From 11fece791fe782946bc45d375970d34f2b8c75c3 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 13 Sep 2024 12:12:52 +0200 Subject: [PATCH] feat(numerica): aggiunge scheda riassuntiva --- Secondo anno/Analisi numerica/README.md | 8 + Secondo anno/Analisi numerica/main.pdf | Bin 0 -> 449625 bytes Secondo anno/Analisi numerica/main.tex | 1874 ++++++++++++++++++ Secondo anno/Analisi numerica/notes_2023.sty | 463 +++++ Secondo anno/Analisi numerica/quiver.sty | 40 + 5 files changed, 2385 insertions(+) create mode 100644 Secondo anno/Analisi numerica/README.md create mode 100644 Secondo anno/Analisi numerica/main.pdf create mode 100644 Secondo anno/Analisi numerica/main.tex create mode 100644 Secondo anno/Analisi numerica/notes_2023.sty create mode 100644 Secondo anno/Analisi numerica/quiver.sty diff --git a/Secondo anno/Analisi numerica/README.md b/Secondo anno/Analisi numerica/README.md new file mode 100644 index 0000000..57f1a05 --- /dev/null +++ b/Secondo anno/Analisi numerica/README.md @@ -0,0 +1,8 @@ +# [Analisi numerica](https://esami.unipi.it/programma.php?c=58005&aa=2023&docente=&insegnamento=ANALISI+NUMERICA&sd=0) + +- [Programma del corso 📘](https://esami.unipi.it/programma.php?c=58005&aa=2023&docente=&insegnamento=ANALISI+NUMERICA&sd=0) +- [Registro del corso 📑](https://unimap.unipi.it/registri/dettregistriNEW.php?re=10336913::::&ri=9222) + +La cartella contiene una scheda riassuntiva, ovverosia un recap di +tutti i risultati principali del corso. Tale scheda è opera originale di [Mario Zito](mailto:m.zito12@studenti.unipi.it), +a cui ho personalmente contribuito. \ No newline at end of file diff --git a/Secondo anno/Analisi numerica/main.pdf b/Secondo anno/Analisi numerica/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5a9f396b5e7b12171c3ea1538883c8b42ae67a9f GIT binary patch literal 449625 zcma&NWptZcvMp+6X145@Su!)l%*@Qp3^Byaj4>rH!1<5X|gN|FYuzZ)*f*Su=YJS4#jFkc%Dgj~Bv+wXLg}^9Rb-$kpt# znTdm`8NvtH#nsu&$PU4C`AknMX}90W|DYQ0#mpy&m*p5jQKquH;4|HBylI9mH<7Ru z`l9_Qnki|I08Ikw``LH}q*{nP(b`Pw4`TnNr5s^MH7&1$w20U9E2fT@Gl{3(355;c zTjpFuzX4M!)2@s4->o}k!_NN2k>n#{UJvGO4|Cfidu@hM)mdsTe*n_I78~!j9GBM- z+qoJxb~aW?)hpkg%I(_hQO|ow6|&4Xm`+|g+MhUI*ix+uM`uX54j!9e$3a6s{dBL- z)avy+Ebm{|+MPIQE{0{(xO*-RJWy-g)!`czHN#BzwmZ%Wl|QF-t2CQmBD?GfhG(T@ zC=MH3re)vwq`1_xQ{on%MLi!`b?_1`%e|SjuC#3Qq)y}(YN_w`J=Q1Re=%VHGL*5} zaA(f>+gHh~tiro7nXaW^5TUf^`X;+cCS|A~v$bl^eutyk--sC+2R`i@BBS2`Ny4>y 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+\usepackage[colorlinks=true,citecolor=blue,linkcolor=blue]{hyperref} +\usepackage{notes_2023} +\usepackage{quiver} + +\setlength{\extrarowheight}{0pt} + +\renewcommand{\sp}{\operatorname{sp}} + +\ifthenelse{\lengthtest { \paperwidth = 11in}} +{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } +{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} + {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } + {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } +} + +\DeclareMathOperator{\DFT}{DFT} +\DeclareMathOperator{\IDFT}{IDFT} + +\newcommand{\ff}{\mathcal{F}} + +\newcommand{\ein}{\varepsilon_{\text{in}}} +\newcommand{\ealg}{\varepsilon_{\text{alg}}} +\newcommand{\ean}{\varepsilon_{\text{an}}} +\newcommand{\etot}{\varepsilon_{\text{tot}}} + +\renewcommand{\eps}{\varepsilon} + +%\pagestyle{empty} +\newcommand{\tx}{\Tilde{x}} +\newcommand{\ty}{\Tilde{y}} +\newcommand{\tz}{\Tilde{z}} +\makeatletter +\renewcommand{\section}{\@startsection{section}{1}{0mm}% + {-1ex plus -.5ex minus -.2ex}% + {0.5ex plus .2ex}%x + {\normalfont\large\bfseries}} +\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% + {-1explus -.5ex minus -.2ex}% + {0.5ex plus .2ex}% + {\normalfont\normalsize\bfseries}} +\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% + {-1ex plus -.5ex minus -.2ex}% + {1ex plus .2ex}% + {\normalfont\small\bfseries}} +\makeatother +\setcounter{secnumdepth}{0} +\setlength{\parindent}{0pt} +\setlength{\parskip}{0pt plus 0.5ex} +% ----------------------------------------------------------------------- + +\title{Schede riassuntive di Analisi Numerica} + +\begin{document} + + \parskip=0.7ex + + \raggedright + \footnotesize + + \begin{center} + \Large{\textbf{Schede riassuntive di Analisi Numerica}} \\ + \end{center} + \begin{multicols}{3} + \setlength{\premulticols}{1pt} + \setlength{\postmulticols}{1pt} + \setlength{\multicolsep}{1pt} + \setlength{\columnsep}{2pt} + + \section{Analisi dell'errore} + + Se $\tilde{x}$ è un'approssimazione di $x$ definiamo $\tilde{x}-x$ come l'\textit{errore assoluto}, $\eps = (\tx-x)/x$ come l'\textit{errore relativo} (rispetto a $x$) e $\eta = (x-\tx)/\tx$ come l'\textit{errore relativo} (rispetto a $\tx$). + In particolare vale che $\tilde{x} = x(1+\eps) = x/(1+\eta)$. + + \subsection{Rappresentazione in base} + Sia $B\ge2$ un numero intero, vale allora il seguente risultato: + + \begin{theorem}[di rappresentazione in base] + Per ogni numero reale $x\neq0$ esistono e sono unici un intero $p$ ed una successione $\{d_i\}_{i\ge1}$ con le seguenti proprietà: + \begin{enumerate}[(1.)] + \item $0\le d_i\le B-1$ + \item $d_1\neq0$ + \item per ogni $k>0$ esiste un $j\ge k$ tale che $d_j\neq B-1$ + (ossia $\{d_i\}_{i \geq 1}$ è frequentemente diversa da $B-1$) + \item $x$ si scrive come: $$x=\text{sgn}(x)B^p\sum_{i\ge1}d_iB^{-i}$$ + \end{enumerate} + \end{theorem} + + L'intero $B$ è detto \textit{base della rappresentazione}, gli interi $d_i$ sono dette \textit{cifre della rappresentazione} ed il numero $\sum_{i\ge1}d_iB^{-i}$ viene chiamato \textit{mantissa}. Si scrive $\pp(x)$ per indicare l'esponente $p$ relativo a $x$. \medskip + + + La condizione ($2$.) è detta di \textit{normalizzazione} e serve a garantire l'unicità della rappresentazione e a memorizzare il numero in maniera più efficiente, mentre la condizione $3$ esclude rappresentazioni con cifre uguali a $B-1$ da un certo punto in poi (per esempio la rappresentazione di $1$ come $0.\overline{9}$ è + esclusa). + + \subsection{Numeri \textit{floating point}} + + Dati gli interi $B\ge2$, $t\ge1$ ed $M$, $m>0$, si definisce l'insieme $\mathcal{F}(t,B,m,M)$ dei \textbf{numeri di macchina} o \textbf{dei numeri in virgola mobile} o ancora dei \textbf{numeri \textit{floating point}} come l'insieme: + $$\zeroset\cup\{\pm B^p\sum_i^td_iB^{-i}|d_1\neq0,0\le d_i\le B-1,-m\le p\le M\},$$ + in particolare $B$ rappresenta la base della rappresentazione, + $t$ la lunghezza della mantissa, $B^{-m}$ il minimo numero che + moltiplica la mantissa e $B^M$ il massimo. + + \subsubsection{Troncamento su \texorpdfstring{$\RR$}{ℝ} e \texorpdfstring{$\RR^n$}{ℝⁿ} e numero di macchina \texorpdfstring{$u$}{u}} + + Sia $x$ un numero reale. Se $-m \leq \pp(x) \leq M$, allora $x$ viene ben rappresentato in $\mathcal{F}$ dal numero $\tx=\fl(x)$, + dove: + \[ x=\sgn(x)B^p\sum_{i\ge1}d_iB^{-i} \implies \tx=\sgn(x)B^p\sum_{i=1}^t d_iB^{-i}, \] + ossia $\tx$ è ottenuto da $x$ troncandone la mantissa al + $t$-esimo termine (\textbf{troncamento}). Facendo così si ottiene + un errore relativo di rappresentazione tale per cui: + \[ \abs{\frac{\tx-x}{x}}M$ il numero non è rappresentabile + e ci ritroviamo rispettivamente in una situazione di \textit{underflow} o di \textit{overflow}. \medskip + + Dato $x\in\RR^n$, si definisce + la rappresentazione $\tx$ rispetto a $x$ in $\mathcal{F}$ come: \[ \tx=(\tx_i)=(\fl(x_i)=(x_i(1+\varepsilon_i)), \quad \abs{\varepsilon_i}(t_2-1)\frac{\log B_2}{\log B_1}+1 \] + Si può definire in modo analogo l'approssimazione per arrotondamento, osservando che in questo caso l'errore relativo di rappresentazione è limitato da $\frac{1}{2}B^{1-t} = \frac{1}2 u$. + + \subsubsection{Aritmetica di macchina} + + Siano $a$, $b\in\mathcal{F}(t,B,m,M)$ e sia $\op$ una delle quattro operazioni aritmetiche ($+$, $-$, $\cdot$ e $/$). Si consideri + $c=a \bop b$. Allora la macchina calcola $c$ con l'approssimazione $\hat{c} := \fl(c)$. Vale pertanto che: + \[ \hat{c}=\fl(a \bop b)=c(1+\delta)=c/(1+\eta)\;\con \abs\delta, \abs\etar_i = \sum_{j=1, \, j\neq i}^n \abs{a_{ij}}$ $\forall i=1$, ..., $n$. Per il primo teorema di + Gershgorin, $A$ non può essere singolare ($0$ non appartiene + per ipotesi a nessun cerchio). Ogni sottomatrice principale di una + matrice fortemente diagonale in senso stretto è allo stesso + modo fortemente diagonale. In particolare, se $A$ è fortemente dominante + diagonale, allora $a_{ii} \neq 0$. \medskip + + Si possono definire in modo analogo le matrici dominanti diagonali (in senso debole) cambiando $>$ in $\geq$, ma non è detto che tali matrici siano non singolari. Un controesempio + è infatti la matrice: \[\begin{pmatrix} + 1 & 1 \\ + 1 & 1 + \end{pmatrix}. \] + + \subsubsection{Secondo teorema di Gershgorin} + + \begin{theorem}[Secondo teorema di Gershgorin] + Sia $A\in \CC^{n \times n}$ e sia $K=\bigcup_{i=1}^n K_i$ l'unione dei + cerchi $K_i$ relativi ad $A$. Sia inoltre + \[ K=M_1\cup M_2\;\con M_1\cap M_2=\emptyset, \] + dove $M_1$ è costituito da $n_1$ cerchi e $M_2$ è costituito da $n_2$ cerchi. Allora $M_1$ contiene $n_1$ autovalori e + $M_2$ ne contiene $n_2$. + \end{theorem} + + \begin{note} + La dimostrazione di questo teorema sfrutta un argomento di + continuità riguarda al segmento $A(t) = D + t(A-D)$ con + $t \in [0, 1]$ e $D = \diag(A)$. Un simile argomento di continuità + può risultare utile in altri contesti. + \end{note} + + Quest'ultimo teorema risulta utile per dimostrare che una matrice + $A \in \RR^{n \times n}$ ha un autovalore reale. Se infatti $K_i$ è un cerchio di $A$ disgiunto dagli altri, allora $K_i$ contiene un unico autovalore, che deve essere dunque necessariamente reale (altrimenti, siccome $A$ è reale, vi sarebbe anche il suo coniugato, e quindi vi sarebbero almeno + $2$ autovalori, assurdo). + + \subsubsection{Terzo teorema di Gershgorin} + + \begin{theorem}[Terzo teorema di Gershgorin] + Sia $\lambda$ un autovalore di $A$. Si ipotizzi che val le seguenti + condizioni + + \begin{enumerate}[(1.)] + \item $A$ è irriducibile, + \item $\lambda \in K_i \implies \lambda \in \partial K_i$ (dove $\partial K_i$ è la frontiera di $K_i$). + \end{enumerate} + + Allora $\lambda \in \bigcap \partial K_i$. + \end{theorem} + + \subsubsection{Matrici irriducibilmente dominanti diagonali (i.d.d.)} + + Una matrice $A\in \CC^{n \times n}$ si dice \textbf{irriducibilmente dominante diagonale} se \begin{enumerate}[(i)] + \item $A$ è irriducibile, + \item $A$ è dominante diagonale \underline{in senso debole}, + \item $\exists h$ t.c.~$\abs{a_{hh}}>r_h = \sum_{j=1, \, j\neq h}^n{a_{hj}}$ (ossia con $0 \notin K_h$). + \end{enumerate} + Per il terzo teorema di Gershgorin le matrici i.d.d.