122 \frametitle{Outline} |
122 \frametitle{Outline} |
123 \tableofcontents |
123 \tableofcontents |
124 % \pausesections |
124 % \pausesections |
125 \end{frame} |
125 \end{frame} |
126 |
126 |
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127 \section{Solving linear equations} |
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128 \begin{frame}[fragile] |
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129 \frametitle{Solution of equations} |
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130 Consider, |
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131 \begin{align*} |
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132 3x + 2y - z & = 1 \\ |
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133 2x - 2y + 4z & = -2 \\ |
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134 -x + \frac{1}{2}y -z & = 0 |
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135 \end{align*} |
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136 Solution: |
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137 \begin{align*} |
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138 x & = 1 \\ |
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139 y & = -2 \\ |
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140 z & = -2 |
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141 \end{align*} |
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142 \end{frame} |
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143 |
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144 \begin{frame}[fragile] |
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145 \frametitle{Solving using Matrices} |
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146 Let us now look at how to solve this using \kwrd{matrices} |
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147 \begin{lstlisting} |
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148 In []: A = matrix([[3,2,-1],[2,-2,4],[-1, 0.5, -1]]) |
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149 In []: b = matrix([[1], [-2], [0]]) |
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150 In []: x = linalg.solve(A, b) |
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151 In []: Ax = dot(A, x) |
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152 In []: allclose(Ax, b) |
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153 Out[]: True |
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154 \end{lstlisting} |
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155 \end{frame} |
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156 |
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157 \begin{frame}[fragile] |
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158 \frametitle{Solution:} |
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159 \begin{lstlisting} |
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160 In []: x |
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161 Out[]: |
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162 array([[ 1.], |
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163 [-2.], |
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164 [-2.]]) |
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165 |
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166 In []: Ax |
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167 Out[]: |
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168 matrix([[ 1.00000000e+00], |
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169 [ -2.00000000e+00], |
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170 [ 2.22044605e-16]]) |
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171 \end{lstlisting} |
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172 \end{frame} |
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173 |
127 \section{Matrices} |
174 \section{Matrices} |
128 \subsection{Initializing} |
175 \subsection{Initializing} |
129 \begin{frame}[fragile] |
176 \begin{frame}[fragile] |
130 \frametitle{Matrices: Initializing} |
177 \frametitle{Matrices: Initializing} |
131 \begin{lstlisting} |
178 \begin{lstlisting} |
142 \end{frame} |
189 \end{frame} |
143 |
190 |
144 \subsection{Basic Operations} |
191 \subsection{Basic Operations} |
145 \begin{frame}[fragile] |
192 \begin{frame}[fragile] |
146 \frametitle{Inverse of a Matrix} |
193 \frametitle{Inverse of a Matrix} |
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194 |
147 \begin{small} |
195 \begin{small} |
148 \begin{lstlisting} |
196 \begin{lstlisting} |
149 In []: linalg.inv(a) |
197 In []: linalg.inv(A) |
150 Out[]: |
198 Out[]: |
151 matrix([[ 3.15221191e+15, -6.30442381e+15, 3.15221191e+15], |
199 matrix([[ 0.07734807, 0.01657459, 0.32044199], |
152 [ -6.30442381e+15, 1.26088476e+16, -6.30442381e+15], |
200 [ 0.09944751, -0.12154696, -0.01657459], |
153 [ 3.15221191e+15, -6.30442381e+15, 3.15221191e+15]]) |
201 [-0.02762431, -0.07734807, 0.17127072]]) |
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202 |
154 \end{lstlisting} |
203 \end{lstlisting} |
155 \end{small} |
204 \end{small} |
156 \end{frame} |
205 \end{frame} |
157 |
206 |
158 \begin{frame}[fragile] |
207 \begin{frame}[fragile] |
174 \begin{frame}[fragile] |
223 \begin{frame}[fragile] |
175 \frametitle{Eigen Values and Eigen Matrix} |
224 \frametitle{Eigen Values and Eigen Matrix} |
176 \begin{small} |
225 \begin{small} |
177 \begin{lstlisting} |
226 \begin{lstlisting} |
178 In []: linalg.eigvals(a) |
227 In []: linalg.eigvals(a) |
179 Out[]: array([ 1.61168440e+01, -1.11684397e+00, -1.22196337e-15]) |
228 Out[]: array([1.61168440e+01, -1.11684397e+00, -1.22196337e-15]) |
180 |
229 |
181 In []: linalg.eig(a) |
230 In []: linalg.eig(a) |
182 Out[]: |
231 Out[]: |
183 (array([ 1.61168440e+01, -1.11684397e+00, -1.22196337e-15]), |
232 (array([ 1.61168440e+01, -1.11684397e+00, -1.22196337e-15]), |
184 matrix([[-0.23197069, -0.78583024, 0.40824829], |
233 matrix([[-0.23197069, -0.78583024, 0.40824829], |
185 [-0.52532209, -0.08675134, -0.81649658], |
234 [-0.52532209, -0.08675134, -0.81649658], |
186 [-0.8186735 , 0.61232756, 0.40824829]])) |
235 [-0.8186735 , 0.61232756, 0.40824829]])) |
187 \end{lstlisting} |
236 \end{lstlisting} |
188 \end{small} |
237 \end{small} |
189 \end{frame} |
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190 |
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191 \section{Solving linear equations} |
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192 \begin{frame}[fragile] |
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193 \frametitle{Solution of equations} |
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194 Example problem: Consider the set of equations |
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195 \vspace{-0.1in} |
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196 \begin{align*} |
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197 3x + 2y - z & = 1 \\ |
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198 2x - 2y + 4z & = -2 \\ |
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199 -x + \frac{1}{2}y -z & = 0 |
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200 \end{align*} |
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201 \vspace{-0.08in} |
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202 To Solve this, |
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203 \begin{lstlisting} |
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204 In []: A = array([[3,2,-1],[2,-2,4],[-1, 0.5, -1]]) |
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205 In []: b = array([1, -2, 0]) |
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206 In []: x = linalg.solve(A, b) |
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207 In []: Ax = dot(A, x) |
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208 In []: allclose(Ax, b) |
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209 Out[]: True |
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210 \end{lstlisting} |
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211 \end{frame} |
238 \end{frame} |
212 |
239 |
213 |
240 |
214 \section{Integration} |
241 \section{Integration} |
215 |
242 |