1 Symbolics with Sage |
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2 ------------------- |
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3 |
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4 Hello friends and welcome to this tutorial on symbolics with sage. |
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5 |
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6 |
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7 .. #[Madhu: Sounds more or less like an ad!] |
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8 |
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9 {{{ Part of Notebook with title }}} |
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10 |
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11 .. #[Madhu: Please make your instructions, instructional. While |
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12 recording if I have to read this, think what you are actually |
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13 meaning it will take a lot of time] |
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14 |
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15 We would be using simple mathematical functions on the sage notebook |
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16 for this tutorial. |
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17 |
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18 .. #[Madhu: What is this line doing here. I don't see much use of it] |
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19 |
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20 During the course of the tutorial we will learn |
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21 |
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22 {{{ Part of Notebook with outline }}} |
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23 |
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24 To define symbolic expressions in sage. Use built-in costants and |
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25 function. Integration, differentiation using sage. Defining |
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26 matrices. Defining Symbolic functions. Simplifying and solving |
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27 symbolic expressions and functions. |
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28 |
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29 .. #[Nishanth]: The formatting is all messed up |
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30 First fix the formatting and compile the rst |
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31 The I shall review |
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32 .. #[Madhu: Please make the above items full english sentences, not |
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33 the slides like points. The person recording should be able to |
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34 read your script as is. It can read something like "we will learn |
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35 how to define symbolic expressions in Sage, using built-in ..."] |
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36 |
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37 Using sage we can perform mathematical operations on symbols. |
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38 |
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39 .. #[Madhu: Same mistake with period symbols! Please get the |
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40 punctuation right. Also you may have to rephrase the above |
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41 sentence as "We can use Sage to perform sybmolic mathematical |
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42 operations" or such] |
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43 |
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44 On the sage notebook type:: |
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45 |
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46 sin(y) |
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47 |
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48 It raises a name error saying that y is not defined. But in sage we |
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49 can declare y as a symbol using var function. |
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50 |
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51 .. #[Madhu: But is not required] |
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52 :: |
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53 var('y') |
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54 |
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55 Now if you type:: |
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56 |
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57 sin(y) |
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58 |
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59 sage simply returns the expression . |
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60 |
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61 .. #[Madhu: Why is this line indented? Also full stop. When will you |
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62 learn? Yes we can correct you. But corrections are for you to |
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63 learn. If you don't learn from your mistakes, I don't know what |
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64 to say] |
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65 |
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66 thus now sage treats sin(y) as a symbolic expression . You can use |
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67 this to do a lot of symbolic maths using sage's built-in constants and |
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68 expressions . |
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69 |
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70 .. #[Madhu: "Thus now"? It sounds like Dus and Nou, i.e 10 and 9 in |
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71 Hindi! Full stop again. "a lot" doesn't mean anything until you |
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72 quantify it or give examples.] |
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73 |
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74 Try out |
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75 |
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76 .. #[Madhu: "So let us try" sounds better] |
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77 :: |
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78 |
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79 var('x,alpha,y,beta') x^2/alpha^2+y^2/beta^2 |
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80 |
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81 Similarly , we can define many algebraic and trigonometric expressions |
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82 using sage . |
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83 |
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84 .. #[Madhu: comma again. Show some more examples?] |
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85 |
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86 |
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87 Sage also provides a few built-in constants which are commonly used in |
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88 mathematics . |
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89 |
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90 example : pi,e,oo , Function n gives the numerical values of all these |
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91 constants. |
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92 |
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93 .. #[Madhu: This doesn't sound like scripts. How will I read this |
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94 while recording. Also if I were recording I would have read your |
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95 third constant as Oh-Oh i.e. double O. It took me at least 30 |
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96 seconds to figure out it is infinity] |
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97 |
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98 For instance:: |
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99 |
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100 n(e) |
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101 |
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102 2.71828182845905 |
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103 |
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104 gives numerical value of e. |
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105 |
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106 If you look into the documentation of n by doing |
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107 |
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108 .. #[Madhu: "documentation of the function "n"?] |
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109 |
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110 :: |
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111 n(<Tab> |
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112 |
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113 You will see what all arguments it can take etc .. It will be very |
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114 helpful if you look at the documentation of all functions introduced |