~sono non singolari. + Inoltre, se $A$ è i.d.d., allora $a_{ii} \neq 0$ per ogni $i$. + %\\Sia $p(x)=x^n+\sum_{i=0}^{n-1}a_ix^i$ un polinomio a coefficienti in $\CC$. Sia $C$ la matrice compagna di $p(x)$ allora trovare gli zeri di $p(x)$ corrispondono agli autovalori di $C$. (si mostra per induzione che $\det(xI-C)=p(x)$). Si possono dunque localizzare gli zeri di un polinomio tramite i teoremi di Gershgorin. Se $a_0\neq0$ si può utilizzare lo stesso risultato a $q(x)=(1/a_0)x^np(x\inv)$ che fornisce inclusioni per i reciproci degli zeri. \\Un'altra localizzazione degli zeri consiste nel costruire la matrice $A=D-ue^t$ dove $D=diag(x_1,\dots,x_n),e^t=(1,\dots,1)\E u=(u_i)\;\con u_i=p(x_i)/\prod_{j=1,j\neq i}(x_i-x_j)$. Il teorema di Gershgorin applicato ad A fornisce delle stime dell'errore con cui i valori $x_i$ approssimano gli zeri. + + \subsection{Forma normale di Schur} + + \begin{theorem} + Data $A\in \CC^{n \times n}$, esiste una matrice $U\in \CC^{n \times n}$ unitaria, ovverosia tale per cui $U^HU=UU^H=I_n$, tale che: \[ U^HAU=T\in T_+(n,\CC), \] + ovverosia con $U^H A U$ triangolare superiore. + Si dice in tal caso che $T$ è una \textbf{forma normale di Schur} + di $A$. + \end{theorem} + + Generalmente esistono più forme normali di Schur per una matrice, + e possono essere ottenute riordinando gli autovalori e/o le + basi scelte. + + \subsubsection{Teorema spettrale} + + \begin{theorem}[spettrale] + Se $A$ è hermitiana, allora una sua forma normale di Schur è + sempre una matrice diagonale con elementi reali, ovverosia + $A$ è ortogonalmente diagonalizzabile con autovalori reali. + + Se invece $A$ è anti-hermitiana ($A^H = -A$), allora una sua + forma normale di Schur è una matrice diagonale con elementi + immaginari puri, ovverosia $A$ è ortogonalmente diagonalizzabile + con autovalori immaginari puri. + \end{theorem} + + \subsubsection{Caratterizzazione delle matrici normali} + + \begin{definition} + Una matrice $A \in \CC^{n \times n}$ si dice \textbf{normale} + se $A A^H = A^H A$. + \end{definition} + + Si enuncia una caratterizzazione delle matrici triangolari + normali: + + \begin{proposition} + Una matrice triangolare $T$ è normale se e solo se è + diagonale. + \end{proposition} + + Dalla precedente proposizione si ottiene facilmente la seguente + fondamentale altra caratterizzazione: + + \begin{theorem} + Una matrice $A \in \CC^{n \times n}$ è normale + se e solo se la sua forma normale di Schur è diagonale (i.e.~è + ortogonalmente diagonalizzabile). + \end{theorem} + + \subsubsection{Autovalori di una matrice unitaria} + + Gli autovalori di una matrice unitaria $A$ hanno modulo $1$. + Se infatti $v$ è un autovettore relativo all'autovalore $\lambda$, + vale che: + \[ A v = \lambda v \implies v^H A^H = \overline\lambda v^H, \] + da cui: + \[ v^H v = v^H A^H A v = \overline\lambda v^H A v = \overline\lambda \lambda v^H v \implies \abs{\lambda} = 1. \] + In particolare, se $A$ è reale, $\lambda \in \{\pm1\}$. + + \subsubsection{Forma normale di Schur reale} + + \begin{definition}[Matrice quasi-triangolare superiore] + Una matrice $T\in \RR^{n \times n}$ si dice \textbf{quasi-triangolare superiore} se si può scrivere nella forma: + \[ \begin{pmatrix} + T_{11} & \dots & T_{1n} \\ + & \ddots & \vdots \\ + & & T_{mm} + \end{pmatrix}, \] + dove i blocchi matriciali $T_{ii}$ possono essere matrici $2 \times 2$ oppure matrici $1 \times 1$, ossia numeri reali. + \end{definition} + + Si definisce analogamente la nozione di matrice quasi-triangolare + \textit{inferiore}. \smallskip + + Gli autovalori di una matrice quasi-triangolare $T$ sono gli autovalori delle sottomatrici $T_{ii}$. Il polinomio caratteristico di $T$ è + il prodotto dei polinomi dei blocchi $T_{ii}$. Il determinante di $T$ + è il prodotto dei determinanti dei blocchi $T_{ii}$, la traccia è + la somma delle tracce dei blocchi $T_{ii}$. \smallskip + + Usando la nozione di matrice quasi-triangolare si può enunciare + un teorema analogo a quello della forma normale di Schur, ma + riadattato per le matrici reali: + + \begin{theorem}[Forma normale di Schur reale] + Per ogni matrice $A\in \RR^{n \times n}$ esistono $T\in \RR^{n \times n}$ quasi-triangolare superiore e $Q \in O(n)$ tali per cui: + \[ Q^tAQ=T. \] + \end{theorem} + + \subsection{Norme di vettori} + + Una \textbf{norma} (vettoriale) su $\CC^n$ è un'applicazione + \[ \norm{\cdot}:\CC^n\longrightarrow\RR \] + tale per cui: + \begin{enumerate}[(1.)] + \item $\forall x\in\CC^n$, $\norm{x} \geq 0$ e $\norm{x}=0\iff x=0$ (definitezza positiva). + \item $\forall\alpha\in\CC$, $\forall x\in\CC^n$, $\norm{\alpha x}=\abs\alpha\norm{x}$ (omogeneità). + \item $\forall x$, $y\in\CC^n$, $\norm{x+y}\le\norm{x}+\norm{y}$ (disuguaglianza triangolare). + \end{enumerate} + + Ogni prodotto hermitiano $\langle \cdot, \cdot \rangle$ definito positivo di $\CC^n$ induce una norma ponendo $\norm{v} = \sqrt{\langle v, v \rangle}$. \medskip + + Per $p \in [1, \infty)$ si dice \textbf{norma di Hölder} di ordine $p$ la norma: + \[ \norm{x}_p=\left(\sum_{i=1}^n|x_i|^p\right)^{\frac{1}{p}}. \] + Alcuni esempi noti di norme di Hölder sono: + \begin{itemize} + \item $\norm{x}_1=\sum_{i=1}^n \abs{x_i}$, la norma $1$, + \item $\norm{x}_2=\left(\sum_{i=1}^nx_i^2\right)^{\frac{1}{2}} = \sqrt{x^H x}$, detta anche \textit{norma euclidea}, indotta dal prodotto hermitiano standard di $\CC^n$, + \item $\norm{x}_{\infty}:=\lim_{p \to \infty} \norm{x}_p = \max_{i}\abs{x_i}$. + \end{itemize} + + Per una norma l'insieme $S = \{x\in\CC^n \mid \norm{x}\le1\}$ è un insieme convesso. Da ciò si deduce che per $0 0$, esiste $\delta > 0$ tale per cui: + \[ \forall x,y\in\CC^n, \norm{x-y}_2 < \delta \implies \abs{\norm{x}-\norm{y}} < \varepsilon, \] + espressione che segue dal fatto che una norma è Lipschitziana (deriva dalla \textit{disuguaglianza triangolare}). + Equivalentemente vale che per ogni $\eps > 0$ esiste $\delta > 0$ tale + per cui: + \[ \forall x,y\in\CC^n, \abs{x_i-y_i} < \delta \implies \abs{\norm{x}-\norm{y}} \leq \varepsilon. \] + A partire da questo risultato si può dimostrare il seguente fondamentale + teorema: + + \begin{theorem}[Equivalenza tra norme] + Per ogni coppia di norme $\norm{\cdot}'$ e $\norm{\cdot}''$ su $\CC^n$, esistono due costanti positive $\alpha$ e $\beta$ (dipendenti da $n$) tali per cui: + \[ \alpha\norm{x}'\le\norm{x}''\le\beta\norm{x}' \quad \forall x\in\CC^n. \] + In altre parole, le norme su $\CC^n$ sono bi-lipschitziane tra loro. + \end{theorem} + + Infatti $S = \{ x \in \CC^n \mid \norm{x}_2 = 1 \}$ è un compatto euclideo (essendo chiuso e limitato, per il teorema di Heine-Borel), + e $\norm{\cdot}$ è continua sulla topologia euclidea. Pertanto tutte le norme di $\CC^n$ inducono la stessa topologia, + ossia quella indotta da $\norm{\cdot}_2$, e quindi coincidono tutti + gli aperti e i chiusi. \smallskip + + In particolare per le norme $1$, $2$ e $\infty$ vale che: + \begin{itemize} + \item $\norm{x}_\infty \leq \norm{x}_1 \leq n \norm{x}_\infty$, + \item $\norm{x}_2 \leq \norm{x}_1 \leq \sqrt{n} \norm{x}_2$, + \item $\norm{x}_\infty \leq \norm{x}_2 \leq \sqrt{n} \norm{x}_\infty$. + \end{itemize} + + Se $U$ è una matrice unitaria, allora $U$ agisce lasciando invariata + la norma $2$, ovverosia $\norm{Ux}_2 = \norm{x}_2$. + + \subsection{Norme di matrici} + + Una \textbf{norma matriciale} è un'applicazione \[ \norm{\cdot}:\CC^{n \times n}\to\RR \] + per la quale valgono le seguenti quattro proprietà: + \begin{enumerate}[(1.)] + \item $\forall A\in \CC^{n \times n}$, $\norm{A}\ge0$ e $\norm{A}=0\iff A=0$ (definitezza positiva), + \item $\forall \lambda\in\CC$, $\forall A\in \CC^{n \times n}$, $\norm{\lambda A}=\abs\lambda\norm{A}$ (omogeneità), + \item $\norm{A+B}\le\norm{A}+\norm{B}$ $\forall A$,$B\in \CC^{n \times n}$ (disuguauglianza triangolare), + \item $\norm{AB}\le\norm{A}\norm{B}$ $\forall A$, $B\in \CC^{n \times n}$ (proprietà submoltiplicativa). + \end{enumerate} + + \subsubsection{Norma di Frobenius} + + Si definisce la \textbf{norma di Frobenius} l'applicazione che + agisce su $A \in \CC^{n \times n}$ in modo tale che: + \[ \norm{A}_F=\left(\sum_{i=1}^n\sum_{j=1}^n\abs{a_{ij}}^2\right)^{\frac{1}{2}}=\sqrt{\tr(A^HA)}. \] + In particolare la norma di Frobenius è esattamente la norma euclidea + dello spazio $\CC^{n \times n}$ immerso in $\CC^{n^2}$ + a cui è naturalmente isomorfo associando ad $A$ il vettore + $(A_i^\top)_i$ ottenuto trasponendo le righe e sovrapponendole + ordinatamente. + + \subsubsection{Norme matriciali indotte} + + \begin{note} + Si ricorda che, data una norma $\norm{\cdot}$, l'insieme $S=\{x\in\CC^n \tc \norm{x}=1\}$ è chiuso e limitato (in tutte le topologie indotte da norme, essendo + equivalenti). Allora, per il teorema di Heine-Borel, $S$ è + compatto, e dunque una funzione continua $f : S \to \RR$ ammette massimo + per il teorema di Weierstrass. + \end{note} + + \begin{note} + Una mappa lineare da $\CC^n$ in sé è continua. Dunque è possibile applicare + il teorema di Weierstrass alla funzione + $\norm{ \cdot } \circ f_A$ ristretta su $S$, dove $f_A$ è l'app.~indotta da una matrice $A$. + \end{note} + + \begin{definition} + Data una norma vettoriale $\norm{\cdot}:\CC^n\to\RR$, si definisce la \textbf{norma matriciale indotta} (o la corrispondente \textit{norma operatore}), come la norma che agisce su $A \in \CC^{n \times n}$ in modo + tale che: + \[ \norm{A}:=\max_{\norm{x}=1} \norm{Ax}=\max_{x \in S} \norm{Ax}=\max_{x \in \CC^n \setminuszero} \frac{\norm{Ax}}{\norm{x}}. \] + \end{definition} + + Per ogni norma operatore valgono le due seguenti aggiuntive + proprietà, \textit{oltre quelle caratterizzanti una norma}: + \begin{enumerate}[(1.)] + \item $\norm{Ax}\le\norm{A}\norm{x}$, $\forall A\in \CC^{n \times n}$, $\forall x\in \CC^n$, + \item $\norm{I}=1$. + \end{enumerate} + + Poiché $\norm{I}_F = \sqrt{n}$, la norma di Frobenius non è in generale indotta + da alcuna norma vettoriale. \smallskip + + Per le norme $1$, $2$, $\infty$ valgono le seguenti identità: \begin{itemize} + \item $\displaystyle \norm{A}_1=\max_{j=1,\dots,n}\sum_{i=1}^n \abs{a_{ij}} = \max_{j=1,\ldots,n} \norm{A^j}_1$, + \item $\displaystyle \norm{A}_\infty=\max_{i=1,\dots,n}\sum_{j=1}^n \abs{a_{ij}} = \max_{i=1,\ldots,n} \norm{A_i}_1$, + \item $\norm{A}_2=\sqrt{\rho(A^HA)}$, dove $\rho(A^H A)$ è l'autovalore di valore assoluto maggiore di $A^H A$ (\textit{raggio spettrale}). + \end{itemize} + Si osserva immediatamente che la norma matriciale $2$ è computazionalmente più difficile + da calcolare rispetto alle norme $1$ e $\infty$. Inoltre, se $A$ è simmetrica, le norme $1$ e $\infty$ coincidono. \smallskip + + Poiché $\norm{A}_F = \sqrt{\tr(A^H A)}$ è la radice della somma degli autovalori di $A^H A$, vale in particolare che $\norm{A}_F \geq \norm{A}_2$. Si osserva che $A^H A$ è semidefinita positiva, e dunque i suoi autovalori sono non negativi. \smallskip + + Se $U$ è