k≥
A.Torres-Hernandez ,a aDepartmentofPhysics,FacultyofScience-UNAM,Mexico
Abstract
1.FractionalPseudo-NewtonMethod
Keywords: FractionalOperators;GroupTheory;FractionalIterativeMethods;RecursiveProgramming.
oα x h(x):= ˆ ek oα k h(x).
Codeofthemultidimensionalfractional pseudo-Newtonmethodusingrecursive programming
ˆ ek
Thefollowingpaperpresentsonewaytodefineandclassifythefractionalpseudo-Newtonmethodthrough agroupoffractionalmatrixoperators,aswellasacodewritteninrecursiveprogrammingtoimplementthis method,whichthroughminormodifications,canbeimplementedinanyfractionalfixed-pointmethodthatallows solvingnonlinearalgebraicequationsystems.
Therefore,denotingby ∂n k thepartialderivativeoforder n
vector x,usingthepreviousoperatoritispossibletodefinethefollowingsetoffractionaloperators On x,α (h):= oα x : ∃oα k h(x)andlim α→n oα k h(x)= ∂n k h(x) ∀k ≥ 1 , (2) whosecomplementmaybedefinedasfollows On,c x,α (h):= oα x : ∃oα k h(x) ∀k ≥ 1andlim α→n oα k h(x) ∂n k h(x)inatleastonevalue k ≥ 1 , (3) asaconsequence,itispossibletodefinethefollowingset On,u c,x,α (h):= On x,α (h) ∪ On,c x,α (h) ∩ oα x : oα k c 0 ∀c ∈ R \ {0} and ∀k ≥ 1 . (4) Ontheotherhand,consideringaconstantfunction h : Ω ⊂ Rm → Rm,itispossibletodefinethefollowingset m On,u c,x,α (h):= oα x : oα x ∈ On,u c,x,α ([h]k ) ∀k ≤ m , (5) E-mail:anthony.torres@ciencias.unam.mx;ORCID:0000-0001-6496-9505 1M´etodopseudo-Newtonfraccional. 1 Applied Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022 DOI : 10.5121/mathsj.2022.9101
Tobeginthissection,itisnecessarytomentionthatduetothelargenumberoffractionaloperatorsthatmay exist[1–6],somesetsmustbedefinedtofullycharacterizethe fractionalpseudo-Newtonmethod1 [7–10].Itis worthmentioningthatcharacterizingelementsoffractionalcalculusthroughsetsisthemainideabehindofthe methodologyknownas fractionalcalculusofsets [11].So,consideringascalarfunction h : Rm → R andthe canonicalbasisof Rm denotedby { } 1,itispossibletodefinethefollowingfractionaloperatoroforder α using Einsteinnotation (1) appliedwithrespecttothe k-thcomponentofthe
o
whichundertheclassicalHadamardproductitisfulfilledthat o0 x ◦ h( ):= h( ) ∀ MO (h (7)
i,x ◦ oqα j,x := o
r α = Aα (orα x ) : r ∈ Z
A◦r α =
m
ConsideringthatwhenusingtheclassicalHadamardproductingeneral opα oqα o(p+q)α x .Itispossibleto definethefollowingmodifiedHadamardproduct[11]: pα pα if i j (Hadamardproductoftypehorizontal) ( if i j (Hadamardproductoftypevertical) , (8)
x
α x
∞,u c,x,α
Proof. Itshouldbenotedthatduetothewaytheset(10)isdefined,justtheHadamardproductoftypeverticalis appliedamongitselements.So, ∀ ,A G (A (o )) itisfulfilledthat (11)
⊂ Rm →
=
∈
)
p+q)α i,x ,
