ch2函數(shù)逼近與曲線擬合

Ch3函數(shù)逼近與曲線擬合

什么是函數(shù)逼近維爾斯特拉斯定理

Weierstrass范數(shù)與賦范線性空間三種常用的范數(shù)內(nèi)積與內(nèi)積空間正交多項式

OrthogonalPolynomials

勒讓德(Legendre)多項式勒讓德多項式的性質(zhì)切比雪夫（Chebyshev）多項式性質(zhì)其它正交多項式最佳一致逼近多項式

optimaluniformapproximatingpolynomial切比雪夫定理xy0yyx=()yyxEn=+()yyxEn=-()yPxn=()證明：反證，設(shè)有2個，分別是Pn

和Qn

。則它們的平均函數(shù)也是一個OUAP。2)()()(xQxPxRnnn+=對于Rn

有Chebyshev交錯組{t1,…,tn+2}使得nkknkknkknnEtytQtytPtytRE

-+-

-=|)()(|21|)()(|21|)()(|nkknkknEtytQtytP=-=-|)()(||)()(|則至少在一個點上必須有)()()()(knkkkntQtytytP-=-

0)()(=-kkntytR0=nE

最佳一次逼近多項式最佳平方逼近法方程組

/*normalequations*/

Hilbert陣！曲線擬合的最小二乘法確定多項式，對于一組數(shù)據(jù)(xi,yi)(i=1,2,…,n)

使得達到極小，這里n

<<

m。naaa10

實際上是a0,a1,…,an

的多元函數(shù)，即[]

=-+++=miinininyxaxaaaaa121010...),...,,(j在

的極值點應(yīng)有kiminjijijxyxa

==-=10][2-=

===+njmikiimikjijxyxa0112記

====mikiikmikikxycxb11,法方程組(或正規(guī)方程組)/*normalequations*/回歸系數(shù)/*regressioncoefficients*/§1L-SApproximatingPolynomials例：xy(xi,yi),i=1,2,…,m方案一：設(shè)baxxxPy+=

)(求a

和b

使得最小。

=-+=miiiiybaxxba12)(),(jButhey,thesystemofequationsforaandbisnonlinear!Takeiteasy!Wejusthavetolinearizeit…線性化

/*linearization*/：令，則bXaY+

就是個線性問題將化為后易解a

和b。),(iiYX),(iiyx§1L-SApproximatingPolynomials方案二：設(shè)xbeaxPy/)(-=

(a>0,b>0)線性化：由可做變換xbay-

lnlnbBaAxXyY-====,ln,1,lnBXAY+

就是個線性問題將化為后易解A

和B),(iiYX),(iiyxMovingLeastSquaresapproximation

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