unitaria e $B = UA$, vale che: + \[ B^H B = (UA)^H UA = A^H U^H U A = A^H A, \] + e quindi $B$ e $A$ condividono la stessa norma di Frobenius e + la stessa norma $2$. Analogamente si vede che $AU$ e $A$ condividono + le stesse due norme. Equivalentemente, la moltiplicazione per matrice unitaria (sia a destra che a sinistra) lascia invariata sia la norma $2$ che quella di Frobenius. In particolare, se $U$ e $V$ sono + unitarie, $\norm{UAV}_2 = \norm{A}_2$ e $\norm{UAV}_F = \norm{A}_F$. \smallskip + + Se $A$ è normale, allora gli autovalori di $A^H A$ sono i moduli + quadrati degli autovalori di $A$. Pertanto, per $A$ normale, + $\norm{A}_2 = \max_{\lambda \in \sp(A)} \abs{\lambda} = \rho(A)$ + e $\norm{A}_F = \sqrt{\sum_{\lambda \in \sp(A)} \abs{\lambda}^2}$. + Si è usato che la forma normale di Schur di $A$ è diagonale e che le + trasformazioni unitarie non variano né $\norm{A}_2$ né + $\norm{A}_F$. \smallskip + + Valgono inoltre le seguenti altre due disuguaglianze: + \begin{itemize} + \item $\norm{A}_F \leq \sqrt{r} \norm{A}_2 \leq \sqrt{n} \norm{A}_2$, dove $r = \rg(A)$, + \item $\norm{A}_2^2 \leq \norm{A}_1 \cdot \norm{A}_\infty$. + \end{itemize} + + \subsubsection{Norma indotta da una matrice non singolare \texorpdfstring{$S$}{S}} + + Sia $\norm{\cdot}$ una norma vettoriale e $S\in \CC^{n \times n}$ una matrice non singolare. Allora, data la + norma vettoriale $\norm{x}_S := \norm{Sx}$, vale che: + \[ \norm{A}_S=\max_{\norm{x}_S=1}\norm{Ax}_S=\max_{\norm{x}_S=1}\norm{SAx}= + \max_{\norm{Sx}=1}{\norm{SAS\inv Sx}}, \] + e quindi, sfruttando che $S$ è non singolare: + \[ \norm{A}_S=\max_{\norm{y}=1}{\norm{SAS\inv y}}, \] + ossia vale che: + \[ \norm{A}_S = \norm{SAS\inv}. \] + + \subsubsection{Norme e raggi spettrali} + + \begin{definition} + Data $A\in \CC^{n \times n}$, si definisce \textbf{raggio spettrale} $\rho(A)$ + il modulo dell'autovalore massimo di $A$, ossia: + \[ \rho(A)=\max\{\abs\lambda \mid \lambda\text{ autovalore di } A\}. \] + \end{definition} + + Se $x$ è un autovettore relativo a $\rho(A)$ di modulo unitario + rispetto a una norma $\norm{\cdot}$, allora vale che: + \[ \norm{Ax} = \norm{\rho(A) x} = \rho(A) \norm{x} = \rho(A), \] + e dunque $\rho(A) \leq \norm{A}$ per ogni $A$. In particolare + $\norm{A} \geq \abs{\lambda}$ per ogni autovalore $\lambda$. \smallskip + + + Si enunciano inoltre i seguenti teoremi: + + \begin{theorem} + Sia $A \in \CC^{n \times n}$. Allora per ogni $\varepsilon > 0$ esiste + una norma indotta $\norm{\cdot}$ tale per cui: + \[ \rho(A) \leq \norm{A} \leq \rho(A) + \varepsilon. \] + Inoltre, se gli autovalori di modulo massimo di $A$ appartengono solo + a blocchi di Jordan di taglia $1$, allora esiste una norma + per cui $\norm{A} = \rho(A)$. + \end{theorem} + + \begin{theorem} + Sia $\norm{\cdot}$ una norma matriciale. Allora vale la seguente identità: + \[ \lim_{k\to+\infty}\norm{A^k}^{\frac{1}{k}}=\rho(A). \] + \end{theorem} + + Se $A\in \CC^{n \times n}$ è tale per cui $\norm{A}<1$ dove $\norm{\cdot}$ è una norma matriciale indotta, allora $1$ non + può essere autovalore di $A$, e dunque $I-A$ è invertibile. Inoltre + vale che:\[ \norm{(I-A)\inv}\le\frac{1}{1-\norm{A}}. \] + + + + + \subsection{Condizionamento di un sistema lineare e numero di condizionamento} + + Data $A\in \CC^{n \times n}$ non singolare e dato un vettore $b\in\CC^n\setminuszero$ si vuole studiare il condizionamento del problema $Ax=b$, ossia di un sistema lineare. \medskip + + + Consideriamo il problema $(A+\delta_A)y=b+\delta_b$, dove perturbiamo il sistema originale mediante dei parametri $\delta_A$ e + $\delta_B$ di cui conosciamo i rapporti $\frac{\norm{\delta_b}}{\norm{b}}$ e $\frac{\norm{\delta_A}}{\norm{A}}$, cercando di ottenere informazioni riguardo + $\frac{\norm{\delta_x}}{\norm{x}}$, sostituendo $y=x+\delta_x$. \medskip + + \begin{definition} + Si dice \textbf{numero di condizionamento} $\mu(A)$ di $A$ nella norma $\norm{\cdot}$ il valore: + \[ \mu(A)=\norm{A}\norm{A\inv}. \] + Si scrive $\mu_p(A)$ per intendere $\norm{A}_p \norm{A\inv}_p$. + \end{definition} + + Studiamo in particolare la perturbazione di $Ax=b$ nel caso in cui $\delta_A=0$. + + \begin{proposition} + Se $\delta_A = 0$, allora vale la seguente disuguaglianza: + \[ \frac{\norm{\delta_x}}{\norm{x}}\le\norm{A}\norm{A\inv}\frac{\norm{\delta_b}}{\norm{b}} = \mu(A) \frac{\norm{\delta_b}}{\norm{b}}. \] + In generale, per $\delta_A \neq 0$ e + $A + \delta_A$ invertibile, vale che: + \[ \varepsilon_x\le\frac{\norm{A}\norm{A\inv}}{1-\varepsilon_A\norm{A}\norm{A\inv}}(\varepsilon_A+\varepsilon_B), \] + dove $\eps_t = \norm{\delta_t}/\norm{t}$. + \end{proposition} + + Pertanto il sistema è \textbf{ben condizionato} se $\mu(A)$ è + relativamente piccolo. \smallskip + + Per il numero di condizionamento valgono le seguenti proprietà: + + \begin{itemize} + \item $\mu(A) \geq \norm{I}$ per la \textit{proprietà submoltiplicativa}, + \item $\mu(A) \geq 1$ per $\norm{\cdot}$ norma operatore ($\norm{I} = 1$), + \item $\mu_2(U)=1$ per $U$ unitaria. + \end{itemize} + + Per $A$ normale vale la seguente identità: + \[\mu_2(A) = \frac{\max_{\lambda \in \sp(A)} \abs{\lambda}}{\min_{\lambda \in \sp(A)} \abs{\lambda}}. \] + Inoltre vale la seguente disuguaglianza: + \[ + \mu_2(A) \leq \mu(A), + \] + dove $\mu(A)$ è riferito a qualsiasi altra norma operatore. + + \section{Metodi diretti per sistemi lineari} + + Ci si propone di risolvere il sistema $Ax=b$ scrivendo $A$ come prodotto di matrici ``consone'' e facilmente invertibili. Se + infatti $A=PQ$, allora il sistema $PQx=b$ può essere risolto + come: + \[ + \begin{cases} + Py=b \\ + Qx=y + \end{cases} + \] + + risolvendo dunque prima $Py=b$ e poi $Qx=y$. + + \subsection{Risoluzione di \texorpdfstring{$Ax=b$}{Ax=b} per \texorpdfstring{$A$}{A} triangolare o unitaria} + + Sono presentati di seguito tre tipi di matrice per le quali + l'invertibilità è garantita e per cui il sistema $Ax=b$ è facilmente + risolvibile. \smallskip + + Se $A$ è \textbf{triangolare inferiore} con $a_{ii}\neq0$ $\forall i$, allora, per risolvere $Ax=b$, è possibile applicare il \textit{metodo di sostituzione in avanti} ponendo: \[ x_1=\frac{b_1}{a_1}, \quad x_i=\frac{1}{a_{ii}}\left(b_i-\sum_{j=1}^{i-1}a_{ij}x_j\right)\;\,\con i\geq 2. \] + + Se $A$ è \textbf{triangolare superiore} con $a_{ii}\neq0$ $\forall i$, allora, per risolvere $Ax=b$, è possibile applicare il \textit{metodo di sostituzione all'indietro} ponendo: \[ x_n=\frac{b_n}{a_n}, \quad x_i=\frac{1}{a_{ii}}\left(b_i-\sum_{j=i+1}^n a_{ij} x_j\right)\;\,\con i < n.\] + + I due algoritmi presentati hanno un costo computazionale di $n^2$ flops e sono entrambi numericamente stabili all'indietro. \smallskip + + Se $A$ è \textbf{unitaria}, $Ax=b \implies x=A^Hb$, e dunque + si verifica che: + \[ x_j=\sum_{i=1}^n \overline{a_{ij}} b_i. \] + Questo algoritmo richiede un costo computazionale di $2n^2-n$ flops ed + è ancora numericamente stabile all'indietro. + + \subsection{Fattorizzazione classiche di matrici} + + In letteratura esistono $4$ fattorizzazioni classiche: + \begin{enumerate} + \item $A=LU$ con $L$ triangolare inferiore con tutti $1$ sulla diagonale e $U$ triangolare superiore (possibile solo per alcune classi di matrici); + \item $A=PLU$ con $L$, $U$ come sopra e $P$ matrice di permutazione (sempre possibile); + \item $A=P_1LUP_2$ con $L$, $U$ come sopra e $P_1$, $P_2$ matrici di permutazione (sempre possibile); + \item $A=QR$ con $Q$ unitaria e $R$ triangolare superiore (sempre possibile). + \end{enumerate} + + \subsubsection{Condizioni per l'esistenza e l'unicità di una fattorizzazione LU} + + \begin{definition} + Si dicono \textbf{sottomatrici principali di testa} le sottomatrici di $A$ di cui prendiamo le prime $k$ righe e colonne. Quando si utilizza + l'aggettivo \textit{proprie}, si esclude $A$ stessa. + \end{definition} + + Vale la seguente condizione sufficiente per l'esistenza e + l'unicità di una + fattorizzazione LU: + + \begin{proposition} + Se tutte le sottomatrici principali di testa proprie sono non singolari allora esiste ed è unica la fattorizzazione $LU$ di $A$. + \end{proposition} + + Se non sono verificate le ipotesi può comunque esistere una fattorizzazione $LU$. Per esempio: + \[ \begin{pmatrix} + 0 & 1 \\ + 0 & 2 + \end{pmatrix}=\begin{pmatrix} + 1 & 0 \\ + 1 & 1 + \end{pmatrix}\begin{pmatrix} + 0 & 1 \\ + 0 & 1 + \end{pmatrix}. \] + + Se $A$ è invertibile, allora la non singolarità delle sottomatrici + principali di testa diventa una condizione necessaria oltre che + sufficiente per l'esistenza e l'unicità della fattorizzazione LU. \smallskip + + Le seguenti classi di matrici ammettono sempre un'unica fattorizzazione + LU: + \begin{itemize} + \item Matrici fortemente dominanti diagonali, le cui sottomatrici + principali di testa sono fortemente dominanti diagonali e dunque + invertibili, + \item Matrici hermitiane definite positive (criterio di Sylvester), + \item Matrici hermitiane definite positive su $\Span(e_1, \ldots, e_{n-1})$ e semidefinite positive su $\CC^n$ (metodo di Jacobi, criterio di Sylvester). + \end{itemize} + + \subsection{Matrici elementari} + + Dati $\sigma\in\CC$, $u$, $v\in\CC^n$ si dice \textbf{matrice elementare} una matrice $M$ del tipo: + \[M=I-\sigma uv^H.\] + Si osserva che $uv^H$ è della seguente forma: + \[ uv^H = (u_i \overline{v_j})_{ij}. \] + Inoltre $\rg(uv^H) \leq \rg(u) \leq 1$. Se + $u$ o $v$ sono nulli $uv^H$ è necessariamente + nullo. \underline{Non} si deve confondere $uv^H$ con + $u^H v$, che invece è il prodotto hermitiano + complesso computato su $u$ e $v$. \medskip + + + Un vettore $x$ è autovettore di $M$ se: + \begin{itemize} + \item $x=u \implies Mu=(1-\sigma(v^Hu))u$, e + dunque $u$ è relativo all'autovalore $1-\sigma(v^Hx)$; + \item $x$ t.c.