) -> RF
    compute_bound : (GEM, F) -> Integer

    convertUPtoUTS(np) ==
        np :: UTS

    convertUTStoUP(ns, n) ==
        univariatePolynomial(ns, qcoerce(n-1))

    convertL3toLL(f) ==
        apf : LL := 0
        ftmp := f
        while ftmp ~= 0 repeat
            lc := leadingCoefficient(ftmp)
            den := convertUPtoUTS(denom lc) :: US
            apf := (apf +
                    monomial((convertUPtoUTS(numer lc) *
                    recip(den) :: US), degree(ftmp)))
            ftmp := reductum ftmp
        apf

-- newtonpolygonPoints takes a LinearOrdinaryDifferentialOperator3 having
-- Laurent series coefficients as input and returns the coordinates of the
-- extreme points of the Newton polygon.
    newtonpolygonPoints(f: LL) : List PO ==
        ftmp := f
        points : List PO := []
        yzero := order(leadingCoefficient(ftmp))
        while ftmp ~= 0 repeat
            xv := degree(ftmp)
            yv := order(leadingCoefficient(ftmp), order(leadingCoefficient(f)))
            if xv ~= 0 then
                points := cons([xv, yv], points)
            yzero := min(yzero, yv)
            ftmp := reductum ftmp
        points := cons([0, yzero], points)
        npg : List PO := [points.2, points.1]
        points3 := rest(rest(points))
        for i in points3 repeat
            ccw : Boolean := true
            while #npg > 1 and ccw repeat
                p1 := npg.2
                p2 := npg.1
                sb1 := (p2.x - p1.x) * (i.y - p1.y)
                sb2 := (p2.y - p1.y) * (i.x - p1.x)
                ccw := (sb1 - sb2) <= 0
                if ccw then
                    npg := rest(npg)
            npg := cons(i, npg)
        reverse!(npg)

-- newtonpolygon takes a LinearOrdinaryDifferentialOperator3 having Laurent
-- series coefficients as input and returns the coordinates of the extreme
-- points of the Newton polygon, the slope of a point to the next point and the
-- Newton polynomial of this slope.
    newtonpolygon(f) ==
        npg : List PO := newtonpolygonPoints(f)
        res : List(Record(point : PO, slope : FZ, npoly : UP)) := []
        restl := rest(npg)
        for i in npg for nxt in restl repeat
            -- now we compute the Newton polynomial of slope number i
            slop : FZ := (nxt.y - i.y) / (nxt.x - i.x)
            npol : UP := 0
            for j in 0..qcoerce((nxt.x - i.x) quo denom(slop)) repeat
                ind : NonNegativeInteger := qcoerce(denom(slop) * j + i.x)
                opc := coefficient(f, ind)
                lsc := coefficient(opc, j * numer(slop) + i.y)
                npol := npol + monomial(lsc, j)
            res := append(res, [[i, slop, npol]])
        res

    testnp(f) ==
        newtonpolygon(convertL3toLL(f))

-- factor_newton takes a monic LinearOrdinaryDifferentialOperator3 having
-- Laurent series coefficients, a factoring routine for polynomials and a
-- string as input and returns the coprime index 1 factorizations.
-- The possible option strings are:
-- "split over k((x))" results in a complete Newton factorization, i.e. the
-- broken Newton polygon and the gcd 1 reducible Newton polynomial cases will
-- be factored.
-- "alg factor" gives only one right factor, the one where the slope is the
-- least steep.
-- "semireg" gives a coprime index 1 LCLM factorization, to be used for
-- computing the semi-regular parts.
-- any other string as the third argument gives all possible safe right factors
-- according to the Newton method.
-- References : Section 5 of
-- http://www.math.fsu.edu/~hoeij/compalg/diffop/versie9_3.pdf
    factor_newton(f, factorizer, option) ==
        if degree(f) <= 1 then
            return [f]
        np := newtonpolygon(f)
        res : List LL := []
        unsafe : List FA
        for k in np repeat
            npol := k.npoly
            v := factors(factorUP(npol, factorizer))
            if k.slope ~= 0 then
                v := [[il.factor ^ il.exponent, 0] for il in v]
            else
                -- regular singular case, we compute the unsafe factors
                unsafe := []
                semi : OneDimensionalArray(UP) := new(#v, 0)
                for i in v for ii in 1..#v repeat
                    semi(ii) := i.factor ^ i.exponent
                for i in v for ii in 1..#v for restl in tails(rest(v)) repeat
                    vif : UP := i.factor
                    dvif := degree(vif)
                    for j in restl for jj in (ii+1)..#v repeat
                        vjf : UP := j.factor
                        if dvif = degree(vjf) then
                            d := qcoerce(dvif - 1)
                            e : F := coefficient(vif, d) - coefficient(vjf, d)
                            e := e / ((dvif :: Integer) :: F)
                            if integer?(e) and e ~= 0 then
                                ie := integer(e)
                                newx : UP := monomial(1,1) + ie :: UP
                                sbt := eval(vjf, monomial(1,1), newx)
                                if (vif - sbt) = 0 then
                                    if ie > 0 then
                                        unsafe := cons(i, unsafe)
                                        semi(jj) := semi(jj) * vif ^ i.exponent
                                    else
                                        unsafe := cons(j, unsafe)
                                        semi(ii) := semi(ii) * vjf ^ j.exponent
                v_old := v
                v := setDifference(v, unsafe)
                if option = "semireg" then
                    v := [[semi(position(i, v_old)), 1] for i in v]
            -- now v contains the safe right factors of slope number k
            slo := k.slope
            if degree(v(1).factor) * denom(slo) = degree(f) then
                -- f allows no coprime index 1 factorization
                return [f]
            for i in v repeat
                jl : F2 := factor_newton2(f, i.factor, k)
                if option = "alg factor" then
                    return [jl.Rt]
                else if option = "split over k((x))" then
                    return append(factor_newton(jl.Qt, factorizer, option),
                                  [jl.Rt])
                res := append(res, [jl.Rt])
        res

    testfn(f, factorizer, option) ==
        factor_newton(convertL3toLL(f), factorizer, option)

    factorUP(np, factorSUP) ==
        snp : SUP := makeSUP(np)
        fsnp : Factored SUP := factorSUP(snp)
        map((x : SUP) : UP +-> unmakeSUP(x), fsnp)$FactoredFunctions2(SUP, UP)

