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| author | Sebastiano Tronto <sebastiano.tronto@gmail.com> | 2021-05-25 17:10:49 +0200 |
|---|---|---|
| committer | Sebastiano Tronto <sebastiano.tronto@gmail.com> | 2021-05-25 17:10:49 +0200 |
| commit | d6c61d988bfa4255baf9cdae42db59ebee38363f (patch) | |
| tree | 118ff3c2424e735149c145524965a4a337e50beb /Lectures | |
| parent | 46eef66b1e1571c77dc828d7e950b129b4c8bfd0 (diff) | |
| download | mathsoftware-d6c61d988bfa4255baf9cdae42db59ebee38363f.tar.gz mathsoftware-d6c61d988bfa4255baf9cdae42db59ebee38363f.zip | |
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| 1 | { | ||
| 2 | "cells": [ | ||
| 3 | { | ||
| 4 | "cell_type": "markdown", | ||
| 5 | "metadata": {}, | ||
| 6 | "source": [ | ||
| 7 | "This lecture's notes are in a different format: the presentations for the $\\LaTeX$ part were made with $\\LaTeX$, so this one is made with Sage, or rather with the [Jupyter Notebook](https://jupyter.org/).\n", | ||
| 8 | "\n", | ||
| 9 | "# The Jupyter Notebook\n", | ||
| 10 | "**Reference:** [[1](https://jupyter.org/documentation)]\n", | ||
| 11 | "\n", | ||
| 12 | "The Jupyter Notebook is one of the default interfaces for SageMath, along with the command line interface. You can access it via web browser, but it is running locally on your device (notice the strange url: `http://localhost:8888/notebooks...`).\n", | ||
| 13 | "\n", | ||
| 14 | "You can create a new notebook by clicking on `New > SageMath 9.2`. You can also create a Python 3 notebook to write Python code.\n", | ||
| 15 | "\n", | ||
| 16 | "Jupyter saves and reads files in the `.ipynb` format. If you download the file for this lecture you can open it and follow the examples interactively.\n", | ||
| 17 | "\n", | ||
| 18 | "## Cells\n", | ||
| 19 | "\n", | ||
| 20 | "The notebook contains one or more *interactive cells* that you can run, like this one below:" | ||
| 21 | ] | ||
| 22 | }, | ||
| 23 | { | ||
| 24 | "cell_type": "code", | ||
| 25 | "execution_count": 2, | ||
| 26 | "metadata": {}, | ||
| 27 | "outputs": [ | ||
| 28 | { | ||
| 29 | "data": { | ||
| 30 | "text/plain": [ | ||
| 31 | "2/5" | ||
| 32 | ] | ||
| 33 | }, | ||
| 34 | "execution_count": 2, | ||
| 35 | "metadata": {}, | ||
| 36 | "output_type": "execute_result" | ||
| 37 | } | ||
| 38 | ], | ||
| 39 | "source": [ | ||
| 40 | "# Exercise: modify this cell to use the print() command\n", | ||
| 41 | "2+2\n", | ||
| 42 | "2/5" | ||
| 43 | ] | ||
| 44 | }, | ||
| 45 | { | ||
| 46 | "cell_type": "markdown", | ||
| 47 | "metadata": {}, | ||
| 48 | "source": [ | ||
| 49 | "If you are reading this from Jupyter rather than from the pdf file, you can edit the cell above and run it again. You can also add more cells by selecting `Insert` from the menu bar.\n", | ||
| 50 | "\n", | ||
| 51 | "Notice that only the last statement produces an output. You can force anything to be written as output with the `print()` command, which works like in Python. As an exercise, try to modify the cell above to provide more output!" | ||
| 52 | ] | ||
| 53 | }, | ||
| 54 | { | ||
| 55 | "cell_type": "markdown", | ||
| 56 | "metadata": {}, | ||
| 57 | "source": [ | ||
| 58 | "## Markdown\n", | ||
| 59 | "\n", | ||
| 60 | "[Markdown](https://en.wikipedia.org/wiki/Markdown) is a simple markup language - think of LaTeX or html, but much simpler.\n", | ||
| 61 | "You can add text to your notebook with Markdown cells by selecting `Cell > Cell Type > Markdown`.\n", | ||
| 62 | "\n", | ||
| 63 | "You can also include some LaTeX code in Markdown cells, with dollar signs $ or align environments:\n", | ||
| 64 | "\n", | ||
| 65 | "\\begin{align*}\n", | ||
| 66 | "\\frac{(x+y)^2}{x+1} = \\frac{x^2+y^2}{x+1}\n", | ||
| 67 | "\\end{align*}\n", | ||
| 68 | "\n", | ||
| 69 | "When you are done writing a Markdown cell, you can run it to see the well-formatted text. To edit the text again, double-click on the cell. Try doing it now to fix the formula above!" | ||
| 70 | ] | ||
| 71 | }, | ||
| 72 | { | ||
| 73 | "cell_type": "markdown", | ||
| 74 | "metadata": {}, | ||
| 75 | "source": [ | ||
| 76 | "# Symbolic expressions\n", | ||
| 77 | "\n", | ||
| 78 | "**Reference:** [[2](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)]\n", | ||
| 79 | "\n", | ||
| 80 | "Now, let's get started with Sage. One thing you might want to do is manipulating symbolic expressions, like the following:" | ||
| 81 | ] | ||
| 82 | }, | ||
| 83 | { | ||
| 84 | "cell_type": "code", | ||
| 85 | "execution_count": 3, | ||
| 86 | "metadata": {}, | ||
| 87 | "outputs": [ | ||
| 88 | { | ||
| 89 | "data": { | ||
| 90 | "text/plain": [ | ||
| 91 | "[x == -sqrt(6) - 1, x == sqrt(6) - 1]" | ||
| 92 | ] | ||
| 93 | }, | ||
| 94 | "execution_count": 3, | ||
| 95 | "metadata": {}, | ||
| 96 | "output_type": "execute_result" | ||
| 97 | } | ||
| 98 | ], | ||
| 99 | "source": [ | ||
| 100 | "f = x^2 + 2*x - 5 == 0\n", | ||
| 101 | "solve(f,x)" | ||
| 102 | ] | ||
| 103 | }, | ||
| 104 | { | ||
| 105 | "cell_type": "markdown", | ||
| 106 | "metadata": {}, | ||
| 107 | "source": [ | ||
| 108 | "Notice that the single `=` is part of an assignment, as in Python: we are *assigning* to the variable `f` the value `x^2 + 2*x - 5 >= 0`, which in this case is an equation, so it contains the symbol `==`. Keep in mind the difference between the two!\n", | ||
| 109 | "\n", | ||
| 110 | "**Exercise:** change the code above to solve the corresponding inequality $x^2+2x-5\\geq 0$." | ||
| 111 | ] | ||
| 112 | }, | ||
| 113 | { | ||
| 114 | "cell_type": "markdown", | ||
| 115 | "metadata": {}, | ||
| 116 | "source": [ | ||
| 117 | "## Mathematical variables\n", | ||
| 118 | "\n", | ||
| 119 | "Last time we saw what *variables* are in Python, and that they are a little bit different from the *Mathematical variables* that you use in Mathematics. In Sage, both concepts are present, but they are still distinct. For example in the cell above `f` is a variable in the sense of computer science, while `x` is a Mathematical variable.\n", | ||
| 120 | "\n", | ||
| 121 | "If you want to use Mathematical variables other than `x`, you first need to *declare* them with the `var()` command:" | ||
| 122 | ] | ||
| 123 | }, | ||
| 124 | { | ||
| 125 | "cell_type": "code", | ||
| 126 | "execution_count": 14, | ||
| 127 | "metadata": {}, | ||
| 128 | "outputs": [ | ||
| 129 | { | ||
| 130 | "data": { | ||
| 131 | "text/plain": [ | ||
| 132 | "[y == -1/2*x - 1/2*sqrt(x^2 + 2*x + 9) - 1/2, y == -1/2*x + 1/2*sqrt(x^2 + 2*x + 9) - 1/2]" | ||
| 133 | ] | ||
| 134 | }, | ||
| 135 | "execution_count": 14, | ||
| 136 | "metadata": {}, | ||
| 137 | "output_type": "execute_result" | ||
| 138 | } | ||
| 139 | ], | ||
| 140 | "source": [ | ||
| 141 | "var('y')\n", | ||
| 142 | "solve(y^2 + (x+1)*y - 2 == 0, y)" | ||
| 143 | ] | ||
| 144 | }, | ||
| 145 | { | ||
| 146 | "cell_type": "markdown", | ||
| 147 | "metadata": {}, | ||
| 148 | "source": [ | ||
| 149 | "Try removing the first line in the cell above and see what error you get!\n", | ||
| 150 | "\n", | ||
| 151 | "Here is another example:" | ||
| 152 | ] | ||
| 153 | }, | ||
| 154 | { | ||
| 155 | "cell_type": "code", | ||
| 156 | "execution_count": 16, | ||
| 157 | "metadata": {}, | ||
| 158 | "outputs": [ | ||
| 159 | { | ||
| 160 | "data": { | ||
| 161 | "text/plain": [ | ||
| 162 | "[x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]" | ||
| 163 | ] | ||
| 164 | }, | ||
| 165 | "execution_count": 16, | ||
| 166 | "metadata": {}, | ||
| 167 | "output_type": "execute_result" | ||
| 168 | } | ||
| 169 | ], | ||
| 170 | "source": [ | ||
| 171 | "var('a', 'b')\n", | ||
| 172 | "f = x^2+a*x+b\n", | ||
| 173 | "solve(f,x)" | ||
| 174 | ] | ||
| 175 | }, | ||
| 176 | { | ||
| 177 | "cell_type": "markdown", | ||
| 178 | "metadata": {}, | ||
| 179 | "source": [ | ||
| 180 | "Some common constants are [already defined](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html) in Sage:" | ||
| 181 | ] | ||
| 182 | }, | ||
| 183 | { | ||
| 184 | "cell_type": "code", | ||
| 185 | "execution_count": 17, | ||
| 186 | "metadata": {}, | ||
| 187 | "outputs": [ | ||
| 188 | { | ||
| 189 | "data": { | ||
| 190 | "text/plain": [ | ||
| 191 | "-1" | ||
| 192 | ] | ||
| 193 | }, | ||
| 194 | "execution_count": 17, | ||
| 195 | "metadata": {}, | ||
| 196 | "output_type": "execute_result" | ||
| 197 | } | ||
| 198 | ], | ||
| 199 | "source": [ | ||
| 200 | "e^(pi*I)" | ||
| 201 | ] | ||
| 202 | }, | ||
| 203 | { | ||
| 204 | "cell_type": "markdown", | ||
| 205 | "metadata": {}, | ||
| 206 | "source": [ | ||
