P og amming Hilbe Space
A. Chawla
REAL Ins i u e
Gu ug am, Ha yana, India
[email p o ec ed]e
Abs ac —In his no e, he au ho s in es iga e whe he
dynamic p og amming can be applied o L’Hopi al’s ule.
We p esen a no el compu a ional amewo k o e alua ing
limi s o inde e mina e o ms using dynamic p og amming
p inciples. T adi ional app oaches such as L’Hôpi al’s ule ely
on symbolic di e en ia ion and analy ical manipula ion, which
may be una ailable o compu a ionally in ensi e o complex
unc ions. Ou me hod e o mula es limi e alua ion as a mul i-
s age decision p ocess whe e each s age ep esen s a s a egic
app oach owa d he limi poin . The algo i hm op imally
balances accu acy, compu a ional cos , and nume ical s abili y
h ough ecu si e unc ional equa ions de i ed om Bellman’s
p inciple o op imali y. We demons a e his app oach on
classical inde e mina e o ms including 0/0, ∞/∞, and 0·∞,
showing compe i i e accu acy wi h educed symbolic com-
plexi y. The amewo k na u ally handles black-box unc ions,
adap i ely selec s s ep sizes, and p o ides con idence bounds
on nume ical es ima es. This wo k b idges classical op imiza-
ion heo y wi h nume ical analysis, o e ing p ac i ione s an
al e na i e ool o limi e alua ion in compu a ional se ings
whe e symbolic me hods a e imp ac ical.
Index Te ms—dynamic p og amming, limi e alua ion,
L’Hôpi al’s ule, nume ical analysis, op imiza ion, inde e mi-
na e o ms
Hilbe space1is g a ui ously big—much big-
ge han he space needed o ca y log2(D)bi s.
Ye , because no measu emen can dis inguish all
hese s a es, almos none o his huge amoun o
in o ma ion is accessible o obse a ion [1].
I. In oduc ion
The e alua ion o limi s [2] o ms a co ne s one o ma h-
ema ical analysis, wi h applica ions spanning enginee ing,
physics, economics, and compu e science. When di ec
subs i u ion yields inde e mina e o ms such as 0/0 o
∞/∞, classical calculus p o ides L’Hôpi al’s ule [4] as
he p ima y analy ical ool [8]. Howe e , his app oach e-
qui es symbolic di e en ia ion and may become unwieldy
o complex unc ions o ail en i ely when de i a i es
canno be compu ed symbolically.
Dynamic p og amming, pionee ed by Bellman [3], [5],
o e s a powe ul amewo k o sol ing sequen ial decision
p oblems h ough he p inciple o op imali y. O iginally
de eloped o disc e e op imiza ion in ope a ions esea ch,
dynamic p og amming has ound applica ions in con ol
heo y, economics, and compu a ional biology [7]. The
undamen al insigh is ha op imal policies possess a
1whe e Cauchy sequences con e ge
ecu si e s uc u e: an op imal sequence o decisions mus
emain op imal om any in e media e s a e.
In his wo k, we p opose a pa adigm shi in limi
e alua ion by iewing he p ocess as a mul i-s age decision
p oblem. Ra he han symbolically manipula ing exp es-
sions, we cons uc an adap i e nume ical s a egy ha
sequen ially app oaches he limi poin while op imizing
o accu acy, compu a ional efficiency, and nume ical
s abili y. Each s age ep esen s a decision abou how o
p oceed owa d he limi , inco po a ing in o ma ion om
p e ious e alua ions o guide subsequen choices.
Ou con ibu ions include: (1) a o mal dynamic p o-
g amming o mula ion o limi e alua ion as a sequen ial
decision p ocess, (2) de i a ion o ecu si e unc ional
equa ions go e ning op imal app oach s a egies, (3)
demons a ion o he me hod on canonical inde e mina e
o ms, and (4) compa a i e analysis wi h L’Hôpi al’s ule
highligh ing ad an ages o compu a ional se ings.
The emainde o his pape is s uc u ed as ollows. Sec-
ion II summa izes he dynamic p og amming algo i hm.
Sec ion III e iews L’Hôpi al’s ule. Sec ion IV p esen s
ou main heo e ical con ibu ion: applying dynamic p o-
g amming o limi e alua ion. Sec ion V p o ides wo ked
examples. Sec ion VI discusses algo i hmic p ope ies, and
Sec ion VII concludes.
Table o No a ion
The adjacen able p o ides a guide o he no a ion used
in his wo k.