~$v^Hx=0$ ($x$ è ortogonale a $v$) $\implies Mx=x$, il cui relativo autovalore è $1$. + \end{itemize} + + Supponiamo che $\sigma(v^H u)$ sia diverso da $0$. + La traccia di $M$ è: + \[ \tr(M)=\tr(I)-\sigma \cdot \tr(uv^H))=(n-1) + (1-\sigma v^Hu). \] + Dal momento che $v^\perp$ ha dimensione $n-1$ -- il + prodotto hermitiano è positivo definito --, allora $\mu_{g}(1) \geq n-1$. \smallskip + + Osservando allora che $\tr(M)$ è la somma degli autovalori di + $M$ e che $\mu_{g}(1-\sigma(v^H u)) \geq 1$, si conclude + che gli unici autovalori di $M$ sono proprio $1$, con molteplicità + algebrica e geometrica $n-1$, e $1-\sigma(v^H u)$, con + molteplicità $1$. In particolare $M$ è sempre diagonalizzabile e + vale la seguente proposizione: + + \begin{proposition} + Se $\sigma(v^H u) \neq 0$, + gli unici autovettori di $M$ sono i vettori ortogonali a + $v$, che sono punti fissi, e i multipli di $u$. + \end{proposition} + + $M$ è non singolare se e solo se $0$ non è autovalore, ossia se + e solo se: + \[ 1-\sigma(v^H u) \neq 0 \iff \sigma(v^H u) \neq 1. \] + + Se $M$ è non singolare, allora vale che: + \[ M\inv=I-\tau uv^H\;\con \tau=\frac{-\sigma}{1-\sigma v^Hu}, \] + e dunque anche $M\inv$ è una matrice elementare. + + \begin{proposition} + Per ogni $x$ e $y \in \CC^n\setminuszero$ esiste $M$ matrice elementare con $\det M \neq0$ tale che $Mx=y$. + In particolare è sufficiente scegliere $v$ non ortogonale + sia a $x$ che a $y$ e porre $\sigma u=\frac{x-y}{v^Hx}$. + \end{proposition} + + \subsubsection{Matrici elementari di Gauss e applicazione alla fattorizzazione LU} + + \begin{definition} + Dato $x\in\CC^n$ con $x_1\neq0$ si definisce la relativa \textbf{matrice elementare di Gauss} come la matrice elementare: + \[ M=I-ue_1^T, \quad u^T=\frac{1}{x_1}(0, x_2, \ldots, x_n). \] + \end{definition} + + Vale sempre $Mx=x_1 \, e_1$. Inoltre + una matrice elementare di Gauss è sempre invertibile, e + $M^{-1}=I+ue_1^T$. Vale sempre $\det(M) = 1$. Una matrice elementare di Gauss corrisponde a uno step di annichilimento + degli elementi sotto il \textit{pivot} + dell'algoritmo di Gauss. \smallskip + + In particolare: + \[ M=\begin{pmatrix} + 1 & 0 & \dots & 0 \\ + -u_2 & 1 & 0 \\ + \vdots & 0 &\ddots \\ + -u_n & 0 & \dots & 1 + \end{pmatrix}\E M\inv=\begin{pmatrix} + 1 & 0 & \dots & 0 \\ + u_2 & 1 & 0 \\ + \vdots & 0 &\ddots \\ + u_n & 0 & \dots & 1 + \end{pmatrix}. \] + + \begin{algorithm}[Fattorizzazione LU con le matrici di Gauss] + Sia $A\in \CC^{n \times n}$ che soddisfa l'esistenza e unicità della fattorizzazione $LU$. Se $M_1$ è la matrice elementare di Gauss associata alla prima colonna di $A$, allora: + \[ M_1A=\begin{pmatrix} + a_{11} & \dots & a_{1n} \\ + 0 \\ + \vdots & {A_1} \\ + 0 + \end{pmatrix}. \] + Se $A_1=(a_{ij}^{(1)})_{i,j>1}$, allora $a_{ij}^{(1)}=a_{ij}-\frac{a_{i1}}{a_{11}} a_{1j}$. Se $\hat{M}_2$ è la matrice elementare di Gauss che annulla la prima colonna di $A_1$, si definisce $M_2$ in modo tale che: + \[ M_2=\begin{pmatrix} + 1 & 0 \\ + 0 & \hat{M}_2 + \end{pmatrix}. \] + Iterando questo procedimento si ottiene il seguente prodotto: + \[ M_{n-1}\dots M_1 A=U, \quad M_1\inv\dots M_{n-1}\inv=L, \] + dove $L$ è triangolare inferiore con $1$ sulla diagonale + e $U$ è triangolare superiore. \smallskip + + In particolare vale che $\diag(U)=(a_{11},a_{22}^{(1)},\dots,a_{nn}^{(n-1)})$ e la $j$-esima colonna di $L=(\ell_{ij})$ è la $j$-esima colonna di $M_j$ cambiata di segno. Sono espresse di seguite le relazioni + di ricorrenza: + \[ \begin{cases} + a_{ij}^{(k+1)}=a_{ij}^{(k)}+m_{ij}^{(k)}a_{jk}^{(k)} & i,j=k+1,\dots,n \\ + m_{ik}^{(k)}=-\frac{a_{ik}^{(k)}}{a_{kk}^{(k)}} & i=k+1,\dots,n \\ + \ell_{ij} = -m_{ij}^{(j)} & i \geq j + \end{cases} \] + + Nel caso della risoluzione di un sistema lineare $Ax=b$, l'algoritmo + può essere esteso aggiornando $b$ ad ogni step; in particolare + $b^{(k+1)} = M_k b^{(k)}$ dove $M_k$ è la $k$-esima matrice di Gauss. In tal caso vale che: + \[ + b_i^{(k+1)} = b_i^{(k)} + m_{ik}^{(k)} b_k^{(k)}, \quad i = k+1, \ldots, n. + \] + + Il costo totale dell'algoritmo è di $\frac{2}{3}n^3+O(n^2)$ operazioni aritmetiche. Se $A$ è anche tridiagonale il costo è $O(n)$. + \end{algorithm} + + Riguardo al precedente algoritmo vale la seguente proposizione: + + \begin{proposition} + Sia $A \in \CC^{n \times n}$ una matrice con sottomatrici principali proprie di testa non singolari per cui esiste ed è unica la fattorizzazione $A = LU$. Se $\tilde{L}$ e $\tilde{U}$ i valori effettivamente calcolati di $L$ e $U$ tramite il precedente algoritmo in aritmetica di macchina e sia $\Delta_A=A-\tilde{L}\tilde{U}$. Allora vale + elemento per elemento di $\Delta_A$ la seguente disuguaglianza: + \[ \abs{\Delta_A}\le 2nu\left(\abs{A}+|\Tilde{L}|\,|\Tilde{U}|\right)+O(u^2), \] + dove $|T| := (|t_{ij}|)$. \smallskip + + Se $\tilde{y}$ è la soluzione del sistema $\tilde{L}y=b$ + calcolato realmente in aritmetica di macchina mediante + l'algoritmo di sostituzione in avanti e $\tx$ è il + vettore effettivamente calcolato risolvendo $\tilde{U}x = \tilde{y}$ mediante sostituzione all'indietro, allora + vale che: + \[ (A + \hat{\Delta}_A) \tx = b, \] + con + \[ + |\hat{\Delta}_A| \leq 4nu\left(\abs{A}+|\Tilde{L}|\,|\Tilde{U}|\right)+O(u^2), \] + dove $|T| := (|t_{ij}|)$. + \end{proposition} + + In particolare questa proposizione mostra che i valori $\tilde{L}$ + e $\tilde{U}$ formano una decomposizione $LU$ di una perturbazione + di $A$, e dunque è possibile effettuare un'analisi all'indietro + dell'algoritmo di Gauss. Se i valori in modulo di $A$ sono troppo + grandi, ci si + può aspettare un mal funzionamento in senso numerico del metodo + di eliminazione gaussiana (anche per il calcolo della soluzione $\tx$). + + \subsubsection{Strategia di \textit{pivoting} parziale per la decomposizione PLU} + + Permutando i pivot è sempre possibile fornire una fattorizzazione + del tipo $PLU$. La \textbf{strategia del \textit{pivoting parziale}} (\textit{massimo pivot parziale}) consiste + nel scegliere al passo $k$ come pivot il termine $a^{(k)}_{hk}$ con $h\ge k$ tale che $|a_{hk}^{(k)}|\ge|a_{ik}^{(k)}|$ per $i=k,\dots,n$. In questo modo se $a^{(k)}_{hk}=0$, allora tutta la parte di colonna + è nulla e si può procedere allo step successivo dell'algoritmo; altrimenti si può scambiare la riga $h$-esima con la riga $k$-esima e + applicare poi l'algoritmo di Gauss. \smallskip + + In questo modo si sta moltiplicando per una matrice di permutazione $P$ relativa alla trasposizione $\tau=(h, k)$. Nel caso in + cui $a^{(k)}_{hk}\neq 0$ vale allora che: + \[ A_{k+1}=M_k(P_kA_k). \] + In particolare vale sempre $m_{ij}^{k} \leq 1$ per $i\geq j$. \smallskip + + Si osserva inoltre che: + \[ + M_k P_k = P_k \tilde{M_k}, + \] + dove, se $M_k = I - u e_q^T$, allora + $\tilde{M_k} = I - P_k u e_q^T$ (infatti $\tilde{M_k} = P_k^T M_k P_k$). + Applicando questa strategia ad ogni passo si ottiene allora una fattorizzazione $PA=LU$ dove $P$ è un'opportuna matrice di permutazione, e dunque si ottiene una fattorizzazione $PLU$. + + \subsubsection{Matrici elementari di Householder e applicazioni alla fattorizzazione QR} + + \begin{definition} + Si definiscono \textbf{matrici elementari di Householder} le + matrici elementari della forma: + \[ M=I-\beta uu^H, \] + con $u\in\CC^n$ e $\beta\in\RR$. + \end{definition} + + Se $u \neq 0$ e $\beta = 0$ o $\beta = 2/(u^Hu) = 2/\langle u, u \rangle$, allora una matrice di Householder è unitaria. Inoltre + una matrice di Householder è sempre hermitiana. + + \begin{proposition} + Per ogni $x\in\CC^n\setminuszero$ esiste $M$ matrice elementare di Householder tale per cui $Mx=\alpha e_1$ per un dato $\alpha = \theta \norm{x}_2$, dove: + \[ \theta=\begin{cases} + \pm 1 & \text{se } x_1=0, \\ + \pm \frac{x_1}{|x_1|} & \text{se } x_1\neq0. + \end{cases}\] + Tale matrice $M$ si ottiene ponendo $u^t=x-\alpha e_1 = (x_1-\alpha,x_2,\dots,x_n)$ e $\beta=\frac{2}{u^Hu}$. + \end{proposition} + + + Per evitare le cancellazioni nell'implementazione del calcolo di una matrice di Householder, e dunque + migliorare la stabilità numerica, è consigliato scegliere sempre $\theta=-\frac{x_1}{\abs{x_1}}$ per $x_1 \neq 0$ (nel caso + in cui $x_1 = 0$, la scelta è indiffernete). \medskip + + \begin{algorithm} + Applicando la stessa filosofia dell'algoritmo di Gauss si possono + utilizzare le matrici di Householder per calcolare la fattorizzazione + $QR$ di una matrice $A$. Sia $u^{(k)}$ il vettore relativo alla $k$-esima matrice di Householder. Allora vale che: + \[ + u_i^{(k)} = \begin{cases} + 0 & \se i < k, \\ + a_{kk}^{(k)}\left(1 + \frac{\sqrt{\sum_{i=k}^n \abs{a_{ik}^{(k)}}^2}}{\abs{a_{kk}^{(k)}}}\right) & \se i = k, \\ + a_{ik}^{(k)} & \altrimenti. + \end{cases} + \] + mentre il parametro $\beta^{(k)}$ della stessa matrice è così + dato: + \[ + \beta^{(k)} = 2\bigg/\sum_{i=k}^n \abs{u_i^{(k)}}^2 + \] + A partire da questi, si ottengono i termini di $A_{k+1}$: + \[ + a_{i, j}^{(k+1)} = \begin{cases} + a_{ij}^{(k)} & \se j < k, \se i \leq k, \\ + 0 & \se j = k, \E i > k, \\ + a_{ij}^{(k)} - \beta^{(k)} u_i^{(k)} \sum_{r=k}^n \overline{u}_r^{(k)} a_{rj}^{(k)} & \se i \geq k, j > k. + \end{cases} + \] + + Nel caso della risoluzione di un sistema lineare $Ax=b$, l'algoritmo + può essere esteso aggiornando $b$ ad ogni step; in particolare + $b^{(k+1)} = M_k b^{(k)}$ dove $M_k$ è la $k$-esima matrice di Householder. In tal caso vale che: + \[ + b_i^{(k+1)} = b_i^{(k)} - \beta^{(k)} u_i^{(k)} \sum_{r=k}^n \overline{u}_r^{(k)} b_r^{(k)}, \quad i=k, \ldots, n. + \] + + Il costo totale è di $\frac{4}{3}n^3 + O(n^2)$ operazioni, il + \textit{doppio} di quello dell'algoritmo gaussiano. + \end{algorithm} + + Riguardo a quest'ultimo algoritmo vale la seguente proposizione: + + \begin{proposition} + Sia $A \in \CC^{n \times n}$ e sia $\tilde{R}$ la matrice triangolare + superiore ottenuta in aritmetica di macchina applicando il metodo di Householder con l'algoritmo + dato in precedenza. Sia inoltre $\tx$ la soluzione ottenuta risolvendo + in aritmetica di macchina il sistema $Ax=b$ con il metodo di Householder, + aggiornando le entrate di $b$. Allora $\tx$ risolve il sistema + $(A + \Delta_A) \tx = b + \delta_b$, dove: + \[ + \norm{\Delta_A}_F \leq u(\gamma n^2 \norm{A}_F + n \norm{\tilde{R}}_F) + O(u^2), + \] + \[ + \norm{\delta_b}_2 \leq \gamma n^2 u \norm{b}_2 + O(u^2), + \] + dove $\gamma$ è una costante positiva. Inoltre $\norm{\tilde{R}}_F$ + differisce da $\norm{\tilde{R}}_F = \norm{\tilde{A}}_F$ ($A = QR$) per un termine proporzionale a $u$, e dunque: + \[ + \norm{\Delta_A}_F \leq \gamma u(n^2 + n) \norm{A}_F + O(u^2). + \] + \end{proposition} + + In particolare, il metodo di Householder è numericamente stabile. + + \section{Metodi stazionari iterativi per sistemi lineari} + + \begin{definition} + Dato il sistema lineare $Ax=b$ con $A\in \CC^{n \times n}$ si definisce + un generico \textbf{partizionamento additivo di $A$} una + partizione della forma $A=M-N$ con $\det M\neq0$. + \end{definition} + + A partire da ciò possiamo riscrivere equivalentemente il sistema come $Mx=Nx+b$, il quale riconduce alla seguente scrittura equivalente del sistema originale: + \[ x=Px+q, \quad P=M\inv N, \, q=M\inv b. \] + Questa formulazione del sistema viene detta \textbf{problema di punto fisso}. + Una volta fissato $x^{(0)}\in\CC^n$ si può generare in modo naturale una successione di vettori: + \[ x^{(k+1)}=Px^{(k)}+q. \] + Se tale successione ammette limite $x^*$, allora il punto limite $x^*$ è un punto fisso di $Px+q$, ed è dunque l'unica soluzione di $Ax=b$. Questo metodo è detto \textbf{metodo iterativo stazionario}. La matrice $P$ ed il vettore $q$ sono indipendenti da $k$, e quindi la matrice $P$ viene detta \textbf{matrice di iterazione} del metodo. \medskip + + Si dice che il metodo iterativo è \textbf{convergente} se per ogni scelta del vettore $x^{(0)}$ la successione $x^{(k)}$ converge alla soluzione $x^*$ del sistema. \smallskip + + Se $A$ è invertibile, definiamo la successione $e^{(k)}=x^{(k)}-x^*$ dove $x^*$ è soluzione del sistema. Allora + $e^{(k+1)} = Px^{(k)} + Q - (Px^* + Q) = P e^{(k)}$. A partire da + questa osservazione si dimostra il seguente teorema: + + \begin{theorem} + Se esiste una norma di matrice indotta $\norm{\cdot}$ tale che $\norm{P}<1$, allora la matrice $A$ è invertibile. In tal caso + il metodo iterativo è convergente ($\lim_{k \to \infty} \norm{e^{(k)}} \leq \lim_{k \to \infty} \norm{P}^k \norm{e^{(0)}} = 0$). + \end{theorem} + + \begin{theorem} + Il metodo iterativo è convergente e $\det A \neq 0$ se e solo se $\rho(P)<1$. + \end{theorem} + + Il quoziente $\norm{e^{(k)}}/\norm{e^{(k-1)}}$ rappresenta la riduzione dell'errore al passo $k$-esimo del metodo iterativo rispetto alla norma scelta. Se si considera la media geometrica $\theta_k\left(e^{(0)}\right)$ delle prime $k$ riduzioni, si ricava che: + \[ \theta_k\left(e^{(0)}\right)=\left(\frac{\norm{e^{(1)}}}{\norm{e^{(0)}}}\dots\frac{\norm{e^{(k)}}}{\norm{e^{(k-1)}}}\right)^{\frac{1}{k}}=\left(\frac{\norm{P^ke^{(0)}}}{\norm{e^{(0)}}}\right)^{\frac{1}{k}}\le\norm{P^k}^{\frac{1}{k}}. \] + + \begin{definition} + Si definisce \textbf{riduzione asintotica media per passo} di un metodo iterativo con errore iniziale $e^{(0)}$ il valore: + \[ \theta(e^{(0)})=\lim_{k \to \infty}\theta_k(e^{(0)}). \] + Tale riduzione rappresenta la velocità di convergenza del metodo: + più è piccolo il valore, più veloce è il metodo. + \end{definition} + + In particolare vale che: + \[ + \theta(e^{(0)}) \leq \lim_{k \to \infty}\norm{P^k}^{\frac{1}{k}}=\rho(P). + \] + + L'uguaglianza è raggiunta per $e^{(0)}$ pari ad un autovettore + relativo al raggio spettrale di $P$. Se $P$ è nilpotente, allora il metodo + converge in un numero finito di passi. \medskip + + + Un esempio di metodo iterativo è dato dal \textbf{metodo di Richardson}, che + pone $P = I - \alpha A$, $q = M\inv b = \alpha A$. In particolare tale + metodo genera la successione $x^{(k+1)}=x^{(k)}-\alpha(Ax^{(k)}-b)$, + dove $\alpha \neq 0$ è un opportuno parametro e $Ax_n - b$ è detto + \textit{errore residuo}. \smallskip + + Poiché il metodo converge se $\rho(P) < 1$, + l'idea naturale è quella di scegliere $\alpha$ in modo tale che $\rho(P)$ sia + minimo, ricordandosi che $\lambda$ è autovalore di $P$ se e solo se + $\alpha (\lambda + 1)$ è autovalore di $A$. Infatti: \[ \det(\lambda I- P) = \det(\lambda I + I - \alpha A) = \frac{1}{\alpha^n} \det(\alpha(\lambda + 1)I - A).\] Se la matrice è hermitiana e definita positiva, allora $\alpha=1/\norm{A}$, dove + $\norm{\cdot}$ è una norma indotta, garantisce la convergenza del metodo. + + \subsubsection{I metodi di Jacobi e Gauss-Seidel} + + Sia $A$ tale per cui $a_{ii} \neq 0$. Si può allora decomporre $A$ nel seguente modo: + \[ A=D-B-C, \] + dove: + \begin{itemize} + \item $D$ è diagonale con $d_{ii}=a_{ii}\neq 0$ ($D = \diag(A)$, $D$ invertibile); + \item $B$ è strettamente triangolare inferiore con $b_{ij}=-a_{ij}$ con $i>j$ ($B = -\tril(A)$); + \item $C$ è strettamente triangolare superiore con $c_{ij}=-a_{ij}$ con $ij\E C$ la matrice triangolare superiore a blocchi tale che $C_{ij}=-A_{ij}\;\con i 0$ (e dunque $f(a) < 0$); + altrimenti è sufficiente applicare l'algoritmo al contrario + a $-f(x)$. \smallskip + + Al passo $(k+1)$-esimo si considera + $c_k = (a_k + b_k)/2$, dove + $a_0 = a$ e $b_0 = b$. Se $f(c_k) = 0$, + l'algoritmo termina; altrimenti, + se $f(c_k) > 0$, $b_{k+1} = c_k$ e + $a_{k+1} = a_k$, e se $f(c_k) < 0$, + $b_{k+1} = b_k$ e $a_{k+1} = c_k$. \smallskip + + Dal momento che gli intervalli $I_k = [a_k, b_k]$ + hanno lunghezza $(b-a)/2^k$, per $k \to \infty$ + la successione $x_k$ ha limite $\alpha$, che è tale + per cui $f(\alpha) = 0$. Inoltre l'intervallo si dimezza + a ogni step, e dunque il metodo converge esponenzialmente + (non per questo è veloce, anzi: come vedremo, ci sono + metodi estremamenti più veloci che convergono in modo + \textit{doppiamente esponenziale}). + + \subsection{Metodi del punto fisso} + + I metodi del punto fisso si ottengono trasformando il problema $f(x)=0$ in un problema del tipo $g(x)=x$. Questa trasformazione si può ottenere in infiniti modi diversi. Ad esempio, data una qualsiasi funzione $h(x)$ (diversa da zero nei punti del + dominio di $f(x)$), si può porre: + \[ g(x)=x-\frac{f(x)}{h(x)}, \] + i cui punti fissi sono gli zeri di $f(x)$. \smallskip + + Se $g(x)$ è continua, la successione + $x_{k+1} = g(x_k)$, con $x_0 \in \RR$, + se convergente, converge ad un punto fisso + di $g(x)$ (e dunque ad uno zero di $f(x)$). + Questo metodo è detto \textbf{metodo del punto fisso} (o \textit{di iterazione funzionale}) associato a $g(x)$. + + \begin{theorem}[del punto fisso] + Sia $\mathcal{I}=[\alpha-\rho,\alpha+\rho]$ e $g(x)\in C^1(\mathcal{I})$ dove $\alpha=g(\alpha)$ e $\rho>0$. Si denoti con $\lambda=\max_{x \in \mathcal{I}}\lvert g'(x) \rvert$. + Se $\lambda<1$ allora per ogni $x_0\in\mathcal{I}$, posto $x_{k+1}=g(x_k)$, vale che: + \[ \lvert x_k-\alpha \rvert \le\lambda^k\rho. \] + Pertanto $x_k\in\mathcal{I}$ e $\lim_k x_k=\alpha$. Inoltre $\alpha$ è l'unico punto fisso di $g(x)$ in $\mathcal{I}$. + \end{theorem} + + + Supponiamo di avere un intervallo $[a,b]$ in cui è presente un punto fisso + $\alpha$ e che $\lvert g'(x) \rvert<1$ su tale intervallo. Sotto queste ipotesi siamo certi che almeno con una delle due scelte $x_0=a$ o $x_0=b$ la successione è ben definita e converge al punto fisso $\alpha$. Infatti basta prendere rispettivamente $\rho=\alpha-a$ o $\rho=b-\alpha$ ed applicare il teorema. Se la successione $x_k$ cade fuori dall'intervallo, allora si arrestano le iterazioni e si riparte con $x_0$ uguale all'altro estremo. \smallskip + + In questo caso la convergenza vale in un intorno opportuno del punto fisso $\alpha$. Denotiamo questo fatto con l'espressione \textbf{convergenza locale}. + Si parla di \textbf{convergenza globale} qualora la convergenza vi sia per + ogni scelta iniziale. \smallskip + + Nell'aritmetica a virgola mobile il teorema diventa: + \begin{theorem} + Nelle ipotesi del teorema del punto fisso sia: + \[ \Tilde{x}_{k+1}=g(\Tilde{x}_k)+\delta_k, \] + dove $\lvert \delta_k \rvert \le\delta$ è l'errore commesso nel calcolo di $g(\Tilde{x}_k)$ e $\delta$ è una quantità nota. Posto $\sigma=\delta/(1-\lambda),\se\sigma<\rho$ allora: \[ \lvert\Tilde{x}_k-\alpha\rvert \le(\rho-\sigma)\lambda^k+\sigma. \] + \end{theorem} + + + Questo teorema ci dice che la distanza di $\Tilde{x}_k$ da $\alpha$ è limitata dalla somma di due parti. La prima converge a zero in modo esponenziale su base $\lambda$. La seconda è costante e rappresenta l'intervallo di incertezza sotto il quale non è consentito andare. \smallskip + + Per la funzione $g(x)=x-f(x)$ l'intervallo di incertezza è: + \[ I=\left[\alpha-\frac{\delta}{\lvert f'(\alpha) \rvert},\alpha+\frac{\delta}{\lvert f'(\alpha) \rvert}\right]. \] + Se $g'(x)>0$, allora $I\subseteq [\alpha-\sigma,\alpha+\sigma]$. \smallskip + + Si ricorda che: + \[ x_{k+1}-\alpha=g'(\xi_k)(x_k-\alpha), \] + dove $\xi_k \in (\alpha, x_k)$ viene dall'applicazione del teorema di Lagrange. Ciò implica che se $0\alpha\Rightarrow x_k>\alpha \; \forall k, \quad \alpha \alpha$ la sottosuccessione $\{x_{2k}\}$ cresce mentre $\{x_{2k+1}\}$ decresce (entrambe al punto $\alpha$). + + \subsubsection{Velocità di convergenza} + + \begin{definition} + Sia $\{x_k\}$ una successione tale che $\lim_kx_k=\alpha$. Si ponga + allora $e_k = x_k - \alpha$ come l'errore assoluto al passo $k$-esimo. Supponiamo esista il limite della riduzione dell'errore al passo $k$-esimo: + \[ \gamma=\lim_k \abs{\frac{x_{k+1}-\alpha}{x_k-\alpha}} = \lim_k \abs{\frac{e_{k+1}}{e_k}}. \] + La convergenza di $\{x_k\}$ a $\alpha$ è detta allora: + \begin{itemize} + \item lineare (o geometrica) se $0<\gamma<1$; + \item sublineare se $\gamma=1$; + \item superlineare se $\gamma=0$. + \end{itemize} + \end{definition} + + Nel caso di convergenza superlineare, se $p>1$ ed esiste il limite: + \[ \lim_k \abs{\frac{x_{k+1}-\alpha}{(x_k-\alpha)^p}} =\sigma, \quad 0<\sigma<\infty \] si dice che la successione \textbf{converge con ordine $p$}. + + \begin{remark} + L'ordine di convergenza non è obbligatoriamente intero. + \end{remark} + + \begin{theorem} + Sia $g(x)\in C^1([a,b])$ e $\alpha\in(a,b)$ t.c.~$g(\alpha)=\alpha$. Se esiste un $x_0\in[a,b]$ tale che la successione $x_{k+1}=g(x_k)$ converge linearmente ad $\alpha$ con fattore $\gamma$ allora: + \[ \lvert g'(\alpha)\rvert=\gamma. \] + Viceversa, se $0<\lvert g'(\alpha)\rvert<1$ allora esiste un intorno $I$ di $\alpha$ contenuto in $[a,b]$ tale che per ogni $x_0\in I$ la successione $\{x_k\}$ converge ad $\alpha$ in modo lineare con fattore $\gamma=\lvert g'(\alpha) \rvert$. + \end{theorem} + + \begin{theorem} + Sia $g(x)\in C^1([a,b])$ e $\alpha\in (a,b)$ t.c.~$g(\alpha)=\alpha$. Se esiste un $x_0\in[a,b]$ tale che la successione $x_{k+1}=g(x_k)$ converge sublinearmente ad $\alpha$ allora: + \[ \abs{g'(\alpha)}=1. \] Viceversa, se $\abs{g'(\alpha)}=1$ esiste un intorno $I$ di $\alpha$ contenuto in $[a,b]$ tale che per ogni $x\in I$, $x\neq\alpha$ vale $\abs{g'(x)}<1$ e $g'(x)$ non cambia segno su $I$ allora tutte le successioni $\{x_k\}$ con $x_0\in I$ convergono ad $\alpha$ in modo sublineare. + \end{theorem} + + \begin{theorem} + Sia $g(x)\in C^p([a,b])$ con $p>1$ intero e $\alpha\in(a,b)$ t.c.~$g(\alpha)=\alpha$. Se esiste un $x_0\in[a,b]$ tale che la successione $x_{k+1}=g(x_k)$ converge superlinearmente ad $\alpha$ con ordine di convergenza $p$ allora: + \[ \abs{g^{(k)}(\alpha)}=0 \;\;\;\, 1 \leq k \leq p-1, \quad g^{(p)}(\alpha)\neq0. \] Viceversa se $\abs{g^{(k)}(\alpha)}=0$ per $k=1,\dots,p-1\E g^{(p)}(\alpha)\neq0$ allora esiste un intorno $I$ di $\alpha$ tale che per ogni $x_0\in I$ tutte le successioni $\{x_k\}$ convergono ad $\alpha$ in modo superlineare con ordine $p$. + \end{theorem} + + \begin{definition} + La successione $\{x_k\}$ converge ad $\alpha$ \textbf{con ordine almeno $p$} se esiste una costante $\beta$ tale che \[ \abs{x_{k+1}-\alpha}\le\beta \, \abs{x_k-\alpha}^p. \] + \end{definition} + + Se una successione converge con ordine $q\ge p$ allora converge anche con ordine almeno $p$. \smallskip + + Se una successione $x_k$ converge ad $\alpha$ in modo che l'errore relativo $\eps_k$ al passo $k$-esimo è limitato nel seguente modo: + \[ \varepsilon_k=\abs{x_k-\alpha/\alpha}\le\beta\gamma^{p^k}\] + allora il numero di cifre significative ($1+\log_2(\varepsilon_k\inv)$) è tale che + \[ 1+\log_2 \, \varepsilon_k\inv \ge1+\log_2 \, \beta\inv+p^k\log_2 \,\gamma\inv. \] + + \subsubsection{Confronto tra metodi} + + Siano dati due metodi iterativi del punto fisso definiti da due funzioni $g_1(x)$ e $g_2(x)$. Si supponga che siano entrambi siano o a convergenza lineare o a convergenza superlineare. Denotiamo con $c_1$, $c_2$ il numero di flops per passo dei due metodi. Se siamo nel caso di convergenza lineare, detti $\gamma_1$, $\gamma_2$ i due fattori di convergenza allora il primo metodo risulta più conveniente se \[ \frac{c_1}{c_2}<\frac{\log{\gamma_1}}{\log{\gamma_2}}. \] + Nel caso di convergenza superlineare, se $p_1$ e $p_2$ sono i due ordini di convergenza, il primo metodo è più efficiente del secondo se \[\frac{c_1}{c_2}<\frac{\log{p_1}}{\log{p_2}}.