| 207 | "We will study symbolic expressions more in detail next time, in the context of calculus/analysis." | ||
| 208 | ] | ||
| 209 | }, | ||
| 210 | { | ||
| 211 | "cell_type": "markdown", | ||
| 212 | "metadata": {}, | ||
| 213 | "source": [ | ||
| 214 | "# Basic rings and fields\n", | ||
| 215 | "\n", | ||
| 216 | "**References:** [[3](https://doc.sagemath.org/html/en/reference/rings_standard/index.html)]\n", | ||
| 217 | "[[4](https://doc.sagemath.org/html/en/reference/rings_numerical/index.html)]\n", | ||
| 218 | "[[5](https://doc.sagemath.org/html/en/reference/finite_rings/index.html)]\n", | ||
| 219 | "\n", | ||
| 220 | "As you should know, a *field* is a Mathematical structure with two operations, addition and multiplication, which respect certain rules (distributivity, associativity, commutativity...). Some examples of fields are the Rational numbers $\\mathbb Q$, the Real numbers $\\mathbb R$ and the Complex numbers $\\mathbb C$, but there are many more. As you should also know, a *(commutative) ring* is like a field, except not all elements different from $0$ need have a multiplicative inverse. For example the integers $\\mathbb Z = \\{ \\dots, -1, 0, 1, 2, \\dots\\}$ are a ring, but not a field.\n", | ||
| 221 | "\n", | ||
| 222 | "These structures are already implemented in Sage. Some of the most common are listed in the following table:\n", | ||
| 223 | "\n", | ||
| 224 | "|Mathematical object|Math symbol|Sage name|\n", | ||
| 225 | "|------------------:|:---------:|:--------|\n", | ||
| 226 | "|Integers|$\\mathbb Z$|`ZZ`|\n", | ||
| 227 | "|Rational numbers|$\\mathbb Q$|`QQ`|\n", | ||
| 228 | "|Real numbers|$\\mathbb R$|`RR`|\n", | ||
| 229 | "|Complex numbers|$\\mathbb C$|`CC`|\n", | ||
| 230 | "|Integers modulo $n$|$\\mathbb Z/n\\mathbb Z$|`Integers(n)`|\n", | ||
| 231 | "|Finite fields|$\\mathbb F_p$|GF(p)|\n", | ||
| 232 | "|$\\dots$|$\\dots$|$\\dots$|" | ||
| 233 | ] | ||
| 234 | }, | ||
| 235 | { | ||
| 236 | "cell_type": "markdown", | ||
| 237 | "metadata": {}, | ||
| 238 | "source": [ | ||
| 239 | "If you write a number or an expression, Sage will figure out where it \"lives\", choosing the most restrictive interpretation possible. For example `3` will be interpreted to be an integer, even if it is also a rational number, a real number and a complex number." | ||
| 240 | ] | ||
| 241 | }, | ||
| 242 | { | ||
| 243 | "cell_type": "markdown", | ||
| 244 | "metadata": {}, | ||
| 245 | "source": [ | ||
| 246 | "## Parents and coercion\n", | ||
| 247 | "**Reference:** [[6](https://doc.sagemath.org/html/en/tutorial/tour_coercion.html)]\n", | ||
| 248 | "\n", | ||
| 249 | "You can check where an object \"lives\" with the `parent()` command. It works more or less like the Python command `type()`, but it gives a more Mathematically inclined answer. Check the reference link [6] above if you want more details." | ||
| 250 | ] | ||
| 251 | }, | ||
| 252 | { | ||
| 253 | "cell_type": "code", | ||
| 254 | "execution_count": 18, | ||
| 255 | "metadata": {}, | ||
| 256 | "outputs": [ | ||
| 257 | { | ||
| 258 | "data": { | ||
| 259 | "text/plain": [ | ||
| 260 | "Rational Field" | ||
| 261 | ] | ||
| 262 | }, | ||
| 263 | "execution_count": 18, | ||
| 264 | "metadata": {}, | ||
| 265 | "output_type": "execute_result" | ||
| 266 | } | ||
| 267 | ], | ||
| 268 | "source": [ | ||
| 269 | "#Edit this cell to find out the type of other objects that we used\n", | ||
| 270 | "parent(3/5)" | ||
| 271 | ] | ||
| 272 | }, | ||
| 273 | { | ||
| 274 | "cell_type": "markdown", | ||
| 275 | "metadata": {}, | ||
| 276 | "source": [ | ||
| 277 | "Sometimes Sage does not give you the best possible interpretation, so you can force something to be interpreted as living in a smaller ring as follows:" | ||
| 278 | ] | ||
| 279 | }, | ||
| 280 | { | ||
| 281 | "cell_type": "code", | ||
| 282 | "execution_count": 4, | ||
| 283 | "metadata": {}, | ||
| 284 | "outputs": [ | ||
| 285 | { | ||
| 286 | "name": "stdout", | ||
| 287 | "output_type": "stream", | ||
| 288 | "text": [ | ||
| 289 | "Symbolic Ring\n", | ||
| 290 | "Integer Ring\n" | ||
| 291 | ] | ||
| 292 | } | ||
| 293 | ], | ||
| 294 | "source": [ | ||
| 295 | "minus_one = e^(pi*I)\n", | ||
| 296 | "minus_one_coerced = ZZ(e^(pi*I)) # coercion\n", | ||
| 297 | "print(parent(minus_one))\n", | ||
| 298 | "print(parent(minus_one_coerced))" | ||
| 299 | ] | ||
| 300 | }, | ||
| 301 | { | ||
| 302 | "cell_type": "markdown", | ||
| 303 | "metadata": {}, | ||
| 304 | "source": [ | ||
| 305 | "**Remark.** Notice that there is a fundamental difference between the rings `RR` and `CC` and all the others in the table above: the real and complex numbers are *approximated*." | ||
| 306 | ] | ||
| 307 | }, | ||
| 308 | { | ||
| 309 | "cell_type": "code", | ||
| 310 | "execution_count": 1, | ||
| 311 | "metadata": {}, | ||
| 312 | "outputs": [ | ||
| 313 | { | ||
| 314 | "name": "stdout", | ||
| 315 | "output_type": "stream", | ||
| 316 | "text": [ | ||
| 317 | "3\n", | ||
| 318 | "3.00000000000000\n" | ||
| 319 | ] | ||
| 320 | } | ||
| 321 | ], | ||
| 322 | "source": [ | ||
| 323 | "print(QQ(3))\n", | ||
| 324 | "print(RR(3))" | ||
| 325 | ] | ||
| 326 | }, | ||
| 327 | { | ||
| 328 | "cell_type": "markdown", | ||
| 329 | "metadata": {}, | ||
| 330 | "source": [ | ||
| 331 | "You can also choose the precision of this approximation using the alternative name `RealField`." | ||
| 332 | ] | ||
| 333 | }, | ||
| 334 | { | ||
| 335 | "cell_type": "code", | ||
| 336 | "execution_count": 4, | ||
| 337 | "metadata": {}, | ||
| 338 | "outputs": [ | ||
| 339 | { | ||
| 340 | "name": "stdout", | ||
| 341 | "output_type": "stream", | ||
| 342 | "text": [ | ||
| 343 | "Real Field with 53 bits of precision\n", | ||
| 344 | "Real Field with 1000 bits of precision\n" | ||
| 345 | ] | ||
| 346 | } | ||
| 347 | ], | ||
| 348 | "source": [ | ||
| 349 | "print(RR)\n", | ||
| 350 | "print(RealField(prec=1000))" | ||
| 351 | ] | ||
| 352 | }, | ||
| 353 | { | ||
| 354 | "cell_type": "markdown", | ||
| 355 | "metadata": {}, | ||
| 356 | "source": [ | ||
| 357 | "# Polynomial rings\n", | ||
| 358 | "\n", | ||
| 359 | "**Reference:** [[7](https://doc.sagemath.org/html/en/reference/polynomial_rings/index.html)]\n", | ||
| 360 | "\n", | ||
| 361 | "If you want to work with polynomials over a certain ring it is better to use this specific construction, rather than the symbolic expressions introduced above." | ||
| 362 | ] | ||
| 363 | }, | ||
| 364 | { | ||
| 365 | "cell_type": "code", | ||
| 366 | "execution_count": 5, | ||
| 367 | "metadata": {}, | ||
| 368 | "outputs": [ | ||
| 369 | { | ||
| 370 | "data": { | ||
| 371 | "text/plain": [ | ||
| 372 | "Multivariate Polynomial Ring in x, y, z over Real Field with 53 bits of precision" | ||
| 373 | ] | ||
| 374 | }, | ||
| 375 | "execution_count": 5, | ||
| 376 | "metadata": {}, | ||
| 377 | "output_type": "execute_result" | ||
| 378 | } | ||
| 379 | ], | ||
| 380 | "source": [ | ||
| 381 | "polring.<x,y,z> = RR[] # Alternative: polring.<x,y,z> = PolynomialRing(RR)\n", | ||
| 382 | "polring" | ||
| 383 | ] | ||
| 384 | }, | ||
| 385 | { | ||
| 386 | "cell_type": "markdown", | ||
| 387 | "metadata": {}, | ||
| 388 | "source": [ | ||
| 389 | "You can use as many variables as you like, and you can replace `RR` with any ring. In the example above `polring` is just the name of the variable (in the computer science sense) associated with this polynomial ring.\n", | ||
| 390 | "\n", | ||
| 391 | "## Operations on polynomials\n", | ||
| 392 | "\n", | ||
| 393 | "The usual Mathematical operations are available on polynomial rings, including Euclidean division `//` and remainder `%`. There is also the single-slash division `/`, but the result may not be a polynomial anymore.\n", | ||
| 394 | "\n", | ||
| 395 | "**Exercise:** use the `parent()` command to find out what the quotient of two polynomials is.\n", | ||
| 396 | "\n", | ||
| 397 | "**Question:** what happens if you remove the first line in the cell below? What if we used the variable `y` instead of `x`?" | ||
| 398 | ] | ||
| 399 | }, | ||
| 400 | { | ||
| 401 | "cell_type": "code", | ||
| 402 | "execution_count": 6, | ||
| 403 | "metadata": {}, | ||
| 404 | "outputs": [ | ||
| 405 | { | ||
| 406 | "name": "stdout", | ||
| 407 | "output_type": "stream", | ||
| 408 | "text": [ | ||
| 409 | "x + 1\n", | ||
| 410 | "-4\n", | ||
| 411 | "(x^2 + 2*x - 3)/(x + 1)\n" | ||
| 412 | ] | ||
| 413 | } | ||
| 414 | ], | ||
| 415 | "source": [ | ||
| 416 | "polring.<x> = QQ[]\n", | ||
| 417 | "p = x^2 + 2*x - 3 # Don't forget * for multiplication!\n", | ||
