From 113700eac3aeb151af42fbab555bde973d5804d4 Mon Sep 17 00:00:00 2001 From: Yadunand Prem Date: Thu, 1 Feb 2024 08:26:59 +0800 Subject: [PATCH] feat: 1522 l2 notes --- cs2104/labs/assignment.pl | 10 +- cs2109s/labs/ps0/ps0.py | 25 ++- ma1522/lec_01.tex | 3 + ma1522/lec_02.tex | 352 ++++++++++++++++++++++++++++++++++++++ ma1522/master.pdf | Bin 43673 -> 66657 bytes ma1522/master.tex | 4 + ma1522/preamble.tex | 7 + 7 files changed, 386 insertions(+), 15 deletions(-) diff --git a/cs2104/labs/assignment.pl b/cs2104/labs/assignment.pl index a36b648..18c5181 100644 --- a/cs2104/labs/assignment.pl +++ b/cs2104/labs/assignment.pl @@ -215,9 +215,7 @@ nth(Idx, [_|Tail], Elem) :- Idx1 is Idx - 1, nth(Idx1, Tail, Elem). -% Taufiq -nth(0, [V | _], V) :- !. -nth(I, [_|T], V) :- nth(I1, T, V), I is I1 + 1, I > 0. + %%%%%%%%%%%%%% @@ -258,9 +256,6 @@ pos_helper(Target, [_|Tail], Acc, Index) :- Acc1 is Acc + 1, pos_helper(Target, Tail, Acc1, Index). -% Taufiq -pos(X, [X | _], 0). -pos(X, [_ | T], I) :- pos(X, T, I1), I is I1 + 1. %%%%%%%%%%%%%% @@ -294,9 +289,6 @@ all_pos_helper(Target, [_|Tail], Acc, Rest) :- Acc1 is Acc + 1, all_pos_helper(Target, Tail, Acc1, Rest). -% Taufiq -all_pos(X, L, []) :- not(member(X, L)). -all_pos(X, L, LI) :- bagof(I, pos(X, L, I), LI). %%%%%%%%%%%%%% diff --git a/cs2109s/labs/ps0/ps0.py b/cs2109s/labs/ps0/ps0.py index 9572cd1..b327756 100644 --- a/cs2109s/labs/ps0/ps0.py +++ b/cs2109s/labs/ps0/ps0.py @@ -538,9 +538,8 @@ def average_increase_in_cases(n_cases_increase, n_adj_entries_avg=7): ''' sd_win = np.lib.stride_tricks.sliding_window_view(n_cases_increase, 2 * n_adj_entries_avg + 1, axis=1) - avg = np.mean(sd_win, axis=2) + avg = np.floor(np.mean(sd_win, axis=2)) res = np.pad(avg, ((0, 0), (n_adj_entries_avg, n_adj_entries_avg)), 'constant', constant_values=(np.nan, np.nan)) - print(res) return res @@ -552,7 +551,7 @@ def test_26(): assert(np.array_equal(actual, expected, equal_nan=True)) -test_26() +# test_26() # Task 2.7 def is_peak(n_cases_increase_avg, n_adj_entries_peak=7): @@ -605,9 +604,23 @@ def is_peak(n_cases_increase_avg, n_adj_entries_peak=7): Hint: to determine `n_adj_entries_avg` from `n_cases_increase_avg`, `np.count_nonzero` and `np.isnan` may be helpful. ''' + # get the number of paddings to add (count nans and / 2) + paddings = int(np.count_nonzero(np.isnan(n_cases_increase_avg[0])) / 2) + nanless = n_cases_increase_avg[:, paddings: -paddings] + # find peaks (get the sliding window) + sd_win = np.lib.stride_tricks.sliding_window_view(nanless, 2 * n_adj_entries_peak + 1, axis=1) + mids = sd_win[:, :, n_adj_entries_peak] + left_vals = np.max(sd_win[:, :, :n_adj_entries_peak], axis=2) + right_vals = np.max(sd_win[:, :, n_adj_entries_peak+1:], axis=2) + peaks = (mids > left_vals) & (mids >= right_vals) - # TODO: add your solution here and remove `raise NotImplementedError` - raise NotImplementedError + means = np.nanmean(nanless, axis=1)[:, np.newaxis] * 0.1 + + significant = mids > means + + result = np.logical_and(peaks, significant) + res = np.pad(result, ((0, 0), (paddings+n_adj_entries_peak, paddings+ n_adj_entries_peak)), 'constant', constant_values=(False, False)) + return res def test_27(): @@ -627,7 +640,7 @@ def test_27(): [False, False, False, False, False, False, False, False, False]]) assert np.all(actual2 == expected2) -#test_27() +test_27() def visualise_increase(n_cases_increase, n_cases_increase_avg=None): ''' diff --git a/ma1522/lec_01.tex b/ma1522/lec_01.tex index 7eda3d9..bbc8715 100644 --- a/ma1522/lec_01.tex +++ b/ma1522/lec_01.tex @@ -166,9 +166,12 @@ To solve a linear system we perform operations: \item add a constant multiple of an equation to another \end{itemize} +Likewise, for a augmented matrix, the operations are on the \textbf{rows} of the augmented matrix \begin{itemize} \item Multiply row by nonzero constant \item Interchange 2 rows \item add a constant multiple of a row to another row \end{itemize} + +To note: all these operations are revertible diff --git a/ma1522/lec_02.tex b/ma1522/lec_02.tex index e69de29..a2e8acc 100644 --- a/ma1522/lec_02.tex +++ b/ma1522/lec_02.tex @@ -0,0 +1,352 @@ +\subsection{Recap} +Given the linear equation $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ + +\begin{enumerate} + \item $a_1 = a_2 = ... = a_n = b = 0$ zero equation + + Solution: $x_1 = t_1, x_2 = t_2, ... = x_n = t_n$ + + \item $a_1 = a_2 = ... = a_n = 0 \neq b$ inconsistent + + No Solution + + \item Not all $a_1 ... a_n$ are zero. + + Set $n-1$ of $x_i$ as params, solve for last variable +\end{enumerate} + +\subsection{Elementary Row Operations Example} + +\begin{center} + + \begin{minipage}{.3\linewidth} + \systeme{x+y+3z = 0, 2x-2y+2z=4, 3x+9y=3} + \end{minipage}% + \begin{minipage}{.3\linewidth} + \begin{equation*} + \begin{amatrix}{3} + 1 & 1 & 3 & 0 \\ + 2 & 2 & 2 & 4 \\ + 3 & 9 & 0 & 3 + \end{amatrix} + \end{equation*} + \end{minipage} +\end{center} + +\subsection{Row Equivalent Matrices} +2 Augmented Matrices are row equivalent if one can be obtained from the other by a series of elementary row operations + +Given a augmented matrix $A$, how to find a row equivalent augmented matrix B of which is of a \textbf{simple} form? + +\subsection{Row Echelon Form} + +\begin{defn}[Row Echelon Form (Simple)] + Augmented Matrix is in row-echelon form if + \begin{itemize} + \item Zero rows are grouped together at the bottom + \item For any 2 successive nonzero rows, The first nonzero number in the lower row appears to the right of the first nonzero number on the higher row + $\begin{amatrix}{4} + 0 & 0 & 1 & 2 & 3 \\ + 0 & 0 & 0 & 1 & 2 \\ + \end{amatrix}$ + \item Leading entry if a nonzero row is a \textbf{pivot point} + \item Column of augmented matrix is called + \begin{itemize} + \item \textbf{Pivot Column} if it contains a pivot point + \item \textbf{Non Pivot Column} if it contains no pivot point + \end{itemize} + \item Pivot Column contains exactly 1 pivot point + + \# of pivots = \# of leading entries = \# of nonzero rows + \end{itemize} +\end{defn} + +Examples of row echlon form: + +\begin{equation*} + \begin{amatrix}{2} + 3 & 2 & 1 \\ + \end{amatrix} + \begin{amatrix}{2} + 1 & -1 & 0 \\ + 0 & 1 & 0 \\ + \end{amatrix} + \begin{amatrix}{2} + 2 & 1 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + \end{amatrix} + \begin{amatrix}{3} + 1 & 2 & 3 & 4 \\ + 0 & 1 & 1 & 2 \\ + 0 & 0 & 2 & 3 \\ + \end{amatrix} + \begin{amatrix}{4} + 