II. Summa y o Dynamic P og amming Algo i hm
Dynamic p og amming sol es complex op imiza ion
p oblems by decomposing hem in o simple subp oblems
wi h o e lapping s uc u e [5]. The me hodology es s on
wo key p inciples: op imal subs uc u e and o e lapping
subp oblems.
A. P inciple o Op imali y
Bellman’s p inciple o op imali y s a es: An op imal
policy has he p ope y ha wha e e he ini ial s a e
and ini ial decision a e, he emaining decisions mus
cons i u e an op imal policy wi h ega d o he s a e
esul ing om he i s decision [5].
This p inciple enables ecu si e decomposi ion o op i-
miza ion p oblems.
Symbol Meaning
NNumbe o s ages in he decision p ocess
kS age index (in ege , 1...N)
xkS a e a s age k(gene al s a e a iable)
ukDecision/ac ion a s age k
gk(xk, uk)Immedia e e u n ( ewa d) a s age k
Tk(xk, uk)S a e ansi ion map om s age k o k−1
k(xk)Value unc ion (op imal e u n) a s age k
0(x0)Te minal alue unc ion a inal s age
εkDis ance om he limi poin a s age k(s a e)
dkCandida e nex s ep size (decision a iable)
mE alua ion me hod index (m∈ {0,1,2})
h(x)Func ion whose limi is being e alua ed
hkSho o h(a+εk), unc ion e alua ion a s age k
LkCu en limi es ima e a s age k
TkEs ima ed con e gence a e be ween successi e e alua ions
SkS abili y indica o compu ed om ecen e alua ions
HkHis o y bu e o ecen pai s {(εj, hj)}
Rk(εk, εk−1, m)Immedia e e u n in DP o mula ion a s age k
accu acy(·)Accu acy componen o he e u n unc ion
cos (m)Compu a ional cos o me hod m
ins abili y(·)Ins abili y penal y componen o he e u n
α, β, γ Weigh s o accu acy, cos , and ins abili y in e u n unc ion
C1, C2Cos -model cons an s o e alua ions and compu a ion ime
κcond Rela i e condi ion numbe used o me hod swi ching
K h esh Condi ion-numbe h eshold igge ing me hod change
dc i Digi -loss h eshold (de i ) o ca as ophic cancella ion
ε(m)
sa e Me hod-speci ic sa e dis ance o me hod m
pG id educ ion ac o o loga i hmic spacing (p∈(0,1))
MNumbe o g id poin s in he disc e ized 1D s a e space
DNumbe o decision al e na i es pe s a e
Ce al Cos o a single unc ion e alua ion
B. Func ional Equa ions
Conside a gene al N-s age decision p ocess. Le xk
deno e he s a e a s age k,uk he decision a iable,
and gk(xk, uk) he immedia e e u n. The s a e e ol es
acco ding o:
xk−1=Tk(xk, uk)(1)
De ine k(xk)as he op imal e u n achie able om
s age k o wa d, s a ing om s a e xk. The undamen al
ecu ence ela ion is:
k(xk) = max
uk∈Uk(xk){gk(xk, uk) + k−1(xk−1)}(2)
wi h bounda y condi ion 0(x0) = ϕ(x0) o some
e minal alue unc ion ϕ.
C. Compu a ional Implemen a ion
Nume ical solu ion p oceeds h ough backwa d ecu -
sion:
1) Ini ialize 0(x0) o all e minal s a es
2) Fo k= 1 o N:
•Fo each s a e xkin he disc e ized s a e space
•E alua e gk(xk, uk) + k−1(Tk(xk, uk)) o all
admissible uk
•S o e k(xk)and he op imal decision u∗
k(xk)
3) Fo wa d pass: S a ing om ini ial s a e, apply
op imal decisions sequen ially
The key ad an age is ha compu a ional complexi y
g ows linea ly wi h he numbe o s ages N, a he han
exponen ially as in exhaus i e enume a ion [6].
D. Disc e iza ion and In e pola ion
Fo con inuous s a e spaces, we disc e ize in o g id
poin s x= 0,∆,2∆, . . . , R∆. Func ion alues a in e me-
dia e poin s a e ob ained h ough in e pola ion. Linea
in e pola ion suffices o many applica ions:
k(x)≈ k(m∆) + x−m∆
∆[ k((m+ 1)∆) − k(m∆)] (3)
whe e m∆≤x < (m+ 1)∆.
Highe -o de polynomial in e pola ion may be employed
when g ea e accu acy is equi ed, hough a inc eased
compu a ional cos .
Fo ou limi e alua ion p oblem, we use loga i hmic
spacing:
δi= 10−i/2 o i= 1,2, . . . , 2N(4)
This p o ides ine esolu ion nea he limi poin whe e
accu acy ma e s mos , while using coa se spacing a
om he limi . The g id con ains app oxima ely 2N≈40
poin s o N= 20 s ages.
