Merge pull request 'Task4' (#5) from _/numerics-2022:main into main
Reviewed-on: https://sciprog.center/git/SPC-education/numerics-2022/pulls/5
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Task 3/Lecture 3.ipynb
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Task 4/Assignment_4_eng.pdf
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Task 4/Lecture 4.ipynb
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Task 4/tex/Assignment_4_eng.tex
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Task 4/tex/Assignment_4_eng.tex
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\documentclass{article}
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\usepackage[utf8]{inputenc}
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\usepackage{biblatex}
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\addbibresource{library.bib}
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\usepackage{listings}
|
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\usepackage{amssymb}
|
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\usepackage{comment}
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\usepackage{graphicx,amsmath}
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\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
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\usepackage{hyperref}
|
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\hypersetup{
|
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colorlinks=true,
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linkcolor=blue,
|
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filecolor=magenta,
|
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urlcolor=cyan,
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pdftitle={Overleaf Example},
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pdfpagemode=FullScreen,
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}
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\title{Numerical Methods: Lecture 4. Conditioning. Floating point arithmetic and stability. Systems of linear equations.}
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\author{Konstantin Tikhonov}
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\begin{document}
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\maketitle
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\section{Suggested Reading}
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\begin{itemize}
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\item Lectures 12-19, 20-23 of \cite{trefethen1997numerical}
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\item Lectures 6-7 of \cite{tyrtyshnikov2012brief}
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\end{itemize}
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\section{Exercises}
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Deadline: 18 Nov
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\begin{enumerate}
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\item (3) Propose a numerically stable way to compute the function $f(x,a)=\sqrt{x+a}-\sqrt{x}$ for positive $x,\;a$.
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\item (2) Consider numerical evaluation $\mathcal{C}=\tan(10^{100})$ with the help of arbitrary-precision arithmetic module \lstinline{mpmath}, which can be called as follows:
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\lstset{language=Python}
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\lstset{frame=lines}
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% \lstset{label={lst:code_direct}}
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\lstset{basicstyle=\ttfamily}
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\begin{lstlisting}
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from mpmath import *
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mp.dps = 64 # precision (in decimal places)
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mp.pretty = True
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+pi
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\end{lstlisting}
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What is the relative condition number of evaluating $\mathcal{C}$ w.r.t the input number $10^{100}$? How many digits do you need to keep at intermediate steps to evaluate $\mathcal{C}$ with 7-digit accuracy?
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\begin{comment}
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\item (3) Check, that the following function
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\lstset{language=Python}
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\lstset{frame=lines}
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\lstset{label={lst:code_direct}}
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\lstset{basicstyle=\ttfamily}
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\begin{lstlisting}
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import math
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def round_to_n(x, n):
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if x == 0:
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return x
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else:
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return round(x, -int(math.floor(math.log10(abs(x)))) + (n - 1))
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\end{lstlisting}
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rounds $x$ to $n$ significant digits.
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A sample program to compute $\sum_{k=1}^{3000}k^{-2}\approx 1.6446$ via consequent summation with rounding of intermediate results to 4 digits looks as follows:
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\lstset{language=Python}
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\lstset{frame=lines}
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\lstset{label={lst:code_direct}}
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\lstset{basicstyle=\ttfamily}
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\begin{lstlisting}
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res = 0
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for k in range(1,3001):
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res = round_to_n(res+1/k**2, 4)
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\end{lstlisting}
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Despite the absence of subtractions (and related precision loss), this code allows to get only two significant digits. Explain, why this happens and propose a more accurate way to compute this sum (maintaining the restriction of keeping only 4 digits of intermediate result).
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\end{comment}
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\item (4) Implement the function \lstinline{solve_quad(b, c)}, receiving coefficients $b$ and $c$ of a quadratic polynomial $x^2 + b x + c$, and returning a pair of equation roots. Your function should always return two roots, even for a degenerate case (for example, a call \lstinline{solve_quad(-2, 1)} should result into \lstinline{(1, 1)}). Additionally, your function is expected to return complex roots.
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After checking ensuring that your algorithm sort of works, try it on the following 5 tests. Make sure that all of them pass.