∀A◦p α ,A◦p α ,A◦r α ∈ m G (Aα (oα x )) itisfulfilledthat A◦p α ◦ A◦q α ◦ A◦r α = A◦p α ◦ A◦q α ◦ A◦r α ∃A◦0 α ∈ m G (Aα (oα x )) suchthat ∀A◦p α ∈ m G (Aα (oα x )) itisfulfilledthat A◦0 α ◦ A◦p α = A◦p α ∀A◦p α ∈ m G (Aα (oα x )) ∃A◦−p α ∈ m G (Aα (oα x )) suchthat A◦p α ◦ A◦−p α = A◦0 α ∀A◦p α ,A◦q α ∈ m G (Aα (oα x )) itisfulfilledthat A◦p α ◦ A◦q α = A◦q α ◦ A◦p α . (12) Fromtheprevioustheorem,itispossibletodefinethefollowinggroupoffractionalmatrixoperators[11]: m GFPN (α):= oα x ∈m MO∞,u c,x,α (h) m G (Aα (oα x )) , (13) where ∀A◦p i,α ,A◦q j,α ∈ m GFPN (α),with i j,thefollowingpropertyisdefined 2 Applied Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022
◦q α ∈ m
x ◦
A◦p α ◦ A◦q α = [A◦p α ]jk ◦ [A◦q α ]jk = o(p+q)α k = [A◦(p+q) α ]jk = A◦(p+q) α ,
Theorem1. Let oα x beafractionaloperatorsuchthat oα x ∈ m MO∞,u c,x,α (h).So,consideringthemodifiedHadamardproduct givenby (8),itispossibletodefinethefollowingsetoffractionalmatrixoperator m G (Aα (oα x )) := A◦ and [ (10)
A◦p α
α x
x
andconsideringthatforeachoperator oα x itispossibletodefinethefollowing fractionalmatrixoperator Aα (oα x )= [Aα (oα x )]jk = oα k , (9) itispossibletoobtainthefollowingtheorem:
where[h]k : Ω R denotesthe k-thcomponentofthefunction h.So,itispossibletodefinethefollowing setoffractionaloperators MO∞,u c,x,α (h):= k Z Ok,u c,x,α (h), (6)
withwhichitispossibletoprovethattheset(10)fulfillsthefollowingproperties,whichcorrespondtothe propertiesofanAbeliangroup:
oα x ∈ m
o
i,x ◦ oqα j,x ,
α
x
A◦r α ]jk := orα k ,
whichcorrespondstotheAbeliangroupgeneratedbytheoperator Aα (o )
m
f : Ω ⊂ R2 → R2 ,
Forsimplicity,atwo-dimensionalvectorfunctionisusedtoimplementthecode,thatis, whichmaybedenotedasfollows: (20)
m +
rqα j,x .
,
(xi )= [A ,β ]jk (xi ) := oβ(α,[xi ]k ) k δjk + δjk xi ,
f (x)= [f ]1(x) [f ]2(x) ,
]i : Ω ⊂ R2 → R ∀i ∈ {1, 2}.Thenconsideringafunction Φ :(R \ Z) × Cn → Cn
x]k |
Therefore,if ΦFPN denotestheiterationfunctionofthefractionalpseudo-Newtonmethod,itispossibleto obtainthefollowingresult: (19)
m,
Let α0 ∈ R \ Z ⇒∀A◦1 α0 ∈ m GFPN (α) ∃ΦFPN = ΦFPN (Aα0 ) ∴ ∀Aα0 ∃ {ΦFPN (Aα ): α ∈ R \ Z}
where A ,β )isamatrixevaluatedinthevalue x ,whichisgivenbythefollowingexpression ,β (22)
Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022
α k,x = Ak,α opα i,x ◦
2.ProgrammingCodeofFractionalPseudo-NewtonMethod
qα j,x ∃A◦r k,α = A◦
Toendthissection,itisworthmentioningthatthefractionalpseudo-Newtonmethodhasbeenusedinthe studyfortheconstructionofhybridsolarreceivers[7,8,12],andthatinrecentyearstherehasbeenagrowing interestinfractionaloperatorsandtheirpropertiesforsolvingnonlinearalgebraicequationsystems[13–22].
xi+1 := Φ(α,xi )= xi A ,β (xi )f (xi ),i =0, 1, 2 ··· ,
(xi
r 1) k,α ◦ A◦1 k,α =
ThefollowingcodewasimplementedinPython3andrequiresthefollowingpackages:
A◦p i,α ◦ A◦q j,α = A◦1 k,α := Ak,α opα i,x ◦ oqα j,x ,p,q ∈ Z \ {0} , (14)
α ∈ m GFPN (α)
A ,β = A ,β A◦1 α : A◦1 α ∈ mGFPN (α)and A ,β (x)= [A ,β ]jk (x) .
