From 3b455a1552dca2df785e9522c23584c8dcccea26 Mon Sep 17 00:00:00 2001 From: Batmaev Date: Wed, 2 Nov 2022 16:10:21 +0300 Subject: [PATCH] =?UTF-8?q?=D1=84=D0=B0=D0=B9=D0=BB=20=D1=81=20=D0=BF?= =?UTF-8?q?=D1=80=D0=B8=D0=BC=D0=B5=D1=80=D0=BE=D0=BC?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Task 3/Assignment_3_eng.pdf | Bin 159921 -> 160115 bytes Task 3/Assignment_3_ru.pdf | Bin 226752 -> 244115 bytes Task 3/tex/Assignment_3_eng.tex | 4 ++-- Task 3/tex/Assignment_3_ru.tex | 28 ++++++++++++++++++---------- 4 files changed, 20 insertions(+), 12 deletions(-) diff --git a/Task 3/Assignment_3_eng.pdf b/Task 3/Assignment_3_eng.pdf index 9d7df3aed888579cdde03ea77ce2d6c0b8df3aef..9c3ef844d5d4de0d0d9e82c60e241da5e1f593fa 100644 GIT binary patch delta 20856 zcmX`yWlSAR6DVLfxVyW%9=rvLyE_zjr$BKyxVs(P-K99iy|}x(yUTrZZ@&9y^K3Gi z`LoIFCbRWB2zR>(@wAX!-~es_eZ3L}BxtaENSHO{r|fR6NLU7gz2H{}#H47XSw2i7 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z?UmaaeTk{Xl%2xhk^U>CLx9xxg_BHad)4u0U?TOAr5Q7hHHz?f0rVV*=}I<;5h$*Y})~Z2y9&cT{|VEq_Py;{{pdu BRV@Gj diff --git a/Task 3/tex/Assignment_3_eng.tex b/Task 3/tex/Assignment_3_eng.tex index be5775a..a690a90 100644 --- a/Task 3/tex/Assignment_3_eng.tex +++ b/Task 3/tex/Assignment_3_eng.tex @@ -114,11 +114,11 @@ Using the method of Lagrange multipliers, and assuming $A^TA$ to be invertible, To be specific, consider $n=3$ points, for which we have approximate locations $r_1=(-1,0),\;r_2=(0, 1),\;r_3=(1,0)$ and $k=1$ measurement $\theta_{123}=9\pi/40$. Clearly, our estimates $r_{1,2,3}$ are not consistent with the measured angle and we have to adjust the estimate: $r_i\to \bar{r}_i= r_i+dr_i$ where $dr_i$ should be found from the condition $$ -(\bar{r}_3-\bar{r}_1)\cdot(\bar{r}_3-\bar{r}_1) = |\bar{r}_3-\bar{r}_1||\bar{r}_3-\bar{r}_1|\cos\theta_{123}, +(\bar{r}_3-\bar{r}_1)\cdot(\bar{r}_2-\bar{r}_1) = |\bar{r}_3-\bar{r}_1||\bar{r}_2-\bar{r}_1|\cos\theta_{123}, $$ which can be linearized in $dr_i$, assuming this correction will end up small. In this approach, one constructs a single (in general, $k$) equation for six (in general, $2n$) variables, so the system will typically by underdetermined. We can consider this system in the least squares sense, which amounts to determining the smallest correction to all $r_i$ which makes the updated locations consistent with observations. In the particular numerical example above, one may find $dr_1=(-h, 0),\;dr_2=(h, -h),\; dr_3=(0, h)$ where $h=\pi/80\approx 0.04$. -Your task is to write the code, which will accept the current estimate of the positions $r_i$ ($n\times 2$, float) and measurement results $\theta_{ijk}$, which are specified by i) indices of points ($k\times 3$, int) and ii) angles ($k$, float); and will output the derived correction to the point positions $dr_i$ ($n\times 2$, float). You can use the numerical example in this exercise to test your code. +Your task is to write the code, which will accept the current estimate of the positions $r_i$ ($n\times 2$, float) and measurement results $\theta_{ijk}$, which are specified by i) indices of points ($k\times 3$, int) and ii) angles ($k$, float); and will output the derived correction to the point positions $dr_i$ ($n\times 2$, float). You can use the numerical example in this exercise to test your code. This dataset is provided here: \href{https://disk.yandex.ru/d/kNjRT_r3S2xk2A}{dataset 1}, (use np.load with the keys \textrm{r}, \textrm{p}, \textrm{theta} to access input data and \textrm{dr} to access output data). You can find additional testing file with the same structure: \href{https://disk.yandex.ru/d/7aGeQvikiUt1AA}{dataset 