| 418 | "q = p // (x+1)\n", | ||
| 419 | "r = p % (x+1)\n", | ||
| 420 | "f = p / (x+1)\n", | ||
| 421 | "print(q)\n", | ||
| 422 | "print(r)\n", | ||
| 423 | "print(f)" | ||
| 424 | ] | ||
| 425 | }, | ||
| 426 | { | ||
| 427 | "cell_type": "markdown", | ||
| 428 | "metadata": {}, | ||
| 429 | "source": [ | ||
| 430 | "You can do more complex operations. Try out `roots()` and `factor` in the cell below.\n", | ||
| 431 | "\n", | ||
| 432 | "**Remark.** Notice how the result can change substantially if you change the base ring.\n", | ||
| 433 | "\n", | ||
| 434 | "**Remark.** [Factorizations](https://doc.sagemath.org/html/en/reference/structure/sage/structure/factorization.html) are a particular object in Sage. They are kinda like a list, but not really. You can get a list of pairs (factor, power) with `list(factor(f))`." | ||
| 435 | ] | ||
| 436 | }, | ||
| 437 | { | ||
| 438 | "cell_type": "code", | ||
| 439 | "execution_count": 7, | ||
| 440 | "metadata": {}, | ||
| 441 | "outputs": [ | ||
| 442 | { | ||
| 443 | "name": "stdout", | ||
| 444 | "output_type": "stream", | ||
| 445 | "text": [ | ||
| 446 | "(t + 1) * (t^2 - 3) * (t^2 + 1)\n", | ||
| 447 | "[(-1, 1)]\n" | ||
| 448 | ] | ||
| 449 | }, | ||
| 450 | { | ||
| 451 | "data": { | ||
| 452 | "text/plain": [ | ||
| 453 | "(y + 1) * x" | ||
| 454 | ] | ||
| 455 | }, | ||
| 456 | "execution_count": 7, | ||
| 457 | "metadata": {}, | ||
| 458 | "output_type": "execute_result" | ||
| 459 | } | ||
| 460 | ], | ||
| 461 | "source": [ | ||
| 462 | "polring_onevar.<t> = QQ[]\n", | ||
| 463 | "\n", | ||
| 464 | "f = t^5 + t^4 - 2*t^3 - 2*t^2 - 3*t - 3\n", | ||
| 465 | "print(factor(f))\n", | ||
| 466 | "print(f.roots()) # Result: list of pairs (root,multiplicity)\n", | ||
| 467 | "\n", | ||
| 468 | "polring_manyvar.<x,y,z> = QQ[]\n", | ||
| 469 | "factor(x*y+x)\n", | ||
| 470 | "\n", | ||
| 471 | "# The following line gives an error, because the polynomial\n", | ||
| 472 | "# is understood to possibly have many variables:\n", | ||
| 473 | "#(x^2-1).roots()" | ||
| 474 | ] | ||
| 475 | }, | ||
| 476 | { | ||
| 477 | "cell_type": "markdown", | ||
| 478 | "metadata": {}, | ||
| 479 | "source": [ | ||
| 480 | "# Matrices and vectors\n", | ||
| 481 | "\n", | ||
| 482 | "**References:** [[8](https://doc.sagemath.org/html/en/reference/matrices/index.html)], but in particular the subections [[9](https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/docs.html)] and [[10](https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix2.html)]\n", | ||
| 483 | "\n", | ||
| 484 | "In Sage you can easily manipulate matrices and vectors" | ||
| 485 | ] | ||
| 486 | }, | ||
| 487 | { | ||
| 488 | "cell_type": "code", | ||
| 489 | "execution_count": 77, | ||
| 490 | "metadata": {}, | ||
| 491 | "outputs": [ | ||
| 492 | { | ||
| 493 | "name": "stdout", | ||
| 494 | "output_type": "stream", | ||
| 495 | "text": [ | ||
| 496 | "[ 1 2 3]\n", | ||
| 497 | "[ 0 0 1]\n", | ||
| 498 | "[ 4 -3 22/7] \n", | ||
| 499 | "\n", | ||
| 500 | "[1/2 0 0]\n", | ||
| 501 | "[ 7 0 0]\n", | ||
| 502 | "[ 1 1 1] \n", | ||
| 503 | "\n", | ||
| 504 | "(3/2, 21, 6) \n", | ||
| 505 | "\n", | ||
| 506 | "[ -7/2 -10 80/7]\n", | ||
| 507 | "[ 17 -4 15/7]\n", | ||
| 508 | "[ 241/7 -18/7 869/49] \n", | ||
| 509 | "\n", | ||
| 510 | "Rank of A = 3\n", | ||
| 511 | "Rank of B = 2\n" | ||
| 512 | ] | ||
| 513 | } | ||
| 514 | ], | ||
| 515 | "source": [ | ||
| 516 | "A = matrix([[1,2,3],[0,0,1],[4,-3,22/7]])\n", | ||
| 517 | "B = matrix([[1/2,0,0],[7,0,0],[1,1,1]])\n", | ||
| 518 | "v = vector([3,4,-1])\n", | ||
| 519 | "\n", | ||
| 520 | "print(A, \"\\n\") # \\n just means \"newline\"\n", | ||
| 521 | "print(B, \"\\n\")\n", | ||
| 522 | "print(B*v, \"\\n\")\n", | ||
| 523 | "print(A^2 + 2*B - A*B, \"\\n\")\n", | ||
| 524 | "\n", | ||
| 525 | "print(\"Rank of A =\", rank(A)) # You can also use A.rank()\n", | ||
| 526 | "print(\"Rank of B =\", rank(B))" | ||
| 527 | ] | ||
| 528 | }, | ||
| 529 | { | ||
| 530 | "cell_type": "markdown", | ||
| 531 | "metadata": {}, | ||
| 532 | "source": [ | ||
| 533 | "**Exercise:** in the cell above, compute the determinant, inverse and characteristic polynomial of the matrix `A`. *Hint: look at the reference [10] above (the functions are listed in alphabetic order).*\n", | ||
| 534 | "\n", | ||
| 535 | "As for polynomials, you can specify where a matrix or a vector lives" | ||
| 536 | ] | ||
| 537 | }, | ||
| 538 | { | ||
| 539 | "cell_type": "code", | ||
| 540 | "execution_count": 57, | ||
| 541 | "metadata": {}, | ||
| 542 | "outputs": [ | ||
| 543 | { | ||
| 544 | "data": { | ||
| 545 | "text/plain": [ | ||
| 546 | "Full MatrixSpace of 2 by 2 dense matrices over Complex Field with 53 bits of precision" | ||
| 547 | ] | ||
| 548 | }, | ||
| 549 | "execution_count": 57, | ||
| 550 | "metadata": {}, | ||
| 551 | "output_type": "execute_result" | ||
| 552 | } | ||
| 553 | ], | ||
| 554 | "source": [ | ||
| 555 | "M = matrix(CC, [[0,1],[1,0]])\n", | ||
| 556 | "parent(M)" | ||
| 557 | ] | ||
| 558 | }, | ||
| 559 | { | ||
| 560 | "cell_type": "markdown", | ||
| 561 | "metadata": {}, | ||
| 562 | "source": [ | ||
| 563 | "You can also solve linear systems and compute eigenvalues and eigenvectors of a matrix\n", | ||
| 564 | "\n", | ||
| 565 | "**Warning.** In linear algebra there are distinct concepts of *left* and *right* eigenvalues (and eigenvector). The one you know is probably that of **right** eigen-{value,vector}, that is an element $\\lambda$ of the base field and a non-zero vector $\\mathbf v$ with $A\\mathbf v=\\lambda\\mathbf v$. The other concept corresponds to the equality $\\mathbf v^TA=\\lambda \\mathbf v$." | ||
| 566 | ] | ||
| 567 | }, | ||
| 568 | { | ||
| 569 | "cell_type": "code", | ||
| 570 | "execution_count": 60, | ||
| 571 | "metadata": {}, | ||
| 572 | "outputs": [ | ||
| 573 | { | ||
| 574 | "data": { | ||
| 575 | "text/plain": [ | ||
| 576 | "(0.289916349448506, 0.0241596957873755)" | ||
| 577 | ] | ||
| 578 | }, | ||
| 579 | "execution_count": 60, | ||
| 580 | "metadata": {}, | ||
| 581 | "output_type": "execute_result" | ||
| 582 | } | ||
| 583 | ], | ||
| 584 | "source": [ | ||
| 585 | "A = Matrix(RR, [[sqrt(59),32],[-1/4,3]])\n", | ||
| 586 | "v = vector(RR, [3,0])\n", | ||
| 587 | "A.solve_right(v) # Solve Ax=v. Alternative: A \\ v" | ||
| 588 | ] | ||
| 589 | }, | ||
| 590 | { | ||
| 591 | "cell_type": "code", | ||
| 592 | "execution_count": 64, | ||
| 593 | "metadata": {}, | ||
| 594 | "outputs": [ | ||
| 595 | { | ||
| 596 | "data": { | ||
| 597 | "text/plain": [ | ||
| 598 | "[\n", | ||
| 599 | "(-0.3722813232690144?, Vector space of degree 2 and dimension 1 over Algebraic Field\n", | ||
| 600 | "User basis matrix:\n", | ||
| 601 | "[ 1 -0.6861406616345072?]),\n", | ||
| 602 | "(5.372281323269015?, Vector space of degree 2 and dimension 1 over Algebraic Field\n", | ||
| 603 | "User basis matrix:\n", | ||
| 604 | "[ 1 2.186140661634508?])\n", | ||
| 605 | "]" | ||
| 606 | ] | ||
| 607 | }, | ||
| 608 | "execution_count": 64, | ||
| 609 | "metadata": {}, | ||
| 610 | "output_type": "execute_result" | ||
| 611 | } | ||
| 612 | ], | ||
| 613 | "source": [ | ||
| 614 | "A = Matrix(QQ, [[1,2],[3,4]])\n", | ||
| 615 | "A.eigenspaces_right() # Also: A.eigenvalues(), A.eigenvectors_right()" | ||
| 616 | ] | ||
| 617 | }, | ||
| 618 | { | ||
| 619 | "cell_type": "markdown", | ||
| 620 | "metadata": {}, | ||
| 621 | "source": [ | ||
| 622 | "We can also extract a specific submatrix by selecting only some rows and columns, with a syntax similar to that of Python's lists. Check out more examples in the reference [9] above, and try them in the cell below." | ||
| 623 | ] | ||
| 624 | }, | ||
| 625 | { | ||
| 626 | "cell_type": "code", | ||
| 627 | "execution_count": 94, | ||
| 628 | "metadata": {}, | ||
| 629 | "outputs": [ | ||
| 630 | { | ||
| 631 | "name": "stdout", | ||
| 632 | "output_type": "stream", | ||
| 633 | "text": [ | ||
| 634 | "[-14 2 0 -1 1 -2 -1]\n", | ||
| 635 | "[ 0 -8 0 9 -2 11 1]\n", | ||
| 636 | "[ 0 3 1 -1 1 1 221]\n", | ||
| 637 | "[ -1 2 1 -25 -10 4 0]\n", | ||
| 638 | "[ -3 0 0 2 16 -1 -2]\n", | ||
| 639 | "[ 1 -3 3 -41 1 0 0]\n", | ||
| 640 | "[ -2 1 0 0 -6 2 12] \n", | ||
| 641 | "\n", | ||
| 642 | "[ 0 9 -2]\n", | ||
| 643 | "[ 1 -1 1] \n", | ||
| 644 | "\n", | ||
| 645 | "[-14 2 0 -1 1 -2 -1] \n", | ||
| 646 | "\n", | ||
| 647 | "[-14 2 0 -1 1]\n", | ||
| 648 | "[ 1 -3 3 -41 1]\n", | ||
| 649 | "[ 0 3 1 -1 1]\n" | ||
| 650 | ] | ||
| 651 | } | ||
| 652 | ], | ||
| 653 | "source": [ | ||
