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zBz=QKC@E6B9C8!#L5eZCY99+LX{~Fo%|*an>(MDyw_q2;Oya5rLbg-7%G@#UGc$7kCF_LwknAuliS0Zty=?;V=+|DNUap0b zqqsB5pmMCoi<|6hteL9_&YX##^?$dSKwfKF;CDVus@YnOpNDFd?J6D50^?taPyf%) dJSTvmBf!=1>$8rNjfIVc3x{\begin{block}{} +$\delta=\B{Ax-b}$. +Приближенные методы: $\mathrm{min}(\delta)$. Точные методы: $\delta=0$.\\ +\end{block} +\begin{block}{Матричный метод} +$\B x = \B A^{-1}\B b$\\ +$\B A \cdot \B A^{-1} = \B A^{-1} \cdot \B A = \B E$. +Нахождение обратной матрицы: +\begin{itemize} +\item с помощью присоединенной: $(\B A | \B E )$ \so $(\B E | \B A^{-1})$; +\item $\B A^{-1} = \dfrac{\mathrm{adj\,}\B A}{|\B A|}$, присоединенная матрица $\mathrm{adj\,}\B A$ +является транспонированной матрицей алгебраических дополнений ($(-1)^{i+j}M_{ij}$, $M_ij$~-- +соответствующий дополнительный минор~--- определитель матрицы с вычеркнутыми $i$-й строкой и $j$-м +столбцом). +\item и т.д., и т.п. +\end{itemize} +Формулы Крамера: $x_j = |A_j|/|A|$, $A_j$ получается из $A$ заменой $j$-го столбца на $\B b$. +\end{block} +} +\only<2>{ +\begin{block}{Метод Гаусса} +$$ +\B A_d\B{x} = \pmb\beta,\quad +\B A_d=\begin{pmatrix} +\alpha_{11}&\alpha_{12}&\alpha_{13}&\cdots&\alpha_{1m}\\ +0&\alpha_{22}&\alpha_{23}&\cdots&\alpha_{2m}\\ +\cdot&\cdot&\cdot&\cdots&\cdot\\ +0&0&0&\cdots&\alpha_{mm} +\end{pmatrix}. +$$ +Прямой ход~--- преобразование к диагональной форме: +$$ +\left(\begin{matrix} +\alpha_{11}&\alpha_{12}&\alpha_{13}&\cdots&\alpha_{1m}\\ +0&\alpha_{22}&\alpha_{23}&\cdots&\alpha_{2m}\\ +\cdot&\cdot&\cdot&\cdots&\cdot\\ +0&0&0&\cdots&\alpha_{mm} +\end{matrix}\middle| +\begin{matrix}\beta_1\\\beta_2\\\cdot\\\beta_m\end{matrix}\right). +$$ +Обратный ход~--- последовательное нахождение $x_m$, $x_{m-1}$, \dots, $x_1$. + +$N\propto n^3$~--- прямой, $N\propto n^2$~--- обратный ход. +\end{block} +} +\only<3>{ +\begin{block}{Метод Зейделя} +$$\B{Bx}_{n+1}+\B{Cx}_n=\B b,$$ +где +$$\B B=\begin{pmatrix} +a_{11}&0&0&\cdots&0\\ +a_{21}&a_{22}&0&\cdots&0\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +a_{m1}&a_{m2}&a_{m3}&\cdots&a_{mm} +\end{pmatrix},\qquad +\B C=\begin{pmatrix} +0&a_{12}&a_{13}&\cdots&a_{1m}\\ +0&0&a_{23}&\cdots&a_{2m}\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\cdots&0 +\end{pmatrix}. +$$ +Отсюда получаем +$$\B x_{n+1}=-\B B^{-1}\B{Cx}_n +\B B^{-1}\B b.