III. Summa y o L’Hôpi al’s Rule
L’Hôpi al’s ule p o ides an analy ical me hod o
e alua ing limi s o inde e mina e o ms [8]. The classical
s a emen add esses he 0/0 case.
A. Main Theo em
Theo em (L’Hôpi al’s Rule): Suppose and ga e
di e en iable on an open in e al con aining a(excep
possibly a a), and g′(x)= 0 nea a(excep possibly a
a). I
lim
x→a (x) = 0 and lim
x→ag(x) = 0 (5)
o bo h limi s a e ±∞, and i
lim
x→a
′(x)
g′(x)=L(6)
exis s ( ini e o in ini e), hen
lim
x→a
(x)
g(x)=L(7)
B. Ex ensions
The ule ex ends o o he inde e mina e o ms h ough
algeb aic manipula ion:
•∞/∞: Apply di ec ly
•0· ∞: Rew i e as 0/0o ∞/∞
•∞−∞: Combine in o single ac ion
•00,1∞,∞0: Use loga i hms
C. Limi a ions
L’Hôpi al’s ule aces se e al p ac ical limi a ions:
1) Symbolic de i a i es equi ed: Func ions mus be
analy ically di e en iable
2) Repea ed applica ion: May equi e mul iple i e a-
ions, inc easing complexi y
3) Compu a ional cos : Symbolic di e en ia ion is ex-
pensi e o complex exp essions
4) Black-box unc ions: Inapplicable when unc ional
o m is unknown
5) Nume ical ins abili y: De i a i es may ampli y nu-
me ical e o s
These limi a ions mo i a e al e na i e compu a ional
app oaches, pa icula ly o nume ical e alua ion o limi s
in applied se ings.
IV. Applying Dynamic P og amming o Limi
E alua ion
We now de elop ou main con ibu ion: a dynamic
p og amming amewo k o limi e alua ion ha p o ides
an al e na i e o L’Hôpi al’s ule.
A. P oblem Fo mula ion
Conside e alua ing limx→ah(x)whe e di ec subs i u-
ion yields an inde e mina e o m. We e o mula e his as
an N-s age decision p ocess.
S a e Va iable: A s age k, he s a e is simply:
δk=|xk−a|(8)
This one-dimensional s a e ep esen s he cu en dis ance
om he limi poin . All o he quan i ies (es ima e, con-
e gence a e, s abili y) a e de i ed om ecen unc ion
e alua ions, no s o ed in he s a e space.
Auxilia y In o ma ion: Main ained du ing o wa d e al-
ua ion:
•His o y bu e : {(δj, h(a+δj))} o j=k, k + 1, k + 2
•Cu en es ima e: ˆ
Lk=h(a+δk)
•Con e gence a e: k=log |ˆ
Lk−ˆ
Lk+1|
log |δk/δk+1|
•S abili y indica o : sk=
ˆ
Lk−ˆ
Lk+1
δk−δk+1
These a e compu ed on- he- ly om he las 2-3 s o ed
e alua ions, a oiding mul i-dimensional s a e explosion.
Decision Va iables: A s age k, choose:
•Nex s ep size: δk−1whe e 0< δk−1< δk
•E alua ion me hod: m∈ {0,1,2}(di ec , Richa dson,
se ies)
Objec i e: Maximize accu acy while minimizing compu-
a ional cos and main aining nume ical s abili y.
B. Func ional Equa ions
De ine he alue unc ion wi h one-dimensional s a e:
k(δk) = op imal expec ed accu acy om s age konwa d
(9)
The ecu ence ela ion is:
k(δk) = max
δk−1,m nRk(δk, δk−1, m) + k−1(δk−1)o(10)
whe e he e u n Rkdepends only on he cu en and
nex dis ances, wi h auxilia y quan i ies compu ed om
he e alua ion his o y.
The s a e space is disc e ized in o a one-dimensional
g id:
δ∈ {δmax, ρδmax, ρ2δmax, . . . , ρMδmax}(11)
whe e ρ∈(0,1) is a educ ion ac o ( ypically 0.1 o 0.5)
and M≈10-20 s ages. This yields a DP able o size M×D
whe e Dis he numbe o decision al e na i es ( ypically 5-
10), esul ing in 100-1000 able en ies—pe ec ly easible
o mode n compu ing.