| 654 | "A = MatrixSpace(ZZ, 7).random_element()\n", | ||
| 655 | "print(A, \"\\n\")\n", | ||
| 656 | "print(A[1:3,2:5], \"\\n\") # Rows from 1 to 3, columns from 2 to 5\n", | ||
| 657 | "print(A[0,0:], \"\\n\") # First row, all columns\n", | ||
| 658 | "print(A[[0,5,2],0:5]) # Rows 0, 5 and 2 (in this order) and columns 0 to 5" | ||
| 659 | ] | ||
| 660 | }, | ||
| 661 | { | ||
| 662 | "cell_type": "markdown", | ||
| 663 | "metadata": {}, | ||
| 664 | "source": [ | ||
| 665 | "**Exercise:** write a sage function that computes the determinant of an $n\\times n$ matrix $A=(a_{ij})$ using Laplace's rule by the first row, that is \n", | ||
| 666 | "\\begin{align*}\n", | ||
| 667 | " \\operatorname{det}A = \\sum_{j=1}^n (-1)^ja_{0j}M_{0j}\n", | ||
| 668 | "\\end{align*}\n", | ||
| 669 | "where $M_{0j}$ is the determinant of the $(n-1)\\times(n-1)$ matrix obtained by removing the $0$-th row and the $j$-th column from $A$." | ||
| 670 | ] | ||
| 671 | }, | ||
| 672 | { | ||
| 673 | "cell_type": "code", | ||
| 674 | "execution_count": 91, | ||
| 675 | "metadata": {}, | ||
| 676 | "outputs": [], | ||
| 677 | "source": [ | ||
| 678 | "def my_det(A):\n", | ||
| 679 | " if not A.is_square():\n", | ||
| 680 | " print(\"Error: matrix is not square\")\n", | ||
| 681 | " \n", | ||
| 682 | " n = A.nrows() # size of the matrix\n", | ||
| 683 | " \n", | ||
| 684 | " # Continue from here!" | ||
| 685 | ] | ||
| 686 | }, | ||
| 687 | { | ||
| 688 | "cell_type": "markdown", | ||
| 689 | "metadata": {}, | ||
| 690 | "source": [ | ||
| 691 | "# Number Theory\n", | ||
| 692 | "\n", | ||
| 693 | "**Reference:** [[11](https://doc.sagemath.org/html/en/reference/rings_standard/sage/rings/integer.html)]\n", | ||
| 694 | "\n", | ||
| 695 | "Sage includes a large library of functions for computing with the integers, see the link above." | ||
| 696 | ] | ||
| 697 | }, | ||
| 698 | { | ||
| 699 | "cell_type": "code", | ||
| 700 | "execution_count": 8, | ||
| 701 | "metadata": {}, | ||
| 702 | "outputs": [ | ||
| 703 | { | ||
| 704 | "name": "stdout", | ||
| 705 | "output_type": "stream", | ||
| 706 | "text": [ | ||
| 707 | "3^2 * 3607 * 3803\n", | ||
| 708 | "True\n", | ||
| 709 | "True\n", | ||
| 710 | "619703040\n", | ||
| 711 | "9\n", | ||
| 712 | "13548070123626141\n" | ||
| 713 | ] | ||
| 714 | } | ||
| 715 | ], | ||
| 716 | "source": [ | ||
| 717 | "n = 123456789\n", | ||
| 718 | "m = 987654321\n", | ||
| 719 | "p = 3607\n", | ||
| 720 | "\n", | ||
| 721 | "print(factor(n))\n", | ||
| 722 | "print(is_prime(p))\n", | ||
| 723 | "print(p.divides(n))\n", | ||
| 724 | "print(euler_phi(m))\n", | ||
| 725 | "print(gcd(n, m))\n", | ||
| 726 | "print(lcm(n, m))" | ||
| 727 | ] | ||
| 728 | }, | ||
| 729 | { | ||
| 730 | "cell_type": "markdown", | ||
| 731 | "metadata": {}, | ||
| 732 | "source": [ | ||
| 733 | "## Primes\n", | ||
| 734 | "\n", | ||
| 735 | "**Reference:** [[12](https://doc.sagemath.org/html/en/reference/sets/sage/sets/primes.html)]\n", | ||
| 736 | "\n", | ||
| 737 | "The set of prime numbers is called `Primes()`. It is like an infinite list: for example you can get the one-millionth prime number or you can use this list to create other lists. You can also check what the first prime number larger than a given number is." | ||
| 738 | ] | ||
| 739 | }, | ||
| 740 | { | ||
| 741 | "cell_type": "code", | ||
| 742 | "execution_count": 9, | ||
| 743 | "metadata": {}, | ||
| 744 | "outputs": [ | ||
| 745 | { | ||
| 746 | "name": "stdout", | ||
| 747 | "output_type": "stream", | ||
| 748 | "text": [ | ||
| 749 | "Set of all prime numbers: 2, 3, 5, 7, ...\n", | ||
| 750 | "31 15485867\n", | ||
| 751 | "47\n", | ||
| 752 | "[79, 83, 89, 97]\n" | ||
| 753 | ] | ||
| 754 | } | ||
| 755 | ], | ||
| 756 | "source": [ | ||
| 757 | "PP = Primes()\n", | ||
| 758 | "print(PP)\n", | ||
| 759 | "print(PP[10], PP[10^6])\n", | ||
| 760 | "print(PP.next(44))\n", | ||
| 761 | "\n", | ||
| 762 | "First_Thousand_Primes = PP[0:1000]\n", | ||
| 763 | "print([p for p in First_Thousand_Primes if p < 100 and p > 75])" | ||
| 764 | ] | ||
| 765 | }, | ||
| 766 | { | ||
| 767 | "cell_type": "markdown", | ||
| 768 | "metadata": {}, | ||
| 769 | "source": [ | ||
| 770 | "## The Chinese remainder theorem (CRT)\n", | ||
| 771 | "\n", | ||
| 772 | "We say that two integers $a$ and $b$ are *congruent* modulo another integer $n>0$ if they have the same remainder when divided by $n$. We denote this by $a\\equiv b\\pmod n$, or in Python/Sage syntax `a % n == b % n`.\n", | ||
| 773 | "\n", | ||
| 774 | "The Chinese remainder theorem states that if $a,b\\in\\mathbb Z$ and $n,m\\in \\mathbb Z_{>0}$ are such that $\\gcd(n,m)=1$ then the system of congruences\n", | ||
| 775 | "\n", | ||
| 776 | "\\begin{align*}\n", | ||
| 777 | "\\begin{cases}\n", | ||
| 778 | " x \\equiv a \\pmod n\\\\\n", | ||
| 779 | " x \\equiv b \\pmod m\n", | ||
| 780 | "\\end{cases}\n", | ||
| 781 | "\\end{align*}\n", | ||
| 782 | "\n", | ||
| 783 | "has exactly one solution modulo $mn$. This means that there is one and only one number $x$ with $0\\leq x<mn$ such that $x\\equiv a\\pmod n$ and $x\\equiv b\\pmod m$.\n", | ||
| 784 | "\n", | ||
| 785 | "The procedure to find such a number is not too hard to describe (you might see it in an algebra or number theory course), but it can be a bit long. Luckily, Sage can do this for you:" | ||
| 786 | ] | ||
| 787 | }, | ||
| 788 | { | ||
| 789 | "cell_type": "code", | ||
| 790 | "execution_count": 10, | ||
| 791 | "metadata": {}, | ||
| 792 | "outputs": [ | ||
| 793 | { | ||
| 794 | "name": "stdout", | ||
| 795 | "output_type": "stream", | ||
| 796 | "text": [ | ||
| 797 | "74306 2 798\n" | ||
| 798 | ] | ||
| 799 | } | ||
| 800 | ], | ||
| 801 | "source": [ | ||
| 802 | "a = 2\n", | ||
| 803 | "b = -1\n", | ||
| 804 | "n = 172\n", | ||
| 805 | "m = 799\n", | ||
| 806 | "\n", | ||
| 807 | "if gcd(n,m) != 1:\n", | ||
| 808 | " print(\"The numbers are not comprime, I can't solve this!\")\n", | ||
| 809 | "else:\n", | ||
| 810 | " x = crt(a, b, n, m)\n", | ||
| 811 | " print(x, x%n, x%m)" | ||
| 812 | ] | ||
| 813 | }, | ||
| 814 | { | ||
| 815 | "cell_type": "markdown", | ||
| 816 | "metadata": {}, | ||
| 817 | "source": [ | ||
| 818 | "**Exercise.** There is a more general version of the Chinese remainder theorem which says that if $a_0, a_1, \\dots, a_k\\in\\mathbb Z$ and $n_0, n_2, \\dots, n_k\\in\\mathbb Z_{>0}$ are such that $\\gcd(n_i, n_j)=1$ for $i\\neq j$, then the system of congruences\n", | ||
| 819 | "\n", | ||
| 820 | "\\begin{align*}\n", | ||
| 821 | "\\begin{cases}\n", | ||
| 822 | " x \\equiv a_0 \\pmod {n_0}\\\\\n", | ||
| 823 | " x \\equiv a_1 \\pmod {n_1}\\\\\n", | ||
| 824 | " \\dots \\\\\n", | ||
| 825 | " x \\equiv a_k \\pmod {n_k}\n", | ||
| 826 | "\\end{cases}\n", | ||
| 827 | "\\end{align*}\n", | ||
| 828 | "\n", | ||
| 829 | "has exactly one solution modulo $\\prod_{i=0}^kn_i$. Use the `crt()` function to find a solution to such a system.\n", | ||
| 830 | "*Hint: start by running the command `help(crt)`." | ||
| 831 | ] | ||
| 832 | }, | ||
| 833 | { | ||
| 834 | "cell_type": "code", | ||
| 835 | "execution_count": 127, | ||
| 836 | "metadata": {}, | ||
| 837 | "outputs": [], | ||
| 838 | "source": [ | ||
| 839 | "#help(crt)" | ||
| 840 | ] | ||
| 841 | }, | ||
| 842 | { | ||
| 843 | "cell_type": "markdown", | ||
| 844 | "metadata": {}, | ||
| 845 | "source": [ | ||
| 846 | "# Cryptography: RSA\n", | ||
| 847 | "\n", | ||
| 848 | "[Cryptography](https://en.wikipedia.org/wiki/Cryptography) is the discipline that studies methods to communicate secrets in such a way that any unauthorized listener would not be able to understand the message.\n", | ||
| 849 | "\n", | ||
| 850 | "A simple cryptographic protocol could be changing every letter of your text following a fixed scheme (or *cypher*), for example by turning every A into a B, every B into a C and so on. However this is not a very secure method, for many reasons. One of them is that at some point the people who want to communicate need to agree on what method to use, and anyone listening to that conversation would be able to decypher every subsequent conversation. A public-key cryptographic protocol solves this problem.\n", | ||
| 851 | "\n", | ||
| 852 | "## Public-key cryptography\n", | ||
| 853 | "\n", | ||
| 854 | "Public-key cryptographic protocols, such as RSA, work like this: there are two keys, a *private* key that is only known to person A (traditionally called Alice in every example), and a *public* key that does not need to be secret.\n", | ||
| 855 | "\n", | ||
