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zX*k#ux8g{!p6;0BXFF;7ay@@u^F439i;8?|iewH;uyM&HhD9vEDuctIP+AgCk_j=y zQiaC>xbnC&IK1`?BQ$_2HiC)m zxY}31J={lucJDv&W1IBN5ZVuehI|AKbWj6ONdyFcV?}*}7<&(QA;jiD?T_#9U*8#^ z@#=v8*$RZwb8|$Lq4ztT-nj*J-vfgnG~8xkboYHwpN_j9th&oLIO3&|UZ?O-9g~;6 zEAyUCKTvOaPDdwDkX4hhQknE&qAq>&Tq&f8SIELaU)+m%hl0aWoJv7z0mX+0e7_dV Mu;k>T@?x<61@!)(hyVZp diff --git a/sose2020/num/uebungen/num3.tex b/sose2020/num/uebungen/num3.tex index a017ecf..937c499 100644 --- a/sose2020/num/uebungen/num3.tex +++ b/sose2020/num/uebungen/num3.tex @@ -75,6 +75,43 @@ \[ k = \frac{x - 1}{1} = x - 1 \xrightarrow{k \to \infty} \infty ,\] also $f$ schlecht konditioniert. + \item Für die Rundungsfehler der einzelnen Operationen gilt für + $|e_1|, |e_2|, |e_3| < \text{eps}$ + \begin{align*} + a &= \cos(x) (1 + e_1) \\ + b &= (1 - a)(1+e_2) \stackrel{\cdot}{=} 1 - \cos x + e_2 - (e_1 + e_2) \cos x + \intertext{Damit folgt} + f_a(x) &\stackrel{\cdot}{=} \frac{1 - \cos x + e_2 - (e_1 + e_2 \cos x) + e_3 - e_3 \cos(x)}{x} + \intertext{Also} + \frac{f_a(x) - f(x)}{f(x)} &\stackrel{\cdot }{=} + - \frac{\cos x }{1 - \cos x} e_1 + e_2 + e_3 + .\end{align*} + Wegen $\frac{\cos x}{1 - \cos x} \xrightarrow{x \to 0} \infty \gg k$ ist + dieser Algorithmus numerisch instabil. + \item Durch Umformungen, lässt sich die auslöschende Subtraktion $1 - \cos x$ vermeiden: + \[ + f(x) = \frac{\sin^2(x)}{x + x \cos(x)} + .\] Mit $|e_1|, |e_2|, |e_3|, |e_4|, |e_5|, |e_6| < \text{eps}$ folgt für die + Rundungsfehler: + \begin{align*} + a &= \sin(x)(1 + e_1) \\ + b &= a^2(1+e_2) \stackrel{\cdot }{=} \sin^2(x) + \sin^2(x)(2 e_1 + e_2) \\ + c &= \cos(x) (1 + e_3) \\ + d &= x c (1+e_4) \stackrel{\cdot }{=} x \cos x + (e_3 + e_4) x \cos(x) \\ + e &= x + d (1 + e_5) \stackrel{\cdot }{=} x + x \cos(x) + (e_3 + e_4 + e_5)x \cos(x) + e_4 x + \intertext{Damit folgt} + f_a(x) &\stackrel{\cdot }{=} + \frac{\sin^2(x) + \sin^2(x) (2 e_1 + e_2) + e_6 \sin^2(x)} + {x + x \cos(x) + (e_3 + e_4 + e_5)x \cos (x) + e_4 x} + \intertext{Also gilt} + \frac{f_a(x) - f(x)}{f(x)} + &\stackrel{\cdot }{=} \frac{2 e_1 + e_2 + e_6 - e_4 - (e_2 + e_3) \frac{\cos(x)}{1 + \cos(x)}} + {1 + e_4 + (e_2 + e_3) \frac{\cos(x)}{1 + \cos(x)}} + \; \stackrel{x \ll 1}{\approx} \; + \frac{\frac{3}{2} e_1 + e_2 + e_6 - e_4 - \frac{1}{2}e_3}{1 + e_4 + \frac{1}{2} e_2 + \frac{1}{2}e_3} + .\end{align*} + Wegen $1 + e_4 + \frac{1}{2} e_2 + \frac{1}{2}e_3 > 1$, folgt, dass die Verstärkungsfaktoren + der Rundungsfehler für $x \ll 1$ nahe $1$ sind, also ist der Algorithmus numerisch stabil. \end{enumerate} \end{aufgabe} @@ -144,6 +181,28 @@ \iff &16 - \alpha^2 > 0 \\ \iff &|\alpha| < 4 .\end{align*} + \item Zunächst ist $\overline{A}^{T} + = \overline{\overline{H}^{T}H}^{T} = \left( H^{T}\overline{H} \right)^{T} + = \overline{H}^{T} H = A$. Also ist $A$ hermitesch. + + Sei nun $A$ positiv definit. Dann sind, wegen $A$ hermitesch, alle Eigenwerte reell + und positiv. Also gilt $\text{det}(A) > 0$, damit $\text{Rg}(A) = n$. Also + \[ + n = \text{Rg}(A) = \text{Rg}(\overline{H}^{T}H) \le \min \{\text{Rg}(\overline{H}^{T}), H\} + = \text{Rg}(H) \le \min \{n, m\} = n + .\] Also $\text{Rg}(H) = n$. + + Sei nun $\text{Rg}(H) = n$. Es ist $A$ semidefinit, denn $\forall x \in \mathbb{C}^{n}$ gilt + \[ + (\overline{H}^{T}Hx, x)_2 = x^{T}\left( \overline{H}^{T}H \right)^{T} \overline{x} + = x^{T}H^{T}\overline{H}\overline{x} = \left( Hx \right)^{T} \overline{\left( Hx \right) } \ge 0 + \qquad (*) + .\] Mit dem Rangsatz ist außerdem: + \[ + n = \text{dim}(\mathbb{C}^{n}) = \text{Rg}(H) + + \text{dim } \text{ker } H = n + \text{dim } \text{ker } H \implies \text{dim } \text{ker } H = 0 + .\] Also folgt $\forall x \in \mathbb{C}^{n} \setminus \{0\}$: + $Hx \neq 0$, also folgt mit ($*$) $A$ positiv definit. \end{enumerate} \end{aufgabe}