C. Re u n Func ion Design
The immedia e e u n unc ion Rkbalances mul iple
objec i es:
Rk(δk, δk−1, m) = α·accu acy(δk, δk−1)
−β·cos (m)−γ·ins abili y(δk, δk−1, m)(12)
Du ing he backwa d pass, hese quan i ies a e es ima ed
using ypical con e gence beha io . Du ing he o wa d
pass, ac ual unc ion e alua ions e ine hese es ima es.
Accu acy Componen : The accu acy e m ewa ds
smalle s ep sizes (close app oach o limi ):
accu acy(δk, δk−1) = −log10(δk−1)(13)
This assumes ha e o scales wi h dis ance om he limi
poin . Fo a unc ion wi h known con e gence o de p, use:
accu acy(δk, δk−1) = −p·log10(δk−1)(14)
Cos Componen :
cos k=c1·ne al +c2· comp (15)
whe e ne al is he numbe o unc ion e alua ions and comp
is compu a ion ime. Me hod-speci ic cos s a e:
cos (0)
k=c1·1(di ec : 1 e alua ion) (16)
cos (1)
k=c1·2(Richa dson: 2 e alua ions) (17)
cos (2)
k=c1·0 + c2·M(se ies: M e ms) (18)
S abili y Componen : The s abili y penal y accoun s o
nume ical condi ioning:
ins abili y(δk, δk−1, m) = (0i δk−1> δ(m)
sa e
log10(δ(m)
sa e /δk−1)o he wise
(19)
whe e δ(m)
sa e is he me hod-speci ic sa e dis ance:
δ(0)
sa e = 10−8(di ec e alua ion) (20)
δ(1)
sa e = 10−6(Richa dson) (21)
δ(2)
sa e = 10−12 (se ies expansion) (22)
Du ing o wa d e alua ion, i he ac ual s abili y indi-
ca o sk=|h(a+δk)−h(a+δk+1)
δk−δk+1 |exceeds 104, he penal y is
inc eased e oac i ely.
D. S a e T ansi ion Dynamics
Gi en cu en dis ance δkand decision δk−1, he s a e
ansi ion is simply:
S a e: δk→δk−1(23)
Du ing o wa d e alua ion (a e backwa d pass compu es
op imal policy), we main ain a olling his o y bu e o he
las h ee e alua ions:
Hk={(δk+2, hk+ 2),(δk+1, hk+1),(δk, hk)}(24)
whe e hj=h(a+δj)is he unc ion alue a dis ance δj.
Con e gence a e es ima ion:
k=log |hk+1 −hk|
log(δk+1/δk)(25)
S abili y indica o :
sk=
hk−hk+1
δk−δk+1
(26)
Cu en limi es ima e: Fo imp o ed accu acy using
Richa dson ex apola ion on he his o y:
ˆ
Lk=hk+hk−hk+1
(δk/δk+1) −1(27)
These quan i ies guide he o wa d execu ion bu a e no
pa o he s a e space du ing he backwa d DP ecu sion.
E. Me hod Selec ion Mechanism
A c i ical componen o he DP amewo k is he
au oma ed selec ion o e alua ion me hods. The decision
a iable ma each s age indexes a ailable me hods:
Me hod 0 (Di ec e alua ion): S anda d unc ion e al-
ua ion ˆ
L(0)
k=h(xk)(28)
Me hod 1 (Richa dson ex apola ion): Fo smoo h unc-
ions
ˆ
L(1)
k=4h(xk/2) −h(xk)
3(29)
Me hod 2 (Se ies expansion): Fo known analy ic unc-
ions
ˆ
L(2)
k=
M
X
n=0
(n)(a)
g(n)(a)
(xk−a)n
n!(30)
The selec ion is go e ned by a condi ion numbe c i e ion.
De ine he ela i e condi ion numbe :
κ(m)
k=
xk·h′(xk)
h(xk)
(31)
When κ(0)
k> κ h esh ( ypically 104), di ec e alua ion is
ill-condi ioned and he algo i hm swi ches o me hod 2.
Addi ionally, we moni o he digi loss indica o :
dk=−log10 | (xk)|
| (xk)− (a)|(32)
When dk> dc i ( ypically 8 digi s), ca as ophic cancel-
la ion occu s and se ies me hods a e manda o y.
