truncation errors: using taylor series to approximate ...let’s say we want to approximate a...
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Truncation errors: using Taylor series to approximate functions
Let’s say we want to approximate a function !(#) with a polynomial
For simplicity, assume we know the function value and its derivatives at #% = 0 (we will later generalize this for any point). Hence,
Approximating functions using polynomials:
! # = (% + (* # + (+ #+ + (, #, + (- #- + ⋯
! 0 = (%
!/ # = (* + 2 (+ # + 3 (, #+ + 4 (- #, +⋯
!′ 0 = (*
!//(#) = 2 (+ + 3×2 (, # + (4×3)(- #+ + ⋯
!′′ 0 = 2 (+!′′′ 0 = (3×2) (,
!/5 0 = (4×3×2) (-
!///(#) = 3×2 (, + (4×3×2)(- # +⋯!/5(#) = (4×3×2)(- +⋯
or f" '
ei ! ai
L4ai=f"Yi!T
f "
fixCi - it x Ci - r ) x. . . x1) ai
Taylor SeriesTaylor Series approximation about point !" = 0
% ! =&'()
*+' !'
% ! = +" + +- ! + +. !. + +/ !/ + +0 !0 + ⋯
⇒fas÷÷÷÷TI→ approximate function values
→ approximate derivatives
→ estimating errors
Taylor Series
! " = ! "$ + !& "$ (" − "$) +!&& "$2! (" − "$),+
!&&& 03! (" − "$)/+⋯
In a more general form, the Taylor Series approximation about point "$ is given by:
! " =1234
5 ! 2 ("$)6! (" − "$)2
Example:Assume a finite Taylor series approximation that converges everywhere for a given function !(#) and you are given the following information:
! 1 = 2; !)(1) = −3; !))(1) = 4; ! - 1 = 0 ∀ 0 ≥ 3
Evaluate ! 4 _fI x) = f Go) t f'
ED G - Xo ) tf "z G - xo5tfHMake x = A and Xo = I
f- (4) = fu ) t f 'Ll) C4 - D t ft ) (4 - D'
= 2 t f - 3) ( 4- Dt 2414-D2
= 2 - 9 t 18 ⇒/fl4)=HJ
Taylor SeriesWe cannot sum infinite number of terms, and therefore we have to truncate.
How big is the error caused by truncation? Let’s write ℎ = # − #%x = ht Xo
-
ffxoth)=fGDtf'GDhtf¥hit h't -- -
t÷t"fh+÷i.
hi
exact - -
truncated part what we are neglecting(Taylor approximation error
ffx) of degree n )
thx )
Taylor series with remainderLet ! be (# + 1)-times differentiable on the interval ('(, ') with !(*)continuous on ['(, '], and ℎ = ' − '(error = exact - approximation
error = fat - tn Cx ) = ¥÷, f"f hi
ht 2
=f h
" "
+ f"o ) h + . . .
( ht2) !
-dominant term when h → o Cor x → to )
error f M h" "
or error = O I hit'
)
Taylor series with remainderLet ! be (# + 1)-times differentiable on the interval ('(, ') with !(*)continuous on ['(, '], and ℎ = ' − '(error = exact - approximation
Remainder Theorem : Rnlx) =FG ) - tncx )
=÷:S . ii
/RnH=fII,Its*1-where BE ( Xo ,
x )
}since 15-xd E h
#×
iris it'ttII¥D
Graphical representation:
tix )f f' Go)=d¥ → dy = f
'
GD ( x - Xo )
\•Ferrier t
,=f Gd t f' Go )( X - Xo ) V
A
go.ua#iIIdYerror=fCD-tiCDa--Remainder
YI [email protected] )I I
f' GD'
i !/error=0(h#tangentatxoi
, -I I
" !>
supposeinterval is reduced by half .
% Xwhat happens to the error ?
÷ ÷ '¥¥I
Example:Given the function
! " = 1(20" − 10)
Write the Taylor approximation of degree 2 about point "* = 0
found
Given the function : FG ) = 120X - LO
write the Taylor approximation of degree 2 about to = o
¥47joking;tho ) =
. 's
G) =
-1204%1726,492=¥g÷o ,
f' 'lol = -9%7=-45
ftdxk-f-shx-EIE.TT trackf"z x'I errors OH )
-
-3- to
. .