-- factor_newton2 is used by factor_newton. It factors a monic
-- LinearOrdinaryDifferentialOperator3 with Laurent series coefficients, given
-- the operator, the Newton polynomial of the right factor (which must be monic)
-- and the term of the Newton polygon to whose slope the polynomial corresponds.
-- The output of this function is a Record of 2 factors.
    factor_newton2(f, r, l) ==
        slop := l.slope
        ycd : Integer := l.point.y
        xcd : Integer := l.point.x
        npol := l.npoly
        shift := min(0, ycd :: FZ + (degree(f) - xcd) * slop)
        -- we will compute the newton polynomial of the left factor
        npo : UP := monomial(1, qcoerce(xcd))
        mlr : UP := eval((npol quo r),monomial(1,1),monomial(1,qcoerce(denom(slop))))
        npo := npo * mlr
        npor : UP := eval(r, monomial(1, 1), monomial(1, qcoerce(denom(slop))))
        ordr := degree(r) * denom(slop)
        ordl := degree(f) - ordr
        -- this number (20) has been arbitrarily chosen : will be modified soon
        -- lnr := lift_newton(slop, f, npo, npor, shift, ordr, 20)
        -- [lnr.Qt, lnr.Rt]
        lwr := ln_wrapper(slop, f, npo, npor, shift, ordr)
        lcr := listream_coefs(lwr, ordl, ordr, shift, slop)
        llr := list_laurent(lcr, ordl, ordr)
        left := laurent_op(llr.laurl, ordl)
        right := laurent_op(llr.laurr, ordr)
        [left, right]

    testf2(f, r, slop) ==
        np := newtonpolygon(convertL3toLL(f))
        l : Record(point : PO, slope : FZ, npoly : UP)
        for i in np repeat
            if i.slope = slop then
                l := i
        factor_newton2(convertL3toLL(f), r, l)

-- laurent_op takes a list of Laurent series which are the coefficients of an
-- operator f of the given degree (the second input) in order and produces f.
    laurent_op(llaur, dgf) ==
        f : LL := 0
        for i in 0..dgf repeat
            f := f + monomial(llaur.(i + 1), i)
        f

-- list_laurent takes a pair of lists of streams of coefficients of a Laurent
-- series in the form required by the series function and produces the
-- corresponding pair of Laurent series. The number of terms in the 2 lists are
-- given by dgl+1 and dgr+1 respectively.
    list_laurent(lsc, dgl, dgr) ==
        laurel := list_laurentop(lsc.llc, dgl)
        laurer := list_laurentop(lsc.rlc, dgr)
        [laurel, laurer]

    testll(slop, f, l_low, r_low, shift, dgr) ==
        lwr := ln_wrapper(slop, convertL3toLL(f), l_low, r_low, shift, dgr)
        lcr := listream_coefs(lwr, degree(f) - dgr, dgr, shift, slop)
        list_laurent(lcr, degree(f) - dgr, dgr)

-- list_laurentop takes a list of stream of coefficients of a Laurent series in
-- the form required by the series function and produces the corresponding
-- Laurent series. The number of terms in the list is given by dgf+1.
    list_laurentop(lsc, dgf) ==
        res : List US := []
        for i in 0..dgf repeat
            res := append(res, [series(lsc.(i + 1))])
        res

-- listream_coefs takes a stream of pairs of operators [l_extra, r_extra] and
-- returns a pair of lists of streams of coefficients in the form required by
-- the series function (i.e. of type Stream TERM where TERM is as defined). The
-- number of terms in the 2 lists is given by dgl+1 and dgr+1 respectively.
    listream_coefs(slr, dgl, dgr, shift, slop) ==
        sle := map((landr : F2) : LL +-> landr.Qt, slr)
        le := retract(denom(slop) * shift)@Integer
        le := le - numer(slop) * dgl
        lsc := listream_coefsop(sle, dgl, le, slop)
        sre := map((landr : F2) : LL +-> landr.Rt, slr)
        re := -(numer(slop) * dgr)
        rsc := listream_coefsop(sre, dgr, re, slop)
        [lsc, rsc]

    testlc(slop, f, l_low, r_low, shift, dgr) ==
        lwr := ln_wrapper(slop, convertL3toLL(f), l_low, r_low, shift, dgr)
        listream_coefs(lwr, degree(f) - dgr, dgr, shift, slop)

-- listream_coefsop takes a stream of operators and returns a list of streams
-- of coefficients in the form required by the series function (i.e. of type
-- Stream TERM where TERM is as defined). The number of terms in the list is
-- given by dgf+1.
    listream_coefsop(sfe, dgf, fe, slop) ==
        res : List Stream TERM := []
        for i in 0..dgf-1 repeat
            nl_st : Stream Integer := stream((x : Integer) : Integer +-> x + 1, 1)
            new_term := map((x : LL, n_l : Integer) : TERM +-> (
                            degn := (n_l-1)*denom(slop) + fe + i * numer(slop);
                            deg : Union(Integer, "failed");
                            deg := retractIfCan(degn /$FZ denom(slop));
                            while deg = "failed" repeat
                                (degn := degn + 1;
                                deg := retractIfCan(degn /$FZ denom(slop)););
                            cx := coefficient(x, i);-- extract coefficient of op
                            coefd := coefficient(cx, deg :: Integer);
                            [deg :: Integer, coefd]), sfe, nl_st)
            res := append(res, [new_term])
        dgc := [[[0$Integer, 1$F]] :: Stream TERM]
        append(res, dgc)

