162 lines
6.4 KiB
TeX
162 lines
6.4 KiB
TeX
\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}
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\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}
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\item (6) Consider computing the function $f(n, \alpha)$ defined by $f(0,\alpha)=\ln(1+1/\alpha)$ and recurrent relation
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\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:
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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 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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