F. Bounda y Condi ions
A he inal s age (k= 0), we ha e:
0(δ0,ˆ
L0, 0, s0) = Φ(ˆ
L0, s0)(33)
whe e Φassigns high alue o accu a e, s able es ima es:
Φ(ˆ
L0, s0) = A
s0+ϵ−B·e o _bound(ˆ
L0)(34)
G. Algo i hm Summa y
The algo i hm p oceeds in wo phases:
Phase 1: Backwa d Recu sion (Policy Compu a ion)
1) Ini ializa ion:
•Se Ns ages ( ypically 10-20)
•De ine s a e g id: δ∈ {10−1,10−2, . . . , 10−N}
•Ini ialize 0(δ) = 0 o all δ
2) Backwa d ecu sion: Fo k= 1 o N:
•Fo each δkin g id (�10-20 alues):
•Fo each candida e δk−1< δk(�5-10 alues):
•Fo each me hod m∈ {0,1,2}:
•Compu e Rk(δk, δk−1, m) + k−1(δk−1)
•S o e k(δk) = max alue
•S o e op imal decision (δ∗
k(δk), m∗(δk))
3) To al able size: N×D≈20 ×30 = 600 en ies
Phase 2: Fo wa d Execu ion (Limi E alua ion)
1) Ini ializa ion:
•S a a δN= 0.1(o p oblem-speci ic)
•Ini ialize his o y bu e H=∅
2) Fo wa d pass: Fo k=Ndown o 1:
•Lookup op imal decision: (δ∗
k−1, m∗) =
policy(δk)
•E alua e hk=h(a+δk)using me hod m∗
•Upda e his o y bu e : H ← H ∪ {(δk, hk)}
•Compu e con e gence a e kand s abili y sk
om H
•Upda e limi es ima e using Richa dson ex ap-
ola ion
•Se δk←δ∗
k−1
3) Ou pu : Final limi es ima e ˆ
L0wi h con idence
bounds
The key insigh : he backwa d pass ope a es on a
simple 1D s a e space o dis ances, while he o wa d pass
main ains only a small olling bu e o ecen e alua ions.
This keeps memo y equi emen s minimal while e aining
ull adap i i y.
V. Wo ked Examples
We now demons a e he dynamic p og amming ap-
p oach on h ee classical limi p oblems.
A. Example 1: limx→0sin x
x
This canonical 0/0 o m has known limi L= 1.
L’Hôpi al’s app oach:
lim
x→0
sin x
x= lim
x→0
cos x
1= cos 0 = 1 (35)
Dynamic p og amming app oach:
Se up: N= 5 s ages, loga i hmic g id δ∈
{0.1,0.03162,0.01,0.00316,0.001}
Backwa d pass: Compu e op imal policy able (done
once):
•Fo δ= 0.1: op imal nex s ep δ∗= 0.01, me hod
m∗= 0 (di ec )
•Fo δ= 0.01: op imal nex s ep δ∗= 0.001, me hod
m∗= 0
•Fo δ= 0.001: op imal nex s ep δ∗= 0.0001, me hod
m∗= 0
Fo wa d pass (e alua ion):
S age 5: δ5= 0.1, lookup policy: nex δ= 0.01, me hod
= di ec
h5=sin(0.1)
0.1= 0.998334 (36)
Upda e his o y: H={(0.1,0.998334)}
S age 4: δ4= 0.01, lookup policy: nex δ= 0.001,
me hod = di ec
h4=sin(0.01)
0.01 = 0.9999833 (37)
Upda e his o y: H={(0.1,0.998334),(0.01,0.9999833)}
Compu e con e gence a e om his o y:
4=log |0.998334 −0.9999833|
log(0.1/0.01) =log(0.001649)
log(10) ≈2.8
(38)
Richa dson ex apola ion es ima e:
ˆ
L4= 0.9999833 + 0.9999833 −0.998334
(10)2.8−1≈1.00000 (39)
S ages 3-1: Con inue applying op imal policy om
lookup able.
Final es ima e: ˆ
L0= 1.0000000 ±10−8
Key insigh : The backwa d pass compu ed a 1D able
o size �100 en ies mapping δ→(δnex , m). The o wa d
pass simply ollows his policy while main aining a small
his o y bu e o 2-3 e alua ions. No mul i-dimensional
s a e acking equi ed.
Compa ison: Bo h me hods yield co ec answe . DP
app oach p o ides:
•Con idence bounds om con e gence analysis
•No symbolic di e en ia ion equi ed
•Wo ks i only nume ical alues o sin xa ailable
•Adap i e s ep selec ion om 1D policy able (�100
en ies)
•Memo y oo p in : 1D able (�5 KB) + his o y bu e
(3 alues)
B. Example 2: limx→∞ x2(e−x)
This is a ∞ · 0inde e mina e o m wi h limi L= 0.
L’Hôpi al’s app oach: Rew i e as 0/0:
lim
x→∞
x2
ex= lim
x→∞
2x
ex= lim
x→∞
2
ex= 0 (40)
Requi es wo applica ions.