1 import mathasmt 2 import numpyasnp 3 from numpy import linalgasla
k,α orpα i,x ◦
α
A
◦
where[f ,thefractionalpseudoNewtonmethodmaybedenotedasfollows[11,23]: (21)
α ◦
3
i
asaconsequence,itisfulfilledthat ∀A◦1 k,α ∈ GFPN ( )suchthat Ak,α o o ( A o (15)
m
α, [x]k ):= α,
thefractionalpseudo-Newtonmethodmaybedefinedandclassifiedthroughthefollowingsetofmatrices: (18)
A ,α
Applied
Then,itispossibletoobtainthefollowingresult: ∀A◦1 ∃ := A 1 I I (16) where Im denotestheidentitymatrixof m × m and isapositiveconstant 1.So,definingthefollowing function β( if |[ 0 1 if |[x]k | =0 , (17)
x→ξ Φ
suchthatitfulfillsthefollowingcondition: lim i→∞ Pi → p, andtherefore,thereexistsatleastonevalue k ≥ m suchthat Pk ∈ B(p; ). (26)
Corollary1. Let Φ :(R \ Z) × Cn → Cn beaniterationfunctionsuchthat Φ ∈ Convδ (ξ).So,if Φ hasanorderof convergenceoforder(atleast) p in B(ξ;1/2),forsome m ∈ N,thereexistsasequence {Pi }i≥m ∈ B(p; δK ) givenbythe followingvalues
Applied Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022
≥
Beforecontinuing,itisnecessarytomentionthatwhatisshownbelowisanextremelysimplifiedwayofhow afractionaliterativemethodshouldbeimplemented.Amoredetaileddescription,aswellassomeapplications, maybefoundinthereferences[11,20–23].Consideringtheparticularcasewith Φ :(R\Z)×Rn → Rn,anddefining thefollowingnotation: ErrDom := xi xi 1 2 i 1,ErrIm := f (xi ) 2 i≥1,X := xi i≥1, (28)
fromwhichitfollowsthattheset(23)isgeneratedbyanuncountablefamilyoffractionalfixed-pointmethods. Beforecontinuing,itisnecessarytodefinethefollowingcorollary[11]:
with δjk theKroneckerdelta.Itisworthmentioningthatoneofthemainadvantagesoffractionaliterative methodsisthattheinitialcondition x0 canremainfixed,withwhichitisenoughtovarytheorder α ofthe fractionaloperatorsinvolveduntilgeneratingasequenceconvergent {xi }i≥1 tothevalue ξ ∈ Ω.Sincetheorder α ofthefractionaloperatorsisvaried,differentvaluesof α cangeneratedifferentconvergentsequencestothesame value ξ butwithadifferentnumberofiterations.So,itispossibletodefinethefollowingset
Corollary2. Let Φ :(R \ Z) × Cn → Cn beaniterationsuchthat Φ ∈ Convδ (ξ).So,if Φ hasanorderofconvergenceof order(atleast) p in B(ξ; δ),itisfulfilledthat: if lim (1)( if lim (1)( ) =0 (27)
p := 1,
α,x
Pi = log ( xi xi 1 ) log ( xi 1 xi 2 ) , (25)
whichmaybeinterpretedasthesetoffractionalfixed-pointmethodsthatdefineaconvergentsequence {xi }i≥1 tosomevalue ξα ∈ B(ξ; δ).So,denotingbycard (·) thecardinalityofaset,undercertainconditionsitispossibleto provethefollowingresult(seereference[11],proofof Theorem2):
x→ξ Φ
Convδ (ξ):= Φ :lim x→ξ Φ(α,x)= ξα ∈ B(ξ; δ) , (23)
itispossibletoimplementaparticularcaseofthemultidimensionalfractionalpseudo-Newtonmethodthrough recursiveprogrammingusingthefollowingfunctions[10]:
card (Convδ (ξ)) =card (R) , (24)
α,x) 0 2,
Thepreviouscorollaryallowsestimatingnumericallytheorderofconvergenceofaniterationfunction Φ that generatesatleastoneconvergentsequence {xi }i≥1.Ontheotherhand,thefollowingcorollaryallowscharacterizing theorderofconvergenceofaniterationfunction Φ throughits Jacobianmatrix Φ (1) [11,22]:
4
16
3
24
28
5
23
32 33
Toimplementtheabovefunctions,itisnecessarytofollowthestepsshownbelow: i) Afunctionmustbeprogrammed(informationofthefollowingnonlinearfunctionmaybefoundinthe reference[9]).