\] + + \subsection{Alcuni metodi del punto fisso} + + \subsubsection{Metodo delle secanti} + + Sia $f(x)\in C^1([a,b])\E\alpha\in[a,b]\tc f(\alpha)=0$. Il metodo definito dalla funzione $$g(x)=x-\frac{f(x)}{m}$$ + dove $m$ è un'opportuna costante è detto \textbf{metodo delle costanti}. \smallskip + + Il metodo consiste nel tracciare la retta passante per il punto $(x_k,f(x_k))$ di coefficiente angolare $m$ e considerare come $x_{k+1}$ l'ascissa dell'intersezione di tale retta con l'asse delle ascisse, reiterando. \smallskip + + Una condizione sufficiente di convergenza è che valga $\abs{1-f'(x)/m}<1$ in un intorno di circolare di $\alpha$. È sufficiente quindi scegliere $m$ in modo che abbia lo stesso segno di $f'(x)$ e $\abs{m}>\frac{1}{2}\abs{f'(x)}$. Se $f'(\alpha)$ fosse nota, la scelta $m=f'(\alpha)$ darebbe una convergenza superlineare. + + \subsubsection{Metodo delle tangenti di Newton} + + La funzione $g(x)$ del \textbf{metodo di Newton} è definita nel modo seguente: \[ g(x)=x-\frac{f(x)}{f'(x)}. \] + Per tale metodo vale: + \[ + g'(x)=1-\frac{f'(x)^2 - f''(x)f(x)}{f'(x)^2} = \frac{f''(x) f(x)}{f'(x)^2}. + \] + Il metodo consiste nel tracciare la retta passante per il punto $(x_k,f(x_k))$ e tangente a esso e considerare come $x_{k+1}$ l'ascissa dell'intersezione di tale retta con l'asse delle ascisse, reiterando. + + \begin{theorem} + Sia $f(x)\in C^2([a,b])$ e sia $\alpha\in(a,b)$ t.c.~$f(\alpha)=0$. Se $f'(\alpha)\neq0$, allora esiste un intorno $I=[\alpha-\rho,\alpha+\rho]\subset[a,b]$ tale che per ogni $x_0\in I$ la successione generata dal metodo di Newton converge ad $\alpha$. + + Inoltre se $f''(\alpha)\neq0$ la convergenza è superlineare di ordine $2$, se $f''(\alpha)=0$ la convergenza è di ordine almeno $2$. + \end{theorem} + + \begin{theorem} + Sia $f(x)\in C^p([a,b])$ con $p>2$ e $\alpha\in(a,b)$ t.c.~$f(\alpha)=0$. Se $f'(\alpha)=\dots=f^{(p-1)}(\alpha)=0$, $f^{(p)}(\alpha)\neq0$ e $f'(x)\neq0$ per $x\neq\alpha$, allora esiste un intorno $I=[\alpha-\rho,\alpha+\rho]\subset[a,b]$ in cui $f'(x)\neq0$ per $x\in I,x\neq\alpha$ e tale che per ogni $x_0\in I$, la successione generata dal metodo di Newton converge ad $\alpha$. Inoltre $\alpha$ è l'unico zero di $f(x)$ in $I$. La convergenza è lineare con fattore di convergenza $\gamma = 1-1/p$. + \end{theorem} + + \begin{theorem} + Se la funzione $f(x)$ è di classe $C^2$ sull'intervallo $I=[\alpha,\alpha+\rho]$ ed è tale che $f'(x)f''(x)>0$ per $x\in I$ allora per ogni $x_0\in I$ la successione generata dal metodo di Newton applicato ad $f(x)$ converge decrescendo ad $\alpha$ con ordine $2$. + Un risultato analogo vale su intervalli del tipo $[\alpha-\rho,\alpha]$, su + cui invece la successione cresce. + \end{theorem} + + \begin{example}[Calcolo del reciproco di un numero $a$] + Per calcolare $1/a$ si può porre $f(x)=a-1/x$ e poi applicare il metodo di + Newton. + \end{example} + + \begin{example}[Calcolo della radice $p$-esima di un numero $a$] + Per calcolare $\sqrt[p]{a}$ è sufficiente applicare il metodo di Newton + con $f(x) = (x^p-a)x^{-q}$, dove $q\geq 0$. + \end{example} + + \begin{example}[Metodo di Aberth] + Se si hanno delle approssimazione $t_1$, ..., $t_n$ di zeri + di $p(x)$, si può applicare il metodo di Newton su + $f(x) = p(x) \big/ \prod_{i=1}^n (x-t_i)$, in modo tale + che $f(t_i)$ sia sempre più grande, diminuendo gli errori e migliorando + le approssimazioni. + \end{example} + + \section{Interpolazione di funzioni} + Siano $\varphi_0(x)$, ..., $\varphi_n(x) : [a,b] \to \RR$ funzioni + linearmente indipendenti. Siano $(x_i,y_i)$ $n+1$ valori assegnati (\textit{nodi}) tali che $x_i\in[a,b]$ e $x_i\neq x_j$ se $i\neq j$. \smallskip + + Il problema dell'interpolazione consiste nel determinare i coefficienti + $a_i \in \RR$ tali per cui: + \[ + f(x)=\sum_{i=0}^na_i \, \varphi_i(x), \qquad f(x_i)=y_i\;\forall i. + \] + Tali condizioni sono dette \textbf{condizioni di interpolazione}. + + \subsection{Interpolazione polinomiale} + + L'interpolazione polinomiale ricerca il polinomio $p$ di grado $n+2$ che + soddisfi le condizioni di interpolazione di $n+1$ nodi. Sia pertanto $\varphi_i(x)=x^i$. Siano $(x_i, y_i)$ per $i = 0$, ..., $n+1$ i nodi + dell'interpolazione. + + \subsubsection{Interpolazione monomiale} + + \begin{definition} + Si dice \textbf{matrice di Vandermonde} nei nodi $x_0$, ..., $x_n$ la matrice di taglia $n+2$: + \[ + V_n=\begin{pmatrix} + 1 & x_0 & \dots & x_0^n \\ + 1 & x_1 & \dots & x_1^n \\ + \vdots & \vdots & &\vdots \\ + 1 & x_n & \dots & x_n^n + \end{pmatrix} + \] + \end{definition} + + \begin{theorem} + Vale $\det V_n=\prod_{j 0$, ovverosia: + \[ + \Pi_n = \begin{pmatrix} + 1 & 0 \\ + 0 & P_{n-1} + \end{pmatrix}, + \] + dove $P_{n-1}$ è la matrice identità specchiata orizzontalmente (ossia con soli $1$ sull'antidiagonale), che è tale per cui $P_{n-1}^2 = I$. + \end{itemize} + \end{proposition} + + Come corollario della precedente proposizione, si ottiene che + $\Omega_n^{-1} = \frac{1}{n} \Omega_n^H$ e che + $\Omega_n^{-1} = \frac{1}{n} \Pi_n \Omega_n = \frac{1}{n} \Omega_n \Pi_n$. + + Dunque il problema dell'interpolazione si risolve ponendo $z= \Omega_n^{-1} y$ e applicando una delle formule proposte precedentemente. + + \begin{remark} + La matrice $F_n=\Omega_n/\sqrt{n}$ è unitaria, e dunque $\norm{F_n}_2=1$. Ciò implica che $\norm{\Omega_n}_2=\sqrt{n}$, dunque il numero di condizionamento di $\Omega_n$ è $1$, ovverosia il problema dell'interpolazione ai nodi di Fourier è ben condizionato. + \end{remark} + + \subsection{Calcolo di (I)DFT: \textit{Fast Fourier Transform} (FFT)} + + Consideriamo il caso del calcolo di $\IDFT(y)$, dove abbiamo i coefficienti $z_0$, ..., $z_{n-1}$ e vogliamo trovare i valori $y_i$ tali per cui + $\Omega_n (z_i) = (y_i)$. Il calcolo di $\DFT(z)$ si può poi fare + eseguendo $\IDFT(z)$ e applicando le relazioni introdotte nella proposizione precedente, aggiungendo + $n$ divisioni e permutando gli indici. \medskip + + Consideriamo il caso in cui $n=2^q$. Allora: + \[ + y_i = \sum_{j=0}^{n-1} \omega_n^{ij} z_j. + \] + Ricordando che $\omega_n^2 = \omega_{\frac{n}{2}}$, allora, separando + gli indici pari da quelli dispari, vale che: + \[ + y_i =\sum_{j=0}^{\frac{n}{2}-1}\omega_{\frac{n}{2}}^{ij}z_{2j}+\omega_n^i \sum_{j=0}^{\frac{n}{2}-1}\omega_{\frac{n}{2}}^{ij}z_{2j+1}. + \] + Pertanto, detti $Y = \left(y_0,\dots,y_{\frac{n}{2}-1}\right)^\top$, $Y' = \left(y_{\frac{n}{2}},\dots,y_{n-1}\right)^\top$ e $D_n = \diag\left(1,\omega_n,\dots,\omega_n^{\frac{n}{2}-1}\right)$, valgono le seguenti due identità: + \[ Y=\IDFT_{\frac{n}{2}}(z_{\text{pari}})+D_n \, \IDFT_{\frac{n}{2}}(z_\text{disp}), \] + \[ Y'=\IDFT_{\frac{n}{2}}(z_\text{pari})-D_n \, \IDFT_{\frac{n}{2}}(z_\text{disp}). \] + + In forma matriciale le due identità si scrivono infine come: \[ y=\begin{pmatrix} + \Omega_{\frac{n}{2}}z_\text{pari} \\ + \Omega_{\frac{n}{2}}z_\text{pari} + \end{pmatrix}+\begin{pmatrix} + D_n\Omega_{\frac{n}{2}}z_\text{disp} \\ + -D_n\Omega_{\frac{n}{2}}z_\text{disp} + \end{pmatrix}. \] + + Poiché $n$ è potenza di $2$ possiamo ripetere questa strategia per calcolare le due trasformate di ordine $\frac{n}{2}$ mediante quattro trasformate di ordine $\frac{n}{4}$, ecc... fino ad avere trasformate di ordine $1$ che non richiedono alcuna operazione. \medskip + + Il costo $c(n)$ di flops di una IDFT su $n$ nodi con questo metodo è ricorsivamente: + \[ + c(n)=2 \, c\!\left(\frac{n}{2}\right)+\frac{n}{2}+n=2 \, c\!\left(\frac{n}{2}\right)+\frac{3}{2}n, + \] + dove $n/2$ sono le moltiplicazioni effettuate e $n$ le addizioni. \smallskip + + Dacché $c(1)=0$ e $n=2^q$ vale che: + \[ c(n)=\frac{3}{2}n\log_2n=3q2^{q-1}. \] + + In generale, l'algoritmo ha dunque complessità $O(n \log_2(n))$. + L'algoritmo per il calcolo della IDFT che si ottiene in questo modo è noto come \textbf{algoritmo di Cooley-Tukey}. + + \subsubsection{Applicazioni della FFT: calcolo del prodotto di polinomi} + + Dati $a(t)$, $b(t)\in\CC[x]$ allora $c(t)=a(t)b(t)$ si può calcolare trovando i coefficienti $c_i=\sum_{j+k=i}a_jb_k$. Se $a(t)$, $b(t)$ hanno grado rispettivamente $p$, $q$, questo algoritmo ha complessità $O((p+q)^2)$. \medskip + + + Sia $n = 2^q \geq p+q$. Con la FFT posso valutare $a(t)$, $b(t)$ nelle radici $n$-esime dell'unità. Dopo aver moltiplicato i valori ottenuti, si può applicare una DFT, ricavando i coefficienti di $c(t)$. Questo algoritmo utilizza $2$ IDFT, $n$ moltiplicazioni ed una DFT; pertanto ha un costo di $O(n\log_2(n))$ flops. + + \section{Metodi dell'integrazione approssimata} + + In questa sezione si illustrano alcuni metodi per approssimare + gli integrali su un intervallo. + + \begin{definition} + Data $f : [a, b] \to \RR$ continua e $n+1$ nodi $x_0$, ..., $x_n$ si definiscono le due quantità: + \[ S[f]=\int_a^bf(x)\,dx, \quad S_{n+1}[f]=\sum_{i=0}^nw_if(x_i), \] + dove i termini $w_i$ sono positivi reali detti \textit{pesi}, possibilmente + variabili. Una scelta di pesi corrisponde a una \textbf{formula di integrazione + approssimata}. + \end{definition} + + \begin{definition} + Si dice \textbf{resto} il valore + \[ r_{n+1}=S[f]-S_{n+1}[f]. \] + \end{definition} + + \begin{definition} + Si dice che una formula di integrazione approssimata ha \textbf{grado di precisione} (massimo) $k\ge0$ se vale \[ r_{n+1}=0 \impliedby f(x)=x^{j}, \quad 0 \leq j \leq k \] e se \[ r_{n+1}\neq0 \impliedby f(x)=x^{k+1}. \] + \end{definition} + + \subsection{Formule di integrazione dell'interpolazione di Lagrange} + + Si può approssimare un integrale utilizzando l'interpolazione di Lagrange sui nodi, ponendo + \[ S_{n+1}[f] = \int_a^b \tilde{f}(x)\,dx = \sum_{i=0}^n \left[ f(x_i) \int_a^b L_i(x)\,dx\right], \] + dove $\tilde{f}(x)$ è il polinomio ottenuto applicando l'interpolazione + di Lagrange su $f$ dati i nodi $(x_0, f(x_0))$, ..., $(x_n, f(x_n))$ e i pesi sono tali per cui $w_i = \int_a^b L_i(x)\,dx$. \smallskip + + Le formule di integrazione interpolatorie hanno grado di precisione almeno $n$ + (infatti se $j \leq n$, $x^j$ è il polinomio che interpola i nodi dati) ed hanno grado di precisione massimo $2n+1$. Se una formula di integrazione ha grado di precisione almeno $n$, allora è interpolatoria. \smallskip + + Se $f\in C^{(n+1)}([a,b])$ e $\abs{f^{(n+1)}(x)}\le M$ allora + \[ \abs{r_{n+1}}\leq\frac{M}{(n+1)!