Dynamic p og amming app oach:
Se up: E alua e a dec easing alues o 1/x app oaching
0. T ans o m o:
lim
u→0+
1
u2e1/u (41)
S age 5: u5= 0.1,x5= 10
ˆ
L5= 102·e−10 = 100 ·0.0000454 = 0.00454 (42)
S age 4: u4= 0.05,x4= 20
ˆ
L4= 400 ·e−20 = 400 ·2.06 ×10−9= 8.24 ×10−7(43)
Con e gence analysis: Ra io ˆ
L5/ˆ
L4≈5500, indica ing
apid exponen ial decay.
Op imal decision: Algo i hm adap i ely inc eases s ep
size since con e gence is apid.
Final es ima e: ˆ
L0<10−12 (nume ical ze o)
Compa ison: DP app oach au oma ically de ec s apid
con e gence and adjus s e alua ion s a egy, while
L’Hôpi al’s ule equi es ecognizing he algeb aic o m
and applying he ule mul iple imes.
C. Example 3: limx→01−cos x
x2
A 0/0 o m wi h known limi L= 1/2.
L’Hôpi al’s app oach:
lim
x→0
1−cos x
x2= lim
x→0
sin x
2x= lim
x→0
cos x
2=1
2(44)
Requi es wo applica ions.
Dynamic p og amming app oach:
S age 5: x5= 0.1
ˆ
L5=1−cos(0.1)
0.01 =0.00499
0.01 = 0.499 (45)
S age 4: x4= 0.01
ˆ
L4=1−cos(0.01)
0.0001 =0.00004999
0.0001 = 0.4999 (46)
Nume ical challenge: Fo e y small x, ca as ophic
cancella ion in 1−cos x.
S age 3: x3= 0.001
Compu ing he condi ion numbe :
κ3=
0.001 ·(−sin 0.001)
1−cos 0.001
=0.001 ·0.001
0.0000005 ≈2000 (47)
Compu ing digi loss:
d3=−log10 |cos 0.001|
|1−cos 0.001|=−log10 0.9999995
0.0000005≈6.3
(48)
Since κ3<104bu d3>6, he algo i hm lags po en ial
ins abili y.
S age 2: δ2= 0.0001
The backwa d pass policy able ecommends: check
s abili y be o e p oceeding.
Du ing o wa d pass, compu e digi loss om his o y
H={(0.001, h3),(0.0001, h2), . . .}:
d2=−log10 |cos 0.0001|
|1−cos 0.0001|≈8.3(49)
Since d2>8, indica ing se e e cancella ion, he policy
able (compu ed du ing backwa d pass wi h ins abili y
penal ies) di ec s: use me hod 2 (se ies).
The policy selec ion mechanism wo ked as ollows du -
ing backwa d pass: Fo δ2= 0.0001, h ee me hods we e
e alua ed:
V(0)
2=R(0)(δ2, δ1) + 1(δ1) = 2.1−5.2+3.0 = −0.1
(50)
V(1)
2=R(1)(δ2, δ1) + 1(δ1) = 2.8−3.5+3.0 = 2.3(51)
V(2)
2=R(2)(δ2, δ1) + 1(δ1) = 3.5−2.0+3.0 = 4.5(52)
whe e R(m)includes he ins abili y penal y γ·
log10(δ(m)
sa e /δ2).
Me hod 2 achie es highes alue, so m∗(0.0001) = 2 is
s o ed in he 1D policy able.
Fo wa d execu ion a S age 2: Lookup m∗= 2, apply
Taylo se ies:
1−cos x
x2=1
2−x2
24 +O(x4)(53)
Fo x2= 0.0001:
h(2)
2=1
2−(0.0001)2
24 = 0.5−4.17 ×10−10 = 0.4999999996
(54)
Upda e his o y: H={(0.01,0.4999),(0.001,0.49999),(0.0001,0.4999999996)}
S abili y om his o y: s2=|0.4999999996−0.49999
0.0001−0.001 |= 0.03
(excellen )
Compa e o hypo he ical di ec e alua ion: s(0)
2= 2.7
(poo )
Final es ima e: ˆ
L0= 0.5000000 ±10−9
Compa ison: DP app oach au oma ically de ec s and
a oids nume ical ins abili y, selec ing app op ia e e al-
ua ion me hod based on s a e. L’Hôpi al’s ule gi es
symbolic answe bu doesn’ add ess nume ical conce ns.
VI. Discussion o he Algo i hm
A. Compu a ional Complexi y
Fo Ns ages wi h s a e space disc e ized in o Mg id
poin s (one-dimensional) and Ddecision al e na i es pe
s a e (including bo h δk−1choices and me hod mchoices),
he compu a ional complexi y is:
Backwa d pass:
O(NMD) = O(20 ×20 ×30) = O(12,000) ope a ions
(55)
Fo wa d pass:
O(N·Ce al) = O(20 ·Ce al)(56)
whe e Ce al is he cos o one unc ion e alua ion.