$$ +\end{block} +} +\only<4>{ +\begin{block}{LU-метод} +$$\B A=\B L\cdot \B U,$$ +где +$$\B L=\begin{pmatrix} +l_{11}&0&0&\cdots&0\\ +l_{21}&l_{22}&0&\cdots&0\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +l_{m1}&l_{m2}&l_{m3}&\cdots&l_{mm} +\end{pmatrix},\qquad +\B U=\begin{pmatrix} +1&u_{12}&u_{13}&\cdots&u_{1m}\\ +0&1&u_{23}&\cdots&u_{2m}\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\cdots&1 +\end{pmatrix}. +$$ +Прямой ход: $\B L\cdot \B U\cdot \B x \equiv \B L\cdot\B y= \B b$, находим $\B y$, +из $\B U\cdot \B x =\B y$ находим $\B x$. +$$\begin{cases} +l_{ij}=a_{ij}-\Sum_{s=1}^{j-1}l_{is}u_{sj},& i\ge j;\\ +u_{ij}=\frac1{l_{ii}}\Bigl(a_{ij}-\Sum_{s=1}^{j-1}l_{is}u_{sj}\Bigr), & i < j. +\end{cases} +$$ +LU-разложение возможно для матриц с преобладанием диагональных элементов +\end{block} +} +\only<5>{ +\begin{block}{Разложение Холецкого} +$\B A=\B L\cdot \B L^T$, либо $\B A=\B U^T\cdot \B U$, где $\B L$~-- нижняя треугольная матрица +со строго положительными элементами на диагонали, $\B U$~-- верхняя треугольная матрица. +Разложение Холецкого всегда существует и единственно для любой симметричной (относительно главной +диагонали) положительно-определенной матрицы (все диагональные миноры положительны). + +$$\begin{cases} +l_{ii}=\sqrt{a_{ii} - \Sum_{s=1}^{i-1}l^2_{is}}; \\ +l_{ij} = \frac1{l_{ii}}\Bigl( a_{ij} - \Sum_{s=1}^{j-1}l_{is}l_{js}\Bigr), & j < i. +\end{cases} +$$ + +Прямой и обратный ходы аналогичны LU-разложению. +\end{block} +} +\only<6>{ +\begin{block}{} +Если $\B A$ содержит~$m$ строк и~$n$ столбцов, то: +\begin{description} +\item[$m=n$] квадратная матрица, возможно существование точного решения; +\item[$mn$] переопределенная система, приближенное решение которой находится +при помощи метода наименьших квадратов (в случае линейной зависимости строк +данной системы может существовать и точное решение). +\end{description} +\end{block} +} +\only<7>{ +\begin{block}{Метод наименьших квадратов ($m > n$)} +$\B{x=A\backslash b}$, %), псевдообратная матрица, \dots +$S=\sum_i(\sum_j a_{ij}x_j - b_i)^2$, $\partder{S}{x_j}=0$ \so +$\B C\B x = \B d$, где +$c_{kj} = \sum_i a_{ik}a_{ij}$, $k,j=\overline{1,n}$, $d_k = \sum_i a_{ik}b_i$. +Т.о. $\B C = \B A^T\cdot \B A$, $\B d=\B A^T\B b$. +\end{block} +\begin{block}{Метод простой итерации} +$\B{x=Bx+c}$, $\B x_{n+1}=\B B\B x_n+\B c$.\\ +Сложностью метода простой итерации при решении матриц больших размерностей является особое свойство +таких матриц~--- существование\к почти собственных значений\н, $\lambda$: +$||\B{Ax}-\lambda\B x||\le\epsilon||\B x||$ при $||\B x||\ne0$.\\ +\end{block} +} +\end{frame} + +\begin{frame}{Число обусловленности матрицы} +\begin{block}{Оценка ошибки решения} +Пусть $\B x'$~-- приближенное решение. Абсолютная и относительная ошибки: $||\B x-\B x'||$ +и $\frc{||\B x-\B x'||}{||\B x||}$. Нам известна невязка $\B r=\B b-A\B x'$: +$$\B r=A\B x-A\B x'=A(\B x-\B x')\so ||\B x-\B x'||=||A^{-1}\B r||\le ||A^{-1}||\,||\B r||,$$ +а т.