-- lift_newton is a procedure that lifts "coprime index 1 factorizations".
-- Meaning of the variables:
-- slop : the concerned slope of the Newton polygon.
-- ff, li, ri : exact operators ff = li * ri is the final goal.
-- l_low, r_low : the lowest line of li and ri. These have gcd 1.
-- shift : the shift of left term.
-- dgr : the degree of ri / v.Rt
-- v : the pair of operators li, ri
-- ei : the error term = li * ri - ff
-- n_l : the lift number that is n_l-1 gives number of lifts performed so far.
-- l_extra, r_extra : the new terms we computed of li and ri.
-- References : Section 4 of
-- http://www.math.fsu.edu/~hoeij/compalg/diffop/versie9_3.pdf
    lift_newton(slop, ff, l_low, r_low, shift, dgr, v, ei, n_l) ==
        dgl := degree(ff) - dgr
        li : LL := v.Qt
        ri : LL := v.Rt
        if slop = 0 then
            -- regular singular case
            i := n_l-1
            r_extra : LL
            l_extra : LL
            if i = 0 then
                r_extra := plug_delta(r_low)
                l_extra := (plug_delta(l_low) * monomial(1,
                                                retract(shift)@Integer)$US)
            else
                sl_low := eval(l_low, monomial(1,1), monomial(1,1)+i :: UP)
                s := extendedEuclidean(r_low, sl_low)
                if s.generator ~= 1 then
                    error "unsafe factor"
                c1 := s.coef1
                c2 := s.coef2
                pi := coeffx(ei, i :: FZ + shift)
                pr_extra := -((c2 * pi) rem r_low)
                r_extra := plug_delta(pr_extra)
                pl_extra := (pi + pr_extra * sl_low) quo r_low
                l_extra := plug_delta(pl_extra)
                l_extra := -(monomial(1, i + retract(shift)@Integer)$US *
                             l_extra)
                r_extra := monomial(1, i)$US * r_extra
            li := li + l_extra
            ei := ei + l_extra * ri + li * r_extra
            ri := ri + r_extra
        else
            s := extendedEuclidean(r_low, l_low, 1) :: Record(coef1 : UP, coef2 : UP)
            c1 := s.coef1
            c2 := s.coef2
            le : Integer := retract(denom(slop) * shift)@Integer
            le := le - numer(slop) * dgl
            re : Integer := -(numer(slop) * dgr)
            fe : Integer := retract(denom(slop) * shift)@Integer
            fe := fe - numer(slop) * degree(ff)
            for i in (n_l-1)*denom(slop)..n_l*denom(slop)-1 repeat
                r_extra : LL
                l_extra : LL
                if i = 0 then
                    r_extra := coefs_operator(r_low, slop, re)
                    l_extra := coefs_operator(l_low, slop, le)
                else
                    pi := coefs_poly(ei, slop, i + fe)
                    pr_extra := (c2 * pi) rem r_low
                    pl_extra := pi - pr_extra * l_low
                    pl_extra := -(pl_extra quo r_low)
                    l_extra := coefs_operator(pl_extra, slop, i + le)
                    r_extra := coefs_operator(-(pr_extra), slop, i + re)
                li := li + l_extra
                ei := ei + l_extra * ri + li * r_extra
                ri := ri + r_extra
        [[li - v.Qt, ri - v.Rt], [li, ri], ei]

    testln(slop, f, l_low, r_low, shift, dgr) ==
        v : F2 := [0, 0]
        ei := -convertL3toLL(f)
        for i in 1..5 repeat
            lnr := lift_newton(slop, convertL3toLL(f), l_low, r_low, shift,
                               dgr, v, ei, qcoerce(i))
            print(lnr.main.Qt :: OutputForm)$OutputForm
            print(lnr.main.Rt :: OutputForm)$OutputForm
            v := lnr.nf
            ei := lnr.error
        lnr

-- ln_wrapper is a wrapper function for lift_newton and is used by
-- factor_newton2. It generates a stream of the extra terms (l_extra, r_extra)
-- that are generated by repeatedly applying lift_newton on its inputs.
    ln_wrapper(slop, f, l_low, r_low, shift, dgr) ==
        lni : NL := [[0, 0], [0, 0], -f]
        n_li : PositiveInteger := 1
        lni := lift_newton(slop, f, l_low, r_low, shift, dgr, lni.nf,
                           lni.error, n_li)
        all : Stream LW := stream((rec : LW) : LW +-> (
        lnr := rec.lr;
        n_l := rec.n_lifts + 1;
        lnr := lift_newton(slop, f, l_low, r_low, shift, dgr, lnr.nf,
                           lnr.error, n_l);
        [lnr, n_l]), [lni, n_li])
        res := map((rec : LW) : F2 +-> rec.lr.main, all)
        res

    testlw(slop, f, l_low, r_low, shift, dgr) ==
        ln_wrapper(slop, convertL3toLL(f), l_low, r_low, shift, dgr)

-- coefs_poly takes an operator f as input, takes coefficients based on slop and
-- i (extracts part of given valuation from f) and gives them as a polynomial.
-- slop must be > 0.
    coefs_poly(f, slop, i) ==
        nsi := extendedEuclidean(numer(slop), denom(slop)).coef1
        start_D := positiveRemainder(-i * nsi, denom(slop))
        start_x := start_D * slop + i /$FZ denom(slop)
        res : UP := 0
        for j in 0..floor(degree(f) /$FZ denom(slop)) repeat
            fc := coeffx(f, start_x + numer(slop) * j :: FZ)
            npc := coefficient(fc, qcoerce(start_D + j * denom(slop)))
            res := res + monomial(npc, qcoerce(start_D + j * denom(slop)))
        res

    testcp(f, slop, i) ==
        coefs_operator(coefs_poly(convertL3toLL(f), slop, i), slop, i)

-- coefs_operator constructs an operator form a given set of coefficients (given
-- as a polynomial np) based on slop and i which together determine valuation.
-- slop must be > 0.
    coefs_operator(np, slop, i) ==
        nsi := extendedEuclidean(numer(slop), denom(slop)).coef1
        start_D := positiveRemainder(-i * nsi, denom(slop))
        start_x := retract(start_D * slop + i /$FZ denom(slop))@Integer
        res : LL := 0
        for j in 0..ceiling(degree(np) /$FZ denom(slop)) repeat
            npc := coefficient(np, qcoerce(start_D + j * denom(slop)))
            sc := monomial(npc, start_x + numer(slop) * j)
            res := res + monomial(sc, qcoerce(start_D + j * denom(slop)))
        res

    testco(f, slop, i) ==
        np := coefs_poly(convertL3toLL(f), slop, i)
        coefs_poly(coefs_operator(np, slop, i), slop, i) = np

-- coeffx takes an operator f and an exponent e as input and returns the
-- coefficient of x^e in f while also replacing xD with x, so that the final
-- result is a polynomial.
    coeffx(f, e) ==
        ftmp := f
        ex := retract(e)@Integer
        res : UP := 0
        while ftmp ~= 0 repeat
            lc := leadingCoefficient(ftmp)
            dg := degree(ftmp)
            res := res + monomial(coefficient(lc, ex), dg)
            ftmp := reductum ftmp
        res

    testcx(f, e) ==
        coeffx(convertL3toLL(f), e)

-- plug_delta takes a polynomial as input and returns the operator obtained by
-- replacing x with xD in it.
    plug_delta(np) ==
        f : LL := 0
        nptmp := np
        while nptmp ~= 0 repeat
            lc := leadingCoefficient(nptmp)
            dg := degree(nptmp)
            f := f + monomial(lc :: US, dg)
            nptmp := reductum nptmp
        f

    testpd(np) ==
        plug_delta(np)

