{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import numpy.linalg as LA"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solve Matrix Equation : $\\large{Ax = b}$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example: 1 Modelling "
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>no.</th>\n",
" <th>f</th>\n",
" <th>x</th>\n",
" <th>y</th>\n",
" <th>z</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>1</td>\n",
" <td>0.720482</td>\n",
" <td>0.737451</td>\n",
" <td>4.10</td>\n",
" <td>0.8695</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>2</td>\n",
" <td>0.812391</td>\n",
" <td>0.884248</td>\n",
" <td>3.65</td>\n",
" <td>0.5750</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>3</td>\n",
" <td>0.813530</td>\n",
" <td>0.863871</td>\n",
" <td>3.55</td>\n",
" <td>0.5125</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>4</td>\n",
" <td>0.532572</td>\n",
" <td>0.872902</td>\n",
" <td>3.85</td>\n",
" <td>0.6520</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>5</td>\n",
" <td>0.697305</td>\n",
" <td>0.891665</td>\n",
" <td>4.70</td>\n",
" <td>0.5400</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>6</td>\n",
" <td>0.733407</td>\n",
" <td>0.880975</td>\n",
" <td>0.70</td>\n",
" <td>0.7745</td>\n",
" </tr>\n",
" <tr>\n",
" <th>6</th>\n",
" <td>7</td>\n",
" <td>0.654187</td>\n",
" <td>0.733362</td>\n",
" <td>2.25</td>\n",
" <td>0.7130</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7</th>\n",
" <td>8</td>\n",
" <td>0.709219</td>\n",
" <td>0.839979</td>\n",
" <td>-1.10</td>\n",
" <td>0.9320</td>\n",
" </tr>\n",
" <tr>\n",
" <th>8</th>\n",
" <td>9</td>\n",
" <td>0.750032</td>\n",
" <td>0.917038</td>\n",
" <td>-1.25</td>\n",
" <td>0.7335</td>\n",
" </tr>\n",
" <tr>\n",
" <th>9</th>\n",
" <td>10</td>\n",
" <td>0.713197</td>\n",
" <td>0.915913</td>\n",
" <td>0.90</td>\n",
" <td>0.7345</td>\n",
" </tr>\n",
" <tr>\n",
" <th>10</th>\n",
" <td>11</td>\n",
" <td>0.521182</td>\n",
" <td>0.885547</td>\n",
" <td>-0.15</td>\n",
" <td>0.7685</td>\n",
" </tr>\n",
" <tr>\n",
" <th>11</th>\n",
" <td>12</td>\n",
" <td>0.824820</td>\n",
" <td>0.884369</td>\n",
" <td>0.60</td>\n",
" <td>0.7090</td>\n",
" </tr>\n",
" <tr>\n",
" <th>12</th>\n",
" <td>13</td>\n",
" <td>0.770785</td>\n",
" <td>0.873444</td>\n",
" <td>1.25</td>\n",
" <td>0.4890</td>\n",
" </tr>\n",
" <tr>\n",
" <th>13</th>\n",
" <td>14</td>\n",
" <td>0.772823</td>\n",
" <td>0.726831</td>\n",
" <td>3.40</td>\n",
" <td>0.5375</td>\n",
" </tr>\n",
" <tr>\n",
" <th>14</th>\n",
" <td>15</td>\n",
" <td>0.600803</td>\n",
" <td>0.638667</td>\n",
" <td>1.80</td>\n",
" <td>0.8985</td>\n",
" </tr>\n",
" <tr>\n",
" <th>15</th>\n",
" <td>16</td>\n",
" <td>0.792857</td>\n",
" <td>0.753639</td>\n",
" <td>1.10</td>\n",
" <td>0.5705</td>\n",
" </tr>\n",
" <tr>\n",
" <th>16</th>\n",
" <td>17</td>\n",