0 & 1 & 2 & 8 & 1 \\ + 0 & 0 & 0 & 0 & 3 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + \end{amatrix} +\end{equation*} + +Examples of NON row echlon form: + +\begin{equation*} + \begin{amatrix}{2} + 0 & \textbf{1} & 0 \\ + \textbf{1} & 0 & 0 \\ + \end{amatrix} + \begin{amatrix}{2} + 0 & 0 & \textbf{1} \\ + \textbf{1} & -1 & 0 \\ + 0 & 0 & 1 \\ + \end{amatrix} + \begin{amatrix}{3} + \textbf{1} & 0 & 2 & 1 \\ + 0 & \textbf{1} & 0 & 2 \\ + 0 & \textbf{1} & 1 & 3 \\ + \end{amatrix} + \begin{amatrix}{4} + \textbf{0} & \textbf{0} & \textbf{0} & \textbf{0} & \textbf{0} \\ + 1 & 0 & 2 & 0 & 1 \\ + 0 & 0 & 0 & 1 & 3 \\ + 0 & 0 & 0 & 0 & 0 \\ + \end{amatrix} +\end{equation*} +\subsection{Reduced Row-Echelon Form} + +\begin{defn}[Reduced Row-Echelon Form] + Suppose an augmented matrix is in row-echelon form. It is in \textbf{reduced row-echelon form} if + \begin{itemize} + \item Leading entry of every nonzero row is 1 + \subitem Every pivot point is one + \item In each pivot column, except the pivot point, all other entries are 0. + \end{itemize} +\end{defn} + +Examples of reduced row-echelon form: + +\begin{equation*} + \begin{amatrix}{2} + 1 & 2 & 3 \\ + \end{amatrix} + \begin{amatrix}{2} + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + \end{amatrix} + \begin{amatrix}{2} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + \end{amatrix} + \begin{amatrix}{3} + 1 & 0 & 0 & 1 \\ + 0 & 1 & 0 & 2 \\ + 0 & 0 & 1 & 3 \\ + \end{amatrix} + \begin{amatrix}{4} + 0 & 1 & 2 & 0 & 1 \\ + 0 & 0 & 0 & 1 & 3 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + \end{amatrix} +\end{equation*} + +Examples of row-echelon form but not reduced: (pivot point is not 1 / all other elements \textbf{in pivot column} must be zero) + +\begin{equation*} + \begin{amatrix}{2} + \textbf{3} & 2 & 1 \\ + \end{amatrix} + \begin{amatrix}{2} + 1 & \textbf{-1} & 0 \\ + 0 & 1 & 0 \\ + \end{amatrix} + \begin{amatrix}{2} + \textbf{2} & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + \end{amatrix} + \begin{amatrix}{3} + \textbf{-1} & 2 & 3 & 4 \\ + 0 & 1 & 1 & 2 \\ + 0 & 0 & 2 & 3 \\ + \end{amatrix} + \begin{amatrix}{4} + 0 & 1 & 2 & \textbf{8} & 1 \\ + 0 & 0 & 0 & \textbf{4} & 3 \\ + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + \end{amatrix} +\end{equation*} + +To note: 2nd matrix has -1 in the pivot column, but 5th matrix has 2 in a non-pivot column so its fine + +\subsection{Solving Linear System} + +If Augmented Matrix is in reduced row-echelon form, then solving it is easy + +\begin{equation*} + \begin{amatrix}{3} + 1 & 0 & 0 & 1 \\ + 0 & 1 & 0 & 2 \\ + 0 & 0 & 1 & 3 \\ + \end{amatrix} + \text{then } x_1 = 1, x_2 = 2, x_3 = 3 +\end{equation*} + +\begin{note} + \begin{itemize} + \item If any equations in the system is inconsistent, the whole system is inconsistent + \end{itemize} +\end{note} + +\subsubsection{Examples} + +Augmented Matrix: $\begin{amatrix}{4} + 1 & -1 & 0 & 3 & -2 \\ + 0 & 0 & 1 & 2 & 5 \\ + 0 & 0 & 0 & 0 & 0 \\ +\end{amatrix}$ + +\begin{itemize} + \item The zero row can be ignored. + \systeme{x_1 - x_2 + 3x_4 = -2, x_3 + 2x_4 = 5} + \item Degree of freedom(\# cols): 4, number of restrictions (\# pivot cols): 2, arbitrary vars(\# non pivot