-- factor_riccati takes a monic LinearOrdinaryDifferentialOperator3 having
-- Laurent series coefficients, a factoring routine for polynomials and a
-- string as input. The input operator should have only 1 slope and the Newton
-- polynomial should be the power of an irreducible polynomial. For an
-- explanation about what the possible option strings are, please refer to the
-- documentation for the factor_op function. This subroutine handles the cases
-- where algebraic extensions or ramifications are needed.
-- References : Steps 3, 4, 5 in Algorithm Riccati solution and Steps 3, 4, 5, 6
-- in Algorithm semi-regular parts of
-- http://www.math.fsu.edu/~hoeij/compalg/diffop/versie9_3.pdf
    factor_riccati(f, factorizer, option) ==
        np := newtonpolygon(f)
        np1 := np.1
        slop := np1.slope
        option = "semireg" and slop = 0 =>
            va := value(factor_op(f, factorizer, "alg factor") :: RS)
            vr : LA :=
                va case LL => [va :: LL, 1, 0, monomial(1, 1)]
                va::LA
            xx : UP := monomial(1, 1)
            vr_ope := vr.ope
            i := xx - coeffx(vr_ope, 0)
            n : Integer := degree(f) quo vr.dext
            if n = 1 then
                il := convertUPtoUTS(i) :: US
                return [[vr.ope + il :: LL, vr.rami, il]]
            npf1 := factorList(factorUP(eval(np1.npoly, monomial(1, 1),
                                           monomial(1, 1) + i), factorizer))
            r : UP := 1
            sols : List Integer := []
            for l in npf1 repeat
                diff : Union(Integer, "failed")
                diff := retractIfCan(xx - l.factor)
                if diff ~= "failed" then
                    fact : UP := l.factor
                    r := r * fact ^ l.exponent
                    sols := cons(diff :: Integer, sols)
            if degree(r) ~= n then
                error "bug"
            n := reduce(min, sols)
            r := eval(r, monomial(1, 1), monomial(1, 1) + n :: UP)
            subs := substitute(i + n :: UP, f) :: LL
            np := newtonpolygon(subs)
            lns : Record(point : PO, slope : FZ, npoly : UP)
            for ip in np repeat
                if ip.slope = 0 then
                    lns := ip
            f2r := factor_newton2(subs, r, lns).Rt
            return [[f2r, vr.rami, convertUPtoUTS(i + n :: UP) :: US]]
        npf := factorList(factorUP(np1.npoly, factorizer)).1
        irred_f := degree(f) <= 1 or (npf.exponent = 1 and
           (option = "split over k((x))" or option = "all right factors"))
        irred_f =>
            option = "semireg" =>
                -- degree(f) = 1 and slope ~= 0
                il : US := -truncate(coefficient(f, 0), 0)
                return [[f + il :: LL, monomial(1, 1), il]]$SP
            [f]$RS
        degree(npf.factor) = 1 and denom(slop) = 1 =>
            -- now slope ~= 0
            cxs : US := -coefficient(npf.factor, 0) * monomial(1, -numer(slop))
            -- apply a homomorphism xD -> xD + cxs to simplify the problem.
            v : FRI := factor_op(substitute(cxs, f) :: LL, factorizer, option)
            if option = "semireg" then
                vsp := v :: SP
                sym : Symbol := variable(cxs)
                retv : SP := []
                for vi in vsp repeat
                    xsubsr := (convertUPtoUTS(vi.ram) :: US) ^ (-numer(slop))
                    fullxr := degree(vi.ram)*(-coefficient(npf.factor, 0))*xsubsr
                    retv := append(retv, [[vi.op, vi.ram, vi.expart + fullxr]])
                return retv
            vrs := v :: RS
            return [substitute(-cxs, i) for i in vrs]$RS
        option = "split over k((x))" =>
            -- an order 1 factor over the alg. closure of k((x))
            ric := factor_riccati(f, factorizer, "alg factor") :: RS
            r1 := value(ric) :: LA
            -- a righthand factor over k((x))
            rk := make_rightfactor(f, r1)
            rk = f => return [f]
            lk := rightQuotient(f, rk)
            return append(factor_riccati(lk, factorizer, option) :: RS, [rk])
        option = "all right factors" =>
            ric := factor_riccati(f, factorizer, "all alg factors") :: RS
            res : RS := []
            for i in ric repeat
                rk := make_rightfactor(f, i :: LA)
                if rk = f then
                    res := [f]
                else
                    res := append(res, [rk])
            return res
        if degree(npf.factor) > 1 then
            -- now we need an algebraic extension on k
            gr : F := zeroOf(makeSUP(npf.factor))
            npgr : UP := (monomial(1, 1) - gr :: UP) ^ npf.exponent
            f2r := factor_newton2(f, npgr, np1).Rt
            v : FRI := factor_riccati(f2r, factorizer, option)
            if v case SP then
                return v
            else
                vrs := v :: RS
                res : RS := []
                for i in vrs repeat
                    if i case LL then
                        nt : LA
                        nt := [i :: LL, qcoerce(degree(npf.factor)), gr,
                               monomial(1, 1)]
                        res := append(res, [nt])
                    else
                        res := append(res, [i])
                return res
        else
            -- now we need a ramification
            n := extendedEuclidean(numer(slop), denom(slop)).coef1
            coefr := -coefficient(npf.factor, 0)
            ror : LL := ramification_of(f, coefr ^ n, qcoerce(denom(slop)))
            exp := retract((1 - n * numer(slop)) /$FZ denom(slop))@Integer
            tosub : F := denom(slop) * coefr ^ exp
            r : UP := (monomial(1, 1) - tosub :: UP) ^ npf.exponent
            np := newtonpolygon(ror)
            lns : Record(point : PO, slope : FZ, npoly : UP)
            for i in np repeat
                if i.slope = (numer(slop) :: FZ) then
                    lns := i
            f2r := factor_newton2(ror, r, lns).Rt
            v : FRI := factor_riccati(f2r, factorizer, option)
            r := monomial(coefr ^ n, qcoerce(denom(slop)))
            if v case SP then
                vsp := v :: SP
                res_sp := [[i.op, eval(r, monomial(1, 1), i.ram), i.expart]
                        for i in vsp] :: SP
                return res_sp
            else
                vrs := v :: RS
                res : RS := []
                for i in vrs repeat
                    nt : LA
                    if i case LL then
                        nt := [i :: LL, 1, 0, r]
                    else
                        ir := i :: LA
                        nt := [ir.ope, ir.dext, ir.alpha, eval(r, monomial(1,1), ir.rami)]
                    res := append(res, [nt])
                return res

    testfr(f, factorizer, option) ==
        factor_riccati(convertL3toLL(f), factorizer, option)