" <td>0.800259</td>\n",
" <td>0.819985</td>\n",
" <td>0.80</td>\n",
" <td>0.4965</td>\n",
" </tr>\n",
" <tr>\n",
" <th>17</th>\n",
" <td>18</td>\n",
" <td>0.537606</td>\n",
" <td>0.847556</td>\n",
" <td>-0.25</td>\n",
" <td>0.8340</td>\n",
" </tr>\n",
" <tr>\n",
" <th>18</th>\n",
" <td>19</td>\n",
" <td>0.575679</td>\n",
" <td>0.884614</td>\n",
" <td>2.85</td>\n",
" <td>0.7870</td>\n",
" </tr>\n",
" <tr>\n",
" <th>19</th>\n",
" <td>20</td>\n",
" <td>0.821320</td>\n",
" <td>0.863913</td>\n",
" <td>0.75</td>\n",
" <td>0.6245</td>\n",
" </tr>\n",
" <tr>\n",
" <th>20</th>\n",
" <td>21</td>\n",
" <td>0.325209</td>\n",
" <td>0.757005</td>\n",
" <td>-0.20</td>\n",
" <td>0.7630</td>\n",
" </tr>\n",
" <tr>\n",
" <th>21</th>\n",
" <td>22</td>\n",
" <td>0.790763</td>\n",
" <td>0.899136</td>\n",
" <td>5.30</td>\n",
" <td>0.4470</td>\n",
" </tr>\n",
" <tr>\n",
" <th>22</th>\n",
" <td>23</td>\n",
" <td>0.803798</td>\n",
" <td>0.897501</td>\n",
" <td>3.45</td>\n",
" <td>0.3915</td>\n",
" </tr>\n",
" <tr>\n",
" <th>23</th>\n",
" <td>24</td>\n",
" <td>0.785717</td>\n",
" <td>0.895834</td>\n",
" <td>4.40</td>\n",
" <td>0.4190</td>\n",
" </tr>\n",
" <tr>\n",
" <th>24</th>\n",
" <td>25</td>\n",
" <td>0.754932</td>\n",
" <td>0.808529</td>\n",
" <td>3.80</td>\n",
" <td>0.5265</td>\n",
" </tr>\n",
" <tr>\n",
" <th>25</th>\n",
" <td>26</td>\n",
" <td>0.793982</td>\n",
" <td>0.892413</td>\n",
" <td>5.10</td>\n",
" <td>0.4350</td>\n",
" </tr>\n",
" <tr>\n",
" <th>26</th>\n",
" <td>27</td>\n",
" <td>0.789723</td>\n",
" <td>0.881186</td>\n",
" <td>3.40</td>\n",
" <td>0.4320</td>\n",
" </tr>\n",
" <tr>\n",
" <th>27</th>\n",
" <td>28</td>\n",
" <td>0.797951</td>\n",
" <td>0.936163</td>\n",
" <td>3.20</td>\n",
" <td>0.5280</td>\n",
" </tr>\n",
" <tr>\n",
" <th>28</th>\n",
" <td>29</td>\n",
" <td>0.796906</td>\n",
" <td>0.934264</td>\n",
" <td>2.80</td>\n",
" <td>0.4445</td>\n",
" </tr>\n",
" <tr>\n",
" <th>29</th>\n",
" <td>30</td>\n",
" <td>0.800149</td>\n",
" <td>0.941757</td>\n",
" <td>3.00</td>\n",
" <td>0.4490</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" no. f x y z\n",
"0 1 0.720482 0.737451 4.10 0.8695\n",
"1 2 0.812391 0.884248 3.65 0.5750\n",
"2 3 0.813530 0.863871 3.55 0.5125\n",
"3 4 0.532572 0.872902 3.85 0.6520\n",
"4 5 0.697305 0.891665 4.70 0.5400\n",
"5 6 0.733407 0.880975 0.70 0.7745\n",
"6 7 0.654187 0.733362 2.25 0.7130\n",
"7 8 0.709219 0.839979 -1.10 0.9320\n",
"8 9 0.750032 0.917038 -1.25 0.7335\n",
"9 10 0.713197 0.915913 0.90 0.7345\n",
"10 11 0.521182 0.885547 -0.15 0.7685\n",
"11 12 0.824820 0.884369 0.60 0.7090\n",
"12 13 0.770785 0.873444 1.25 0.4890\n",
"13 14 0.772823 0.726831 3.40 0.5375\n",
"14 15 0.600803 0.638667 1.80 0.8985\n",
"15 16 0.792857 0.753639 1.10 0.5705\n",
"16 17 0.800259 0.819985 0.80 0.4965\n",
"17 18 0.537606 0.847556 -0.25 0.8340\n",