| 856 | "The public key is used to *encrypt* the message (that is to \"lock\" it, or \"hyde\" it), but one needs the private key to *decrypt* it. Imagine having two keys for your door, but one can only be used to lock it, while the other only to open it.\n", | ||
| 857 | "\n", | ||
| 858 | "The message exchange works like this: suppose that person B (Bob) wants to send a secret message to Alice. Then Alice secretely generates a private and a public key and sends only the public one to Bob. Now Bob encrypts the message and sends it to Alice, who can use her private key to decrypt it. Even if Eve (short for *eavesdropper*, an unauthorized listener) listens to every message exchanged, she won't be able to decypher the secret: the private key has never left Alice's house!\n", | ||
| 859 | "\n", | ||
| 860 | "Notice that such a protocol is *asymmetric*: if Alice wanted to send a secret to Bob in reply, Bob would need to generate a pair of keys of his own.\n", | ||
| 861 | "\n", | ||
| 862 | "Let's see how we can do this in practice, using number theory!\n", | ||
| 863 | "\n", | ||
| 864 | "## RSA\n", | ||
| 865 | "\n", | ||
| 866 | "As many other cryptography protocols, RSA is based on a Mathematical process that is easy to do in one direction, but very hard to invert. In this case the hard process is integer factorization, that is decomposing an integer number as a product of primes." | ||
| 867 | ] | ||
| 868 | }, | ||
| 869 | { | ||
| 870 | "cell_type": "code", | ||
| 871 | "execution_count": 2, | ||
| 872 | "metadata": {}, | ||
| 873 | "outputs": [ | ||
| 874 | { | ||
| 875 | "name": "stdout", | ||
| 876 | "output_type": "stream", | ||
| 877 | "text": [ | ||
| 878 | "True True False\n" | ||
| 879 | ] | ||
| 880 | } | ||
| 881 | ], | ||
| 882 | "source": [ | ||
| 883 | "p = 100003100019100043100057100069\n", | ||
| 884 | "q = 100144655312449572059845328443\n", | ||
| 885 | "n = p*q\n", | ||
| 886 | "print(is_prime(p), is_prime(q), is_prime(p*q))\n", | ||
| 887 | "\n", | ||
| 888 | "# Use the command below to see how long it takes\n", | ||
| 889 | "#timeit(\"factor(n)\", number=1, repeat=1)" | ||
| 890 | ] | ||
| 891 | }, | ||
| 892 | { | ||
| 893 | "cell_type": "markdown", | ||
| 894 | "metadata": {}, | ||
| 895 | "source": [ | ||
| 896 | "In order to generate the keys, Alice picks a number $n$ which is the product of two large primes $p$ and $q$ of more or less the same size. Finding such primes is relatively easy compared to factoring the number $n$ she obtained. Then she computes the Euler totient $\\varphi(n)=(p-1)(q-1)$ of $n$, which she can do because she knows that $n=pq$ - it would be impossible otherwise!\n", | ||
| 897 | "\n", | ||
| 898 | "Then Alice can compute two integers $(d,e)$ such that $de\\equiv 1\\pmod{\\varphi(n)}$. She will send the numbers $n$ and $d$ to Bob and keep $e$ secret. In this case the public key is the pair $(n,d)$, while $e$ is the private key.\n", | ||
| 899 | "\n", | ||
| 900 | "Of course, she does all of this using Sage!" | ||
| 901 | ] | ||
| 902 | }, | ||
| 903 | { | ||
| 904 | "cell_type": "code", | ||
| 905 | "execution_count": 105, | ||
| 906 | "metadata": {}, | ||
| 907 | "outputs": [ | ||
| 908 | { | ||
| 909 | "data": { | ||
| 910 | "text/plain": [ | ||
| 911 | "(419199544978969, 235530823946467, 80799425863927)" | ||
| 912 | ] | ||
| 913 | }, | ||
| 914 | "execution_count": 105, | ||
| 915 | "metadata": {}, | ||
| 916 | "output_type": "execute_result" | ||
| 917 | } | ||
| 918 | ], | ||
| 919 | "source": [ | ||
| 920 | "def two_large_primes():\n", | ||
| 921 | " p, q = 0, 0\n", | ||
| 922 | " # We make sure that they are different\n", | ||
| 923 | " while p == q:\n", | ||
| 924 | " p = Primes()[randint(10^6, 2*10^6)]\n", | ||
| 925 | " q = Primes()[randint(10^6, 2*10^6)]\n", | ||
| 926 | " return p, q\n", | ||
| 927 | "\n", | ||
| 928 | "def random_unit_mod(N):\n", | ||
| 929 | " R = Integers(N)\n", | ||
| 930 | " d = R(0)\n", | ||
| 931 | " # We make sure that it is invertible\n", | ||
| 932 | " while not d.is_unit():\n", | ||
| 933 | " d = R.random_element()\n", | ||
| 934 | " return d\n", | ||
| 935 | "\n", | ||
| 936 | "def Alice_generate_keys():\n", | ||
| 937 | " p, q = two_large_primes()\n", | ||
| 938 | " n = p*q\n", | ||
| 939 | " phi_n = (p-1)*(q-1) # euler_phi(n) is slow!\n", | ||
| 940 | " \n", | ||
| 941 | " d = random_unit_mod(phi_n)\n", | ||
| 942 | " e = d^-1\n", | ||
| 943 | " return n, d, e\n", | ||
| 944 | "\n", | ||
| 945 | "Alice_generate_keys()" | ||
| 946 | ] | ||
| 947 | }, | ||
| 948 | { | ||
| 949 | "cell_type": "markdown", | ||
| 950 | "metadata": {}, | ||
| 951 | "source": [ | ||
| 952 | "Now, how does Bob encrypt his message? Let's say he wants to send to Alice the number $m$ with $1<m<n$ (In practice he would like to send her some text with emojis, or maybe a voice message; but for computers everything is a number, and there are different ways to translate any sort of information to a number. He just chooses one of the many standard methods that already exist, no cryptography is needed in this step. If the message $m$ is too long, he can split it up in some pieces and repeat the process multiple times.)\n", | ||
| 953 | "\n", | ||
| 954 | "Now he computes $m^d\\pmod n$ and sends it back to Alice." | ||
| 955 | ] | ||
| 956 | }, | ||
| 957 | { | ||
| 958 | "cell_type": "code", | ||
| 959 | "execution_count": 3, | ||
| 960 | "metadata": {}, | ||
| 961 | "outputs": [ | ||
| 962 | { | ||
| 963 | "data": { | ||
| 964 | "text/plain": [ | ||
| 965 | "149461597163501" | ||
| 966 | ] | ||
| 967 | }, | ||
| 968 | "execution_count": 3, | ||
| 969 | "metadata": {}, | ||
| 970 | "output_type": "execute_result" | ||
| 971 | } | ||
| 972 | ], | ||
| 973 | "source": [ | ||
| 974 | "def Bob_encrypt(m, n, d):\n", | ||
| 975 | " R = Integers(n)\n", | ||
| 976 | " return R(m)^d # Assume that n is large enough\n", | ||
| 977 | " \n", | ||
| 978 | "message = 42424242\n", | ||
| 979 | "Bob_encrypt(message, 419199544978969, 235530823946467)" | ||
| 980 | ] | ||
| 981 | }, | ||
| 982 | { | ||
| 983 | "cell_type": "markdown", | ||
| 984 | "metadata": {}, | ||
| 985 | "source": [ | ||
| 986 | "Since $de\\equiv 1\\pmod{\\varphi(n)}$, it follows that $(m^d)^e\\equiv m\\pmod n$ (see [Wikipedia: Euler's theorem](https://en.wikipedia.org/wiki/Euler%27s_theorem)). So for Alice it is very easy to get back the original message:" | ||
| 987 | ] | ||
| 988 | }, | ||
| 989 | { | ||
| 990 | "cell_type": "code", | ||
| 991 | "execution_count": 108, | ||
| 992 | "metadata": {}, | ||
| 993 | "outputs": [ | ||
| 994 | { | ||
| 995 | "data": { | ||
| 996 | "text/plain": [ | ||
| 997 | "42424242" | ||
| 998 | ] | ||
| 999 | }, | ||
| 1000 | "execution_count": 108, | ||
| 1001 | "metadata": {}, | ||
| 1002 | "output_type": "execute_result" | ||
| 1003 | } | ||
| 1004 | ], | ||
| 1005 | "source": [ | ||
| 1006 | "def Alice_decrypt(m_encrypted, n, e):\n", | ||
| 1007 | " R = Integers(n)\n", | ||
| 1008 | " return R(m_encrypted)^e\n", | ||
| 1009 | "\n", | ||
| 1010 | "Alice_decrypt(149461597163501, 419199544978969, 80799425863927)" | ||
| 1011 | ] | ||
| 1012 | }, | ||
| 1013 | { | ||
| 1014 | "cell_type": "markdown", | ||
| 1015 | "metadata": {}, | ||
| 1016 | "source": [ | ||
| 1017 | "Another assumption on which RSA relies is that even if one knows $M=m^e$ and $e$, extracting the $e$-th root of $M$ modulo $n$ (and thus obtaining $m$) is very hard. Currently the best known way to do this is by factorizing $n$ first, which is considered to be a very hard problem. However, there is no proof that faster algorithms can't be devised.\n", | ||
| 1018 | "\n", | ||
| 1019 | "Moreover, one day we will overcome the current technological difficulties and quantum computers will be available. Quantum computers are not just \"more powerful\" than classical hardware, but they work based on completely different logical foundations and they make the factorization problem much easier to solve: for example [Shor's algorithm](https://en.wikipedia.org/wiki/Shor%27s_algorithm) takes advantage of this different logic and can factorize numbers quickly, if run on a quantum computer.\n", | ||
| 1020 | "\n", | ||
| 1021 | "To this day the largest number factorized with a quantum computer is $21=3\\times 7$. Nonetheless, quantum-safe cryptography protocols (i.e. based on problems that are hard to solve also with quantum computers) have already been developed." | ||
| 1022 | ] | ||
| 1023 | } | ||
| 1024 | ], | ||
| 1025 | "metadata": { | ||
| 1026 | "kernelspec": { | ||
| 1027 | "display_name": "SageMath 9.2", | ||
| 1028 | "language": "sage", | ||
