math 174: numerical analysis i -...
TRANSCRIPT
CHEBYSHEV, 1853Given a continuous function f defined on a closed interval [a,b] and a positive integer n, can we “represent” f by a polynomial p(x), of degree at most n, in such a way that the maximum error at any point x in [a,b] is controlled?
In particular, is it possible to construct p so that the error
is minimized?
)()(max xpxfbxa -££
QUESTIONS?• Why should such a polynomial even exist?• If it does, can we hope to construct it?• If it exists, is it also unique?• What happens if we change the measure of
error to, say,
• Can we approximate f by a rational function? What about by a trigonometric function?
ò -b
adxxpxf 2)()(
APPROXIMATION THEORY
• In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errorsintroduced thereby. Note that what is meant by best and simpler will depend on the application.
WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885
THEOREM: Let
For every
there exists a polynomial p such that
].,[ baCf Î
,0>e
. ,)()( bxaxpxf ££<- e
For Math 174, suppose f and p are real-valued
Karl Weierstrass
In other words: Every continuous real-valued function defined on an interval [a,b] can be uniformly approximated as closely as desired by a real-valued polynomial function.
In other words: For every continuous function f:[a,b]àR there exists a sequence of real-valued polynomials p1, p2, p3,… uniformly converging to f on the [a,b].
WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885
Why polynomials?
Because polynomials are the simplest functions, and computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance.
WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885
The Weierstrass approximation theorem was generalized by Marshall H. Stone in the
STONE-WEIERSTRASS THEOREM (1937).
There are hundreds of proofs of the WeierstrassApproximation Theorem. One is the constructive proof using the Bernstein Polynomials.
Bernstein polynomial is not the only approximating polynomial. We just need this to prove existence.
WEIERSTRASS (FIRST) APPROXIMATION THEOREM, 1885
Bernstein Basis Polynomials
vnvnv xx
vn
xb --÷÷ø
öççè
æ= )1()(,
( )( )
44,4
34,3
23,1
1,1
)(
14)(
13)(
)(
xxb
xxxb
xxxb
xxb
=
-=
-=
=
( )22,0
1,0
0,0
1)(
1)(1)(
xxb
xxbxb
-=
-=
=
EXAMPLES:
Bernstein Polynomials
åå=
-
=
-÷÷ø
öççè
æ==
n
v
vnvv
n
vnvvn xx
vn
xbxB00
, )1()();( bbb
Bernstein coefficients or Bézier coefficients
INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912
Why create “new” polynomials instead of using Taylor polynomials?
Taylor polynomials are not appropriate, they are applicable only to functions that are infinitely differentiable and not to all continuous functions.
WLOG, assume [a,b]=[0,1]. Change of variable:
å=
--÷÷ø
öççè
æ÷øö
çèæ=
NÎ÷øö
çèæ=
n
v
vnvn
v
xxvn
nvfxfB
nvf
0)1();(
n ,Let b
abaxx old
new --
=
INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912
Notice the similarity to binomial distribution:
1)1(0
=å -÷÷ø
öççè
æ=
-n
v
vnv ppvn
np=value expected [ ]1,0Îp
INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912
Also the median and mode
INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912
v=Number of successes
b v,n
(x)
)()}(max{, and Fix
,, xbxbxn
nnxnv =
)()}(max{, and Fix
,, xbxbxn
nnxnv =
INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912
negligible if n is large
INSIGHT ON A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM, 1912
Recall: Weak law of large numbers
INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912
úû
ùêë
é=÷
øö
çèæ=
å -÷÷ø
öççè
æ÷øö
çèæ=
=
-
nvnvfE
xxvn
nvfxfB
n
v
vnvn
,...,1,0,
)1();(0
We can treat Bn(f;x) as the expected value of the numbers f(v/n), v=0,1,…,n.
INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912
( )xfnnxf
nnxfEnv
nvfE
xxvn
nvfxfB
n
v
vnvn
=÷øö
çèæ=
úû
ùêë
é÷øö
çèæ»ú
û
ùêë
é=÷
øö
çèæ=
å -÷÷ø
öççè
æ÷øö
çèæ=
=
-
,...,1,0,
)1();(0
If n is large…
In summary, for large n
)();( xfxfBn »
INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912
What we have done is a naïve way of suspecting that
Actually, the above limit is true for any point x in [a,b].
INSIGHT ON HOW BERNSTEIN PROVED THE WEIERSTRASS APPROXIMATION THEOREM, 1912
( ) ¥®® nxfxfBn as ,);(