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\lstset{language=Python}
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\lstset{frame=lines}
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\lstset{label={lst:code_direct}}
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||||
\lstset{basicstyle=\ttfamily}
|
||||
\begin{lstlisting}
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tests = [{'b': 4.0, 'c': 3.0},
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{'b': 2.0, 'c': 1.0},
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{'b': 0.5, 'c': 4.0},
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{'b': 1e10, 'c': 3.0},
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{'b': -1e10, 'c': 4.0}]
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\end{lstlisting}
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\item (5) Consider the polynomial $$
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w(x)=\Pi_{r=1}^{20}(x-r)=\sum_{i=0}^{20} a_i x^i
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$$ and investigate the condition number of roots of this polynomial w.r.t the coefficients $a_i$. Perform the following experiment, using \texttt{numpy} root-finding algorithm. Randomly perturb $w(x)$ by replacing the coefficients $a_i\to n_i a_i$, where $n_i$ is drawn from a normal distribution of mean $1$ and variance $\exp(-10)$. Show the results of $100$ such experiments in a single plot, along with the
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roots of the unperturbed polynomial $w(x)$. Using one of the experiments, estimate the relative and absolute condition number of the problem of finding the roots of $w(x)$ w.r.t. polynomial coefficients.
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\item (10)
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Consider the least squares problem $Ax\approx b$ at
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$$
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A = \begin{bmatrix}
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1 & 1\\
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1 & 1.00001\\
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1 & 1.00001
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\end{bmatrix},\quad b = \begin{bmatrix}
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2 \\
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0.00001 \\
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4.00001
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\end{bmatrix}.
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$$
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\begin{itemize}
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\item
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Formally, solution is given by
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\begin{equation}
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\label{ex}
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x = ( A^T A )^{-1} A^T b.
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\end{equation}
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Using this equation, compute the solution analytically.
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\item Implement Eq. (\ref{ex}) in \lstinline{numpy} in single and double precision; compare the results to the analytical one.
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\item Instead of Eq. (\ref{ex}), implement SVD-based solution to least squares. Which approach is numerically more stable?
|
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\item Use \lstinline{np.linalg.lstsq} to solve the same equation. Which method does this function use?
|
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\item
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What are the four condition numbers of this problem, mentioned in Theorem 18.1 of Ref. \cite{trefethen1997numerical}? Give examples of perturbations $\delta b$ and $\delta A$ that approximately attain those condition numbers?
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\end{itemize}
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\item (7)
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Let $$A = \begin{bmatrix}
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\epsilon & 1 & 0\\
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1 & 1 & 1\\
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0 & 1 & 1
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\end{bmatrix}$$
|
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\begin{itemize}
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\item Find analytically LU decomposition with and without pivoting for the matrix $A$.
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\item Explain, why can the LU decomposition fail to approximate factors $L$ and $U$ for $|\epsilon|\ll 1$ in finite-precision arithmetic?
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\end{itemize}
|
||||
|
||||
\item (6) Consider computing the function $f(n, \alpha)$ defined by $f(0,\alpha)=\ln(1+1/\alpha)$ and recurrent relation
|
||||
\begin{equation}
|
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f(n,\alpha)=\frac{1}{n}-\alpha f(n-1,\alpha).
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\end{equation}
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Compute $f(20, 0.1)$ and $f(20, 10)$ in standard (double) precision. Now, do the same exercise in arbitrary
|
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precision arithmetic:
|
||||
\lstset{language=Python}
|
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\lstset{frame=lines}
|
||||
\lstset{label={lst:code_direct}}
|
||||
\lstset{basicstyle=\ttfamily}
|
||||
\begin{lstlisting}
|
||||
from mpmath import mp, mpf
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mp.dps = 64 # precision (in decimal places)
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f = mp.zeros(1, n)
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f[0] = mp.log(1+1/mpf(alpha))
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for i in range(1, n):
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f[i] = 1/mpf(i) - mpf(alpha)*f[i-1]
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||||
\end{lstlisting}
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||||
\end{enumerate}
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||||
Plot the relative difference between exact and approximate results, in units of machine epsilon \texttt{np.finfo(float).eps} for $\alpha=0.1$ and $\alpha=10$ as function of $n$. How would you evaluate $f(30, 10)$ without relying on the arbitrary precision arithmetic?