}\abs{\int_a^b\prod_{j=0}^n(x-x_j) \, dx}. \] + + \subsection{Formule di Newton-Cotes (semplici)} + Scegliendo nodi equispaziati, con $h=\frac{b-a}{n}$, si ottengono le + formule di Newton-Cotes. nel caso $n=1$ vale \[ w_0=\int_a^bL_0(x)\,dx=\frac{h}{2}=w_1\Rightarrow S_2[f]=\frac{h}{2}(f(x_0)+f(x_1)), \] + ovverosia corrisponde all'area del trapezio con di base $\abs{x_0}$ e + $\abs{x_1}$ e con altezza $h$. Il resto $r_2$ corrisponde invece a \[ \int_a^bf(x)\,dx-S_2[f]=-\frac{1}{12}h^3f''(\xi), \quad \xi\in(a,b). \] + Nel caso $n=2$ invece \[ w_0=w_2=\frac{h}{3}, \quad w_1=\frac{4}{3}h \] ed il resto è dunque + \[ \int_a^bf(x)\,dx-S_3[f]=-\frac{1}{90}h^5f^{(4)}(\xi),\quad \xi\in(a,b) \] + + \subsection{Formule di Newton-Cotes composte (trapezi, Cavalieri-Simpson)} + Si può rendere più precisa la formula di integrazione introducendo altri nodi equispaziati $z_0$, ..., $z_N$ e risolvendo \[ \int_a^bf(x)\,dx=\sum_{i=0}^{N-1}\int_{z_i}^{z_{i+1}}f(x)\,dx, \] + approssimando ciascuno degli integrali con le formule di Newton-Cotes semplici per $n=1$ o $2$. \smallskip + + Scegliendo sempre $n=1$ si ottiene la \textbf{formula dei trapezi}, per la quale: \[ J_2^{(n)}[f]=\sum_{i=0}^{n-1}S_2^{(i)}[f]=\frac{b-a}{2n}\left[f(z_0)+f(z_n)+2\sum_{i=1}^{n-1}f(z_i)\right] \] + Inoltre vale che: + \[ S[f]-J_2^{(nN)}[f]=-\frac{(b-a)h^2}{12}f''(\xi). \] + + Scegliendo invece sempre $n=2$ si ottiene la \textbf{formula di Cavalieri-Simpson}, per la quale: + \begin{multline*} + J_3^{(n)}[f] = \frac{b-a}{6n} \!\left[f(z_0) + f(z_n) + 2 \sum_{k=1}^{n-1} f(z_k) + \right. \\ \left.+ 4 \sum_{k=0}^{n-1} f\!\left(\frac{z_k + z_{k+1}}{2}\right) \right]. + \end{multline*} + + \section{Risoluzione di problemi di Cauchy con metodi a un passo} + + Sia $I \subseteq \RR$ un intervallo e consideriamo una funzione + $f(t, y) : I \times \RR \to \RR$ continua. Dati i parametri + iniziali $t_0 \in I$, $y_0 \in \RR$ si associa a questi il seguente + \textit{problema di Cauchy}: + \[ + \begin{cases} + y'(t) = f(t, y(t)), \\ + y(t_0) = y_0, + \end{cases} + \] + su cui si ipotizza che l'incognita $y$ è $C^1(I, \RR)$. \smallskip + + Se $f(t, y)$ è Lipschitziana rispetto a $y$, ovverosia esiste + $L \in \RR$ t.c.~$\abs{f(t, y_1) - f(t, y_2)} \leq L \abs{y_1 - y_2}$ per + ogni $t \in I$, $y_1$, $y_2 \in \RR$, allora esiste ed è unica + la soluzione $y$ al problema di Cauchy proposto. + + \begin{notation}[Intervallo destro di $t_0$ e successione per passi] + Per intervallo destro di un $t_0 \in \RR$ intendiamo + un intervallo della forma $[t_0, t_0 + T]$ con $t > 0$. \smallskip + + Dato $h > 0$, definiamo $t_n := t_0 + nh$ per $n = 0$, ..., + $N_h$, dove $N_h$ è il massimo naturale $p$ per cui + $ph \leq T$ (ossia il massimo naturale per cui la successione + definita sta dentro l'intervallo destro definito). + \end{notation} + + \begin{notation}[Approssimazioni] + Scriveremo $y_n$ per indicare $y(t_n)$ e $u_n$ per indicare + l'approssimazione ricavata di $y(t_n)$, mentre + $f_n := f(t_n, u_n)$. + \end{notation} + + \begin{definition}[Metodo a un passo] + Si dice \textbf{metodo a un passo} un metodo di + risoluzione approssimato di un problema di Cauchy + che imposti una sequenza della forma: + \[ + \begin{cases} + u_{n+1} = u_n + h \phi(t_n, u_n, f_n, h) & 0 \leq n \leq N_h - 1, \\ + u_0 = y_0. + \end{cases} + \] + \end{definition} + + \begin{definition} + Sia $\eps_{n+1} := y_{n+1} - (y_n + h \phi(t_n, y_n, f(t_n, y_n), h))$ la differenza + tra il valore esatto $y_{n+1} = y(t_{n+1})$ e quello dato dal + metodo applicato al valore esatto $y_{n+1} = y(t_{n+1})$. + Si definisce allora \textbf{errore locale di troncamento} al + nodo $(n+1)$-esimo il valore: + \[ + \tau_{n+1}(h) = \frac{\eps_{n+1}}{h}. + \] + Si definisce inoltre l'\textbf{errore globale di troncamento} come + massimo del modulo degli errori locali: + \[ + \tau(h) = \max_{n=0,\ldots,N_h - 1} \abs{\tau_{n+1}(h)}. + \] + \end{definition} + + \begin{definition}[Consistenza del metodo] + Si dice che un metodo è consistente di ordine $p$ per il problema se $\tau(h) = o(h)$ ($\lim_{h \to 0} \tau(h) = 0$) e + se $\tau(h) = O(h^p)$ per $h \to 0$. + \end{definition} + + \begin{definition}[Convergenza del metodo] + Posto $e_n = y_n - u_n$, detto \textit{errore globale}, un + metodo si dice \textbf{convergente} di ordine $p$ se esiste + $C(h) = o(h)$ ($\lim_{h \to 0} C(h) = 0$), $C(h) = O(h^p)$ per + $h \to 0$ tale per cui + $\abs{y_n - u_n} \leq C(h)$ per ogni $n$ della successione. + \end{definition} + + \subsection{Metodo di Eulero} + + Si definisce \textbf{metodo di Eulero} il metodo a un passo + che si ottiene imponendo $\phi(t_n, u_n, f_n, h) = f_n$, ovverosia: + \[ + \begin{cases} + u_{n+1} = u_n + h f_n & 0 \leq n \leq N_h - 1, \\ + u_0 = y_0. + \end{cases} + \] + L'idea del metodo di Eulero è quella di approssimare innanzitutto + i reali con una successione $t_n$ prendendo $h \ll 1$ (in questo + modo $N_h \gg 1$ e si ottiene una sequenza arbitrariamente lunga + di reali). In questo modo l'equazione + \[ + y'(t) = f(t, y(t)) + \] + può essere approssimata ponendo l'equazione vera sui $t_n$ con $y'(t) \approx \frac{u_{n+1} - u_n}{h}$ e $f(t, y(t)) \approx f_n$. La + seconda equazione $y(t_0) = y_0$ si riconduce invece facilmente + a $u_0 = y_0$. \smallskip + + Per ottenere dunque un'approssimazione è dunque necessario + risolvere la relazione di ricorrenza tra $u_{n+1}$ e $u_n$, + ponendo $u_0 = y_0$, o calcolare direttamente la sequenza. + + Per il metodo di Eulero vale che: + \[ + \eps_{n+1} = y_{n+1} - (y_n + h f(t_n, y_n)). + \] + Il metodo di Eulero è sempre consistente di ordine $1$ + ed è sempre convergente, ancora di ordine $1$. + + \subsubsection{Metodo di Eulero implicito} + + Si definisce \textbf{metodo di Eulero implicito} il metodo + che si ottiene sostituendo a $\phi(t_n, u_n, f_n, h)$ il valore $f_{n+1}$, ovverosia: + \[ + \begin{cases} + u_{n+1} = u_n + h f_{n+1} & 0 \leq n \leq N_h - 1, \\ + u_0 = y_0. + \end{cases} + \] + Tale metodo è consistente di ordine $1$ e convergente, ancora + di ordine $1$. + + \begin{example}[Problema test di Dahlquist] + Si consideri il seguente problema di Cauchy: + \[ + \begin{cases} + y' = \lambda y, \\ + y(0) = 1, + \end{cases} + \] + con $I = [0, \infty)$. Applicando il metodo di Eulero implicito + si deve risolvere il sistema: + \[ + \begin{cases} + u_{n+1} = u_n + \lambda u_{n+1}, + u_0 = 1, + \end{cases} + \] + che ha come soluzione $u_n = \left(\frac{1}{1-h\lambda}\right)^n$. + \end{example} + + \vfill + \hrule + ~\\ + Opera originale di Mario Zito, modifiche e aggiornamenti + a cura di Gabriel Antonio Videtta. + ~\\Reperibile su + \url{https://notes.hearot.it}. + \end{multicols} +\end{document} diff --git a/Secondo anno/Analisi numerica/notes_2023.sty b/Secondo anno/Analisi numerica/notes_2023.sty new file mode 100644 index 0000000..19d4675 --- /dev/null +++ b/Secondo anno/Analisi numerica/notes_2023.sty @@ -0,0 +1,463 @@ +\ProvidesPackage{notes_2023} + +\usepackage{amsmath,amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsopn} +\usepackage{bookmark} +\usepackage{faktor} +\usepackage[utf8]{inputenc} +\usepackage{mathtools} +\usepackage{nicefrac} +\usepackage{stmaryrd} +\usepackage{marvosym} +\usepackage{float} +\usepackage{enumerate} +\usepackage{scalerel} +\usepackage{stackengine} +\usepackage{wasysym} + +\usepackage{tikz-cd} + +\usepackage{quiver} + +\usepackage[italian]{babel} + +\usepackage{tabularx} + +% Setup preliminari +\setlength{\extrarowheight}{4pt} + +\newcommand{\system}[1]{\begin{cases} #1 \end{cases}} + +\newcommand{\wip}{\begin{center}\textit{Questo avviso sta ad indicare che questo documento è ancora una bozza e non è + da intendersi né completo, né revisionato.}\end{center}} + +\newcommand\hr{\vskip 0.05in \par\vspace{-.5\ht\strutbox}\noindent\hrulefill\par} + +% Modalità matematica/fisica +\newcommand{\SMatrix}[1]{\begin{psmallmatrix}#1\end{psmallmatrix}} + +\let\oldvec\vec +\renewcommand{\vec}[1]{\underline{#1}} + +\newcommand{\con}{\text{con }} +\newcommand{\dove}{\text{dove }} +\newcommand{\E}{\text{ e }} +\newcommand{\altrimenti}{\text{altrimenti}} +\newcommand{\se}{\text{se }} +\newcommand{\tc}{\text{ t.c. }\!} +\newcommand{\epari}{\text{ è pari}} +\newcommand{\edispari}{\text{ è dispari}} + +\newcommand{\nl}{\ \\} + +\newcommand{\bigmid}{\;\middle\vert\;} + +\newcommand{\lemmaref}[1]{\textit{Lemma \ref{#1}}} +\newcommand{\thref}[1]{\textit{Teorema \ref{#1}}} + +\newcommand{\li}[0]{$\blacktriangleright\;\;$} + +\newcommand{\tends}[1]{\xrightarrow[\text{$#1$}]{}} +\newcommand{\tendsto}[1]{\xrightarrow[\text{$x \to #1$}]{}} +\newcommand{\tendstoy}[1]{\xrightarrow[\text{$y \to #1$}]{}} +\newcommand{\tendston}[0]{\xrightarrow[\text{$n \to \infty$}]{}} + +\setlength\parindent{0pt} + +% Principio di induzione e setup dimostrativi. +\newcommand{\basestep}{\mbox{(\textit{passo base})}\;} +\newcommand{\inductivestep}{\mbox{(\textit{passo induttivo})}\;} + +\newcommand{\rightproof}{\mbox{($\implies$)}\;} +\newcommand{\leftproof}{\mbox{($\impliedby$)}\;} + +% Spesso utilizzati al corso di Fisica 1. +\newcommand{\dx}{\dot{x}} +\newcommand{\ddx}{\ddot{x}} +\newcommand{\dv}{\dot{v}} + +\newcommand{\del}{\partial} +\newcommand{\tendstot}[0]{\xrightarrow[\text{$t \to \infty$}]{}} + +\newcommand{\grad}{\vec{\nabla}} +\DeclareMathOperator{\rot}{rot} + +\newcommand{\ihat}{\hat{i}} +\newcommand{\jhat}{\hat{j}} +\newcommand{\khat}{\hat{k}} + +\newcommand{\der}[1]{\frac{d#1}{dx}} +\newcommand{\parx}{\frac{\del}{\del x}} +\newcommand{\pary}{\frac{\del}{\del y}} +\newcommand{\parz}{\frac{\del}{\del z}} + +% Spesso utilizzati al corso di Analisi 1. +%\newcommand{\liminf}{\lim_{x \to \infty}} +\newcommand{\liminfty}{\lim_{x \to \infty}} +\newcommand{\liminftym}{\lim_{x \to -\infty}} +\newcommand{\liminftyn}{\lim_{n \to \infty}} +\newcommand{\limzero}{\lim_{x \to 0}} +\newcommand{\limzerop}{\lim_{x \to 0^+}} +\newcommand{\limzerom}{\lim_{x \to 0^-}} + +\newcommand{\xbar}{\overline{x}} +\newcommand{\ybar}{\overline{y}} +\newcommand{\tbar}{\overline{t}} +\newcommand{\zbar}{\overline{z}} +\newcommand{\RRbar}{\overline{\RR}} + +% Spesso utilizzati al corso di Geometria 2 + +\newcommand{\PP}{\mathbb{P}} +\newcommand{\PPGL}{\mathbb{P}\!