## Download e-book for iPad: Numerische Mathematik I by Moeller H.M.

By Moeller H.M.

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Example text

Bn−1 bn−2 − x0 bn−1 ⇒ ⇒ a1 a0 = = b0 − x0 b1 p(x0 ) − x0 b0 ⇒ ⇒ an pn (x)−pn (x0 ) x−x0 = x0 x0 x0 n−1 bk xk −−−−→ n−1 x→x0 k=0 k=0 bn−2 = = .. a(1) n a(1) n−1 b0 p(x0 ) = = a(1) 1 a(1) 0 bn bk x0k = pn (x0 ) an an−1 ··· a2 a1 – x0 a(1) n ··· x0 a(1) 3 x0 a(1) 2 a(1) n a(1) n−1 ··· a(1) 2 a(1) 1 – x0 a(2) n ··· x0 a(2) 3 x0 a(2) 2 a(2) n a(2) n−1 ··· a(2) 2 a(2) 1 – x0 a(3) n ··· x0 a(3) 3 x0 a(3) 2 a(3) n .. a(3) n−1 .. ··· .. a(3) 2 .. a(3) 1 .. ··· ··· ··· ··· ··· a0 x0 a(1) 1 pn (x0 ) x0 a(2) 1 pn (x0 ) x0 a(3) 1 ..

0 ... 0 0 ..        .......... 0 1  −a1 . . . . −an−1 Frobenius-Begleitmatrix Zum Eigenwert λ von A gehört der Eigenvektor (1, λ, λ2 , . . , , λn−1 ) B. Mit vollständiger Induktion über n zeigt man daß det(A − λE) = (−1)n pn (λ) n = 1: p1 (x) = x + a0 , (A − λE) = −a0 − λ = −p1 (λ) 40 P n−1⇒n:   0 1 0 . .  ·  A =  ·  A˜  · −a0  0        A˜ ist die Frobenius-Begleitmatrix zum Polynom p˜ n−1 (x) = a1 + a2 x + · · · + an−1 x1 n − 2 + xn−1    −λ 1 0 .

1 0 1 −λ 0 0 · · · · 1 ............ 0 −a2 . . . . . −an−1 − λ = −λ · (−1)n−1 p˜ n−1 (λ) − (−1)n (−a0 ) · 1 −λ 0 0 ...... 1 · .. . . −λ 0 0 0 1 = (−1)n (a1 λ + a2 λ2 + · · · + λn ) + a0 (−1)n = (−1)n pn (λ)       1   z   0  z   z2   0           z  ...  =  ...  =  ...      zn−2  zn−1   0  n−1   n   −a0 z z 0 . . . 1 0 ...        1   0    z   0         ..