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JlXg5Qhkz8pm3Vos5T0Qcpj(iwF6BGC(v1aD@JpuB8b- zfs9(|CV0yioi{J&6K07Y(!}@OX6c<6FY0Hq;rp9Y{uOr_0xAp~Cxpr$of%n+O{0XmlN+2R6IORy>)wTN?wA0mV;|q3e~2rd7k4>9G80Are=h_ zA97O9;@)Yd|B!7=mu4 znx2Za2lf!I<;Y-eym(@KOZHA6t`xy*#pO^}GO|xjt;NhNg#UeZ$E#ycU(#uupCphu z05A~Ea9PF#+v6<~ZyK-priIm|`sZpI-Mjt+1L1n{${6)Y{FidVo%nQZcq)d(rLC0D zrA3N})*uwFW7ER^&wwfBbqmErD4`Ax3`@2V!aJyVlU!wo`zHr*I@1Ki5XE%L4R*ZM zd@O>4PW>09-^x${U`_oud{8KatJx)vB~J~d&FL%axOG4`&Ay>$HL52(4KXmhD@`b< zB-n+LOrb)_ExBg^jUgCI*{!Jx3$<(NX+bz)YG87-qYE~_K}E?CDYAU`Fww4==;XHm%E{lsby_p^;G|>FbV_w14mD7jW3E|C7Un9oye#c)5f? zp+gwOUn`Ew;^Esq^Zv2OVO3OzHgxkCI+)L2_E z!gHWCd!J};QqO4^bgTqL6gv)&adsMlZJ6{W!Tev-lUX$f=y>D_2bLj_m*2Bd2yUjP z)rE#=EmNyOIaqPUmpY5lFtGFAguL;oOXG+acl`O50f z6VzxXk)3kdeDo^y5bC~~(BxYPzb zx?6(XZm|OudHmLhfE_+EtCjA+CI!Z$&mue814gqugL-#{*Gcg8nPF*-o-a}E6Cpjf z@mfn?Ub-iT=OetX?{#kycpY(A3s>yo(FzYoOJu#Q;-@pMC^t$-)C?eH-CtDPY2wFi z=>m;%nZ1%@^7U9?t9%dDK0=~j{w;n!kZC;azNl1dFL6RgqJTPrI$^QYXw<>l3 diff --git a/semester3/numcs/numcs-summary.tex b/semester3/numcs/numcs-summary.tex index 555a9ad..d422199 100644 --- a/semester3/numcs/numcs-summary.tex +++ b/semester3/numcs/numcs-summary.tex @@ -33,7 +33,6 @@ \vspace{4cm} \begin{center} \begin{Large} - \quote{Denken vor Rechnen} % FIXME: Marked for removal \quote{Wer in Python Type annotation benötigt, der soll kein Python verwenden} (2025-10-09T10:43) % FIXME: Marked for removal \quote{Wenn ich keine Lust habe, das zu berechnen, dann wende ich einfach Gewalt an} \end{Large} diff --git a/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex b/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex index 1a56597..7360883 100644 --- a/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex +++ b/semester3/numcs/parts/05_curve-fitting/01_linear/01_normal-equations.tex @@ -36,4 +36,7 @@ $B_a = -aI & A \\ A^H & 0 \end{bmatrix} -$ +$ +% TODO: What the f does the kappa mean??? +ersetzen, wobei wir $a$ so wählen, dass $\kappa(B_a)$ minimal wird. +% TODO: Consider adding the code and the example, but likely skip diff --git a/semester3/numcs/parts/05_curve-fitting/01_linear/02_orthogonal-transforms.tex b/semester3/numcs/parts/05_curve-fitting/01_linear/02_orthogonal-transforms.tex index e69de29..de4ebf1 100644 --- a/semester3/numcs/parts/05_curve-fitting/01_linear/02_orthogonal-transforms.tex +++ b/semester3/numcs/parts/05_curve-fitting/01_linear/02_orthogonal-transforms.tex @@ -0,0 +1,54 @@ +\subsubsection{Lösung mittels orthogonaler Transformation} +Nicht nur die Normalengleichungen, aber auch das LU-Verfahren kann für gewisse Matrizen (im Falle von LU sind es Matrizen mit $m > n$) ungeeignet sein. + +Wir versuchen wieder $||r||_2^2$ zu minimieren, mit $r = Ax - b$. +Mithilfe der $QR$-Zerlegung lässt sich ein Lösungsansatz herleiten, der höhere numerische Stabilität aufweist, als die Normalengleichungen. +Sei $A = QR = Q \begin{bmatrix} + \tilde{R} \\ 0 + \end{bmatrix}$. +Dann, nach Umformungen erhalten wir $||Rx - \tilde{b}||_2^2$ mit $\tilde{b} = Q^Hb$. + +Nutzt man beispielsweise Housholder-Spiegelungen zur Berechnung der $QR$-Zerlegung, so kann man die Transformationen direkt auf $b$ anwenden +und so kann man sich das Abspeichern der Matrix $Q$ komplett sparen. + +Falls jedoch die Matrix $A$ nicht vollen Rang hat (was sehr oft der Fall ist), dann ist es besser, die Singulärwertzerlegung zu verwenden. +Dann ist: +\rmvspace +\begin{align*} + ||Ax - b||_2 = ||U \Sigma V^H x - b||_2 = ||\Sigma V^H x - U^H b||_2 +\end{align*} + +\drmvspace +\innumpy verwendet \texttt{numpy.linalg.lstsq} die SVD für das Lösen + +\setLabelNumber{all}{15} +\fancydef{Pseudoinverse} $A^+ = (A^H A)^{-1} A^H = V_1 \Sigma_r^+ U_1^H$ + +Die drei bisher besprochenen Verfahren lassen sich in zwei Kategorien einordnen: +\begin{enumerate} + \item $A \in K^{m \times n}$ ist voll besetzt und $n$ ist klein ($m \gg n$) + \item $A \in K^{m \times n}$ ist dünn besetzt und $m, n$ sind gross +\end{enumerate} +Im ersten Fall wird aufgrund der numerischen Stabilität die $QR$ oder SVD-Methode verwendet. +Im zweiten Fall verwendet man die Normalengleichungen, da diese die Struktur der dünn besetzten Matrizen verwenden können. + +\begin{code}{python} +import numpy as np + +A = np.array([[98.269, 1.0], [0.0, 1.0], [-194.96, 1.0]]) +b = np.array([852.7, 624.5, 172.7]) + +def least_squares_svd(A, b, epsilon=1e-6): + U, s, Vh = np.linalg.svd(A) + r = 1 + np.where(s / s[0] > epsilon)[0].max() # numerical rank + y = np.dot(Vh[:r, :].T, np.dot(U[:, :r].T, b) / s[:r]) + return y + +# qr-decomposition: +def least_squares_qr(A, b): + Q, R = np.linalg.qr(A) + b_tilde = np.dot(Q.T, b) + return np.linalg.solve(R, b_tilde) + +np.linalg.lstsq(A, b) +\end{code} diff --git a/semester3/numcs/parts/05_curve-fitting/01_linear/03_total-curve-fitting.tex b/semester3/numcs/parts/05_curve-fitting/01_linear/03_total-curve-fitting.tex index e69de29..6c63585 100644 --- a/semester3/numcs/parts/05_curve-fitting/01_linear/03_total-curve-fitting.tex +++ b/semester3/numcs/parts/05_curve-fitting/01_linear/03_total-curve-fitting.tex @@ -0,0 +1,40 @@ +\subsubsection{Totale Ausgleichsrechnung} +Es kann vorkommen, dass sowohl die Matrix $A$, wie auch der Vektor $b$ fehlerhaft sind. +Dann ersetzen wir das System $Ax = b$ durch ein neues System $\hat{A}\hat{x} = \hat{b}$, +welches so nah wie möglich am ursprünglichen System liegt und so für welches gilt $\hat{b} \in \text{Bild}(\hat{A})$. + +Wir versuchen also die folgende Norm zu minimieren: +\begin{align*} + ||C - \hat{C}||_F + = + \left|\left| + \begin{bmatrix} + A & b + \end{bmatrix} + - + \begin{bmatrix} + \hat{A} & \hat{b} + \end{bmatrix} + \right|\right|_F +\end{align*} + +\drmvspace +Das Problem lässt sich umschreiben als +\rmvspace +\begin{align*} + \min_{\text{Rang}(\hat{C}) = n} ||C - \hat{C}||_F +\end{align*} + +\drmvspace +Theorem \ref{all:7-1-50} liefert die Lösung. Die Singulärwertzerlegung +\rmvspace +\begin{align*} + C = U\Sigma V^H = \sum_{j = 1}^{n + 1} \sigma_j (u)_j (v)_j^H +\end{align*} + +\drmvspace +gibt das Optimum +\rmvspace +\begin{align*} + \hat{C} = \sum_{j = 1}^{n} \sigma_j (u)_j (v)_j^H +\end{align*} diff --git a/semester3/numcs/parts/05_curve-fitting/02_non-linear/00_intro.tex b/semester3/numcs/parts/05_curve-fitting/02_non-linear/00_intro.tex index a6825c2..6834e37 100644 --- a/semester3/numcs/parts/05_curve-fitting/02_non-linear/00_intro.tex +++ b/semester3/numcs/parts/05_curve-fitting/02_non-linear/00_intro.tex @@ -1 +1,2 @@ +\newsection \subsection{Nichtlineare Ausgleichsrechnung} diff --git a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex index e606dd2..116e6b7 100644 --- a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex +++ b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex @@ -43,8 +43,6 @@ Dies gilt jedoch nicht, weil für jedes $z = 1^i$ zwar jedes $0^i 1^i \in L$ gil Um die Nichtregularität konkreter Sprachen zu beweisen, sucht man nach einfach verifizierbaren Eigenschaften, denn wenn eine Sprache eine dieser Eigenschaften \textit{nicht} erfüllt, so ist sie nicht regulär. -% TODO: For Kolmogorov complexity elaborate some more, i.e. how to do proofs properly / how to derive a word more easily -% -> TA Slides explain that really well \vspace{0.3cm} \hrule