13
8
6
18
14
11 12
11
13
9
26
17
18
14 15
9
8
10
10
ii) Threeemptyvectors,afractionalorder α,andaninitialcondition x0 mustbedefinedbeforeimplementing thefunctionFractionalPseudoNewton. 1 ErrDom=[] 2
29 30
9
3 4
12
21
3
1 def Dfrac(α ,x): return pow (x,-α )/mt.gamma(1-α ) if abs (1-α )>0 else 0 def β (α ,x): return α if abs (x)>0 else 1 def A β (α ,x): N= len (x) y=np.zeros((N,N)) = pow (10,-4) for i in range (0,N): y[i][i]=Dfrac(β (α ,x[i]),x[i])+ return y def FractionalPseudoNewton(ErrDom,ErrIm,X,α ,x0): Tol= pow (10,-5) Lim= pow (10,2) 19 x1=x0-np.matmul(A β (α ,x0),f(x0)) ED=la.norm(x1-x0) 22 if ED>0: EI=la.norm(f(x1)) 25 ErrDom.append(ED) ErrIm.append(EI) X.append(x1) N= len (X) if max (ED,EI)>Tol and N<Lim: ErrDom,ErrIm,X=FractionalPseudoNewton(ErrDom,ErrIm,X,α ,x1) return ErrDom,ErrIm,X
2
17
27
4 5
7
31
20
6 7
6 7
2
8
5
1 def f(x): y=np.zeros((2,1)) 4 a1=0.5355 a2=1.5808 a3=1.5355 a4=0.5808 a5=18.9753 a6=451474 a7=396499 d1= pow (x[0],a3)- pow (x[1],a3) d2= pow (x[0],a4)- pow (x[1],a4) d3= pow (x[0],a3+a4)- pow (x[1],a3+a4) y[0]=x[0]-(a6/a5)+(a2 * x[0] * pow (x[1],a3) * d2-a1 * pow (x[0],a2) * d1)/(a1 * a2 * d3) y[1]=x[1]-(a7/a5)+(a2 * pow (x[0],a3) * x[1] * d2-a1 * pow (x[1],a2) * d1)/(a1 * a2 * d3) return y ErrIm=[] X=[] α =-0.02705 x0=np.ones((2,1)) x0[0]=1 x0[1]=2
10 11 ErrDom,ErrIm,X=FractionalPseudoNewton(ErrDom,ErrIm,X,α ,x0) 5 Applied Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022
15 16
=
P
References
61 41844.5708618494611857.3212862052066.11497121228039e 052.0557739187213006e 05
1 24154.689005572621615.77056522465532412.278084455753173.9427518435878
i [xi ]1 [xi ]2 xi xi 1 2 f (xi ) 2
. . . . . . . . . .