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115 |
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116 .. #[Madhu: What does etc .. mean in a script?] |
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117 |
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118 Also we can define the no of digits we wish to use in the numerical |
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119 value . For this we have to pass an argument digits. Type |
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120 |
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121 .. #[Madhu: "no of digits"? Also "We wish to obtain" than "we wish to |
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122 use"?] |
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123 :: |
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124 |
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125 n(pi, digits = 10) |
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126 |
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127 Apart from the constants sage also has a lot of builtin functions like |
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128 sin,cos,sinh,cosh,log,factorial,gamma,exp,arcsin,arccos,arctan etc ... |
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129 lets try some out on the sage notebook. |
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130 |
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131 .. #[Madhu: Here "a lot" makes sense] |
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132 :: |
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133 |
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134 sin(pi/2) |
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135 |
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136 arctan(oo) |
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137 |
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138 log(e,e) |
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139 |
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140 |
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141 Given that we have defined variables like x,y etc .. , We can define |
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142 an arbitrary function with desired name in the following way.:: |
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143 |
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144 var('x') function(<tab> {{{ Just to show the documentation |
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145 extend this line }}} function('f',x) |
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146 |
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147 .. #[Madhu: What will the person recording show in the documentation |
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148 without a script for it? Please don't assume recorder can cook up |
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149 things while recording. It is impractical] |
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150 |
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151 Here f is the name of the function and x is the independent variable . |
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152 Now we can define f(x) to be :: |
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153 |
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154 f(x) = x/2 + sin(x) |
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155 |
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156 Evaluating this function f for the value x=pi returns pi/2.:: |
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157 |
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158 f(pi) |
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159 |
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160 We can also define functions that are not continuous but defined |
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161 piecewise. We will be using a function which is a parabola between 0 |
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162 to 1 and a constant from 1 to 2 . type the following as given on the |
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163 screen |
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164 |
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165 .. #[Madhu: Instead of "We will be using ..." how about "Let us define |
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166 a function ..."] |
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167 :: |
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168 |
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169 |
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170 var('x') h(x)=x^2 g(x)=1 f=Piecewise(<Tab> {{{ Just to show the |
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171 documentation extend this line }}} |
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172 f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) f |
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173 |
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174 Checking f at 0.4, 1.4 and 3 :: f(0.4) f(1.4) f(3) |
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175 |
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176 .. #[Madhu: Again this doesn't sound like a script] |
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177 |
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178 for f(3) it raises a value not defined in domain error . |
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179 |
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180 |
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181 Apart from operations on expressions and functions one can also use |
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182 them for series . |
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183 |
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184 .. #[Madhu: I am not able to understand this line. "Use them as |
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185 .. series". Use what as series?] |
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186 |
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187 We first define a function f(n) in the way discussed above.:: |
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188 |
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189 var('n') function('f', n) |
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190 |
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191 .. #[Madhu: Shouldn't this be on 2 separate lines?] |
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192 |
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193 To sum the function for a range of discrete values of n, we use the |
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194 sage function sum. |
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195 |
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196 For a convergent series , f(n)=1/n^2 we can say :: |
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197 |
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198 var('n') function('f', n) |
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199 |
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200 f(n) = 1/n^2 |
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201 |
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202 sum(f(n), n, 1, oo) |
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203 |
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204 For the famous Madhava series :: var('n') function('f', n) |
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205 |
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206 .. #[Madhu: What is this? your double colon says it must be code block |
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207 but where is the indentation and other things. How will the |
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208 recorder know about it?] |
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209 |
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210 f(n) = (-1)^(n-1)*1/(2*n - 1) |
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211 |
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212 This series converges to pi/4. It was used by ancient Indians to |
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213 interpret pi. |
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214 |
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215 .. #[Madhu: I am losing the context. Please add something to bring |
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216 this thing to the context] |
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217 |
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218 For a divergent series, sum would raise a an error 'Sum is |
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219 divergent' :: |
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220 |
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221 var('n') |
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222 function('f', n) |
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223 f(n) = 1/n sum(f(n), n,1, oo) |
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224 |
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225 |
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226 |
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227 |
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228 We can perform simple calculus operation using sage |
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229 |
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230 .. #[Madhu: When you switch to irrelevant topics make sure you use |