к. $||\B b||\le||A||\,||\B x||$, $\frc{1}{||\B x||}\le\frc{||A||}{||\B b||}$: +$$\frac{||\B x-\B x'||}{||\B x||}\le||A^{-1}||\,||\B r||\,\frac{||A||}{||\B b||}=k(A)\frac{||\B +r||}{||\B b||}.$$ +\ж Число обусловленности\н: $k(A)=||A||\,||A^{-1}||$. Чем оно больше, тем больше флуктуации $\B x$ +влияют на общее решение. У хорошо обусловленных матриц $K(A)\equiv1$ (напр., ортогональные матрицы, +у которых $A^T=A^{-1}$). +\end{block} +\end{frame} + +\section{Степенные уравнения} +\begin{frame}{Степенные уравнения} +\begin{defin} +\ж Степенное уравнение\н имеет вид $p_n(x)=0$, где $p_n(x)$~-- полином~$n$~-й степени вида +$p_n(x)=\sum_{i=0}^n C_nx^n$. +\end{defin} +\begin{block}{Методы решения} +Точные~--- до третьей степени включительно (в общем случае) и итерационные: +\begin{description} +\item[бисекция] деление пополам отрезка, где находится корень; +\item[метод хорд] замена полинома хордой, проходящей через точки $(x_1, p_n(x_1)$ и $(x_2, +p_n(x_2)$; +\item[метод Ньютона] имеет быструю сходимость, но требует знакопостоянства $f'(x)$ и $f''(x)$ на +выбранном интервале $(x_1, x_2)$. +\end{description} +\end{block} +\end{frame} + +\begin{frame}{Бисекция (дихотомия)} +\img{bisect} +Отрезок делится вплоть до заданной точности $b_n-a_n\le\epsilon$, корень $x\approx(b_n+a_n)/2$. + +Применяется и для поиска значений в упорядоченном ряду. +\end{frame} + +\begin{frame}{Метод хорд (секущих)} +\begin{block}{} +$$x_{i+1}=x_{i-1}+\frac{y_{i-1}\cdot(x_i-x_{i-1})}{y_i-y_{i-1}}.$$ +\end{block} +\begin{pict}\smimg[0.5]{chords1}\,\smimg[0.5]{chords2}\end{pict} +\end{frame} + +\begin{blueframe}{Метод Ньютона} +\begin{block}{} +$$x_{i+1}=x_i+\frac{y_i}{y'_i}.$$ +\end{block} +\begin{pict}\smimg[0.5]{newton1}\,\smimg[0.5]{newton2}\end{pict} +\end{blueframe} + +\section{Численное интегрирование и дифференцирование} +\begin{frame}{Численное интегрирование} +\begin{block}{} +Для численного решения уравнения $\displaystyle I=\Int_a^b f(x)\,dx$ наиболее популярны: +\begin{description} +\item[метод прямоугольников] $I\approx\sum_{i=1}^n f(x_i)[x_i-x_{i-1}]$; +\item[метод трапеций] $I\approx\sum_{i=1}^n \frac{f(x_{i-1})+f(x_i)}{2}[x_i-x_{i-1}]$; +\item[метод Симпсона] $\Int_{-1}^1 f(x)\,dx\approx\frac13\bigl(f(-1)+4f(0)+f(1)\bigr)$ \so +$I\approx\frac{b-a}{6n}\Bigl(f(x_0)+f(x_n)+2\sum_{i=1}^{n/2-1} +f(x_{2i}) + 4\sum_{i=1}^{n/2}f(x_{2i-1})\Bigr)$. +\end{description} +и многие другие. +\end{block} +\end{frame} + +\begin{frame}{Метод прямоугольников} +\begin{columns} +\column{0.45\textwidth} +\begin{block}{} +$$\int_a^b f(x)\,dx\approx$$ +\begin{list}{}{} +\item $\displaystyle\sum_{i=0}^{n-1}f(x_{i})(x_{i+1}-x_{i})$; +\item $\displaystyle\sum_{i=1}^{n}f(x_{i})(x_i-x_{i-1})$; +\item $\displaystyle\sum_{i=0}^{n-1}f\bigl(\frac{x_{i}+x_{i+1}}{2}\bigr)(x_{i+1}-x_{i})$. +\end{list} +Для равномерных сеток: + +$\displaystyle h\sum_{i=0}^{n-1} f_i$; +$\displaystyle h\sum_{i=1}^{n} f_i$; +$\displaystyle h\bigl(\sum_{i=1}^{n-1} f_i + \frac{f_0+f_n}{2}\bigr)$. + +\end{block} +\column{0.45\textwidth} +\img{rectangmeth} +\end{columns} +\end{frame} + +\begin{lightframe}{Метод трапеций} +\begin{columns} +\column{0.6\textwidth} +\begin{block}{} +$$\int_a^b f(x)\,dx\approx +\sum_{i=0}^{n-1}\frac{f(x_i)+f(x_{i+1})}{2}(x_{i+1}-x_i).$$ + +Для равномерных сеток~--- формула Котеса: +$$\int_a^b f(x)\,dx = +h\left(\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right) + E_n(f),$$ +$$E_n(f)=-\frac{f''(\xi)}{12}(b-a)h^2, \xi\in[a,b].$$ +\end{block} +\column{0.4\textwidth} +\img{trapezmeth} +\end{columns} +\end{lightframe} + +\begin{blueframe}{Метод Симпсона} +\begin{columns} +\column{0.6\textwidth} +\begin{block}{} +$$\int_a^b f(x)\,dx\approx \frac{b-a}{6}\left(f(a)+4f\bigl(\frac{a+b}{2}\bigr)\right)$$ +\end{block} +\column{0.4\textwidth} +\img{Simpsons_method_illustration} +\end{columns} +\begin{block}{} +Формула Котеса: +$$I\approx \frac{h}{3}\Bigl(f(x_0)+ +2\sum_{i=1}^{N/2-1}f(x_{2i}) + 4\sum_{i=1}^{N/2}f(x_{2i-1} + f(x_N)\Bigr).$$ + +\end{block} +\end{blueframe} + + +\begin{frame}{Численное дифференцирование} +\begin{block}{} +Аппроксимация функции интерполяционным многочленом (Ньютона, Стирлинга и т.п.), разделенные +разности. +\end{block} +\begin{block}{Аппроксимация многочленом} +$$f(x)\approx P_N(x)\quad\Arr\quad f^{(r)}(x)\approx P_N^{(r)}(x).$$ +Полином Ньютона: +$$P_N(x)=\sum_{m=0}^{N}C_x^m\sum _{k=0}^{m}(-1)^{m-k}\,C_m^k\,f(k).$$ +Полином Лагранжа: +$$P_N(x) = \sum_{k=0}^N y_k \frac {(x-x_0)(x-x_1) \ldots (x-x_{k-1})(x-x_{k+1}) \ldots (x-x_n)} {(x_k-x_0)(x_k-x_1) \ldots +(x_k-x_{k-1})(x_k-x_{k+1}) \ldots (x_k-x_n)}.$$ +А также: интерполяция кубическими сплайнами, разложение по базису тригонометрических функций и т.п. +\end{block} +\end{frame} + +\begin{frame}{} +\begin{block}{Разделенные разности} + В нулевом приближении можно заменить производную $f^{(n)}$ разделенной разностью $n$-го порядка. +$$f(x_{0};\;x_{1};\;\ldots ;\;x_{n})=\sum _{j=0}^{n}{\frac {f(x_j)}{\prod \limits _{i=0 \atop i\neq j}^{n}(x_{j}-x_{i})}}.