| 1029 | "name": "sagemath" | ||
| 1030 | }, | ||
| 1031 | "language_info": { | ||
| 1032 | "codemirror_mode": { | ||
| 1033 | "name": "ipython", | ||
| 1034 | "version": 3 | ||
| 1035 | }, | ||
| 1036 | "file_extension": ".py", | ||
| 1037 | "mimetype": "text/x-python", | ||
| 1038 | "name": "python", | ||
| 1039 | "nbconvert_exporter": "python", | ||
| 1040 | "pygments_lexer": "ipython3", | ||
| 1041 | "version": "3.8.5" | ||
| 1042 | } | ||
| 1043 | }, | ||
| 1044 | "nbformat": 4, | ||
| 1045 | "nbformat_minor": 4 | ||
| 1046 | } | ||
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| 1 | { | ||
| 2 | "cells": [ | ||
| 3 | { | ||
| 4 | "cell_type": "markdown", | ||
| 5 | "metadata": {}, | ||
| 6 | "source": [ | ||
| 7 | "# Symbolic expressions\n", | ||
| 8 | "\n", | ||
| 9 | "**Reference:** [[1](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)]\n", | ||
| 10 | "\n", | ||
| 11 | "Last time we saw the basics of symbolic expressions:\n", | ||
| 12 | "* How to define and manipulate symbolic expressions\n", | ||
| 13 | "* How to introduce new variables (in the Mathematical sense) with `var()`\n", | ||
| 14 | "* How to solve equations and inequalities\n", | ||
| 15 | "* Some of the Mathematical constants that are included in Sage, and how to approximate them using `n()`\n", | ||
| 16 | "\n", | ||
| 17 | "Here are some examples to remind you of these basic things:" | ||
| 18 | ] | ||
| 19 | }, | ||
| 20 | { | ||
| 21 | "cell_type": "code", | ||
| 22 | "execution_count": 2, | ||
| 23 | "metadata": {}, | ||
| 24 | "outputs": [ | ||
| 25 | { | ||
| 26 | "name": "stdout", | ||
| 27 | "output_type": "stream", | ||
| 28 | "text": [ | ||
| 29 | "[\n", | ||
| 30 | "x == -sqrt(-pi),\n", | ||
| 31 | "x == sqrt(-pi)\n", | ||
| 32 | "]\n", | ||
| 33 | "[\n", | ||
| 34 | "z == -sqrt(pi + x^2),\n", | ||
| 35 | "z == sqrt(pi + x^2)\n", | ||
| 36 | "]\n", | ||
| 37 | "[[y < -2], [y > 1]]\n", | ||
| 38 | "2*pi + e is approximately 9.00146713563863\n" | ||
| 39 | ] | ||
| 40 | } | ||
| 41 | ], | ||
| 42 | "source": [ | ||
| 43 | "var('y', 'z') # Define new variables (x is already defined by Sage)\n", | ||
| 44 | "f = x^2 + pi\n", | ||
| 45 | "g = y^2 + y - 2 > 0\n", | ||
| 46 | "print( solve(f==0, x) )\n", | ||
| 47 | "print( solve(z^2 - f, z) )\n", | ||
| 48 | "print( solve(g, y) )\n", | ||
| 49 | "print( 2*pi + e, \"is approximately\", n(2*pi + e) )" | ||
| 50 | ] | ||
| 51 | }, | ||
| 52 | { | ||
| 53 | "cell_type": "markdown", | ||
| 54 | "metadata": {}, | ||
| 55 | "source": [ | ||
| 56 | "Now we will see some more details about solving equations and manipulating their solutions." | ||
| 57 | ] | ||
| 58 | }, | ||
| 59 | { | ||
| 60 | "cell_type": "markdown", | ||
| 61 | "metadata": {}, | ||
| 62 | "source": [ | ||
| 63 | "## Solving equations and inequalities\n", | ||
| 64 | "\n", | ||
| 65 | "**Reference** [[1](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)] for the details of `solve()` and `find_root()`, [[2](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/relation.html#solving)] for examples.\n", | ||
| 66 | "\n", | ||
| 67 | "Other than equations and inequalities, we can also solve systems: it is enough to give Sage a list of expressions and a list of variables with respect to which we want to solve. For example the system\n", | ||
| 68 | "\n", | ||
| 69 | "\\begin{align*}\n", | ||
| 70 | " \\begin{cases}\n", | ||
| 71 | " x + y = 2 \\\\\n", | ||
| 72 | " 2x - y = 6\n", | ||
| 73 | " \\end{cases}\n", | ||
| 74 | "\\end{align*}\n", | ||
| 75 | "\n", | ||
| 76 | "Can be solved as" | ||
| 77 | ] | ||
| 78 | }, | ||
| 79 | { | ||
| 80 | "cell_type": "code", | ||
| 81 | "execution_count": 40, | ||
| 82 | "metadata": {}, | ||
| 83 | "outputs": [ | ||
| 84 | { | ||
| 85 | "data": { | ||
| 86 | "text/plain": [ | ||
| 87 | "[[x == (8/3), y == (-2/3)]]" | ||
| 88 | ] | ||
| 89 | }, | ||
| 90 | "execution_count": 40, | ||
| 91 | "metadata": {}, | ||
| 92 | "output_type": "execute_result" | ||
| 93 | } | ||
| 94 | ], | ||
| 95 | "source": [ | ||
| 96 | "solve([x+y == 2, 2*x - y == 6], [x,y])" | ||
| 97 | ] | ||
| 98 | }, | ||
| 99 | { | ||
| 100 | "cell_type": "markdown", | ||
| 101 | "metadata": {}, | ||
| 102 | "source": [ | ||
| 103 | "**Exercise.** Find the intersection of the circle of radius $2$ centered in the origin and the parabula of equation $y=x^2-2x^2+1$." | ||
| 104 | ] | ||
| 105 | }, | ||
| 106 | { | ||
| 107 | "cell_type": "markdown", | ||
| 108 | "metadata": {}, | ||
| 109 | "source": [ | ||
| 110 | "### The set of solutions\n", | ||
| 111 | "\n", | ||
| 112 | "One would expect the result of `solve()` to be a list of solutions, but it is actually a list of expressions (technically it is not a list but a different type of Python collection, but this is not so important)" | ||
| 113 | ] | ||
| 114 | }, | ||
| 115 | { | ||
| 116 | "cell_type": "code", | ||
| 117 | "execution_count": 37, | ||
| 118 | "metadata": {}, | ||
| 119 | "outputs": [ | ||
| 120 | { | ||
| 121 | "data": { | ||
| 122 | "text/plain": [ | ||
| 123 | "x == -3" | ||
| 124 | ] | ||
| 125 | }, | ||
| 126 | "execution_count": 37, | ||
| 127 | "metadata": {}, | ||
| 128 | "output_type": "execute_result" | ||
| 129 | } | ||
| 130 | ], | ||
| 131 | "source": [ | ||
| 132 | "solutions = solve(x^2-9 == 0, x)\n", | ||
| 133 | "solutions[0] # This is the expression 'x == -3'" | ||
| 134 | ] | ||
| 135 | }, | ||
| 136 | { | ||
| 137 | "cell_type": "markdown", | ||
| 138 | "metadata": {}, | ||
| 139 | "source": [ | ||
| 140 | "To read the actual solution without the `x ==` part you can use the `rhs()` or `lhs()` functions, which can be applied to any expression containing a relation operator (like `==`, `<`, `>=`...) and return the *right hand side* and *left hand side* of the expression, respectively" | ||
| 141 | ] | ||
| 142 | }, | ||
| 143 | { | ||
| 144 | "cell_type": "code", | ||
| 145 | "execution_count": 41, | ||
| 146 | "metadata": {}, | ||
| 147 | "outputs": [ | ||
| 148 | { | ||
| 149 | "name": "stdout", | ||
| 150 | "output_type": "stream", | ||
| 151 | "text": [ | ||
| 152 | "rhs: 2\n", | ||
| 153 | "lhs: x\n" | ||
| 154 | ] | ||
| 155 | } | ||
| 156 | ], | ||
| 157 | "source": [ | ||
| 158 | "f = x == 2\n", | ||
| 159 | "print(\"rhs:\", f.rhs())\n", | ||
| 160 | "print(\"lhs:\", f.lhs())" | ||
| 161 | ] | ||
| 162 | }, | ||
| 163 | { | ||
| 164 | "cell_type": "markdown", | ||
| 165 | "metadata": {}, | ||
| 166 | "source": [ | ||
| 167 | "When you solve an inequality or a system, the set of solutions can be more complicated to describe. In this case the result is a list containing lists of expressions that have to be `True` at the same time. It is easier to explain with an example:" | ||
| 168 | ] | ||
| 169 | }, | ||
| 170 | { | ||
| 171 | "cell_type": "code", | ||
| 172 | "execution_count": 38, | ||
| 173 | "metadata": {}, | ||
| 174 | "outputs": [ | ||
| 175 | { | ||
| 176 | "name": "stdout", | ||
| 177 | "output_type": "stream", | ||
| 178 | "text": [ | ||
| 179 | "Simple inequality: [[x < -3], [x > 3]]\n", | ||
| 180 | "System of inequalities:\n", | ||
| 181 | " [\n", | ||
| 182 | "[3 < x, x < 6],\n", | ||
| 183 | "[x < -3]\n", | ||
| 184 | "]\n" | ||
| 185 | ] | ||
| 186 | } | ||
| 187 | ], | ||
| 188 | "source": [ | ||
| 189 | "print(\"Simple inequality:\", solve(x^2-9 > 0, x))\n", | ||
| 190 | "print(\"System of inequalities:\\n\", solve([x^2-9 > 0, x < 6], x))" | ||
| 191 | ] | ||
| 192 | }, | ||
| 193 | { | ||
| 194 | "cell_type": "markdown", | ||
| 195 | "metadata": {}, | ||
| 196 | "source": [ | ||
| 197 | "In the last example (system of inequalities), Sage is telling us that the system\n", | ||
| 198 | "\\begin{align*}\n", | ||
| 199 | " \\begin{cases}\n", | ||
| 200 | " x^2-9 > 9 \\\\\n", | ||
| 201 | " x < 6\n", | ||
| 202 | " \\end{cases}\n", | ||
| 203 | "\\end{align*}\n", | ||
| 204 | "has two solutions:\n", | ||
| 205 | "* $x$ is between $3$ and $6$;\n", | ||
| 206 | "* $x$ is less than $-3$.\n", | ||
| 207 | "\n", | ||
| 208 | "Since in Sage (and in Python) expressions can have at most on relational operator like `<`, the first solution requires two expressions to be described. Hence the \"list of lists\".\n" | ||
| 209 | ] | ||
| 210 | }, | ||
| 211 | { | ||
| 212 | "cell_type": "markdown", | ||
| 213 | "metadata": {}, | ||
| 214 | "source": [ | ||
| 215 | "**Exercise.** In the first exercise you were asked to solve a system of equations, but some of its solutions were complex numbers. Select only the real solutions and print them as pairs $(x,y)$." | ||
| 216 | ] | ||
| 217 | }, | ||
| 218 | { | ||