-- substitute(a, f) substitutes xD + a for xD in an operator f.
    substitute(a, f) ==
        new_mon : LL := monomial(1, 1)
        if a case UP then
            new_mon := new_mon + convertL3toLL(a :: UP :: RF :: L3)
        else
            new_mon := new_mon + a :: US :: LL
        if f case LL then
            ftmp := f :: LL
        else
            fr := f :: LA
            ftmp := fr.ope
            new_mon := ramification_of(new_mon, leadingCoefficient(fr.rami),
                                       qcoerce(degree(fr.rami)))
        res : LL := 0
        while ftmp ~= 0 repeat
            lc := leadingCoefficient(ftmp)
            dg := degree(ftmp)
            res := res + lc * new_mon ^ dg
            ftmp := reductum ftmp
        if f case LL then
            fl := f :: LL
            return res
        else
            fr := f :: LA
            return [res, fr.dext, fr.alpha, fr.rami]

    testsb(a, f) ==
        substitute(a, convertL3toLL(f)) :: LL

-- ramification_of(f, coefr, n) gives the ramification of an expression; it maps
-- xD to xD / n, and maps x to coefr * x^n.
    ramification_of(f, coefr, n) ==
        if coefr = 1 and n = 1 then
            return f
        res : LL := 0
        ftmp : LL := f
        while ftmp ~= 0 repeat
            lc := leadingCoefficient(ftmp)
            dg := degree(ftmp)
            mlc := multiplyCoefficients((x : Integer) : F +-> coefr ^ x, lc)
            mlc := multiplyExponents(mlc, n)
            extra := monomial(mlc, dg) * ((1 /$FZ n) ^ dg) :: F :: US
            res := res + extra
            ftmp := reductum ftmp
        n ^ degree(f) * res

    testro(f, coefr, n) ==
        ramification_of(convertL3toLL(f), coefr, n)

-- factor_op takes a monic LinearOrdinaryDifferentialOperator3 having Laurent
-- series coefficients, a factoring routine for polynomials and a string as
-- input and produces different kinds of local factorizations.
-- The possible option strings are:
-- "split over k((x))" gives a factorization over k((x))
-- "all right factors" gives all safe right factors over k((x))
-- "all alg factors" gives all safe right factors over alg. closure of k((x))
-- "alg factor" gives one safe right factor over alg. closure of k((x))
-- References : Algorithm Riccati solution and Algorithm semi-regular parts of
-- http://www.math.fsu.edu/~hoeij/compalg/diffop/versie9_3.pdf
    factor_op(f, factorizer, option) ==
        fnr := factor_newton(f, factorizer, option)
        if option = "semireg" then
            res : SP := []
            for i in fnr repeat
                res := append(res,
                              factor_riccati(i, factorizer, option) :: SP)
            return res
        else
            res : RS := []
            for i in fnr repeat
                res := append(res,
                              factor_riccati(i, factorizer, option) :: RS)
            return res

    testfo(f, factorizer, option) ==
        factor_op(convertL3toLL(f), factorizer, option)

-- make_righfactor(f, ric) produces an irreducible right factor of f over k((x))
-- from a 1st order right-factor over the algebraic closure of k((x)) where f is
-- an operator. This subroutine computes LCLM(xD - r, "and conjugates over
-- k((x))") where r is a Riccati solution. In this way, a local irreducible
-- factor is obtained. This is only necessary if the coprime index is > 1,
-- otherwise one can factor via the Newton method.
-- References : Section 7 of
-- http://www.math.fsu.edu/~hoeij/compalg/diffop/versie9_3.pdf
    make_rightfactor(f, ric) ==
        d : PositiveInteger := qcoerce(degree(ric.rami))
        -- now we multiply by the degree of the algebraic extension
        d := d * ric.dext
        if d = degree(f) then
            return f
        -- now, f has an irreducible right-hand factor R in k((x))[xD] of order
        -- d, with corresponding right-hand factor in alg. closure of k((x))[xD]
        -- being ric. So, we need to get R using lift_rightfactor(d, ric).
        lift_rightfactor(d, ric)

-- lift_righfactor is a function that lifts "coprime index > 1 factorizations".
-- Meaning of the variables:
-- ric : solution of the Riccati, i.e. 1st order factor over alg. clos. k((x))
-- ram : ramification is a map x -> c * x^(ram) for a constant c.
-- This procedure generates linear equations stating ric is a righthand factor
-- of R. The coefficients of R are then computed by solving these equations.
-- References : Section 7 of
-- http://www.math.fsu.edu/~hoeij/compalg/diffop/versie9_3.pdf
    lift_rightfactor(order, ric) ==
        cric := leadingCoefficient(ric.rami)
        ram := degree(ric.rami)
        r := -coefficient(ric.ope, 0)
        rp : US := 1
        s : Vector US := new(order + 1, 0)
        s.1 := 1
        mindg : Integer := 0
        for i in 1..order repeat
            rp := (apply(D()$LL, 0, rp) + r * rp) / ram :: F
            s.(i+1) := rp
            mindg := min(mindg, order(rp, 0))
        map!((x : US) : US +-> x / monomial(1, mindg), s)
        sysram : List Vector US := [new(order+1, 0) for i in 1..ram]
        nn : Stream NonNegativeInteger
        nn := stream((x : NonNegativeInteger) : NonNegativeInteger +-> x+1, 0)
        for i in 1..ram repeat
            for j in 1..order+1 repeat
                ser : US := s.j
                cst := map((x : NonNegativeInteger) : TERM +-> (
                            dg := i - 1 + x * ram;
                            [x, coefficient(ser, dg) / (cric ^ x)]),
                           nn)$StreamFunctions2(NonNegativeInteger, TERM)
                sysram(i)(j) := series(cst)
        system : List Vector US := [new(order+1, 0) for i in 1..order]
        pa : Vector F := [ric.alpha ^ i for i in 0..ric.dext-1]
        for i in 1..ram repeat
            for j in 1..order+1 repeat
                ser : US := sysram(i)(j)
                for k in 1..ric.dext repeat
                    nsr := map((x : F) : F +-> (
                                psq := particularSolutionOverQ(pa,
                                                 x)$IntegerLinearDependence(F);
                                psq = "failed" => x;
                                (psq :: Vector FZ).k :: F), ser)
                    system((i - 1) * ric.dext + k)(j) := nsr
        ll : List List US := [[0 for i in 1..order] for j in 1..order]
        for i in 1..order repeat
            for j in 1..order repeat
                ll(i)(j) := system(i)(j)
        denm : SquareMatrix(order, US) := squareMatrix(matrix(ll))
        nums : List SquareMatrix(order, US) := [empty() for i in 1..order]
        for i in 1..order repeat
            for j in 1..order repeat
                for k in 1..order repeat
                    if k = i then
                        ll(j)(k) := system(j)(order+1)
                    else
                        ll(j)(k) := system(j)(k)
            nums.i := squareMatrix(matrix(ll))
        den := minordet(denm)
        coefs : List US := [0 for i in 1..order]
        for i in 1..order repeat
            num := minordet(nums.i)
            coefs.i := -num / den
        res : LL := monomial(1, order)
        for i in 0..order-1 repeat
            res := res + monomial(coefs.(i+1), i)
        res