2}. \end{enumerate} \printbibliography diff --git a/Task 3/tex/Assignment_3_ru.tex b/Task 3/tex/Assignment_3_ru.tex index 2902bd7..9b1084a 100644 --- a/Task 3/tex/Assignment_3_ru.tex +++ b/Task 3/tex/Assignment_3_ru.tex @@ -40,7 +40,7 @@ \title{\texorpdfstring{ Численные методы, осень 2022\\ Задание 3 [Проекторы. Задача наименьших квадратов. QR-разложение.] \\ - Всего баллов: 47 \ Срок сдачи: 4 ноября + Всего баллов: 47 \ Срок сдачи (после переноса): 11 ноября }{} } @@ -134,27 +134,35 @@ $$ $$ Предполагая, что матрица $A^T A$ обратима, воспользуйтесь методом множителей Лагранжа и получите выражение для точки минимума $x$. -\item (20) Эта задача посвящена локализации точек на плоскости. Рассмотрим $n$ точек с \emph{неточными} координатами: $A_i = \brk{x_i,\, y_i}$. Измерим $m$ углов между некоторыми из них: $\theta_{ijk} = \angle\, A_i A_j A_k$. Наша цель --- используя результаты измерений, уточнить координаты точек. +\item (20) Эта задача посвящена локализации точек на плоскости. Рассмотрим $n$ точек с \emph{неточными} координатами: $A_i = \brk{x_i,\, y_i}$. Измерим $m$ углов между некоторыми из них: $\theta_{ijk} = \angle \brk!{\vv{A_i A}_j,\ \vv{A_i A}_k}$. Наша цель --- используя результаты измерений, уточнить координаты точек. -В качестве примера рассмотрим $n = 3$ точки с приближенными координатами: $A_1 = (1,\, 0),\ A_2 = (-1,\, 0),\ A_3 = (0,\, 1)$ --- и один измеренный угол $\theta_{123} = 9\pi/40$. Так как наши оценки координат не согласуются с измеренным углом, мы должны их скорректировать: +В качестве примера рассмотрим $n = 3$ точки с приближенными координатами: $A_1 = (-1,\, 0),\ A_2 = (0,\, 1),\ A_3 = (1,\, 0)$ --- и один измеренный угол $\theta_{123} = 9\pi/40$. Так как наши оценки координат не согласуются с измеренным углом, мы должны их скорректировать: $$ x_i \to x_i + dx_i, \quad y_i \to y_i + dy_i, $$ где $dx_i$ и $dy_i$ находятся из условия $$ -\vv{A_1 A}_2 \cdot \vv{A_3 A}_2 = -\abs{A_1 A_2} \cdot \abs{A_3 A_2} \cdot \cos(\theta_{123}) +\vv{A_1 A}_2 \cdot \vv{A_1 A}_3 = +\abs{A_1 A_2} \cdot \abs{A_1 A_3} \cdot \cos(\theta_{123}) $$ Это условие может быть линеаризовано в предположении, что поправки окажутся малыми. При таком подходе мы конструируем $m = 1$ уравнений для $2n = 6$ переменных. Система обычно оказывается недоопределённой. Но мы можем рассматривать её в смысле метода наименьших квадратов, т.е. пытаться определить наименьшие поправки к координатам, которые делали бы их согласованными с измерениями. В рассмотренном примере можно найти решение $$ -dr_1 = (dx_1,\, dy_1) = (0, h), \quad -dr_2=(-h,\, 0), \quad -dr_3=(h,\, -h), \qquad -\text{ где } h = \pi/80 \approx 0.04 +dr_1 = (dx_1,\, dy_1) = (-h,\, 0), \quad +dr_2=(h,\, -h), \quad +dr_3=(0,\, h), \qquad +\text{ где } h \approx \pi/80 \approx 0.04 $$ Ваша задача --- написать код, который принимает текущие оценки координат точек $(n \times 2)$ и измеренные углы $\theta_{ijk}$ ($m$ значений и $m \times 3$ индексов). Возвращать он должен найденные поправки $(n \times 2)$. -Вы можете использовать пример из этого упражнения, чтобы протестировать свой код. + +Чтобы протестировать свой код, вы можете использовать \href{https://disk.yandex.ru/d/kNjRT_r3S2xk2A}{пример} из этого упражнения, а также другой \href{https://disk.yandex.ru/d/7aGeQvikiUt1AA}{пример} с б\'{о}льшим числом точек. Загрузить данные можно следующим образом: +\begin{minted}[frame = lines, framesep = \framesep]{python} + with np.load('./data_1.npz') as data: + r = data['r'] # исходные точки + p = data['p'] - 1 # индексы измеренных углов + theta = data['theta'] # углы в радианах + dr = data['dr'] # предполагаемый ответ + \end{minted} \end{enumerate}