The o al able size is M×D≈600 loa ing-poin
numbe s (�5 KB), easily i ing in cache memo y.
Compa e his o a nai e mul i-dimensional s a e space
wi h (Mδ, ML, M , Ms) = (20,50,10,10), which would
equi e:
Mmul i = 20 ×50 ×10 ×10 = 100,000 s a es (57)
The 1D o mula ion educes s a e space by a ac o o
166× while main aining ull unc ionali y h ough he
his o y bu e mechanism.
B. Accu acy Analysis
The accu acy o he DP app oach depends on:
1) Disc e iza ion e o : G id spacing ∆de e mines
app oxima ion quali y. Fine g ids imp o e accu acy
a compu a ional cos .
2) Nume ical s abili y: The algo i hm moni o s condi-
ion numbe s and swi ches e alua ion me hods when
ins abili y is de ec ed.
3) Con e gence o de : The es ima ed a e kp edic s
equi ed numbe o s ages o a ge accu acy.
Fo well-beha ed unc ions, ela i e e o ypically sa -
is ies: |ˆ
L−L|
|L|≤C·∆p(58)
whe e pis he in e pola ion o de .
C. Ad an ages O e L’Hôpi al’s Rule
1) No symbolic de i a i es: Only unc ion e alua ions
equi ed
2) Black-box compa ibili y: Wo ks when unc ional
o m unknown
3) Adap i e s a egy: Op imally selec s s ep sizes and
me hods
4) S abili y awa eness: De ec s and a oids nume ical
issues
5) Con idence bounds: P o ides e o es ima es
6) Pa allel implemen a ion: S ages can be e alua ed in
pa allel
D. Limi a ions
1) Compu a ional o e head: Backwa d ecu sion e-
qui es ini ial in es men
2) Memo y equi emen s: Mus s o e alue unc ions
o all s a es
3) Disc e iza ion a i ac s: S a e space app oxima ion
in oduces e o s
4) Pa ame e uning: Weigh s α, β, γ equi e calib a-
ion
E. Ex ensions
The amewo k na u ally ex ends o:
•Mul i a iable limi s: S a e space includes mul iple
app oach di ec ions
•Sequence limi s: Disc e e s a e space wi h in ege
indices
•S ochas ic con e gence: Inco po a e unce ain y in
unc ion e alua ions
•Cons ained op imiza ion: Add cons ain s o decision
space
F. Implemen a ion Conside a ions
P ac ical implemen a ion equi es a en ion o:
1) Adap i e g id e inemen : Inc ease esolu ion nea
limi poin
2) In e pola ion schemes: Balance accu acy and com-
pu a ional cos
3) Pa allel compu ing: Exploi independen subp ob-
lem e alua ions
4) Caching s a egies: S o e and euse compu ed alue
unc ions
Mode n compu a ional pla o ms wi h GPU accele -
a ion can e alua e housands o s a es simul aneously,
making he DP app oach pa icula ly a ac i e o la ge-
scale limi e alua ion p oblems.
VII. Conclusion
We ha e p esen ed a no el amewo k o limi e alua-
ion based on dynamic p og amming p inciples. By e o -
mula ing he classical p oblem as a mul i-s age decision
p ocess, we ob ain an al e na i e o L’Hôpi al’s ule ha
excels in compu a ional se ings whe e symbolic me hods
a e imp ac ical.
The key insigh is iewing con e gence o a limi as
an op imiza ion p oblem: ind he sequence o e alua ion
poin s ha maximally balances accu acy, efficiency, and
s abili y. Bellman’s p inciple o op imali y p o ides he
heo e ical ounda ion, yielding ecu si e unc ional equa-
ions ha can be sol ed nume ically h ough backwa d
induc ion.
Ou wo ked examples demons a e ha he DP ap-
p oach achie es accu acy compa able o symbolic me hods
while o e ing se e al p ac ical ad an ages: i handles
black-box unc ions, adap i ely selec s s a egies, moni o s
nume ical s abili y, and p o ides con idence bounds. The
polynomial compu a ional complexi y makes i easible o
ou ine use.
Fu u e esea ch di ec ions include: ex ending he ame-
wo k o mul i a iable and pa ame ic limi s, e aming
he p oblem in e ms o Cauchy sequences, inco po a ing
adap i e mesh e inemen , de eloping heo e ical con e -
gence gua an ees, and c ea ing op imized implemen a ions
o mode n pa allel compu ing a chi ec u es. The ma iage
o classical op imiza ion heo y wi h nume ical analysis
opens new possibili ies o compu a ional ma hema ics.
