chapter 3: interpolation and polynomial approximationmagan/math3311_2013/notenew/ch3.1_short.pdf ·...
TRANSCRIPT
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Chapter 3: Interpolation and
Polynomial Approximation
-
x
y Known data
Unknown
Can we get unknown data from those known? How?
Yes!
By interpolation: find a polynomial that gives function y(x)
which fits all known data and is relatively accurate in the
whole data domain, so that unknown y(x) can be found.
-
3.1:Interpolation and the Lagrange
Polynomial
• The polynomial that passes two known data
points (x0,y0) and (x1,y1) can be expressed as
01
0)(1
and
10
1)(0
;1
)1
(,0
)0
(
where
),()()()()( 1100
xx
xxxL
xx
xxxL
yxfyxf
xfxLxfxLxP
−
−=
−
−=
==
+=
-
points.data known two thethrough
passing function One) (Degreelinear unique theis
)( ,)( 1100
P
yxPyxP ==Q
-
For case with n+1 known data points,
4444444 34444444 21pointsdata 1
1100 ))(,()),...,(,()),(,((
+n
nn xfxxfxxfx
-
)).......()((
)......)((numerator where
,...2,1,0,)(
11
10
,
nkk
k
kkn
xxxxxx
xxxx
nkxxnumerator
numeratorLL
−−−
−−=
==
==
+−
10
1 case points for two :example For .except all contain i.e
0 xx
xxLkxxixx −
−=−−
∑=
=++=n
k
knkknnn xLxfLxfxLxfxP0
,,0,0 )()()()()()( L
-
Theorem 3.2
)()(
)()( where
)()()(
,....1,0),()( whichin
exists )( uniqueTHEN
........,at given are )( of values
and numbers,distinct 1 are ......., If
0,
0
k
10
10
xLxx
xxxL
xLxfxP
nKxPxf
xP
xxxxf
nxxx
k
ik
in
kii
kn
n
k
k
kK
n
n
=−
−Π=
=
==
+
≠=
=
∑
P(x) Satisfies given data
-
Example 1
Three points data:
,25.0)(,4.0)( ,5)(
4 ,5.2 ,2
210
210
===
===
xfxfxf
xxx
15.1)425.005.0()()()(
3
5)5.4(
))((
))(()(
3
32)244(
))((
))(()(
10)5.6())((
))(()(
0
1202
102
2101
201
2010
210
+−==
+−=
−−
−−=
−+−=
−−
−−=
+−=−−
−−=
∑=
xxxLxfxP
xx
xxxx
xxxxxL
xx
xxxx
xxxxxL
xxxxxx
xxxxxL
n
k
kk
Degree two polynomial
-
[ ] well.)( esapproximat )( , withinfound isIt ).(for )(
construct to1
)( of pts 3 uses (example) case thefact, In
20 xfxPxx
xfxP
xxf =
=1/x
-
Theorem 3.3 (Error of interpolation using Lagrange polynomial )
[ ] [ ][ ] [ ]
withexists
,in )(number a , each for :THEN
, and ,,......, If 110
baxbax
baCfbaxxx nn
ζ
+∈∈
))....()(()!1(
)()()(or
))....()(()!1(
))(()()(
10
)1(
10
1
n
n
n
n
xxxxxxn
fxPxf
xxxxxxn
xfxPxf
−−−+
=−
−−−+
+=
+
+
ζ
ζ
-
available. be should
bonds its and ion,interpolat theof
error theestimate to3.3 Theorem use To
)1( +nf
)(
)()(),()()(
where
00 ik
in
Kii
kk
n
K
kxx
xxxLxLxfxP
−
−Π==≠=
=
∑
-
•Recursively Lagrange Polynomial
k
k
mmm
mmm
xxxxf
xP
......, pointsat )( with
agrees valueits that defined is )(
:Define
21
21 ...,
-
xe
xxxxx
for
.6 ,4 ,3 ,2 ,1 have weif e.g. 43210 =====
43
210
43
21
40302010
4321
443322
1100
4
0
))()()((
))()()((
)()()(
)()()()(
)()(
xx
xxx
ik
i
kii
eLeL
eLeLexxxxxxxx
xxxxxxxx
xfLxfLxfL
xfLxfLxfxx
xxxP
++
++−−−−
−−−−=
+++
+=−
−Π=≠=
-
421 ..........))((
))(()(
4121
424,2,1
xxxeee
xxxx
xxxxxP ++
−−
−−=
421 ,, points 3 useonly i.e. xxx
Theorem 3.5
).....,( of numbersdistinct two
be and (2) ;.....,at defined be (1) If
10
10
k
jik
xxx
xxxxxf
But,
-
THEN:
)(
)()()()(
..1,1,....1,0..1,1,....1,0
ji
kiiikjjj
xx
PxxxPxxxP
−
−−−= +−+−
-
Theorem 3.5 Says,
We can construct a higher-degree (including
more given-data points.) polynomials [P(x)
with all data points including points. i, j ] from a
lower-degree Polynomials without including
points i,j.