| 219 | "cell_type": "markdown", | ||
| 220 | "metadata": {}, | ||
| 221 | "source": [ | ||
| 222 | "When solving a system of equations (not inequalities), you can use the option `solution_dict=True` to have the solutions arranged as a *dictionary*, which is a type of Python collection that we did not treat in this course" | ||
| 223 | ] | ||
| 224 | }, | ||
| 225 | { | ||
| 226 | "cell_type": "code", | ||
| 227 | "execution_count": 44, | ||
| 228 | "metadata": {}, | ||
| 229 | "outputs": [ | ||
| 230 | { | ||
| 231 | "data": { | ||
| 232 | "text/plain": [ | ||
| 233 | "[{x: 8/3, y: -2/3}]" | ||
| 234 | ] | ||
| 235 | }, | ||
| 236 | "execution_count": 44, | ||
| 237 | "metadata": {}, | ||
| 238 | "output_type": "execute_result" | ||
| 239 | } | ||
| 240 | ], | ||
| 241 | "source": [ | ||
| 242 | "solve([x+y == 2, 2*x - y == 6], [x,y], solution_dict=True)" | ||
| 243 | ] | ||
| 244 | }, | ||
| 245 | { | ||
| 246 | "cell_type": "markdown", | ||
| 247 | "metadata": {}, | ||
| 248 | "source": [ | ||
| 249 | "### Alternative method for real roots: `find_root()`\n", | ||
| 250 | "\n", | ||
| 251 | "The `solve()` method is very useful when solving *symbolic* equations, for example when you have two variables and you want to solve for one of them in terms of the other. However, it does not always find explicit solutions.\n", | ||
| 252 | "\n", | ||
| 253 | "When you want to find an explicit, even if approximate, solution, it can be better to use `find_root()`. This function works *numerically*, which means that it finds an approximation of the root. It only works for real solutions and you need to specify an interval where you want the root to be searched:" | ||
| 254 | ] | ||
| 255 | }, | ||
| 256 | { | ||
| 257 | "cell_type": "code", | ||
| 258 | "execution_count": 52, | ||
| 259 | "metadata": {}, | ||
| 260 | "outputs": [ | ||
| 261 | { | ||
| 262 | "name": "stdout", | ||
| 263 | "output_type": "stream", | ||
| 264 | "text": [ | ||
| 265 | "Using solve():\n", | ||
| 266 | " [\n", | ||
| 267 | "x == -e^x + 10\n", | ||
| 268 | "]\n", | ||
| 269 | "Using find_root(): 2.070579904980303\n" | ||
| 270 | ] | ||
| 271 | } | ||
| 272 | ], | ||
| 273 | "source": [ | ||
| 274 | "f = e^x + x - 10\n", | ||
| 275 | "print(\"Using solve():\\n\", solve(f, x))\n", | ||
| 276 | "print(\"Using find_root():\", f.find_root(0,100))" | ||
| 277 | ] | ||
| 278 | }, | ||
| 279 | { | ||
| 280 | "cell_type": "markdown", | ||
| 281 | "metadata": {}, | ||
| 282 | "source": [ | ||
| 283 | "## Evaluating functions\n", | ||
| 284 | "\n", | ||
| 285 | "If an expression contains only one variable you can evaluate it easily, even if it is not a function." | ||
| 286 | ] | ||
| 287 | }, | ||
| 288 | { | ||
| 289 | "cell_type": "code", | ||
| 290 | "execution_count": 21, | ||
| 291 | "metadata": {}, | ||
| 292 | "outputs": [ | ||
| 293 | { | ||
| 294 | "name": "stdout", | ||
| 295 | "output_type": "stream", | ||
| 296 | "text": [ | ||
| 297 | "1\n", | ||
| 298 | "y + 3 > (y + 3)^2\n" | ||
| 299 | ] | ||
| 300 | } | ||
| 301 | ], | ||
| 302 | "source": [ | ||
| 303 | "var('y')\n", | ||
| 304 | "f = x^2-3\n", | ||
| 305 | "g = x > x^2\n", | ||
| 306 | "\n", | ||
| 307 | "print(f(2))\n", | ||
| 308 | "print(g(3+y))" | ||
| 309 | ] | ||
| 310 | }, | ||
| 311 | { | ||
| 312 | "cell_type": "markdown", | ||
| 313 | "metadata": {}, | ||
| 314 | "source": [ | ||
| 315 | "If an expression contains more than one variable, you can specify a value for each of them and they will be substituted in alphabetic order. You can also specify a value only for some of the variables." | ||
| 316 | ] | ||
| 317 | }, | ||
| 318 | { | ||
| 319 | "cell_type": "code", | ||
| 320 | "execution_count": 38, | ||
| 321 | "metadata": {}, | ||
| 322 | "outputs": [ | ||
| 323 | { | ||
| 324 | "name": "stdout", | ||
| 325 | "output_type": "stream", | ||
| 326 | "text": [ | ||
| 327 | "-2 == 0\n", | ||
| 328 | "3*y == 2\n" | ||
| 329 | ] | ||
| 330 | } | ||
| 331 | ], | ||
| 332 | "source": [ | ||
| 333 | "var('y','z')\n", | ||
| 334 | "\n", | ||
| 335 | "f = y*z^2 - y == z\n", | ||
| 336 | "print(f(2, 0))\n", | ||
| 337 | "print(f(z=2))" | ||
| 338 | ] | ||
| 339 | }, | ||
| 340 | { | ||
| 341 | "cell_type": "markdown", | ||
| 342 | "metadata": {}, | ||
| 343 | "source": [ | ||
| 344 | "## Symbolic computations\n", | ||
| 345 | "\n", | ||
| 346 | "Sage can understand and simplify symbolic expressions such as sums (finite or infinite) and products. In the following cell, we compute the following sums using the [`sum()`](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.sum) function:\n", | ||
| 347 | "\n", | ||
| 348 | "\\begin{align*}\n", | ||
| 349 | " \\begin{array}{llcc}\n", | ||
| 350 | " (1) & \\sum_{k=0}^nk &=&\\frac{n^2+n}{2}\\\\\n", | ||
| 351 | " (2) & \\sum_{k=0}^nk^4 &=&\\frac{6n^5+15n^4+10n^3-n}{30}\\\\\n", | ||
| 352 | " (3) & \\sum_{k=0}^n\\binom nk &=& 2^n\\\\\n", | ||
| 353 | " (4) & \\sum_{k=0}^\\infty \\frac1{k^2} &=& \\frac{\\pi^2}{6}\n", | ||
| 354 | " \\end{array}\n", | ||
| 355 | "\\end{align*}" | ||
| 356 | ] | ||
| 357 | }, | ||
| 358 | { | ||
| 359 | "cell_type": "code", | ||
| 360 | "execution_count": 22, | ||
| 361 | "metadata": {}, | ||
| 362 | "outputs": [ | ||
| 363 | { | ||
| 364 | "name": "stdout", | ||
| 365 | "output_type": "stream", | ||
| 366 | "text": [ | ||
| 367 | "(1) 1/2*n^2 + 1/2*n\n", | ||
| 368 | "(2) 1/5*n^5 + 1/2*n^4 + 1/3*n^3 - 1/30*n\n", | ||
| 369 | "(3) 2^n\n", | ||
| 370 | "(4) 1/6*pi^2\n" | ||
| 371 | ] | ||
| 372 | } | ||
| 373 | ], | ||
| 374 | "source": [ | ||
| 375 | "var('k', 'n') # Remember to declare all variables\n", | ||
| 376 | "\n", | ||
| 377 | "s = []\n", | ||
| 378 | "s.append( sum(k, k, 0, n) )\n", | ||
| 379 | "s.append( sum(k^4, k, 0, n) )\n", | ||
| 380 | "s.append( sum(binomial(n,k), k, 0, n) )\n", | ||
| 381 | "s.append( sum(1/k^2, k, 1, infinity) )\n", | ||
| 382 | "\n", | ||
| 383 | "for i in range(len(s)):\n", | ||
| 384 | " print(\"({}) {}\".format(i+1, s[i]))" | ||
| 385 | ] | ||
| 386 | }, | ||
| 387 | { | ||
| 388 | "cell_type": "markdown", | ||
| 389 | "metadata": {}, | ||
| 390 | "source": [ | ||
| 391 | "An alternative notation is `expression.sum(k, a, b)`. There is an analogous [`prod()`](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.prod) for products." | ||
| 392 | ] | ||
| 393 | }, | ||
| 394 | { | ||
| 395 | "cell_type": "markdown", | ||
| 396 | "metadata": {}, | ||
| 397 | "source": [ | ||
| 398 | "Sometimes Sage tries to keep an expression in its original form without expanding out sums and products. To change this behavior you can use the [`expand()`](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.expand) function:" | ||
| 399 | ] | ||
| 400 | }, | ||
| 401 | { | ||
| 402 | "cell_type": "code", | ||
| 403 | "execution_count": 30, | ||
| 404 | "metadata": {}, | ||
| 405 | "outputs": [ | ||
| 406 | { | ||
| 407 | "name": "stdout", | ||
| 408 | "output_type": "stream", | ||
| 409 | "text": [ | ||
| 410 | "(x + 1)^2 - (x - 1)^2\n", | ||
| 411 | "4*x\n" | ||
| 412 | ] | ||
| 413 | } | ||
| 414 | ], | ||
| 415 | "source": [ | ||
| 416 | "f = (x+1)^2 - (x-1)^2\n", | ||
| 417 | "print(f)\n", | ||
| 418 | "print(f.expand())" | ||
| 419 | ] | ||
| 420 | }, | ||
| 421 | { | ||
| 422 | "cell_type": "markdown", | ||
| 423 | "metadata": {}, | ||
| 424 | "source": [ | ||
| 425 | "### The Symbolic Ring\n", | ||
| 426 | "**Reference:** [[3](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/ring.html)]\n", | ||
| 427 | "\n", | ||
| 428 | "The symbolic expressions that we have seen so far live in a ring called *symbolic ring* and denoted by `SR` in Sage. This ring works like the ring `ZZ` of integers or `RR` of reals numbers. In particular, you can define matrices and other objects using it as a \"basis\"." | ||
| 429 | ] | ||
| 430 | }, | ||
| 431 | { | ||
| 432 | "cell_type": "code", | ||
| 433 | "execution_count": 45, | ||
| 434 | "metadata": {}, | ||
| 435 | "outputs": [ | ||
| 436 | { | ||
| 437 | "name": "stdout", | ||
| 438 | "output_type": "stream", | ||
| 439 | "text": [ | ||
| 440 | "-b*c + a*d\n", | ||
| 441 | "[(-a, 2)]\n" | ||
| 442 | ] | ||
| 443 | } | ||
| 444 | ], | ||
| 445 | "source": [ | ||
| 446 | "var('a', 'b', 'c', 'd')\n", | ||
| 447 | "\n", | ||
| 448 | "M = matrix([[a,b], [c,d]])\n", | ||
| 449 | "print(M.determinant())\n", | ||
| 450 | "\n", | ||
| 451 | "polring.<x> = SR[]\n", | ||
| 452 | "f = x^2 + 2*a*x + a^2\n", | ||
| 453 | "print(f.roots())" | ||
| 454 | ] | ||
| 455 | }, | ||
| 456 | { | ||
| 457 | "cell_type": "markdown", | ||
| 458 | "metadata": {}, | ||