\GL} + +% Spesso utilizzati al corso di Geometria 1. + +\DeclareMathOperator{\OO}{O} % gruppo ortogonale +\DeclareMathOperator{\SOO}{SO} % gruppo ortogonale speciale + +\newcommand{\proj}[1]{\Matrix{#1 \\[0.03in] \hline 1}} +\newcommand{\projT}[1]{\Matrix{#1 & \rvline & 1}^\top} + +\newcommand{\cc}{\mathcal{C}} + +\let\AA\undefined +\DeclareMathOperator{\Iso}{Iso} +\newcommand{\AA}{\mathcal{A}} +\newcommand{\MM}{\mathcal{M}} +\newcommand{\KKxn}{\KK[x_1, \ldots, x_n]} + +\let\ext\faktor +\newcommand{\quot}[1]{/{#1}} + +\newcommand{\Aa}{\mathcal{A}} +\newcommand{\Ad}[1]{\mathcal{A}_{#1}} +\DeclareMathOperator{\An}{\mathcal{A}_n} +\DeclareMathOperator{\AnK}{\mathcal{A}_n(\KK)} +\DeclareMathOperator{\Giac}{Giac} + +\DeclareMathOperator{\IC}{IC} +\DeclareMathOperator{\Aff}{Aff} +\DeclareMathOperator{\Orb}{Orb} +\DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\Gr}{Gr} +\newcommand{\vvec}[1]{\overrightarrow{#1}} + +\newcommand{\conj}[1]{\overline{#1}} + +\DeclareMathOperator{\PH}{PH} +\DeclareMathOperator{\PS}{PS} + +\let\imm\Im +\let\Im\undefined +\DeclareMathOperator{\Im}{Im} +\DeclareMathOperator{\Rad}{Rad} +\newcommand{\restr}[2]{ + #1\arrowvert_{#2} +} + +\newcommand{\innprod}[1]{\langle #1 \rangle} + +\newcommand{\zerovecset}{\{\vec 0\}} +\newcommand{\bigzero}{\mbox{0}} +\newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} + +\newcommand{\Idv}{\operatorname{Id}_V} +\newcommand{\Idw}{\operatorname{Id}_W} +\newcommand{\IdV}[1]{\operatorname{Id}_{#1}} +\DeclareMathOperator{\CI}{CI} +\DeclareMathOperator{\Bil}{Bil} +\DeclareMathOperator{\Mult}{Mult} +\DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\adj}{adj} +\DeclareMathOperator{\Cof}{Cof} +\DeclareMathOperator{\pr}{pr} +\DeclareMathOperator{\Sp}{sp} + +\newcommand{\dperp}{{\perp\perp}} + +\newcommand{\Eigsp}[0]{V_{\lambda}} +\newcommand{\Gensp}[0]{\widetilde{V_{\lambda}}} +\newcommand{\eigsp}[1]{V_{\lambda_{#1}}} +\newcommand{\gensp}[1]{\widetilde{V_{\lambda_{#1}}}} +\newcommand{\genspC}[1]{\widetilde{V_{#1}}} + +\DeclareMathOperator{\val}{val} +\DeclareMathOperator{\Span}{Span} +\newcommand{\charpoly}[1]{p_{#1}} +\newcommand{\charpolyrestr}[2]{p_{#1\arrowvert_#2}\hspace{-1pt}(\lambda)} +\newcommand{\minpoly}[1]{\varphi_{#1}} +\newcommand{\valf}{\val_f} +\newcommand{\valfv}{\val_{f,\V}} +\newcommand{\e}[1]{\vec{e_{#1}}} +\newcommand{\V}{\vec{v}} +\newcommand{\VV}[1]{\vec{v_{#1}}} +\newcommand{\basisdual}{\dual{\basis}} +\newcommand{\vecdual}[1]{\vec{\dual{#1}}} +\newcommand{\vecbidual}[1]{\vec{\bidual{#1}}} +\newcommand{\NMatrix}[1]{\begin{matrix} #1 \end{matrix}} +\newcommand{\Matrix}[1]{\begin{pmatrix} #1 \end{pmatrix}} +\newcommand{\Vector}[1]{\begin{pmatrix} #1 \end{pmatrix}} + +\DeclareMathOperator{\rg}{rg} + +\let\v\undefined +\newcommand{\v}{\vec{v}} +\newcommand{\vv}[1]{\vec{v_{#1}}} +\newcommand{\w}{\vec{w}} +\newcommand{\U}{\vec{u}} +\newcommand{\ww}[1]{\vec{w_{#1}}} +\newcommand{\uu}[1]{\vec{u_{#1}}} +\newcommand{\x}{\vec{x}} +\newcommand{\xx}[1]{\vec{x_{#1}}} +\newcommand{\y}{\vec{y}} +\newcommand{\yy}[1]{\vec{y_{#1}}} + +\newcommand{\mapstoby}[1]{\xmapsto{#1}} +\newcommand{\oplusperp}{\oplus^\perp} + +\newcommand{\tauindis}{\tau_{\text{indis}}} +\newcommand{\taudis}{\tau_{\text{dis}}} + +% Spesso utilizzati durante il corso di Algebra 1 + +\newcommand{\Det}[1]{\begin{vmatrix} + #1 +\end{vmatrix}} +\DeclareMathOperator{\disc}{disc} +\newcommand{\Frob}{\mathcal{F}} + +\newcommand{\mono}{\hookrightarrow} + +\newcommand{\pev}{\nu_p} +\newcommand{\exactdiv}{\mathrel\Vert} +\newcommand{\pset}{\mathcal{P}} + +\newcommand{\Dn}{D_n} +\newcommand{\Sn}{S_n} + +\newcommand{\mulgrp}[1]{\left(#1\right)^*} + +\newcommand{\ZZmulmod}[1]{\mulgrp{\ZZmod{#1}}} + +\newcommand{\bij}{\leftrightarrow} +\newcommand{\ZZpmod}[1]{\ZZ \quot {\left(#1\right)} \ZZ} +\newcommand{\ZZmod}[1]{\ZZ \quot #1 \ZZ} +\newcommand{\cleq}[1]{\overline{#1}} + +\newcommand{\rotations}{\mathcal{R}} +\newcommand{\gen}[1]{\langle #1 \rangle} +\DeclareMathOperator{\Cl}{Cl} + +\newcommand{\actson}{\circlearrowleft} +\newcommand{\Cyc}[1]{\left<#1\right>} + +% Comandi personali. + +\newcommand{\card}[1]{\left|#1\right|} +\newcommand{\nsqrt}[2]{\!\sqrt[#1]{#2}\,} +\newcommand{\zeroset}{\{0\}} +\newcommand{\setminuszero}{\setminus \{0\}} + +\newenvironment{solution} +{\textit{Soluzione.}\,} + +\theoremstyle{definition} + +\let\abstract\undefined +\let\endabstract\undefined + +\newtheorem*{abstract}{Abstract} +\newtheorem*{algorithm}{Algoritmo} +\newtheorem*{corollary}{Corollario} +\newtheorem*{definition}{Definizione} +\newtheorem*{example}{Esempio} +\newtheorem{exercise}{Esercizio} +\newtheorem*{notation}{Notazione} +\newtheorem{lemma}{Lemma} +\newtheorem*{nlemma}{Lemma} +\newtheorem*{note}{Nota} +\newtheorem*{remark}{Osservazione} +\newtheorem*{proposition}{Proposizione} +\newtheorem*{summary}{Sommario} +\newtheorem*{theorem}{Teorema} +\newtheorem*{scheme}{Schema della dimostrazione} + +\newcommand{\basis}{\mathcal{B}} +\newcommand{\BB}{\mathcal{B}} + +\newcommand{\basisC}{\mathcal{B}} + +\newcommand{\HH}{\mathbb{H}} + +\newcommand{\FFp}[1]{\mathbb{F}_p} +\newcommand{\FFpx}[1]{\mathbb{F}_p[x]} + +\newcommand{\CCx}{\mathbb{C}[x]} + +\newcommand{\KK}{\mathbb{K}} +\newcommand{\KKx}{\mathbb{K}[x]} + +\newcommand{\QQx}{\mathbb{Q}[x]} +\newcommand{\RRx}{\mathbb{R}[x]} + +\newcommand{\ZZi}{\mathbb{Z}[i]} +\newcommand{\ZZp}{\mathbb{Z}_p} +\newcommand{\ZZpx}{\mathbb{Z}_p[x]} +\newcommand{\ZZx}{\mathbb{Z}[x]} + +\newcommand{\ii}{\mathbf{i}} +\newcommand{\jj}{\mathbf{j}} +\newcommand{\kk}{\mathbf{k}} + +\newcommand{\bidual}[1]{#1^{**}} +\newcommand{\dual}[1]{#1^{*}} +\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} % L(V, W) +\newcommand{\Ll}{\mathcal{L}} + +\newcommand{\nsg}{\triangleleft} % sottogruppo normale proprio +\newcommand{\nsgeq}{\trianglelefteqslant} % sottogruppo normale + +% evan.sty original commands +\newcommand{\cbrt}[1]{\sqrt[3]{#1}} +\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} +\newcommand{\ceiling}[1]{\left\lceil #1 \right\rceil} +\newcommand{\mailto}[1]{\href{mailto:#1}{\texttt{#1}}} +\newcommand{\eps}{\varepsilon} +\newcommand{\vocab}[1]{\textbf{\color{blue}\sffamily #1}} +\providecommand{\alert}{\vocab} +\newcommand{\catname}{\mathsf} +\providecommand{\arc}[1]{\wideparen{#1}} + +% From H113 "Introduction to Abstract Algebra" at UC Berkeley +\newcommand{\CC}{\mathbb C} +\newcommand{\FF}{\mathbb F} +\newcommand{\NN}{\mathbb N} +\newcommand{\QQ}{\mathbb Q} +\newcommand{\RR}{\mathbb R} +\newcommand{\ZZ}{\mathbb Z} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} + +% From Kiran Kedlaya's "Geometry Unbound" +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +\newcommand{\norm}[1]{\left\lVert #1 \right\rVert} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} + +% From M275 "Topology" at SJSU +\newcommand{\Id}{\mathrm{Id}} +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} +\newcommand{\defiff}{\overset{\mathrm{def}}{\iff}} + +% From the USAMO .tex files +\newcommand{\dg}{^\circ} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} + +% Alcuni degli operatori più comunemente utilizzati. + +\DeclareMathOperator{\Char}{char} +\DeclareMathOperator{\Dom}{Dom} +\DeclareMathOperator{\Fix}{Fix} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\existsone}{\exists !} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Imm}{Imm} +\DeclareMathOperator{\Ker}{Ker} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\MCD}{MCD} +\DeclareMathOperator{\Mor}{Mor} +\DeclareMathOperator{\mcm}{mcm} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\tr}{tr} + +% Reimposta alcuni simboli presenti di default in LaTeX con degli analoghi +% più comuni. + +\let\oldemptyset\emptyset +\let\emptyset\varnothing + +% Trasforma alcuni simboli in operatori matematici. + +\let\oldcirc\circ +\let\circ\undefined +\DeclareMathOperator{\circ}{\oldcirc} + +\let\oldexists\exists +\let\exists\undefined +\DeclareMathOperator{\exists}{\oldexists} + +\let\oldforall\forall +\let\forall\undefined +\DeclareMathOperator{\forall}{\oldforall} + +\let\oldnexists\nexists +\let\nexists\undefined +\DeclareMathOperator{\nexists}{\oldnexists} + +\let\oldland\land +\let\land\undefined +\DeclareMathOperator{\land}{\oldland} + +\let\oldlnot\lnot +\let\lnot\undefined +\DeclareMathOperator{\lnot}{\oldlnot} + +\let\oldlor\lor +\let\lor\undefined +\DeclareMathOperator{\lor}{\oldlor} + +\DeclareOption{counter}{ + \let\algorithm\@undefined + \let\endalgorithm\@undefined + \let\corollary\@undefined + \let\endcorollary\@undefined + \let\c@lemma\@undefined + \let\lemma\@undefined + \let\endlemma\@undefined + \let\proposition\@undefined + \let\endproposition\@undefined + \let\theorem\@undefined + \let\endtheorem\@undefined + + \newtheorem{algorithm}{Algoritmo}[chapter] + \newtheorem{corollary}{Corollario}[chapter] + \newtheorem{lemma}{Lemma}[chapter] + \newtheorem{proposition}{Proposizione}[chapter] + \newtheorem{theorem}{Teorema}[chapter] + \newtheorem{notation}{Notazione}[chapter] +} + +\newcommand{\pp}{\mathtt{p}} +\newcommand{\fl}{\mathtt{fl}} +\newcommand{\op}{\mathtt{op}} +\newcommand{\bop}{\mathbin{\op}} +\DeclareMathOperator{\argmax}{argmax} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\triu}{triu} +\DeclareMathOperator{\tril}{tril} +\newcommand{\innp}[1]{\langle #1 \rangle} + +\ProcessOptions\relax + +\author{di Gabriel Antonio Videtta} +\date{\vspace{-0.5cm}} diff --git a/Secondo anno/Analisi numerica/quiver.sty b/Secondo anno/Analisi numerica/quiver.sty new file mode 100644 index 0000000..d1d7d07 --- /dev/null +++ b/Secondo anno/Analisi numerica/quiver.sty @@ -0,0 +1,40 @@ +% *** quiver *** +% A package for drawing commutative diagrams exported from https://q.uiver.app. +% +% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ +% libraries, and defining a new TikZ style for curves of a fixed height. +% +% Version: 1.3.0 +% Authors: +% - varkor (https://github.com/varkor) +% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c) + +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{quiver}[2021/01/11 quiver] + +% `tikz-cd` is necessary to draw commutative diagrams. +\RequirePackage{tikz-cd} +% `amssymb` is necessary for `\lrcorner` and `\ulcorner`. +\RequirePackage{amssymb} +% `calc` is necessary to draw curved arrows. +\usetikzlibrary{calc} +% `pathmorphing` is necessary to draw squiggly arrows. +\usetikzlibrary{decorations.pathmorphing} + +% A TikZ style for curved arrows of a fixed height, due to AndréC. +\tikzset{curve/.style={settings={#1},to path={(\tikztostart) + .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + .. (\tikztotarget)\tikztonodes}}, + settings/.code={\tikzset{quiver/.cd,#1} + \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}}, + quiver/.cd,pos/.initial=0.35,height/.initial=0} + +% TikZ arrowhead/tail styles. +\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}} +\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}} +\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}} +% TikZ arrow styles. +\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}} + +\endinput