"18 19 0.575679 0.884614 2.85 0.7870\n",
"19 20 0.821320 0.863913 0.75 0.6245\n",
"20 21 0.325209 0.757005 -0.20 0.7630\n",
"21 22 0.790763 0.899136 5.30 0.4470\n",
"22 23 0.803798 0.897501 3.45 0.3915\n",
"23 24 0.785717 0.895834 4.40 0.4190\n",
"24 25 0.754932 0.808529 3.80 0.5265\n",
"25 26 0.793982 0.892413 5.10 0.4350\n",
"26 27 0.789723 0.881186 3.40 0.4320\n",
"27 28 0.797951 0.936163 3.20 0.5280\n",
"28 29 0.796906 0.934264 2.80 0.4445\n",
"29 30 0.800149 0.941757 3.00 0.4490"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df = pd.read_csv('data/data1.csv')\n",
"df"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{equation}\n",
"f=a+bx+cy+dz\n",
"\\end{equation}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Normal equation are\n",
"\n",
"$\\sum_{}^{}{f}=na+b\\sum{x}+c\\sum{y}+d\\sum{z}$\n",
"\n",
"$\\sum_{}^{}{xf}=a\\sum{x}+b\\sum{x^2}+c\\sum{xy}+d\\sum{xz}$\n",
"\n",
"$\\sum_{}^{}{yf}=a\\sum{y}+b\\sum{xy}+c\\sum{y^2}+d\\sum{yz}$\n",
"\n",
"$\\sum_{}^{}{zf}=a\\sum{z}+b\\sum{xz}+c\\sum{zy}+d\\sum{z^2}$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{equation}\n",
"A=\n",
"\\begin{pmatrix}\n",
"a11 & a12 & a13 &a14\\\\\n",
"a21 & a22 & a23 &a24\\\\\n",
"a31 & a32 & a33 &a34\\\\\n",
"a41 & a42 & a43 &a44\\\\\n",
"\\end{pmatrix}\n",
"\\end{equation}\n",
"\\begin{equation}\n",
"b=\n",
"\\begin{pmatrix}\n",
"b1 \\\\\n",
"b2 \\\\\n",
"b3 \\\\\n",
"b4\\\\\n",
"\\end{pmatrix}\n",
"\\end{equation}"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"n=30\n",
"a11=n\n",
"a12=np.sum(df['x']) #sum of x\n",
"a13=np.sum(df['y']) #sum of y\n",
"a14=np.sum(df['z']) #sum of z\n",
"a21=np.sum(df['x']) #sum of x\n",
"a22=np.sum(df['x']**2) #sum of x2\n",
"a23=np.sum(np.dot(df['x'],df['y'])) #sum of xy\n",
"a24=np.sum(np.dot(df['x'],df['z'])) #sum of xz\n",
"a31=np.sum(df['y']) #sum of y\n",
"a32=np.sum(np.dot(df['y'],df['x'])) #sum of xy\n",
"a33=np.sum(df['y']**2) #sum of y2 \n",
"a34=np.sum(np.dot(df['y'],df['z'])) #sum of yz\n",
"a41=np.sum(df['z']) #sum of z\n",
"a42=np.sum(np.dot(df['z'],df['x'])) #sum of xz\n",
"a43=np.sum(np.dot(df['z'],df['y'])) #sum of yz\n",
"a44=np.sum(df['z']**2) #sum of z2\n",
"b1=np.sum(df['f']) #sum of f\n",
"b2=np.sum(np.dot(df['f'],df['x'])) #sum of fx\n",
"b3=np.sum(np.dot(df['f'],df['y'])) #sum of fy\n",
"b4=np.sum(np.dot(df['f'],df['z'])) #sum of fz"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0.85507492, 0.14860913, -0.00131447, -0.41792768])"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = np.array([[a11,a12,a13,a14],[a21,a22,a23,a24],[a31,a32,a33,a34],[a41,a42,a43,a44]])\n",
"b = np.array([b1,b2,b3,b4])\n",
"x = LA.solve(A,b)\n",
"x"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" a= 0.855\n",
"\n",
" b= 0.149\n",
"\n",
" c= -0.00131\n",
"\n",
" d= -0.418\n"
]
}
],
"source": [
"print('\\n a=',\"{:10.3}\".format(x[0]))\n",