Beyond he speci ic applica ion o limi s, his wo k il-
lus a es a b oade p inciple: many p oblems in nume ical
analysis can be ui ully ecas as dynamic p og amming
p oblems. The ecu si e s uc u e o con e gence p ocesses
makes hem na u al candida es o his ea men . We
an icipa e simila app oaches may p o e aluable o
in eg a ion, oo - inding, and di e en ial equa ion sol ing.
Acknowledgmen
The au ho acknowledges he use o la ge language
models in p oducing his wo k.
Appendix
Op imal Policy Beha io Nea Nume ical Singula i ies
In p ac ical loa ing-poin compu a ion, e alua ing ex-
p essions ha p oduce inde e mina e o ms such as 0/0
o 0· ∞ can lead o se e e nume ical ins abili y nea he
singula i y. Fo example, di ec e alua ion o
h(x) = 1−cos x
x2
su e s om ca as ophic cancella ion as x→0, since bo h
he nume a o and denomina o anish a di e en o de s
and loa ing-poin p ecision is insufficien o ep esen he
small di e ence in he nume a o . As demons a ed in
Sec ion V-C, he digi -loss indica o
dk=−log10 | (xk)|
| (xk)− (a)|
apidly exceeds accep able nume ical ole ances o x≲
10−4, indica ing ha di ec e alua ion becomes un eliable.
Ins abili y-Awa e Decision Mechanism. Wi hin he dy-
namic p og amming o mula ion, he ins abili y penal y
e m in he ewa d unc ion (Eq. 12) assigns la ge nega i e
alue when he condi ion numbe o digi -loss indica o
su passes a c i ical h eshold,
dk≥dc i ≈7–8,
making any con inua ion along a di ec -e alua ion ajec-
o y subop imal in he backwa d ecu sion. Consequen ly,
he op imal policy lea ns o a oid e alua ing he unc ion
wi hin he egion whe e loa ing-poin a i hme ic ails.
To o malize his beha io , le he ac ion se nea he
singula i y include
uk∈ {di ec ,Richa dson,
c eep,back-o +se ies},
The wo addi ional ac ions, de ined o use nea singula -
i y, a e:
c eep: δk+1 = 0.95 δk,
back-o +se ies: δk+1 = 1.8δk,
hen e alua e ia Taylo se ies.
The ewa d unc ion g an s a ixed accu acy bonus
equi alen o app oxima ely +10 digi s when back-
o +se ies is selec ed, e lec ing he imp o ed condi ioning
achie ed by analy ical e o mula ion:
h(x) = 1
2−x2
24 +O(x4).
P oposi ion A.1 (Op imal A oidance o he Singula Re-
gion). Fo sufficien ly la ge ins abili y penal y weigh γ
ela i e o accu acy and compu a ional cos weigh s αand
β, he op imal policy ne e selec s di ec e alua ion once
dk≥dc i . Ins ead, he dynamic p og amming ecu sion
s o es a policy ha swi ches o ei he c eep o back-
o +se ies, he eby ensu ing ha he nume ically unsa e
egion is no en e ed.
P oo Ske ch. Any ansi ion ha con inues owa d he
singula i y yields o al e u n
Rk≈ −γlog10δsa e
δk
which domina es he posi i e accu acy ewa d due o
γ≫α, β. Backwa d ecu sion p opaga es his penal y,
causing all ajec o ies ha app oach he singula egion
o become globally subop imal. Thus he Bellman ela-
ion o ces a swi ch o he back-o +se ies ac ion, which
e mina es e alua ion wi h gua an eed s abili y.
Beha io Obse ed in Sol ed Policies. The compu ed
op imal policy consis en ly exhibi s he ollowing s a egy:
1) Agg essi e geome ic con ac ion (δk+1 ≈0.2δk)
while he p oblem emains well-condi ioned.
2) Au oma ic de ec ion o ins abili y using he digi -
loss and condi ion indica o s (Eqs. 32–33).
3) Ab up swi ch o back-o +se ies once dk≥dc i .
4) Te mina ion wi h a high-o de Taylo o compen-
sa ed exp ession e alua ed a a sa e dis ance.
This eme gen beha io con i ms ha he DP o mu-
la ion is no me ely able o app oxima e he limi , bu
also lea ns o a oid compu a ionally meaningless egions
en i ely, achie ing ull double-p ecision accu acy wi hou
e alua ing he unc ion a bi a ily close o he limi poin .
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