%        File: formulario.tex
%     Created: Mon Jun 23 09:00 PM 2008 C
% Last Change: Mon Jun 23 09:00 PM 2008 C
%
\documentclass[a4paper,landscape,10pt]{article}
\usepackage{multicol}
\usepackage{amath}
%\setlength{\oddsidemargin}{1pt}
\usepackage[in,empty]{fullpage}


\begin{document}
\def\R{{\mathbb{R}}}
\def\rd{{\mathbb{R}^2}}
\def\C{{\mathbb{C}}}
\def\V{{\textbf{V}}}
\def\K{{\mathbb{K}}}
\def\.{{\cdot}}
\def\Sym{{\operatorname{Sym}}}
\def\rn{{\R^n}}
\def\Let{{\textbf{Sia} }\;}
\def\Sia{\Let}
\def\Se{{\textbf{Se }\;}}
\def\While{{\textbf{While }\;}}
\newcommand{\Ll}[1]{\mathcal{L}\pton{#1}}


\begin{footnotesize}
\begin{multicols*}{4}
\textbf{Sistemi lineari}\\
\begin{enumerate}
	\item Norme:
		\eal{&
		\rho(A)=\lambda_M(A)=\max\{|\lambda| \;|\;\lambda\T{
		autoval }A\}\NN
		\norm{A}_{\8}=\max_{1\le i\le n}\sumjm|a_{ij}|\NN
		\norm{A}_1=\norm{\TR
A}_{\8}\NN
		\norm{A}_2 =\sqrt{\lambda_M(\TR AA)}\NN
		\norm{\inv A}_2= \frac{1}{\sqrt{\lambda_m(\TR A A)}}\NN
		A=\TR A \> \norm{A}_2 = |\lambda_M(A)|\NN
		A=\TR A \> \norm{\inv
		A}_2=\frac{1}{|\lambda_m(A)|}\NN
		\rho(A)=\inf_{p,q\in \mathbb{N}^*} \norm{A}_{p,q}
		}
	\item Triangolare superiore:
		\eal{&
 x_i = \frac{b_i - \sum_{j=i+1}^n a_{ij}x_j}{a_{ii}} \;\forall
 i=n,..,1\NN
\T{Triangolare inferiore: }\NN
	x_i = \frac{b_i - \sum_{j=1}^{i-1} a_{ij}x_j}{a_{ii}} \;\forall
	i=1,..,n
		}
	\item Condizionamento:
		\def\D{{\Delta}}
		\eal{&A(x+\D x)= b+\D b\NN
		\frac{\norm{\D x}}{\norm x}\le
		\mu(A)\frac{\norm{\D b}}{\norm{b}}\NN
		\mu_2(A)=\frac{\sqrt{\lambda_M(\TR AA)}}{\sqrt{\lambda_m(\TR A A)}}\NN
A=\TR A \> \mu_2(A)=\frac{|\lambda_M|}{|\lambda_m|}
		}
	\item Gauss (pivot):
	\eal{&
 \oname{pivot\_riga} k:\\
	&\qq \Let \ove i \T{ t.c. }\;\;|a_{\ove i k}|:=\max_{i:=k,..,n}|a_{ik}|\\
	&\qq \oname{scambio\_righe} \ove i, k\\
& \oname{pivot\_scalato} k:\\
	&\qq d_i:=\frac{|a_{ik}|}{\max_{j=k,..,n}|a_{ij}|}\\
	&\qq \Let \ove i \T{ t.c. }\;d_{\ove i}=\max_{i=1,..,n}d_i\\
&\forall k=1,..,n-1\\
	&\qq \oname{pivot} k\\
&\qq \forall i=k+1,..,n \\
&\qq \qq a_{ik}:=\frac{a_{ik}}{a_{kk}}\\
&\qq\qq \forall j=k+1,..,n+1\\
&\qq\qq\qq a_{ij} := a_{ij} -a_{ik}a_{kj}
	}
	\item Fattorizzazione Cholesky
		\eal{&A\in \R^{n\*n}, \T{ simmetrica, definita pos.}\NN
		A=R\TR R,\;\;R\in \R^{n\*n}\T{ triang. inf.}\NN
		a_{ij}= \sum_{k=1}^{\min\pgra{i,j}}R_{ik}R_{jk}\;\forall
		i,j\NN
	\forall j=1,\dots,n\\
	&\qq r_{jj}:=\sqrt{a_{jj}-\sum_{k=1}^{j-1}r_{jk}^2 }\\
	&\qq \forall i=j+1,\dots,n\\
	&\qq\qq r_{ij} := \frac{a_{ij} - \sum_{k=1}^{j-1}r_{ik}r_{jk}
	}{r_{jj}}
		}
%\tiny{
%\begin{verbatim}
%1.#.#.