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\printbibliography
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\end{document}
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Task 4/tex/Assignment_4_ru.tex
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Task 4/tex/Assignment_4_ru.tex
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\documentclass[prb, notitlepage, aps, 11pt]{revtex4-2}%
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\usepackage[utf8]{inputenc}
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\usepackage[T2A]{fontenc}
|
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\usepackage[english, russian]{babel}
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\usepackage{amsmath}
|
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\usepackage{enumitem}
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\usepackage{amsmath}
|
||||
\usepackage{delimset}
|
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\usepackage[pdftitle = a]{hyperref}
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\usepackage{datetime}
|
||||
\usepackage{minted}
|
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\usemintedstyle{friendly}
|
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\usepackage[a]{esvect}
|
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\hypersetup{
|
||||
colorlinks=true,
|
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linkcolor=blue,
|
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filecolor=magenta,
|
||||
urlcolor=cyan,
|
||||
pdfpagemode=FullScreen,
|
||||
}
|
||||
\usepackage{microtype}
|
||||
|
||||
\newcommand{\framesep}{0.6em}
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\BeforeBeginEnvironment{minted}{\vspace{-1.6em}}
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||||
\AfterEndEnvironment{minted}{\vspace{-0.5em}}
|
||||
|
||||
\begin{document}
|
||||
|
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\begin{center}
|
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версия от \today \quad \currenttime
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\end{center}
|
||||
|
||||
\title{\texorpdfstring{
|
||||
Численные методы, осень 2022\\
|
||||
Задание 4 [Число обусловленности. Числа с плавающей точкой и вычислительная устойчивость] \\
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||||
Всего баллов: 37 \ Срок сдачи: 18 ноября
|
||||
}{}
|
||||
}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section*{Рекомендованная литература}
|
||||
|
||||
\begin{itemize}
|
||||
\item Лекции 12--19, 20--23 из \cite{trefethen1997numerical}
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||||
\item Лекции 6--7 из \cite{tyrtyshnikov2012brief}
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||||
\end{itemize}
|
||||
|
||||
\section*{Упражнения}
|
||||
|
||||
\begin{enumerate}
|
||||
|
||||
\item (3) Предложите вычислительно устойчивый способ вычислить функцию
|
||||
$f(x,a)=\sqrt{x+a}-\sqrt{x}$
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||||
при положительных $x$ и $a$.
|
||||
|
||||
\item (2) Вычислите $\mathcal{C}=\tan(10^{100})$ с помощью модуля \mintinline{python}{mpmath}, предназначенного для арифметики произвольной точности. Пример использования:
|
||||
%
|
||||
\begin{minted}[frame = lines, framesep = \framesep]{python}
|
||||
from mpmath import *
|
||||
mp.dps = 64 # точность (число десятичных цифр)
|
||||
mp.pretty = True
|
||||
+pi # pi — переменная из mpmath
|
||||
\end{minted}
|
||||
%
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||||
Чему равно относительное число обусловленности при вычислении $\mathcal{C} = \mathcal{C}(10^{100})$? Сколько цифр нужно хранить в памяти при промежуточных вычислениях, чтобы получить $\mathcal{C}$ с~точностью в 7 значащих цифр?
|
||||
|
||||
\item (4) Реализуйте функцию \mintinline{python}{solve_quad(b, c)}, возвращающую корни приведённого квадратного уравнения $x^2 + b x + c = 0$. Корни могут повторяться или быть комплексными.
|
||||
|
||||
Когда вам покажется, что функция работает, запустите её на следующих пяти тестах. Добейтесь того, чтобы она правильно работала на каждом из них.
|
||||
%
|
||||
\begin{minted}[frame = lines, framesep = \framesep]{python}
|
||||
tests = [{'b': 4.0, 'c': 3.0},
|
||||
{'b': 2.0, 'c': 1.0},
|
||||
{'b': 0.5, 'c': 4.0},
|
||||
{'b': 1e10, 'c': 3.0},
|
||||
{'b': -1e10, 'c': 4.0}]
|
||||
\end{minted}
|
||||
|
||||
\item (5) Рассмотрите многочлен
|
||||
$$
|
||||
w(x) = \prod_{r=1}^{20} (x-r) = \sum_{i=0}^{20} a_i x^i
|
||||
$$
|
||||
и исследуйте число обусловленности его корней, выступающих в роли функций от коэффициентов $a_i$. Проведите эксперимент: случайным образом измените коэффициенты и найдите новые корни с помощью алгоритма из \texttt{numpy}.
|
||||
Коэффициенты изменяйте по правилу $a_i \to n_i a_i$, где $n_i$ подчиняются нормальному распределению с математическим ожиданием, равным 1, и дисперсией, равной $\exp(-10)$. Проведите 100 таких экспериментов и изобразите результаты на одном графике вместе с корнями исходного многочлена.
|
||||
|
||||
Оцените по одному из экспериментов абсолютное и относительное число обусловленности корней многочлена как функций его коэффициентов.