Therefore,fromthe Corollary1,thefollowingresultisobtained: 66 log 66 log 65
[4] MAbu-ShadyandMohammedKAKaabar.Ageneralizeddefinitionofthefractionalderivativewithapplications. MathematicalProblemsinEngineering,2021.
3 27022 68658301532616837 962632984793275 47569502431936 4252930355906
4 28968 49778615837615149 130862413852576 49645595851331988 3049656277824
[1] Jos´eATenreiroMachadoetal.Areviewofdefinitionsforfractionalderivativesandintegral. Mathematical ProblemsinEngineering,pages1–6,2014.
Applied Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022
[6] Jian-GenLiu,Xiao-JunYang,Yi-YingFeng,andPingCui.Newfractionalderivativewithsigmoidfunction asthekernelanditsmodels. ChineseJournalofPhysics,68:533–541,2020.
( x
[8] A.Torres-Hernandez,F.Brambila-Paz,P.M.Rodrigo,andE.De-la-Vega.Reductionofanonlinearsystem anditsnumericalsolutionusingafractionaliterativemethod. JournalofMathematicsandStatisticalScience, 6:285–299,2020.ISSN2411-2518.
Whenimplementingtheprevioussteps,ifthefractionalorder α andinitialcondition x0 areadequatetoapproachazeroofthefunction f ,resultsanalogoustothefollowingareobtained:
x65 )
fromwhichitisconcludedthatthefractionalpseudo-Newtonmethodhasanorderofconvergence(atleast) linearin B(ξ; δ).
Table1:Resultsobtainedusingthefractionalpseudo-Newtonmethod[10].
x64 ) ≈ 1.0655 ∈ B(p; δK ),
6
whichisconsistentwiththe Corollary2,sinceif ΦFPN ∈ Convδ (ξ),ingeneral ΦFPN fulfillsthefollowing condition(seereference[22],proofof Proposition1): lim x→ξ Φ (1) FPN (α,x) 0, (29)
[3] MehmetYavuzandNecati Ozdemir.Comparingthenewfractionalderivativeoperatorsinvolvingexponentialandmittag-lefflerkernel. Discrete&ContinuousDynamicalSystems-S,13(3):995,2020.
5 31513 39590875931414371 1203088337282661 16645024316861670 221737448303
[7] A.Torres-Hernandez,F.Brambila-Paz,P.M.Rodrigo,andE.De-la-Vega.Fractionalpseudo-newtonmethod anditsuseinthesolutionofanonlinearsystemthatallowstheconstructionofahybridsolarreceiver. Applied MathematicsandSciences:AnInternationalJournal(MathSJ),7:1–12,2020.DOI:10.5121/mathsj.2020.7201.
65 41844 5709087758311857 32126968221 964996494386499e 057 37448238916247e 06 66 41844 5709068337211857 321260217379 662028743039773e 068 428415184912125e 06
62 41844.570862911411857.3212593259982.690017756661858e 052.567920334469916e 05 63 41844.5708933431911857.3212755723.449675719137571e 051.2220011320189923e 05 64 41844 5708918826811857 321259645141 5993685206416137e 051 4647204650186194e 05
[2] GSalesTeodoro,JATenreiroMachado,andECapelasDeOliveira.Areviewofdefinitionsoffractional derivativesandotheroperators. JournalofComputationalPhysics,388:195–208,2019.
( x
[5] KhaledMSaad.Newfractionalderivativewithnon-singularkernelforderivinglegendrespectralcollocation method. AlexandriaEngineeringJournal,59(4):1909–1917,2020.
6 33594 702999016313550 0054644163142237 42458904800471489 2609462571957
2 23797 52520777181417409 8674610220264221 0409735515232457 2339691838274
[19] GiroCandelario,AliciaCordero,JuanRTorregrosa,andMar´ıaPVassileva.Anoptimalandlowcomputationalcostfractionalnewton-typemethodforsolvingnonlinearequations. AppliedMathematicsLetters, 124:107650,2022.