"print('\\n b=',\"{:.3}\".format(x[1]))\n",
"print('\\n c=',\"{:.3}\".format(x[2]))\n",
"print('\\n d=',\"{:.3}\".format(x[3]))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[0.5958894003599622,\n",
" 0.7413760360958024,\n",
" 0.7645997139268501,\n",
" 0.7072466193451008,\n",
" 0.7557255547611699,\n",
" 0.6613906839806531,\n",
" 0.6631192534951209,\n",
" 0.5918408249814666,\n",
" 0.68644830702124,\n",
" 0.6830369792734646,\n",
" 0.6656950521790324,\n",
" 0.6894008781179288,\n",
" 0.7788669964212885,\n",
" 0.7339832421031306,\n",
" 0.5721125629994265,\n",
" 0.7271988454495809,\n",
" 0.7683794858588099,\n",
" 0.6328064581873758,\n",
" 0.6538812940618572,\n",
" 0.7214786253986405,\n",
" 0.6489567737789436,\n",
" 0.7949143877809594,\n",
" 0.8202981096648657,\n",
" 0.8073086989485949,\n",
" 0.7501958066178496,\n",
" 0.7991933185714644,\n",
" 0.8010132155323733,\n",
" 0.7693252220065355,\n",
" 0.8044656816412954,\n",
" 0.8034356104392159]"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f_exp=[] # expected value of f\n",
"for i in range(n):\n",
" f_exp.append(x[0]+x[1]*df.iloc[i,2]+x[2]*df.iloc[i,3]+x[3]*df.iloc[i,4])\n",
"f_exp "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{equation}\n",
"RMSE=\\sqrt{\\frac{1}{n}\\sum{(f-\\hat{f})^2}}\\\\\n",
"R^2=1-\\frac{\\sum{(f-\\hat{f})^2}}{\\sum{(f-\\bar{f})^2}}\n",
"\\end{equation}"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"RMSE=np.sqrt(np.sum((df['f']-f_exp)**2)/n)\n",
"R_square=1-(np.sum(np.sum(np.sum((df['f']-f_exp)**2))/np.sum((df['f']-df['f'].mean())**2)))"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" RMSE= 0.093\n",
"\n",
" R square= 0.36\n"
]
}
],
"source": [
"print('\\n RMSE=',\"{:.2}\".format(RMSE))\n",
"print('\\n R square=',\"{:.2}\".format(R_square))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example:2 Electric circuit"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Calculate I1,I2,I3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using Kirchhoff's laws in junction a,loop abdea,aegha"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$I3=I1+I2$\n",
"\n",
"$25.1I1+24+40.5I2=0$\n",
"\n",
"$-18+40.5I2+78.25I3=0$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 18.9867374, -12.3596817, 6.6270557])"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = np.array([[1,1,-1],[25.1,40.5,0],[0,40.5,78.25]]) #arranging I1,I2,I3\n",
"b = np.array([0,-24,18])\n",
"x = LA.solve(A,b)\n",
"x"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" I1= 19.0\n",
"\n",
" I2= -12.4\n",
"\n",
" I3= 6.63\n"
]
}
],
"source": [
"print('\\n I1=',\"{:.3}\".format(x[0]))\n",
"print('\\n I2=',\"{:.3}\".format(x[1]))\n",
"print('\\n I3=',\"{:.3}\".format(x[2]))\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.6"
},
"widgets": {
"state": {},
"version": "1.1.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}