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231 some connectors in English like "Moving on let us see how to |
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232 perform simple calculus operations using Sage" or something like |
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233 that] |
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234 For example lets try an expression first :: |
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235 |
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236 diff(x**2+sin(x),x) 2x+cos(x) |
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237 |
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238 The diff function differentiates an expression or a function . Its |
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239 first argument is expression or function and second argument is the |
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240 independent variable . |
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241 |
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242 .. #[Madhu: Full stop, Full stop, Full stop] |
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243 |
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244 We have already tried an expression now lets try a function :: |
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245 |
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246 f=exp(x^2)+arcsin(x) diff(f(x),x) |
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247 |
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248 To get a higher order differentiation we need to add an extra argument |
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249 for order :: |
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250 |
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251 diff(<tab> diff(f(x),x,3) |
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252 |
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253 .. #[Madhu: Please try to be more explicit saying third argument] |
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254 |
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255 in this case it is 3. |
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256 |
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257 |
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258 Just like differentiation of expression you can also integrate them :: |
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259 |
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260 x = var('x') s = integral(1/(1 + (tan(x))**2),x) s |
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261 |
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262 .. #[Madhu: Two separate lines.] |
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263 |
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264 To find the factors of an expression use the "factor" function |
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265 |
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266 .. #[Madhu: See the diff] |
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267 |
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268 :: |
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269 factor(<tab> y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) f = |
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270 factor(y) |
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271 |
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272 One can also simplify complicated expression using sage :: |
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273 f.simplify_full() |
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274 |
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275 This simplifies the expression fully . You can also do simplification |
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276 of just the algebraic part and the trigonometric part :: |
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277 |
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278 f.simplify_exp() f.simplify_trig() |
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279 |
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280 .. #[Madhu: Separate lines?] |
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281 |
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282 One can also find roots of an equation by using find_root function:: |
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283 |
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284 phi = var('phi') find_root(cos(phi)==sin(phi),0,pi/2) |
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285 |
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286 .. #[Madhu: Separate lines?] |
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287 |
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288 Lets substitute this solution into the equation and see we were |
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289 correct :: |
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290 |
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291 var('phi') f(phi)=cos(phi)-sin(phi) |
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292 root=find_root(f(phi)==0,0,pi/2) f.substitute(phi=root) |
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293 |
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294 .. #[Madhu: Separate lines?] |
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295 |
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296 as we can see the solution is almost equal to zero . |
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297 |
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298 .. #[Madhu: So what?] |
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299 |
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300 We can also define symbolic matrices :: |
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301 |
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302 |
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303 |
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304 var('a,b,c,d') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) A |
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305 |
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306 .. #[Madhu: Why don't you break the lines?] |
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307 |
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308 Now lets do some of the matrix operations on this matrix |
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309 |
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310 .. #[Madhu: Why don't you break the lines? Also how do you connect |
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311 this up? Use some transformation keywords in English] |
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312 :: |
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313 A.det() A.inverse() |
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314 |
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315 .. #[Madhu: Why don't you break the lines?] |
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316 |
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317 You can do :: |
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318 |
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319 A.<Tab> |
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320 |
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321 To see what all operations are available |
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322 |
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323 .. #[Madhu: Sounds very abrupt] |
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324 |
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325 {{{ Part of the notebook with summary }}} |
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326 |
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327 So in this tutorial we learnt how to |
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328 |
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329 |
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330 We learnt about defining symbolic expression and functions . |
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331 And some built-in constants and functions . |
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332 Getting value of built-in constants using n function. |
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333 Using Tab to see the documentation. |
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334 Also we learnt how to sum a series using sum function. |
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335 diff() and integrate() for calculus operations . |
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336 Finding roots , factors and simplifying expression using find_root(), |
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337 factor() , simplify_full, simplify_exp , simplify_trig . |
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338 Substituting values in expression using substitute function. |
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339 And finally creating symbolic matrices and performing operation on them . |
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340 |
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341 .. #[Madhu: See what Nishanth is doing. He has written this as |
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342 points. So easy to read out while recording. You may want to |
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343 reorganize like that] |
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