![Page 1: Sayısal Metotlar - Girişoguzhanoztas.com/sm/sayisal_metotlar1.pdf · 2015-05-07 · Kaynak Kitaplar : Numerical Analysis . Richard L.Burden, J.Douglas Faires . International Thomson](https://reader030.vdocuments.pub/reader030/viewer/2022020215/5b31ca377f8b9aed688b8143/html5/thumbnails/1.jpg)
Sayısal Metotlar
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Kaynak Kitaplar : Numerical Analysis Richard L.Burden, J.Douglas Faires International Thomson Publishing Company Numerical Methods Using Matlab John H. Mathews, Kurtis D. Fink Prentice Hall Mühendisler için Sayısal Yöntemler Steven C. Chapra, Raymond P.Canale Çeviri: Hasan Heperkan, Uğur Kesgin Literatür Yayınevi
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İÇERİK -Tek değişkenli fonksiyonların köklerinin bulunması -Interpolasyon(ara nokta tayini) -Sayısal integral -Sayısal türev -Adi Diferansiyel denklemlerin köklerinin bulunması -LU ayrıştırması -Öz değer, Öz vektör -Lineer olamayan denklemlerin köklerinin bulunması
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Tek değişkenli fonksiyonlar Yarılama Metodu :
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[a,b] aralığında köke bakılır p1 bu aralığın orta noktasıdır. f(p1)=0 ise p=p1 dir. f(p1) ve f(a1) aynı işaretliyse kök [p1,b1], a2=p1, b2=b1 dir. f(p1) ve f(a1) farklı işaretliyse kök [a1,p1], a2=a1, b2=p1 dir.
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Algoritması: Adım 1: i=1 FA=f(a) Adım 2: while i<=N0 do steps 3-6 Adım 3: p=(a+b)/2 FP=f(p) Adım 4: if FP=0 or (b-a)/2<TOL then OUTPUT(p) STOP Adım 5: i=i+1 Adım 6: if FA.FP>0 then a=p; FA=FP else b=p Adım 7: OUTPUT(‘Method failed after N0 iterations, N0=‘,N0) STOP
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104)( 23 −+= xxxfÖrnek:
fonksiyonunun [1,2] aralığında kökünün olup olmadığına bakılacak: f(1)= - 5 , f(2)= 14
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Sabit Nokta Metodu :
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104)( 23 −+= xxxfÖrnek:
fonksiyonunun [1,2] aralığında kökünün olup olmadığına bakılacak: f(1)= - 5 , f(2)= 14
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104)( 23 −+= xxxf
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Newton-Raphson Metodu :
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Secant Metodu :
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Regula-Falsi Metodu :
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Interpolasyon
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Interpolasyon Lagrange Interpolasyonu :
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Newton’un Bölünmüş Farklar Metodu :
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Newton’un Bölünmüş Farklar Metodu :
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Örnek : Aşağıdaki nokta değerleri verilmiş olsun.
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Newton’un İleri Bölünmüş Farklar Metodu :
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Newton’un geri Bölünmüş Farklar Metodu :
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Newton’un orta Bölünmüş Farklar Metodu :
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Geri Fark Formülü :
İleri Fark Formülü :
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Orta Fark Formülü :
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Hermit Metodu :
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Sayısal Türev
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Sayısal İntegral
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Adi Diferansiyel Denklemler Euler Metodu : y’=f(t,y) , [t0,tM] , y(t0)=y0 tk=a+kh , k=0,1,2,…,M , h=(b-a)/M y(t) fonksiyonu t=t0 noktasında Taylor serisine açıldığında : y’(t0)=f(t0,y(t0)), h=t1-t0 , değerlerini yerine yazarak elde edilir.
2))(())(()()(
201
000ttcytttytyty −′′
+−′+=
2)())(,()()(
2
10001hcytythftyty ′′++=
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h yeterince küçük seçilirse ikinci türeve sahip terim ihmal edilebilir. Bu durumda :
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Heun Metodu :
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Taylor Metodu :
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Runge-Kutta Metodu :
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Örnek :