%2.3.#.
%2.4.5.
%2.4.6.
%	\end{verbatim}
%	}
	\item Gershgorin-Hadamard
		\eal{&
		A\in C^{n\*n}\NN
		D_i=\pgra{z\in \C\;|\;|z-a_{ii}|\le R_i}\NN
R_i= \sum_{j \neq i}|a_{ij}|\NN
\pgra{\T{autoval. di $A$ }} \( \bigcup_{i=1}^nD_i
		}
	\item Metodo delle potenze:
		\def\l{{\lambda}}
		\def\s{{\sigma}}
\eal{&
|\l_1| > |\l_2| \ge |\l_3| \ge .. \ge  |\l_n|\NN
x_0\in \C^{n,1},\;\;x_0 \neq 0\\
&y_k=\frac{x_k}{\norm{x_k}},\;\;x_{k+1}=Ay_k\NN
\s_k=\frac{\TR x_kAx_k}{\TR x_k x_k} = \frac{\TR y_kx_{k+1}}{\TR y_ky_k}\NN
z_0=x_0,\;z_{k+1}=Az_k,\;\s_k'=\frac{\TR z_kA z_k}{\TR z_kz_k}\NN
\lim_{k\to \8} \s_k = \lim_{k\to \8} \s_k'= \l_1\NN\NN
\T{Crit. arresto: }\frac{\norm{(A-\s_kI)y_k}}{\s_k\norm {y_k}} < \eps\NN
\T{oppure }|\s_{k+1}-\s_k|<\eps 
}
\end{enumerate}

\textbf{Interpolazione}\\
\begin{enumerate}
	\item Lagrange:
		\def\l{{\lambda}}
	\eal{
	&
	\l_j(x) =\prod_{0\le i\le n,\;\;i \neq j} \frac{x-x_i}{x_j-x_i}\\
	&
	p(x) =\sumzjn y_j\l_j(x)\NN
	\sumzjn \lambda_j(x)(x_j-x)^k\equiv\sist{1 & k=0\\
	0 & k > 0}\NN
	\l_j(x_i)=\delta_{ij}
	}
	\item Differenze divise:
\eal{&f[x_0]=f(x_0)\\
&f[x_0,..,x_j]=\NN
=\frac{f[x_1,..,x_j]-f[x_{0},..,x_{j-1}]}{x_j-x_0}
}
\begin{tiny}
	\eal{&
\begin{matrix}
f[x_0]=f(x_0)   &            &               & \\
        &       & f[x_0,x_1] &               & \\
f[x_1]=f(x_1) & &            & f[x_0,x_1,x_2]& \\
        &       & f[x_1,x_2] &               & \\
f[x_2]=f(x_2) & &           &               & 
\end{matrix}
	}
\end{tiny}
	\item Newton:
	\eal{
&w_0(x)=1,\; w_j(x)=\prod_{i=0}^{j-1}(x-x_i)\\
&p_{n}(x)=\sum_{i=0}^{n}f[x_0,\dots,x_i]w_i(x)\\
&R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\omega_{n+1}(x)
		}
	\item Chebyshev
		\def\f{{\varphi}}
	\eal{&t_n(x)=\cos n \f(x):[-1,1]\longrightarrow \R\\
	&\f(x)=\arcos x:[-1,1]\longrightarrow [0,\pi]\\
	&t_0(x)=1,\; t_1(x)=x\\
	&t_n(x)=2t_{n-1}(x)x -t_{n-2}(x)\;\;\forall n>1\\
	&\deg t_n(x) =n\;\;\forall n\ge 0\NN
	c_n=2^{n-1}\;\;\forall n\ge 1\\
	&\T{radici di $t_n$:} \NN
	\alpha_k=\cos\pton{\frac{\pi}{2n}+k\frac{\pi}{n}}\;\;k=0,..,n-1\NN
	\T{max: }M_k=\cos\pton{k\frac{2\pi}{n}}\;\;k=0,..,n\NN
	\T{min:
	}m_k=\cos\pton{k\frac{2\pi}{n}+\frac{\pi}{n}}\;\;k=0,..,n
	}
\item 
	Polinomio di Bernstein:
	\eal{&B_{n,k}(x)= \binom{n}{k} x^k(1-x)^{n-k}\NN
	\sum_{k=0}^n B_{n,k}(x) =1\NN
	B_n(f)(x)=\sumzkn f\pton{\frac{k}{n}} B_{n,k}(x)\NN
	\ulim{B_n(f)}{f}{[0,1]}\NN
	\T{ dove }
	 f\in
	 \sczn{0}{[0,1]}}
 \item Minimi quadrati
	 \def\f{{\varphi}} 
	 \def\F{{\phi}}
	 \def\a{\textbf{a}}
	 \def\y{\textbf{y}}
	 \def\X{\textbf{v}}
	 \eal{&\f(x,\a)=\sumjm a_j\f_{j}(x),\;\; \F\in  \R^{n\*m}\NN
	 \F_{jk}=\f_k(x_j),\;\;\TR \F \F \a =\TR \F \y,\;\;\det \TR \F \F
	 \neq 0\NN
	 \eps(\X)_i = y_i- \f(x_i,\X),\;\;\norm{\eps(\a)}_2^2\T{ minima}
	 }
\end{enumerate}
\textbf{Zeri e punti fissi}\\
\begin{enumerate}
	\item
	\eal{&\T{Newton: }x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)}\NN
		\T{Secanti: }\NN
		x_{k+1}=x_k-f(x_k)\frac{x_k-x_{k-1}}{f(x_k)-f(x_{k-1})}\NN
	\T{Bisezione:} \\
	&\While:\\
	&\qq c:=\frac{a+b}{2}\\
	&\qq \Se |b-a| < 2\eps  \| f(c)=0:\NN
	\qq \qq \textbf{break}\\
	&\qq \Se f(c)f(a)>0:\qq  a:=c\\
	&\qq \Se f(c)f(a)<0: \qq b:=c\\
	&\alpha:=c
	}
\item Criterio di arresto di Aitken per successioni lineari:
	\def\f{{\varphi}}
	\def\a{{\alpha}}
	\eal{&\T{Hp: } x_n=\f(x_{n-1}) \to \a=\f(\a)\NN
	 \T{Hp: }\f'(\a) \neq 0,\;x_{n} \neq
	\a\;\;\forall n\\
	&\pass{x_n- \frac{x_{n+1}x_{n-1}-x_{n}^2}{x_{n+1}+x_{n-1}-2x_n}
} <\eps 
	}
\item Minimo di $f$ unimodale:
	\eal{&
	\While\\
&\qq x_1:=a+(1-r)(b-a)\\
&\qq x_2:=a+r(b-a)\\
&\qq \Se f(x_1)>f(x_2): \qq a:=x_1\\
&\qq \textbf{Altrimenti: }\qq b:=x_2\\
&\qq \Se b-a < \eps: \\
&\qq \qq x_*:=a\\
&\qq \qq \textbf{break}
	}
\end{enumerate}
\textbf{Integrali}\\
	\begin{enumerate}
		\item Norma
		\eal{&\norm{I} = \intd{a}{b} w(x)\;dx,\;\;\norm{I_n}=\sumzin |w_i|		}
	\item Formula interpolatoria $I_n$:
	\eal{&\deg I_n \ge n\\
	&|I(f)-I_n(f)| \le
\norm{f-L_n}   
\intd{a}{b}w(x)\;dx\NN
w_j=\intd{a}{b} w(x)\lambda_j(x)\;dx\;\;\forall j=0,..,n\\
&\T{Sistema per il calcolo dei pesi: }\NN
\intd{a}{b} w(x)x^i\;dx =\sumzjn w_jx_j^i\;\;\forall i=0,..,n
}
	\item Cambio di pesi e nodi:
		\def\a{{\alpha}}
		\def\b{{\beta}}
	\eal{&x_j,w_j\T{ su }[\alpha,\beta],\;x_j',w_j'\T{ su }[a,b]\\
&w_j'=w_j \frac{b-a}{\b-\a}\\
&x_j'=\frac{b-a}{\b-\a}(x_j-\a)+a}
	\item Newton-Cotes: formula interpolatoria su $n+1$ nodi
			uniformi.