$$ +В частности: +$$f(x_0;\;x_1)={\frac {f(x_1)}{x_1-x_0}}+{\frac {f(x_0)}{x_0-x_1}},$$ +$$f(x_0;\;x_1;\;x_2)={\frac {f(x_2)}{(x_2-x_1)(x_2-x_0)}}+{\frac {f(x_1)}{(x_1-x_2)(x_1-x_0)}}+{\frac +{f(x_0)}{(x_0-x_2)(x_0-x_1)}}\ldots$$ +\end{block}\end{frame} + +\section{Дифференциальные уравнения} +\begin{frame}{Дифференциальные уравнения} +\only<1>{ +\begin{defin} +\ж Обыкновенные дифференциальные уравнения\н~(ОДУ) порядка~$n$ задаются в виде +функции $f(x,y,y',\ldots,y^{(n)})=0$. +\end{defin} +\begin{block}{} +Разделение переменных:\vspace{-2em} +$$y'=f(x,y) \so \phi(y)\,dy=\psi(x)\,dx \so y=y_0+\Int_0^{x}\psi(x)\,dx.$$ + +ОДУ второго порядка: +$$Ay''+By'+Cy+Dx=0.$$ +Если $D\equiv0$, а множители $A$, $B$ и~$C$~--- константы, имеем однородное ОДУ. +$y=\C_1\exp(k_1x)+\C_2\exp(k_2x)$, где~$k_1$ и~$k_2$~-- корни\к +характеристического уравнения\н $Ak^2+Bk+C=0$. +\end{block}} +\only<2>{ +\begin{defin} +\ж Дифференциальные уравнения в частных производных\н~(ЧДУ) для функции +$y=y(x_1,x_2,\cdots,x_n)$ имеют вид +$$f(y,x_1,\ldots,x_n;\partder{y}{x_1},\ldots;\dpartder{y}{x_1},\ldots;\cdots; +\frac{\partial^m y}{\partial x_1^m},\ldots)=0.$$ +\end{defin} +\begin{block}{} +Однако, наиболее часто встречаются ЧДУ первого порядка для функции двух +переменных $z=z(x,y)$ вида +$$f(z,x,y,\partder{z}{x},\partder{z}{y})=0.$$ +\end{block}} +\only<3>{ +\begin{block} +\ж Нелинейные\н дифференциальные уравнения содержат некоторые производные +функции~$y$ не как простые множители, а как аргументы функций (чаще всего~--- +степенных), например: $(y'')^3-\sin y'=\tg(xy)$. Обычные физические задачи +никогда не приводят к таким уравнениям, однако, и их решения вполне можно +найти при помощи численных методов. +\end{block} +\begin{block}{Методы решения} +Рунге--Кутты, Эйлера, Адамса, конечных разностей и т.п. + +Дифференциальные уравнения высших порядков сводят путем замены переменных к системе ОДУ первого +порядка. +\end{block} +} +\end{frame} + +\begin{blueframe}{} +\begin{block}{Метод Эйлера} +Аппроксимация интегральной кривой кусочно-линейной функцией. Задача Коши в простейшем виде: +$\frac{dy}{dx}=f(x,y)$, $y|_{x=x_0}=y_0$. Решение ищется на интервале $(x_0, b]$. +$$y_i=y_{i-1}+(x_i-x_{i-1})f(x_{i-1},y_{i-1}),\qquad i=\overline{1,n}.$$ +\end{block} +\img[0.5]{Euler_method} +\end{blueframe} + +\begin{frame}{} +\begin{block}{Метод Рунге-Кутты} +$$y_{n+1}=y_n+\frac{h}{6}\bigl(k_1+2k_2+2k_3+k_4\bigr),\qquad \text{где}$$ +$k_1=f(x_n,y_n)$, $k_2=f\bigl(x_n+\frc{h}2, y_n+\frc{h}2 k_1\bigr)$, +$k_3=f\bigl(x_n+\frc{h}2, y_n+\frc{h}2 k_2\bigr)$, +$k_4=f(x_n+h,y_n+hk_3)$ ($h$~-- шаг сетки по $x$). +\end{block} +\img[0.5]{Runge-Kutta} +\end{frame} + + +\begin{frame}{Спасибо за внимание!} +\centering +\begin{minipage}{5cm} +\begin{block}{mailto} +eddy@sao.ru\\ +edward.emelianoff@gmail.com +\end{block}\end{minipage} +\end{frame} +\end{document}