cols): 4-2 = 2. Set this to the non-pivot cols +\end{itemize} +\begin{enumerate} + \item Let $x_4 = t$ and sub into 2nd eqn + \subitem $x_3 + 2t = 5 \Rightarrow x_3 = 5-2t$ + \item sub $x_4 = t$ into 1st eqn + \subitem $x_1 - x_2 + 3t = -2$ + \subitem Let $x_2 = s$. Then $x_1 = -2 + s - 3t$ + \item Infinitely many sols with ($s$ and $t$ as arbitrary params) + \subitem $x_1 = -2 + s - 3t, x_2 = s, x_3 = 5-2t, x_4 = t$ + +\end{enumerate} + +Augmented Matrix: $\begin{amatrix}{5} + 0 & 2 & 2 & 1 & -2 & 2 \\ + 0 & 0 & 1 & 1 & 1 & 3 \\ + 0 & 0 & 0 & 0 & 2 & 4 \\ +\end{amatrix}$ +\begin{itemize} + \item \systeme{0x_1 + 2x_2 + 2x_3 + 1x_4 -2x_5 = 2,x_3 + x_4 +x_5 = 3,2x_5 = 4} + \item Degree of freedom: 5, number of restrictions: 3, arbitrary vars: 5-3 = 2 +\end{itemize} + +\begin{enumerate} + \item by 3rd eqn, $2x_5 = 4 \Rightarrow x_5 = 2$ + \item sub $x_5 = 2$ into 2nd eqn + \subitem $x_3 + x_4 + 2 = 3 \Rightarrow x_3 + x_4 = 1$ + \subitem let $x_4 = t$. Then $x_3 = 1-t$ + \item sub $x_5 = 2, x_3 = 1-t, x_4 = t$ into 1st eqn + \subitem $2x_2 + 2(1-t) + t - 2(2) = 2 \Rightarrow 2x_2 -t = 4 \Rightarrow x_2 = \frac{t}{2} + 2$ + \item system has inf many solns: $x_1 = s, x_2 = \frac{t}{2} + 2, x_3 = 1-t, x_4 = t, x_5 = 2$ where $s$ and $t$ are arbitrary +\end{enumerate} + +\subsubsection{Algorithm} +Given the augmented matrix is in row-echelon form. +\begin{enumerate} + \item Set variables corresponding to non-pivot columns to be arbitrary parameters + \item Solve variables corresponding to pivot columns by back substitution (from last eqn to first) +\end{enumerate} + + +\subsection{Gaussian Eliminiation} + +\begin{defn}[Gaussian Elimination]\ \\ + \begin{enumerate} + \item Find the left most column which is not entirely zero + \item If top entry of such column is 0, replace with nonzero number by swapping rows + \item For each row below top row, add multiple of top row so that leading entry becomes 0 + \item Cover top row and repeat to remaining matrix + \end{enumerate} +\end{defn} + +\begin{note}[Algorithm with Example]\ \\ + $\begin{amatrix}{6} + 0 & 0 & 0 & 2 & 4 & 2 & 8 \\ + 0 & 1 & 2 & 4 & 5 & 3 &-9 \\ + 0 &-2 &-4 &-5 &-4 & 3 & 6 \\ + \end{amatrix}$ + \begin{enumerate} + \item Find the left most column which is not all zero (2nd column) + \item Check top entry of the selection. If its zero, replace it by a nonzero number by interchanging the top row with another row below + \subitem $\begin{amatrix}{6} + 0 & 1 & 2 & 4 & 5 & 3 &-9 \\ + 0 & 0 & 0 & 2 & 4 & 2 & 8 \\ + 0 &-2 &-4 &-5 &-4 & 3 & 6 \\ + \end{amatrix}$ + \item For each row below the top row, adda suitable multiple of top row so that leading entry becomes 0. + \subitem $2R_1 + R_3$ will ensure that the -2 turns to 0 + \subitem $\begin{amatrix}{6} + 0 & 1 & 2 & 4 & 5 & 3 &-9 \\ + 0 & 0 & 0 & 2 & 4 & 2 & 8 \\ + 0 & 0 & 0 & 3 & 6 & 9 &-12\\ + \end{amatrix}$ + \item Cover top row and repeat procedure to the remaining matrix + \subitem $\begin{amatrix}{6} + 0 & 1 & 2 & 4 & 5 & 3 &-9 \\ + \hline + 0 & 0 & 0 & 2 & 4 & 2 & 8 \\ + 0 & 0 & 0 & 3 & 6 & 9 &-12\\ + \end{amatrix}$ + \subitem Look at $C_4$. $R_3 \times -1.5R_2$ will set $R_3C_4$ to zero. + \subitem $\begin{amatrix}{6} + 0 & 1 & 2 & 4 & 5 & 3 &-9 \\ + \hline + 0 & 0 & 0 & 2 & 4 & 2 & 8 \\ + 0 & 0 & 0 & 0 & 0 & 6 & -24\\ + \end{amatrix}$ + \subitem This is now in row echelon form. + \end{enumerate} + Only use $R_i \Leftrightarrow R_j or R_i + CR_j$ in this method. +\end{note} + +\subsection{Gauss-Jordan Elimination} +\begin{defn}[Gauss Joran Elimination]\ \\ + \begin{enumerate} + \item[1-4.] Use Gaussian Eliminiation to get row-echelon form + \setcounter{enumi}{4} + \item For each nonzero row, multiply a suitable constant so pivot point becomes 1 + \item Begin with last nonzero row and work backwords + \subitem Add suitable multiple of each row to the rows above to introduce 0 above pivot point + \end{enumerate} + \begin{itemize} + \item Every matrix has a unique reduced row-echelon form. + \item Every nonzero matrix has infinitely many row-echelon ofrm + \end{itemize} +\end{defn} +\begin{note}[Gauss Jordan Elimination Example] Suppose an augmented matrix is in row-echelon form. + $\begin{amatrix}{5} + 1 & 2 & 4 & 5 & 3 & -9 \\ + 0 & 0 & 2 & 4 & 2 & 8 \\ + 0 & 0 & 0 & 0 & 6 & -24 \\ + \end{amatrix}$ + \begin{enumerate} + \item All pivot points must be 1 + \subitem multiply $R_2$ by $\frac{1}{2}$ and $R_3$ by $\frac{1}{6}$ + \subitem $\begin{amatrix}{5} + 1 & 2 & 4 & 5 & 3 & -9 \\ + 0 & 0 & 1 & 2 & 1 & 4 \\ + 0 & 0 & 0 & 0 & 1 & -4 \\ + \end{amatrix}$ + \item In each pivot col, all entries other than pivot point must be 0. Work backwards + \subitem $R_1 + -3R_1$, $R_2 + -R_1$ + \subitem $\begin{amatrix}{5} + 1 & 2 & 4 & 5 & 0 & 3 \\ + 0 & 0 & 1 & 2 & 0 & 8 \\ + 0 & 0 & 0 & 0 & 1 & -4 \\ + \end{amatrix}$ + \subitem $R_1 + -4R_2$ + \subitem $\begin{amatrix}{5} + 1 & 2 & 0 & -3 & 0 & -29 \\ + 0 & 0 & 1 & 2 & 0 & 8 \\ + 0 & 0 & 0 & 0 & 1 & -4 \\ + \end{amatrix}$ + + \end{enumerate} +\end{note} + + diff --git a/ma1522/master.pdf b/ma1522/master.pdf index 21a01dbd44e9b21018af5d799bb37e4fb9b93e57..09968546fc0066039987cd7ba035a3bbda5585bb 100644 GIT binary patch delta 58182 zcmZs?Q>-pb(6+g3+qP}nwr$(bUbeN@Ubb!9wr$&*_nVo6Z!-TuC!Id%N;;jYtE%qG zSO(dT0IT~049u1s2uuNBW9JeOfN^zmF*mk{@!Gh~oOU?kfctan4?~EGR+uU!lk&$6 z0lWj;hc-cXb5L$!T@|sr+0k$!?UyW`)UiRj#TG1h4OP3SC>~9kdwMt4y6Nk0SiUi4 z%4?wg<6XMntF(Y%>1^r#Yy9hUczg+SQ9>4UO?xj}-vqJmB3=wYVl0mmrZcnQ-719Y>&u;`5H@{0BC)wKJ<|IPb%#}jVxE` z4Z(5Yq30;5&(($jx=@(<9Izj8f3}6<{uQ;-3hv2?kOB8;Huh3d{OuAe4$-;fbpkK{ zwCIwe+hAEHaZxJ*n4v6!b*(??sQ;eC5VhAE02*EKYCW!0@p?G1Q8Y=OgJde4;Lqq5ZMlWzei 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one more part: Examples. % Set style and colour later. diff --git a/ma1522/preamble.tex b/ma1522/preamble.tex index 4a068b9..146e28e 100644 --- a/ma1522/preamble.tex +++ b/ma1522/preamble.tex @@ -9,7 +9,10 @@ \usepackage[english]{babel} \usepackage{framed} \usepackage[dvipsnames]{xcolor} +\usepackage{soul} \usepackage{tcolorbox} +\usepackage{systeme} +\usepackage{multicol} \colorlet{LightGray}{White!90!Periwinkle} \colorlet{LightOrange}{Orange!15} @@ -34,6 +37,10 @@ \declaretheorem[style=notesty,numbered=no]{note} \tcolorboxenvironment{note}{colback=LightRed} +\declaretheoremstyle[name=Definition,]{defnsty} +\declaretheorem[style=defnsty,numbered=no]{defn} +\tcolorboxenvironment{defn}{colback=LightGray} + \setstretch{1.2} \geometry{ textheight=9in,