	\eal{
	&\T{\textbf{Chiusa}}\NN
	x_0=a,\;x_n=b,\;h=\frac{b-a}{n},\;\NN
	x_i=a+ih\;\;i=0,..,n\\
	&\T{\textbf{Aperta}}\NN
	h=\frac{b-a}{n+2},\;x_i=a+(i+1)h
	\forall i=0,..,n\NN
	\deg I_n = \sist{ n &n\T{ dispari}\\
	n+1&n \T{ pari}}
	}
\begin{tiny}
	Resto:\\
\begin{tabular}{rcc}
n & \T{Chiusa } & \T{Aperta}\\
\hline
\strut{6} 0 &  &  $\frac{1}{3} h^3 f^{(2)}(\xi)$ \\
\strut{6}
\rule{0pt}{4ex} 1 & $-\frac{1}{12} h^3 f^{(2)}(\xi)$ &  $\frac{3}{4}h^3 f^{(2)}(\xi)$ \\
\strut{6}
2 & $-\frac{1}{90} h^5 f^{(4)}(\xi)$ &  $\frac{14}{45}h^5 f^{(4)}(\xi)$ \\
\strut{6} 3 & $-\frac{3}{80} h^5 f^{(4)}(\xi)$ &  $\frac{95}{144}h^5 f^{(4)}(\xi)$ \\
\strut{6}
 4 & $-\frac{8}{945} h^7 f^{(6)}(\xi)$ & $\frac{41}{140}h^7 f^{(6)}(\xi)$ 
\end{tabular}
\end{tiny}

		\item Formula dei trapezi
	\eal{&T(f)\definizione{=}\frac{f(a)+f(b)}{2}(b-a)\\
	&w(x)\equiv 1,\;\deg T =1\\
	&\T{Resto: }-\frac{(b-a)^3}{12}f''(\xi),\;\;\xi\in [a,b]
	}
	Composita, con distibuzione uniforme dei nodi:
		\eal{&
C_{N}(f)= h\pton{\frac{f(a)+f(b)}{2} + \sum_{i=1}^{N-1} f(a+ih)}\\
&x_{i}-x_{i-1}=h=\frac{b-a}{N}\\
&\T{Resto: }-\frac{b-a}{12}h^2f''(\xi)\\
&\T{Resto di Eulero-MacLaurin: }\\
&\qq c_1h^2 + c_2h^4 +\dots + c_kh^{2k}+ \dots
		}
	\item Cavalieri-Simpson (formula interpolatoria)
	 \eal{
	 &\T{Nodi }x_0=a, x_1=\frac{a+b}{2}, x_2=b\\
	 &S(f) = \frac{b-a}{6}\pqua{f(a)+4f\pton{\frac{a+b}{2}}+f(b)}\\
	 &w(x)\equiv 1,\;\deg S =2\\
	 &\T{Resto formula composita: }-\frac{b-a}{2880}h^4f^{(4)}(\xi)\\
	 &\T{con }x_{i}-x_{i-1}=h=\frac{b-a}{N}
	 }
 \item Gauss-Chebyshev:
	 \eal{&w(x)\equiv \frac{1}{\sqrt{1-x^2}},\;[a,b]=[-1,1],\;\deg
	 I_n=2n+1\\
	 &w_i=\frac{\pi}{n+1}\;\;\forall i=0,\dots,n\\
	 &x_i=\cos\pqua{\frac{1}{n+1}\pton{\frac{\pi}{2}+i\pi}}\;\;i=0,\dots,n\\
	 }
 \item Estrapolazione di Richardson:
	\eal{
&F_1(f,h)=I_n(f,h),\qq h<1\\
&F_{k+1}(f,h)=\frac{q^{p_k}F_k(f,h)-F_k(f,qh)}{q^{p_k}-1}\\
&\T{Per la formula dei trapezi: }p_k=2k
	 }
 \item Integrazione adattiva:
\def\a{{\alpha}}
\def\b{{\beta}}
\eal{&\oname{IntAdattiva}(\a,\b):\;\;\\
&\qq M=:I_n(\a,\b) \\
&\qq N=:I_n'(\a,\b)\\
&\qq \Se  |M-N| < \frac{\eps}{b-a} (\b-\a)\\
&\qq \qq I=: N\\
&\qq \textbf{Altrimenti}:\\
&\qq \qq I=: \oname{IntAdattiva}\pton{\a,\frac{\a+\b}{2}} +\\
&\qq\qq\qq\qq +
\oname{IntAdattiva}\pton{\frac{\a+\b}{2},\b}\;\;\\
&\qq I=: \oname{IntAdattiva}(a,b)
}

	\end{enumerate}


\end{multicols*}
\end{footnotesize}
\end{document}