|
||||
|
||||
\item (10)
|
||||
Рассмотрим задачу наименьших квадратов --- $Ax\approx b$:
|
||||
$$
|
||||
A = \begin{bmatrix}
|
||||
1 & 1\\
|
||||
1 & 1.00001\\
|
||||
1 & 1.00001
|
||||
\end{bmatrix},\quad b = \begin{bmatrix}
|
||||
2 \\
|
||||
0.00001 \\
|
||||
4.00001
|
||||
\end{bmatrix}
|
||||
$$
|
||||
|
||||
\begin{itemize}
|
||||
\item Формально решение можно найти как
|
||||
%
|
||||
\begin{equation}
|
||||
\label{ex}
|
||||
x = ( A^T A )^{-1} A^T b.
|
||||
\end{equation}
|
||||
%
|
||||
Вычислите его по этой формуле аналитически.
|
||||
|
||||
\item Вычислите (\ref{ex}) с помощью
|
||||
\mintinline{python}{numpy}, используя числа одинарной и двойной точности; сравните результат c аналитическим.
|
||||
|
||||
\item Помимо формулы (\ref{ex}), реализуйте решение, основанное на сингулярном разложении. Какой способ вычислительно более стабильный?
|
||||
|
||||
\item Решите эту же задачу с помощью \mintinline{python}{np.linalg.lstsq}. Какой алгоритм использует эта функция?
|
||||
|
||||
\item Какие четыре числа обусловленности, относящиеся к этой задаче, упоминаются в теореме 18.1 из~\cite{trefethen1997numerical}? (Возможно, их требуется вычислить --- прим. пер.).
|
||||
Приведите примеры таких $\delta b$ и $\delta A$, при которых приблизительно достигаются оценки на $\norm{\delta x}$, даваемые числами обусловленности.
|
||||
\end{itemize}
|
||||
|
||||
\item (7)
|
||||
Пусть
|
||||
$$
|
||||
A = \begin{bmatrix}
|
||||
\epsilon & 1 & 0\\
|
||||
1 & 1 & 1\\
|
||||
0 & 1 & 1
|
||||
\end{bmatrix}
|
||||
$$
|
||||
\begin{itemize}
|
||||
\item Аналитически найдите LU-разложение матрицы $A$ с применением выбора главного элемента и без него.
|
||||
\item Объясните, почему при $|\epsilon|\ll 1$ мы можем неправильно оценить множители $L$ и $U$ в арифметике конечной точности.
|
||||
\end{itemize}
|
||||
|
||||
\item (6) Пусть функция $f(n, \alpha)$ определена следующим образом:
|
||||
%
|
||||
\begin{align*}
|
||||
f(n,\alpha) &= \frac{1}{n} - \alpha f(n-1,\alpha) \\
|
||||
f(0,\alpha) &= \ln(1+1/\alpha)
|
||||
\end{align*}
|
||||
%
|
||||
Вычислите $f(20, 0.1)$ и $f(20, 10)$ с помощью арифметики обычной (двойной) точности. Теперь сделайте то же самое в арифметике произвольной точности:
|
||||
%
|
||||
\begin{minted}[frame = lines, framesep = \framesep]{python}
|
||||
from mpmath import mp, mpf
|
||||
mp.dps = 64 # precision (in decimal places)
|
||||
f = mp.zeros(1, n)
|
||||
f[0] = mp.log(1 + 1/mpf(alpha))
|
||||
for i in range(1, n):
|
||||
f[i] = 1/mpf(i) - mpf(alpha) * f[i-1]
|
||||
\end{minted}
|
||||
%
|
||||
Постройте в единицах машинного эпсилон график относительной разности между точными и приближёнными результатами как функции от $n$. Сделайте это при $\alpha=0.1$ и при $\alpha=10$. Машинный эпсилон можно получить как \mintinline{python}{np.finfo(float).eps}. \\
|
||||
Как бы вы стали вычислять $f(30, 10)$ без арифметики произвольной точности?