µ-
Z10-1
-
y= ax
!error log
bogeys-
- liogcatbwlogcx)
Islope !b=bfg:fg:
= III. =3
,→ error -
- ou 's r
Example:Given the function
! " = −"% + 1
Write the Taylor approximation of degree 2 about point "( = 0
IDENT
AH = ft xo) tf'GDC x - xD tf "f ( x - * It - - -
f- ( o ) = I
f-' G) = ¥C- x 't 15
"
= - X ( I - Ej" '
→ ft Co) = o
f"
A) = C I- Ej"
EH - a - xD" "
→ f''
to ) = -
A
thx) = I - zcx5 or /tdD=I-I
! " = −"% + 1error = tz - Hx )
O
{t2
¥
"
good"
approximation close to xo = o
! '
error increases when x moves away from to
-use log-hog plot to better visualize what is happening
Close to to.
error = thx ) - far )
f- G) =NETT
- no-5
thx ) = t - %
-1-0-9IRat s
ITHI.li/⇐
Olli)
3 !
here h =X - Xo =
X
Let's get Big-0 of error from the plot !
sbr-tfggfj.bg:7?-=-.s.IL--4
→ error -
- och" )
what happened here !
→ I"
G) = o hence the next term that is not
zero is f'
*G)
Example:
A)B)C)D)E)
Demo “Taylor of exp(x) about 2”
IDEMOJ
error = e' I ¥75'ta÷5t - -- ]
t#error = Off-29 )
* ef.MG - 24as x → 2
,
e becomessmaller
* e E M x"
} all are valid options !as × → 2 ,
ears does not
show asymptoticbehavior
- this is .
tightestbound .
Finite difference approximationFor a given smooth function ! " , we want to calculate the derivative !′ " at " = 1.Suppose we don’t know how to compute the analytical expression for !′ " , but we have available a code that evaluates the function value:
Can we find an approximation for the derivative with the available information? YES !
Caldasf'
G) = ¥¥fffGth)h-f#)
Can we simply use f'G) a fCxth)h-f# and make
h small ? How do we choose h ?
what is the error ?
f : IR → IR and f is smooth I h : small change#tur baton )K
to X .
- Taylor Series :
f- ( x the) = HHtf
!x) h tfz h 't.. . Cjcest replaced to with x )
f- ( x th ) flex ) tf'
G) h . to Chi )
ffxth ) - f G)=
f'
G) to Ch ) → f'G) =fGth) - Och)
: - -
-error
he extract -
fdf¥f¥fFITTED# F. D. approx )
/errordue\to FD approx
⇒ et = 01h )-
Demo: Finite Difference
! " = $% − 2
(!$")*+ = $%
(!),,-." = $%/0 − 2 − ($%−2)ℎ
$--.-(ℎ) = )45((!$")*+ − (!),,-.")
We want to obtain an approximation for !′ 1
$--.- < !99 : ℎ2
truncation error
IE IE's
Demo: Finite Difference
! " = $% − 2
(!$")*+ = $%
(!),,-." = $%/0 − 2 − ($%−2)ℎ
$--.-(ℎ) = )45((!$")*+ − (!),,-.")
We want to obtain an approximation for !′ 1
ℎ $--.-
$--.- < !99 : ℎ2
truncation error
Demo
I I
↳ absolute error !
Demo: Finite Difference
Should we just keep decreasing the perturbation ℎ, in order to approach the limit ℎ → 0 and obtain a better approximation for the derivative?
$′ & = lim+→,$ & + ℎ − $(&)
ℎ
~ error =Och )
I
Uh-Oh!What happened here?
! " = $% − 2
!′ " = $% → !′ 1 ≈ 2.7
!′ 1 = lim1→2
! 1 + ℎ − !(1)ℎ①
-managing:Freeman.
D very small h Chiao"
)
f- ( Ith ) - to ) →starts increasing
losingprecision due to cancelation !
2) even smaller Chf to- "
) → f C Ith ) = fu ) → If = O
I machine epsilon C because we are adding
h to I.
If another FP number,
than this would
be the gap .
!""#"~%ℎ2Truncation error:
Rounding error: !""#"~2(ℎ
I
I
'
-8I 10
in
h =
Eh⇒ HIE
h -- NI
f- ( Ith ) - fu ) h =NIF
- -8h he to