    same_charclass?(f, g, factorizer) ==
        if degree(f) ~= degree(g) then
            return false
        if leadingCoefficient(f) ~= 1 then
            f := (1 / leadingCoefficient(f)) * f
        if leadingCoefficient(g) ~= 1 then
            g := (1 / leadingCoefficient(g)) * g
        if degree(f) = 1 then
            il : US := truncate(coefficient(f - g, 0), 0)
            return order(il, 0) = 0 and integer?(coefficient(il, 0))
        r := map(newtonpolygon,
                 [f, g])$ListFunctions2(LL,
                             List(Record(point : PO, slope : FZ, npoly : UP)))
        r11 := r(1)(1)
        r21 := r(2)(1)
        if r11.slope = 0 then
            c := r11.npoly
            c2 := r21.npoly
            d := degree(c)
            dm1 : NonNegativeInteger := qcoerce(d - 1)
            fd := d :: Integer :: F
            cx : F := (coefficient(c, dm1)$UP - coefficient(c2, dm1)) / fd
            if not integer?(cx) then
                return false
            r := [cons([r11.point, r11.slope, eval(r11.npoly, monomial(1, 1),
            monomial(1, 1) - cx :: UP)], [r(1)(j) for j in 2..#(r(1))]), r(2)]
        if r(1) ~= r(2) then
            return false
        gg := value(factor_op(g, factorizer, "alg factor") :: RS) :: LA
        rr := -truncate(coefficient(gg.ope, 0), 0)
        gr : UP := gg.rami
        ff := substitute(rr, ramification_of(f, leadingCoefficient(gr),
                                             qcoerce(degree(gr)))) :: LL
        ffnp := newtonpolygon(ff)
        if ffnp(1).slope ~= 0 then
            return false
        fff := factorList(factorUP(ffnp(1).npoly/leadingCoefficient(ffnp(1).npoly),
                                factorizer))
        for i in fff repeat
            if (degree(i.factor) = 1 and integer?(coefficient(i.factor, 0) /
                                                coefficient(i.factor, 1))) then
                return true
        false

    testsc(f, g, factorizer) ==
        same_charclass?(convertL3toLL(f), convertL3toLL(g), factorizer)

-- input: local factor, maximal order to look for, Newton polygon bound,
-- point of singularity, global operator, extra singularities bound, min order
-- to start with (i.e. max(degree(r), min_order) will be starting index, option
-- is a string which can be "use adjoint" if we are reconstructing left hand
-- global factor.
-- output: "failed" or a global factorization.
    try_factorization(r, max_order, bound, sng, f, eb, min_order, option) ==
        for i in max(degree(r), min_order)..max_order repeat
            fls := flist(r, i)
            n : Integer := 0
            for ff in fls repeat
                for fc in ff repeat
                    n := min(n, order(fc, 0))
            if n < 0 then
                fls := [[monomial(1, -n) * fc for fc in ff] for ff in fls]
            flst := [[retract(fc) for fc in ff] for ff in fls]
            n := #flst
            tf2 := try_factorization2(flst, n, bound, eb)
            if tf2 ~= "failed" then
                flm := tf2 :: List SUP
                sr : L3 := +/[unmakeSUP(flm(j+1))$UP :: RF * monomial(1, j)
                              for j in 0..#flm-1]
                sr := (1 / leadingCoefficient(sr)) * sr
                if option = "use adjoint" then
                    sr := adjoint(sr)
                srl := convert(sr)$LODOConvertions(F, UP, RF)
                zero_sing := sng.point case F and sng.point::F = 0
                if not(zero_sing) then
                    ftmp := srl
                    srl := 0
                    old_x := monomial(1, 1)$UP
                    newx : RF :=
                        sng.point = "infinity" => 1/old_x
                        (old_x - sng.point::F::UP)::RF
                    newd : L :=
                        sng.point = "infinity" =>
                            -(old_x^2)::RF * D()
                        D()
                    while ftmp ~= 0 repeat
                        lc := leadingCoefficient(ftmp)
                        dg := degree(ftmp)
                        lcn := subsup(numer(lc), newx)
                        lcd := subsup(denom(lc), newx)
                        slc := lcn / lcd
                        srl := srl + slc * newd ^ dg
                        ftmp := reductum ftmp
                if option ~= "use adjoint" then
                    re := rightGcd(f, srl)
                    if degree(re) > 0 then
                        return [rightQuotient(f, re), re]
                else
                    re : L
                    if sng.point = "infinity" then
                        re := (1 / monomial(1, degree(f))) :: RF :: L
                    else
                        re := (1 /
                    (monomial(1,1) - sng.point :: F :: UP)^degree(f)) :: RF :: L
                    srl := re * leadingCoefficient(f) * srl
                    srl := srl * (1 / leadingCoefficient(srl))
                    re := leftGcd(f, srl)
                    if degree(re) > 0 then
                        return [re, leftQuotient(f, re)]
        "failed"

    try_factorization2(fl, n, bound, eb) ==
        lv : List Vector SUP
        eta : List NonNegativeInteger
        cb := ceiling(bound(#fl-1))@Integer + eb
        eta := [qcoerce(cb) for i in 1..n]
        cnt := #fl*(cb + 1)
        vdim := #first(fl)
        acc := (cnt + vdim + 3) quo vdim
        nstep := ceiling(acc * 2 / 3)
        repeat
            lv := [vector([makeSUP(convertUTStoUP(g,acc)) for g in f])
                   for f in fl]
            hps := hp_solve(lv, eta, qcoerce(acc))$VectorHermitePadeSolver
            ncols(hps) = 1 => return members(column(hps, 1))
            ncols(hps) = 0 => return "failed"
            -- increase acc
            acc := acc + nstep
            nstep := floor(3 /$FZ 2 * nstep) + 2

    flist(r, order) ==
        res : List List US := []
        for i in 0..order repeat
            p := xDn_modr(i, r)
            res := append(res, [[coefficient(p, j) for j in 0..degree(r)-1]])
        res

    xDn_modr(n, r) ==
        if n < degree(r) then
            return monomial(1, qcoerce(n))
        a := xDn_modr(n-1, r)
        coefa := coefficient(a, qcoerce(degree(r)-1))
        monomial(1, 1)$LL * a + a * monomial(1, 1)$LL - coefa * r

    factor(f, factorizer, y) ==
        if degree(f) <= 1 then
            return [f]
        factorizer2 := ((x : SUP) : Factored(SUP) +-> factorizer(x, y))
        b := factor_global(f, factorizer2)
        if #b = 1 then
            return b
        else
            res := concat(factor(b.1, factorizer, y), factor(b.2, factorizer, y))
            for i in 3..#b repeat
                concat!(res, factor(b.i, factorizer, y))
            return res