], pointsdata includet don'
,,....1,1,......1,0 and ,....1,1,......1,0 e.g.[
ji
kiiPkjjP +−+−
-
Example (recursively generating Polynomial)
Neville’s method
1,0
01
0110
10
1
43210
)0,1 i.e.( )()(
used, are and if 3.5, Theorem From
polynomial degree-first The
)2.2( ),9.1( ),6.1( ),3.1()( ),0.1(
2291613101
pointsdata givenFour
P
jixx
PxxPxxP
xx
ffffxff
,., x., x., x., x.x
=
==−
−−−=
=
=====
(2 pts, minimum required)
)( 1xf )( 0xf
-
02
1,022,10
2,1,0
210
4,33,22,1
433221
)()(
, , , say, (3pts), twodegreefor 3.5, Theorem
using degree,-higher toproceed , , , have we
used, are pairs ),,( and ),( ),,( if Similarly,
xx
PxxPxxP
xxx
PPP
xxxxxx
−
−−−=
lower degree
higher degree
-
jiQ ,∴degree of polynomial j+1 data points
last data point used (note i=0,1,2,
-
.polynomial degree having andpoint
data 1 using is, that iteration, final theis AND
)()(
,.....2,1For
,.....2,1For
ioninterpolat iterated sNeville' so,
)0(
,
1,11,
,
n
nQP
xx
QxxQxxQ
ij
ni
ij
nn
jii
jiijiji
ji
+=
−
−−−=
=
=
≤≤
−
−−−−
Iteration
procedure
is shown
in the next
pages
-
03
2,232,30
33
13
1,231,31
32
23
0,230,32
31
02
1,121,20
22
12
0,120,21
21
11
)()(,3
)()(,2
)()(,13
)()(,2
)()(,12
11
xx
QxxQxxQj
xx
QxxQxxQj
xx
QxxQxxQji
xx
QxxQxxQj
xx
QxxQxxQji
Qji
−
−−−==→
−
−−−==→
−
−−−==→=
−
−−−==→
−
−−−==→=
→=→= )( 22 xfP = )( 1xf
-
)(
)()( ,4
)(
)()( ,3
)(
)()( ,2
)(
)()( ,14
04
3,343,40
4,4
14
2,342,41
3,4
24
1,341,42
2,4
34
0,340,43
1,4
xx
QxxQxxQj
xx
QxxQxxQj
xx
QxxQxxQj
xx
QxxQxxQji
−
−−−==→
−
−−−==→
−
−−−==→
−
−−−==→=
P.degreelow previous from
obtained are circle red with,* jiQ4,4)( QxP =
We get P(x) by using the previous
lower-degree P.
-
Each row is completed before the succeeding rows
are begun.
Data point
Example: Schematic Explanation of Neville’s Iteration
interpolation using five data points (degree four)