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
\bibliography{library.bib}
|
||||
\end{document}
|
||||
118
Task 4/tex/library.bib
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118
Task 4/tex/library.bib
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|
||||
@book{trefethen1997numerical,
|
||||
title={Numerical linear algebra},
|
||||
author={Trefethen, Lloyd N and Bau III, David},
|
||||
volume={50},
|
||||
year={1997},
|
||||
publisher={Siam}
|
||||
}
|
||||
@book{robert2013monte,
|
||||
title={Monte Carlo statistical methods},
|
||||
author={Robert, Christian and Casella, George},
|
||||
year={2013},
|
||||
publisher={Springer Science \& Business Media}
|
||||
}
|
||||
@book{murphy2012machine,
|
||||
title={Machine learning: a probabilistic perspective},
|
||||
author={Murphy, Kevin P},
|
||||
year={2012},
|
||||
publisher={MIT press}
|
||||
}
|
||||
@book{boyd2004convex,
|
||||
title={Convex optimization},
|
||||
author={Boyd, Stephen and Boyd, Stephen P and Vandenberghe, Lieven},
|
||||
year={2004},
|
||||
publisher={Cambridge university press}
|
||||
}
|
||||
@book{trefethen2019approximation,
|
||||
title={Approximation Theory and Approximation Practice, Extended Edition},
|
||||
author={Trefethen, Lloyd N},
|
||||
year={2019},
|
||||
publisher={SIAM}
|
||||
}
|
||||
@book{devroye:1986,
|
||||
author = {Devroye, Luc},
|
||||
date = {1986)},
|
||||
description = {Non-Uniform Random Variate Generation},
|
||||
keywords = {algorithms generation random simulation},
|
||||
location = {New York},
|
||||
publisher = {Springer-Verlag},
|
||||
title = {Non-Uniform Random Variate Generation(originally published with},
|
||||
year = 1986
|
||||
}
|
||||
@book{williams2006gaussian,
|
||||
title={Gaussian processes for machine learning},
|
||||
author={Williams, Christopher K and Rasmussen, Carl Edward},
|
||||
volume={2},
|
||||
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|
||||
year={2006},
|
||||
publisher={MIT press Cambridge, MA}
|
||||
}
|
||||
@book{hairer1993solving,
|
||||
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|
||||
author={Hairer, Ernst and N{\o}rsett, Syvert P and Wanner, Gerhard},
|
||||
year={1993},
|
||||
publisher={Springer-Vlg}
|
||||
}
|
||||
@book{hairer1993solving2,
|
||||
title={Solving ordinary differential equations. 2, Stiff and differential-algebraic problems},
|
||||
author={Hairer, Ernst and N{\o}rsett, Syvert P and Wanner, Gerhard},
|
||||
year={1993},
|
||||
publisher={Springer-Vlg}
|
||||
}
|
||||
@book{tyrtyshnikov2012brief,
|
||||
title={A brief introduction to numerical analysis},
|
||||
author={Tyrtyshnikov, Eugene E},
|
||||
year={2012},
|
||||
publisher={Springer Science \& Business Media}
|
||||
}
|
||||
@book{amosov2003,
|
||||
title={Numerical Methods for Engineers},
|
||||
author={Amosov, AA and Dubinskiy YuA and Kopchenova, NV},
|
||||
year={2003},
|
||||
publisher={Izdatelstvo MEI}
|
||||
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|
||||
@article{arbenz2012lecture,
|
||||
title={Lecture notes on solving large scale eigenvalue problems},
|
||||
author={Arbenz, Peter and Kressner, Daniel and Z{\"u}rich, DME},
|
||||
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|
||||
volume={2},
|
||||
year={2012}
|
||||
}
|
||||
@article{trefethen1996finite,
|
||||
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|
||||
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|
||||
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|
||||
publisher={Cornell University-Department of Computer Science and Center for Applied~…}
|
||||
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|
||||
@book{boyd2018introduction,
|
||||
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|
||||
author={Boyd, Stephen and Vandenberghe, Lieven},
|
||||
year={2018},
|
||||
publisher={Cambridge university press}
|
||||
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|
||||
@article{halko2011finding,
|
||||
title={Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions},
|
||||
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|
||||
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|
||||
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|
||||
number={2},
|
||||
pages={217--288},
|
||||
year={2011},
|
||||
publisher={SIAM}
|
||||
}
|
||||
@book{demmel1997applied,
|
||||
title={Applied numerical linear algebra},
|
||||
author={Demmel, James W},
|
||||
year={1997},
|
||||
publisher={SIAM}
|
||||
}
|
||||
@article{dahlquist198533,
|
||||
title={33 years of numerical instability, Part I},
|
||||
author={Dahlquist, Germund},
|
||||
journal={BIT Numerical Mathematics},
|
||||
volume={25},
|
||||
number={1},
|
||||
pages={188--204},
|
||||
year={1985},
|
||||
publisher={Springer}
|
||||
}
|
||||
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Reference in New Issue
Block a user