    inf_singularity?(f : L) : Boolean ==
        fi : L := 0
        ftmp := f
        ux : RF := -(monomial(1, 2)$UP) :: RF
        yu : L := 1
        for i in 0..degree(f) repeat
            lc := coefficient(f, i)
            lcn := subsup(numer(lc), 1 / monomial(1, 1))
            lcd := subsup(denom(lc), 1 / monomial(1, 1))
            slc := lcn / lcd
            fi := fi + slc * yu
            yu := ux * D() * yu
        fi := (1 / leadingCoefficient(fi)) * fi
        dlc := [denom(coefficient(fi, e))
                for e in minimumDegree(fi)..degree(fi)]
        if eval(lcm(dlc), monomial(1, 1), 0) = 0 then
            return true
        false

    testis(f) ==
        inf_singularity?(f)

-- work in progress.
    factor_global(f, factorizer) ==
        if degree(f) <= 1 then
            return [f]
        if leadingCoefficient(f) ~= 1 then
            a := factor_global((1 / leadingCoefficient(f)) * f, factorizer)
            return cons(leadingCoefficient(f) * a(1), [a(i) for i in 2..#a])
        if degree(f) = 2 then
            -- may be faster
            -- return factor(f)$LinearOrdinaryDifferentialOperatorFactorizer(F, UP)
            print(message("Better to use old factorizer")$OutputForm
                 )$OutputForm
        gem := ge_minimal(f, factorizer, "")
        bound : List FZ := [0 for i in 1..degree(f)-1]
        eb : Integer := -1
        for s in gem repeat
            np : List Record(point : PO, slope : FZ, npoly : UP)
            np := newtonpolygon(s.singularity.lpf)
            np := append(np, [[[degree(s.singularity.lpf), 0], 0, 0]])
            v : List FZ := []
            for i in 1..(#np)-1 repeat
                jiy := np(i).point.y
                ji1y := np(i+1).point.y
                jix := np(i).point.x
                ji1x := np(i+1).point.x
                for l in jix..ji1x-1 repeat
                    v := append(v, [jiy :: FZ + (l - jix) * (ji1y - jiy)
                                    /$FZ (ji1x - jix)])
            for i in 1..degree(f)-1 repeat
                bound(i) := bound(i) + s.singularity.dxt * v(i+1)
            eb := eb + s.singularity.dxt
        bound := [-bound(degree(f) - j) + j * eb :: FZ for j in 1..degree(f)-1]
        eb := compute_bound(gem, 0)
        gema := ge_minimal(f, factorizer, "use adjoint")
        eba : Integer := compute_bound(gema, 0)
        all_one := true
        -- the singularities we have considered
        done_s : FG := []
        while not empty?(gem) repeat
            dl : List PositiveInteger := [s.singularity.dxt for s in gem]
            min_deg := reduce(min, dl)
            for ge in gem repeat
                if ge.singularity.dxt = min_deg then
                    gem := remove(ge, gem)
                    vrs := factor_op(ge.singularity.lpf, factorizer,
                                     "all right factors") :: RS
                    done_s := append(done_s, [[ge.singularity, vrs]])
                    for ssp in ge.fos for sop in vrs repeat
                        if degree(ssp.op) = 1 and degree(sop :: LL) = 1 then
                            return factor_minmult1(bound, ge.singularity, f, eb,
                                                   eba, sop :: LL, factorizer)
                    all_one := all_one and #vrs = 1 and degree(vrs.1 :: LL) = 1
        if all_one then
            -- f has only 1 exponential part
            -- return factor(f)$LinearOrdinaryDifferentialOperatorFactorizer(F, UP)
            print(message("Better to use old factorizer")$OutputForm
                 )$OutputForm
        all_one := true
        done_sa : FG := []
        for s in done_s repeat
            all_one := all_one and #s.arf = 1
            for sop in s.arf repeat
                t := try_factorization(sop :: LL, floor(degree(f) /$FZ 2), bound,
                                       s.sing, f, eb, 0, "")
                if t ~= "failed" then
                    return t :: List L
            vrs := factor_op(adjoint(s.sing.lpf), factorizer,
                             "all right factors") :: RS
            done_sa := append(done_sa, [[s.sing, vrs]])
            for sop in vrs repeat
                t := try_factorization(sop :: LL, floor(degree(f) /$FZ 2), bound,
                                       s.sing, f, eba, 0, "use adjoint")
                if t ~= "failed" then
                    return t :: List L
        -- now try the same but searching for higher order factors
        for s in done_s for sa in done_sa repeat
            for sop in s.arf repeat
                t := try_factorization(sop :: LL, degree(f)-1, bound, s.sing, f,
                                       eb, floor(degree(f) /$FZ 2)+1, "")
                if t ~= "failed" then
                    return t :: List L
            for sop in sa.arf repeat
                t := try_factorization(sop :: LL, degree(f)-1, bound, s.sing, f,
                                       eba, floor(degree(f) /$FZ 2)+1, "use adjoint")
                if t ~= "failed" then
                    return t :: List L
        if all_one then
            -- up to conjugation only 1 exp part in every singularity
            for s in done_s repeat
                iarf1 := first(s.arf) :: LL
                if degree(iarf1) > 1 then
                    j1 := factor_op(iarf1, factorizer, "alg factor") :: RS
                    jj := value(j1) :: LA
                    kk := make_rightfactor(iarf1, jj)
                    t := try_factorization(kk, degree(f)-1, bound, s.sing, f, 0,
                                           0, "")
                    if t ~= "failed" then
                        return t :: List L
                    vrs := factor_op(adjoint(s.sing.lpf), factorizer,
                                     "all right factors") :: RS
                    r_left : LL := 0
                    for l in vrs while r_left = 0 repeat
                        if same_charclass?(l :: LL, adjoint(kk), factorizer) then
                            r_left := l :: LL
                    -- now try if r gives a left hand global factor
                    t := try_factorization(r_left, degree(f)-1, bound, s.sing,
                                           f, 0, 0, "use adjoint")
                    if t ~= "failed" then
                        return t :: List L
        -- may be incomplete
        -- factor(f)$LinearOrdinaryDifferentialOperatorFactorizer(F, UP)
        print(message("Try the old factorizer")$OutputForm)$OutputForm
        [f]

    testfg(f, factorizer) ==
        factor_global(f, factorizer)