7
[14] AliciaCordero,IvanGirona,andJuanRTorregrosa.Avariantofchebyshev’smethodwith3αth-orderof convergencebyusingfractionalderivatives. Symmetry,11(8):1017,2019.
[18] GiroCandelario,AliciaCordero,andJuanRTorregrosa.Multipointfractionaliterativemethodswith(2α+ 1)th-orderofconvergenceforsolvingnonlinearproblems. Mathematics,8(3):452,2020.
[16] KrzysztofGdawiec,WiesławKotarski,andAgnieszkaLisowska.Visualanalysisofthenewton’smethodwith fractionalorderderivatives. Symmetry,11(9):1143,2019.
[15] KrzysztofGdawiec,WiesławKotarski,andAgnieszkaLisowska.Newton’smethodwithfractionalderivatives andvariousiterationprocessesviavisualanalysis. NumericalAlgorithms,86(3):953–1010,2021.
[13] RErfanifar,KSayevand,andHEsmaeili.Onmodifiedtwo-stepiterativemethodinthefractionalsense: someapplicationsinrealworldphenomena. InternationalJournalofComputerMathematics,97(10):2109–2141,2020.
[12] EduardoDe-laVega,AnthonyTorres-Hernandez,PedroMRodrigo,andFernandoBrambila-Paz.Fractional derivative-basedperformanceanalysisofhybridthermoelectricgenerator-concentratorphotovoltaicsystem. AppliedThermalEngineering,193:116984,2021.DOI:10.1016/j.applthermaleng.2021.116984.
[21] A.Torres-Hernandez,F.Brambila-Paz,andE.De-la-Vega.Fractionalnewton-raphsonmethodandsome variantsforthesolutionofnonlinearsystems. AppliedMathematicsandSciences:AnInternationalJournal (MathSJ),7:13–27,2020.DOI:10.5121/mathsj.2020.7102.
Applied Mathematics and Sciences: An International Journal (MathSJ) Vol.9, No.1, March 2022
[11] A.Torres-HernandezandF.Brambila-Paz.Setsoffractionaloperatorsandnumericalestimationoftheorder ofconvergenceofafamilyoffractionalfixed-pointmethods. FractalandFractional,5(4):240,2021.DOI: 10.3390/fractalfract5040240.
[9] A.Torres-Hernandez,F.Brambila-Paz,andJ.J.Brambila.Anonlinearsystemrelatedtoinvestmentunder uncertaintysolvedusingthefractionalpseudo-newtonmethod. JournalofMathematicalSciences:Advances andApplications,63:41–53,2020.DOI:10.18642/jmsaa 7100122150.
[22] A.Torres-Hernandez,F.Brambila-Paz,U.Iturrar´an-Viveros,andR.Caballero-Cruz.Fractionalnewtonraphsonmethodacceleratedwithaitken’smethod. Axioms,10(2):1–25,2021.DOI:10.3390/axioms10020047.
[10] A.Torres-HenandezandF.Brambila-Paz.Anapproximationtozerosoftheriemannzetafunctionusing fractionalcalculus. MathematicsandStatistics,9(3):309–318,2021.DOI:10.13189/ms.2021.090312.
[17] AliAkgul,AliciaCordero,andJuanRTorregrosa.Afractionalnewtonmethodwith2αth-orderofconvergenceanditsstability. AppliedMathematicsLetters,98:344–351,2019.
[23] A.Torres-Hernandez,F.Brambila-Paz,andR.Montufar-Chaveznava.Accelerationoftheorderofconvergenceofafamilyoffractionalfixedpointmethodsanditsimplementationinthesolutionofanonlinear algebraicsystemrelatedtohybridsolarreceivers.2021.arXivpreprintarXiv:2109.03152.
[20] A.Torres-HernandezandF.Brambila-Paz.Fractionalnewton-raphsonmethod. AppliedMathematicsand Sciences:AnInternationalJournal(MathSJ),8:1–13,2021.DOI:10.5121/mathsj.2021.8101.