1 .. Objectives |
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2 .. ---------- |
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3 |
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4 .. By the end of this tutorial you will -- |
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5 |
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6 .. 1. Get an idea of the range of things for which Sage can be used. |
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7 .. #. Know some of the functions for Calculus |
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8 .. #. Get some insight into Graphs in Sage. |
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9 |
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10 |
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11 .. Prerequisites |
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12 .. ------------- |
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13 |
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14 .. Getting Started -- Sage |
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15 |
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16 .. Author : Puneeth |
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17 Internal Reviewer : Anoop Jacob Thomas<anoop@fossee.in> |
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18 External Reviewer : |
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19 Language Reviewer : Bhanukiran |
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20 Checklist OK? : <06-11-2010, Anand, OK> [2010-10-05] |
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21 |
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22 Script |
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23 ------ |
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24 |
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25 {{{ show the welcome slide }}} |
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26 |
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27 Hello Friends. Welcome to this tutorial on using Sage. |
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28 |
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29 {{{ show the slide with outline }}} |
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30 |
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31 In this tutorial we shall quickly look at a few examples of using Sage |
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32 for Linear Algebra, Calculus, Graph Theory and Number theory. |
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33 |
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34 {{{ show the slide with Calculus outline }}} |
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35 |
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36 Let us begin with Calculus. We shall be looking at limits, |
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37 differentiation, integration, and Taylor polynomial. |
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38 |
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39 {{{ show sage notebook }}} |
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40 |
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41 We have our Sage notebook running. In case, you don't have it running, |
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42 start is using the command, ``sage --notebook``. |
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43 |
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44 To find the limit of the function x*sin(1/x), at x=0, we say |
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45 :: |
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46 |
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47 lim(x*sin(1/x), x=0) |
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48 |
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49 We get the limit to be 0, as expected. |
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50 |
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51 It is also possible to the limit at a point from one direction. For |
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52 example, let us find the limit of 1/x at x=0, when approaching from |
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53 the positive side. |
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54 :: |
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55 |
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56 lim(1/x, x=0, dir='above') |
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57 |
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58 To find the limit from the negative side, we say, |
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59 :: |
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60 |
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61 lim(1/x, x=0, dir='below') |
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62 |
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63 Let us now see how to differentiate, using Sage. We shall find the |
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64 differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``. We |
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65 shall first define the expression, and then use the ``diff`` function |
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66 to obtain the differential of the expression. |
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67 :: |
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68 |
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69 var('x') |
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70 f = exp(sin(x^2))/x |
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71 |
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72 diff(f, x) |
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73 |
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74 We can also obtain the partial differentiation of an expression w.r.t |
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75 one of the variables. Let us differentiate the expression |
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76 ``exp(sin(y - x^2))/x`` w.r.t x and y. |
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77 :: |
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78 |
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79 var('x y') |
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80 f = exp(sin(y - x^2))/x |
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81 |
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82 diff(f, x) |
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83 |
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84 diff(f, y) |
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85 |
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86 Now, let us look at integration. We shall use the expression obtained |
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87 from the differentiation that we did before, ``diff(f, y)`` --- |
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88 ``e^(sin(-x^2 + y))*cos(-x^2 + y)/x``. The ``integrate`` command is |
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89 used to obtain the integral of an expression or function. |
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90 :: |
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91 |
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92 integrate(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y) |
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93 |
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94 We get back the correct expression. The minus sign being inside or |
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95 outside the ``sin`` function doesn't change much. |
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96 |
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97 Now, let us find the value of the integral between the limits 0 and |
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98 pi/2. |
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99 :: |
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100 |
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101 integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2) |
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102 |
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103 Let us now see how to obtain the Taylor expansion of an expression |
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104 using sage. Let us obtain the Taylor expansion of ``(x + 1)^n`` up to |
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105 degree 4 about 0. |
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106 :: |
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107 |
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108 var('x n') |
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109 taylor((x+1)^n, x, 0, 4) |
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110 |
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111 This brings us to the end of the features of Sage for Calculus, that |
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112 we will be looking at. For more, look at the Calculus quick-ref from |
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113 the Sage Wiki. |
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114 |
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115 Next let us move on to Matrix Algebra. |
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116 |
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117 {{{ show the equation on the slides }}} |
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118 |
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119 Let us begin with solving the equation ``Ax = v``, where A is the |
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120 matrix ``matrix([[1,2],[3,4]])`` and v is the vector |
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121 ``vector([1,2])``. |
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122 |
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123 To solve the equation, ``Ax = v`` we simply say |
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124 :: |
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125 |
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126 x = solve_right(A, v) |
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127 |
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128 To solve the equation, ``xA = v`` we simply say |
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129 :: |
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130 |
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131 x = solve_left(A, v) |
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132 |
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133 The left and right here, denote the position of ``A``, relative to x. |
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134 |
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135 #[Puneeth]: any suggestions on what more to add? |
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136 |
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137 Now, let us look at Graph Theory in Sage. |
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138 |
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139 We shall look at some ways to create graphs and some of the graph |
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140 families available in Sage. |
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141 |
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142 The simplest way to define an arbitrary graph is to use a dictionary |
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143 of lists. We create a simple graph by |
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144 :: |
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145 |
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146 G = Graph({0:[1,2,3], 2:[4]}) |
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147 |
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148 We say |
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149 :: |
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150 |
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151 G.show() |
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152 |
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153 to view the visualization of the graph. |
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154 |
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155 Similarly, we can obtain a directed graph using the ``DiGraph`` |
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156 function. |
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157 :: |
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158 |
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159 G = DiGraph({0:[1,2,3], 2:[4]}) |
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160 |
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161 |
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162 Sage also provides a lot of graph families which can be viewed by |
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163 typing ``graph.<tab>``. Let us obtain a complete graph with 5 vertices |
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164 and then show the graph. |
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165 :: |
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166 |
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167 G = graphs.CompleteGraph(5) |
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168 |
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169 G.show() |
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170 |
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171 |
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172 Sage provides other functions for Number theory and |
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173 Combinatorics. Let's have a glimpse of a few of them. |
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174 |
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175 |
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176 :: |
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177 |
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178 prime_range(100, 200) |
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179 |
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180 gives primes in the range 100 to 200. |
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181 |
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182 :: |
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183 |
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184 is_prime(1999) |
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185 |
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186 checks if 1999 is a prime number or not. |
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187 |
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188 :: |
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189 |
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190 factor(2001) |
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191 |
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192 gives the factorized form of 2001. |
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193 |
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194 :: |
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195 |
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196 C = Permutations([1, 2, 3, 4]) |
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197 C.list() |
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198 |
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199 gives the permutations of ``[1, 2, 3, 4]`` |
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200 |
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201 :: |
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202 |
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203 C = Combinations([1, 2, 3, 4]) |
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204 C.list() |
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205 |
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206 gives all the combinations of ``[1, 2, 3, 4]`` |
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207 |
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208 That brings us to the end of this session showing various features |
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209 available in Sage. |
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210 |
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211 .. #[[Anoop: I feel we should add more slides, a possibility is to add |
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212 the code which they are required to type in, I also feel we should |
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213 add some review problems for them to try out.]] |
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214 |
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215 {{{ Show summary slide }}} |
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216 |
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217 We have looked at some of the functions available for Linear Algebra, |
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218 Calculus, Graph Theory and Number theory. |
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219 |
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220 This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India |
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221 |
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222 Hope you have enjoyed and found it useful. |
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223 Thank you! |
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224 |
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