    factor_minmult1(bound, sng, f, eb, eba, r, factorizer) ==
        fl := floor(degree(f) /$FZ 2)
        t := try_factorization(r, fl, bound, sng, f, eb, 0, "")
        if t ~= "failed" then
            return t :: List L
        w := factor_op(adjoint(sng.lpf), factorizer, "all right factors") :: RS
        r_left : LL := 0
        for i in w while r_left = 0 repeat
            if same_charclass?(i :: LL, adjoint(r), factorizer) then
                r_left := i :: LL
        if r_left = 0 then
            error "bug"
        -- now try if r gives a left hand global factor
        t := try_factorization(r_left, fl, bound, sng, f, eba, 0, "use adjoint")
        if t ~= "failed" then
            return t :: List L
        -- now try more terms
        t := try_factorization(r, degree(f)-1, bound, sng, f, eb, fl+1, "")
        if t ~= "failed" then
            return t :: List L
        t := try_factorization(r_left, degree(f)-1, bound, sng, f, eba, fl+1,
                               "use adjoint")
        if t ~= "failed" then
            return t :: List L
        [f]

    l_p(f, p) ==
        lpf : L := 0
        if p = 0 then
            lpf := f
        else
            ftmp := f
            while ftmp ~= 0 repeat
                lc := leadingCoefficient(ftmp)
                dg := degree(ftmp)
                newx : RF
                if p = "infinity" then
                    newx := 1 / monomial(1, 1)
                else
                    newx := (monomial(1, 1) + p :: F :: UP) :: RF
                lcn := subsup(numer(lc), newx)
                lcd := subsup(denom(lc), newx)
                slc := lcn / lcd
                newd : L
                if p = "infinity" then
                    newd := -(monomial(1, 1)$UP ^ 2) :: RF * D()
                else
                    newd := D()
                lpf := lpf + slc * newd ^ dg
                ftmp := reductum ftmp
        llr := convertL3toLL(convert(lpf)$LODOConvertions(F, UP, RF))
        (1 / leadingCoefficient(llr)) * llr

    testlp(f, p) ==
        l_p(f, p)

    subsup(np, newx) ==
        res : RF := 0
        nptmp := np
        while nptmp ~= 0 repeat
            lc := leadingCoefficient(nptmp)
            dg := degree(nptmp)
            res := res + (lc :: UP) * newx ^ dg
            nptmp := reductum nptmp
        res

    get_trace(f : F, k : K) : F ==
        min_pol := minPoly(k)
        Sae := SimpleAlgebraicExtension(F, UP, unmakeSUP(min_pol))
        fa := univariate(f, k, min_pol)$POLYCATQ
        trace(reduce(unmakeSUP(fa))$Sae)$Sae

    testgt(f, k) ==
        get_trace(f, k)

    make_poly(f : LL) : SUP ==
        res : SUP := 0
        ftmp := f
        while ftmp ~= 0 repeat
            lc := leadingCoefficient(ftmp)
            dg := degree(ftmp)
            res := res + monomial(coefficient(lc, 0), dg)
            ftmp := reductum ftmp
        res

    compute_bound(gem, rlc) ==
        res : FZ := 0
        for ge in gem repeat
            ma : Union(FZ, "-infinity") := "-infinity"
            for f in ge.mge for fps in ge.fos for ji in 1..#ge.mge repeat
                r := degree(fps.ram)
                -- c = - an exponent; bound = - sum exponents
                c := coefficient(f, 0)
                for jj in rest(ge.mge, ji) for fpsjj in rest(ge.fos, ji) repeat
                    rjj := degree(fpsjj.ram)
                    vpei := order(f, 0) / r
                    vpej := order(jj, 0) / rjj
                    c := c + min(vpei, vpej) :: F
                -- now compute the trace of this algebraic number divided by
                -- the degree of this number over Q
                ks := kernels(c)
                for ke in ks repeat
                    if not(retractIfCan(ke :: F)@Union(
                             AlgebraicNumber, "failed") case "failed") then
                        c := get_trace(c, ke) / (degree(minPoly(ke)) :: F)
                    else
                        if eval(denominator(c), ke, rlc) = 0 then
                            return compute_bound(gem, rlc + 1)
                        c := eval(c, ke, rlc)
                cq := retract(c)@FZ
                if ma = "-infinity" then
                    ma := -(cq :: FZ)
                else
                    ma := max(-(cq :: FZ), ma :: FZ)
            if ma = "-infinity" then
                -- there are no 1st order factors
                return 0
            res := res + ge.singularity.dxt * ma :: FZ
        max(0, floor(res))

    gen_exp(f, p, factorizer, y) ==
        factorizer2 := ((x : SUP) : Factored(SUP) +-> factorizer(x, y))
        v := factor_op(l_p(f, p), factorizer2, "semireg") :: SP
        res : GE := []
        for e in v repeat
            -- ramification index
            r := degree(e.ram)
            -- semi-regular part
            s := e.op
            es := [rootOf(g.factor) for g in factorList(factorizer2(make_poly(s)))]
            -- now add expart, then divide by r
            ss := [(root :: US + e.expart) / (r :: F) for root in es]
            tr : RF
            if p = "infinity" then
                tr := 1 / monomial(1, 1)
            else
                tr := (monomial(1, 1) - p :: F :: UP) :: RF
            res := append(res, [[ss, e.ram, tr]])
        res

    ge_minimal(f, factorizer, option) ==
        dlc := [denom(coefficient(f, e)) for e in minimumDegree(f)..degree(f)]
        (dlc :=
        squareFreeBasis(dlc)$CylindricalAlgebraicDecompositionUtilities(F, UP))
        dnf : List FA := []
        for p in dlc repeat
            dnf := append(dnf, factors(factorUP(p, factorizer)))
        singularities : CB
        if inf_singularity?(f) then
            singularities := [["infinity", l_p(f, "infinity"), 1]]
        else
            singularities := []
        for g in dnf repeat
            root := rootOf(makeSUP(g.factor))
            singularities := cons([root, l_p(f, root),
                                   qcoerce(degree(g.factor))], singularities)
        if option = "use adjoint" then
            singularities := [[sg.point, adjoint(sg.lpf), sg.dxt]
                              for sg in singularities]
        res : GEM := []
        for sg in singularities repeat
            v := factor_op(sg.lpf, factorizer, "semireg") :: SP
            mg : List US := []
            for e in v repeat
                r := degree(e.ram)
                s := e.op
                es := [retract(rootOf(g.factor))@FZ
                       for g in factorList(factorizer(make_poly(s)))]
                ms := reduce(min, es) :: F
                ss := (ms :: US + e.expart) / (r :: F)
                mg := append(mg, [ss])
            res := cons([sg, v, mg], res)
        res

)boot $tryRecompileArguments := true

--Copyright (c) 2015, Abhinav Baid.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