| 459 | "source": [ | ||
| 460 | "**Exercise.** Compute the eigenvalues of the matrix\n", | ||
| 461 | "\\begin{align*}\n", | ||
| 462 | "\\begin{pmatrix}\n", | ||
| 463 | "\\cos \\alpha & \\sin \\alpha\\\\\n", | ||
| 464 | "-\\sin\\alpha & \\cos \\alpha\n", | ||
| 465 | "\\end{pmatrix}\n", | ||
| 466 | "\\end{align*}" | ||
| 467 | ] | ||
| 468 | }, | ||
| 469 | { | ||
| 470 | "cell_type": "markdown", | ||
| 471 | "metadata": {}, | ||
| 472 | "source": [ | ||
| 473 | "# Calculus\n", | ||
| 474 | "**Reference:** [[4](https://doc.sagemath.org/html/en/reference/calculus/index.html)] for an overview, but most functions are described in [[1](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html)]" | ||
| 475 | ] | ||
| 476 | }, | ||
| 477 | { | ||
| 478 | "cell_type": "markdown", | ||
| 479 | "metadata": {}, | ||
| 480 | "source": [ | ||
| 481 | "## Limits and series\n", | ||
| 482 | "\n", | ||
| 483 | "**References:** [[5](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/calculus.html#sage.calculus.calculus.limit)] for limits, [[6](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.series)] for series\n", | ||
| 484 | "\n", | ||
| 485 | "You can compute limits" | ||
| 486 | ] | ||
| 487 | }, | ||
| 488 | { | ||
| 489 | "cell_type": "code", | ||
| 490 | "execution_count": 54, | ||
| 491 | "metadata": {}, | ||
| 492 | "outputs": [ | ||
| 493 | { | ||
| 494 | "name": "stdout", | ||
| 495 | "output_type": "stream", | ||
| 496 | "text": [ | ||
| 497 | "1\n", | ||
| 498 | "0\n" | ||
| 499 | ] | ||
| 500 | } | ||
| 501 | ], | ||
| 502 | "source": [ | ||
| 503 | "f = sin(x)/x\n", | ||
| 504 | "# print(f(0)) # This one gives an error\n", | ||
| 505 | "print( f.limit(x=0) )\n", | ||
| 506 | "\n", | ||
| 507 | "print( (e^(-x)).limit(x=infinity) )" | ||
| 508 | ] | ||
| 509 | }, | ||
| 510 | { | ||
| 511 | "cell_type": "markdown", | ||
| 512 | "metadata": {}, | ||
| 513 | "source": [ | ||
| 514 | "**Exercise.** Compute the constant $e$ using a limit." | ||
| 515 | ] | ||
| 516 | }, | ||
| 517 | { | ||
| 518 | "cell_type": "markdown", | ||
| 519 | "metadata": {}, | ||
| 520 | "source": [ | ||
| 521 | "You can also specify a direction for the limit. If you don't, Sage assumes that you want to take a two-sided limit." | ||
| 522 | ] | ||
| 523 | }, | ||
| 524 | { | ||
| 525 | "cell_type": "code", | ||
| 526 | "execution_count": 55, | ||
| 527 | "metadata": {}, | ||
| 528 | "outputs": [ | ||
| 529 | { | ||
| 530 | "name": "stdout", | ||
| 531 | "output_type": "stream", | ||
| 532 | "text": [ | ||
| 533 | "und\n", | ||
| 534 | "1\n", | ||
| 535 | "-1\n" | ||
| 536 | ] | ||
| 537 | } | ||
| 538 | ], | ||
| 539 | "source": [ | ||
| 540 | "f = abs(x)/x # 1 if x>0, -1 if x<0\n", | ||
| 541 | "print( f.limit(x=0) ) # undefined\n", | ||
| 542 | "print( f.limit(x=0, dir=\"+\") )\n", | ||
| 543 | "print( f.limit(x=0, dir=\"-\") )" | ||
| 544 | ] | ||
| 545 | }, | ||
| 546 | { | ||
| 547 | "cell_type": "markdown", | ||
| 548 | "metadata": {}, | ||
| 549 | "source": [ | ||
| 550 | "There is also the alternative notation `limit(f, x, dir)` which does the same as `f.limit(x, dir)`." | ||
| 551 | ] | ||
| 552 | }, | ||
| 553 | { | ||
| 554 | "cell_type": "markdown", | ||
| 555 | "metadata": {}, | ||
| 556 | "source": [ | ||
| 557 | "You can also compute series expansions up to any order. **Watch out:** the notation uses `==` instead of `=` as `limit()` does." | ||
| 558 | ] | ||
| 559 | }, | ||
| 560 | { | ||
| 561 | "cell_type": "code", | ||
| 562 | "execution_count": 56, | ||
| 563 | "metadata": {}, | ||
| 564 | "outputs": [ | ||
| 565 | { | ||
| 566 | "name": "stdout", | ||
| 567 | "output_type": "stream", | ||
| 568 | "text": [ | ||
| 569 | "1 + 1*x + 1/2*x^2 + Order(x^3)\n", | ||
| 570 | "(-2) + 1*x + 1*x^2 + (-1/6)*x^3 + (-1/12)*x^4 + 1/120*x^5 + 1/360*x^6 + Order(x^7)\n", | ||
| 571 | "1*(x - 1) + (-1/2)*(x - 1)^2 + Order((x - 1)^3)\n" | ||
| 572 | ] | ||
| 573 | } | ||
| 574 | ], | ||
| 575 | "source": [ | ||
| 576 | "f = e^x\n", | ||
| 577 | "g = sin(x) - 2*cos(x)\n", | ||
| 578 | "h = log(x)\n", | ||
| 579 | "\n", | ||
| 580 | "print(f.series(x==0, 3))\n", | ||
| 581 | "print(g.series(x==0, 7))\n", | ||
| 582 | "print(h.series(x==1, 3))" | ||
| 583 | ] | ||
| 584 | }, | ||
| 585 | { | ||
| 586 | "cell_type": "markdown", | ||
| 587 | "metadata": {}, | ||
| 588 | "source": [ | ||
| 589 | "## Derivatives\n", | ||
| 590 | "**References:** [[7](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.derivative)] and [[8](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/functional.html#sage.calculus.functional.derivative)] for derivatives, [[9](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/functions.html#sage.calculus.functions.jacobian)] for the Jacobian matrix and [[10](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/expression.html#sage.symbolic.expression.Expression.hessian)] for the Hessian." | ||
| 591 | ] | ||
| 592 | }, | ||
| 593 | { | ||
| 594 | "cell_type": "markdown", | ||
| 595 | "metadata": {}, | ||
| 596 | "source": [ | ||
| 597 | "When computing derivatives, you need to specify with respect to which variables you want to derive, except in case there is only one." | ||
| 598 | ] | ||
| 599 | }, | ||
| 600 | { | ||
| 601 | "cell_type": "code", | ||
| 602 | "execution_count": 57, | ||
| 603 | "metadata": {}, | ||
| 604 | "outputs": [ | ||
| 605 | { | ||
| 606 | "name": "stdout", | ||
| 607 | "output_type": "stream", | ||
| 608 | "text": [ | ||
| 609 | "8*y^3\n", | ||
| 610 | "6*x^2 - 1\n" | ||
| 611 | ] | ||
| 612 | } | ||
| 613 | ], | ||
| 614 | "source": [ | ||
| 615 | "var('y')\n", | ||
| 616 | "print( (x^2+2*y^4).derivative(y) ) # Alternative: derivative(f, y)\n", | ||
| 617 | "print( (2*x^3-x+2).derivative() )" | ||
| 618 | ] | ||
| 619 | }, | ||
| 620 | { | ||
| 621 | "cell_type": "markdown", | ||
| 622 | "metadata": {}, | ||
| 623 | "source": [ | ||
| 624 | "You can also compute higher order derivatives:" | ||
| 625 | ] | ||
| 626 | }, | ||
| 627 | { | ||
| 628 | "cell_type": "code", | ||
| 629 | "execution_count": 58, | ||
| 630 | "metadata": {}, | ||
| 631 | "outputs": [ | ||
| 632 | { | ||
| 633 | "name": "stdout", | ||
| 634 | "output_type": "stream", | ||
| 635 | "text": [ | ||
| 636 | "6*x\n", | ||
| 637 | "84*x^5*y + 10*y^4 + 24*x^2*y\n", | ||
| 638 | "1680*x^3 + 48\n" | ||
| 639 | ] | ||
| 640 | } | ||
| 641 | ], | ||
| 642 | "source": [ | ||
| 643 | "print( (x^3).derivative(x, x) ) # Same as (x^3).derivative(x, 2)\n", | ||
| 644 | "\n", | ||
| 645 | "f = x^7*y^2 + x^4*y^2 - 2*x^3 + x^2*y^5 + y + 2\n", | ||
| 646 | "print( f.derivative(x, x, y) ) # Twice in x, once in y\n", | ||
| 647 | "print( f.derivative(x, 4, y, 2) ) # 4 times in x, twice in y" | ||
| 648 | ] | ||
| 649 | }, | ||
| 650 | { | ||
| 651 | "cell_type": "markdown", | ||
| 652 | "metadata": {}, | ||
| 653 | "source": [ | ||
| 654 | "Jacobian and Hessian matrices are also easy to compute:" | ||
| 655 | ] | ||
| 656 | }, | ||
| 657 | { | ||
| 658 | "cell_type": "code", | ||
| 659 | "execution_count": 59, | ||
| 660 | "metadata": {}, | ||
| 661 | "outputs": [ | ||
| 662 | { | ||
| 663 | "name": "stdout", | ||
| 664 | "output_type": "stream", | ||
| 665 | "text": [ | ||
| 666 | "[-2*x + 2*y 2*x]\n", | ||
| 667 | "[ 0 3*y^2]\n", | ||
| 668 | "[ y + 1 x + 1] \n", | ||
| 669 | "\n", | ||
| 670 | "[ 2 -4*y + 1]\n", | ||
| 671 | "[ -4*y + 1 -4*x + 6*y]\n" | ||
| 672 | ] | ||
| 673 | } | ||
| 674 | ], | ||
| 675 | "source": [ | ||
| 676 | "f = (-x^2 + 2*x*y, y^3, x+y+x*y)\n", | ||
| 677 | "print( jacobian(f, [x,y]), \"\\n\" )\n", | ||
| 678 | "\n", | ||
| 679 | "g = x^2 + x*y + y^3 -2*x*y^2 -3\n", | ||
| 680 | "print( g.hessian() )" | ||
| 681 | ] | ||
| 682 | }, | ||
| 683 | { | ||
| 684 | "cell_type": "markdown", | ||
| 685 | "metadata": {}, | ||
| 686 | "source": [ | ||
| 687 | "*Note:* the notation `f.jacobian([x,y])` is also valid, but only if you specify that `f` is vector by declaring it as `f = vector([...])`." | ||
| 688 | ] | ||
| 689 | }, | ||
| 690 | { | ||
| 691 | "cell_type": "markdown", | ||
| 692 | "metadata": {}, | ||
| 693 | "source": [ | ||
| 694 | "## Integrals\n", | ||
| 695 | "**References:** [[11](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/integration/integral.html)] for symbolic integration and [[12](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/integration.html)] for numerical methods.\n", | ||
| 696 | "\n", | ||
| 697 | "You should remember from high school or from your first calculus/analysis course that derivatives are easy, but integrals are hard.\n", | ||
| 698 | "When using a computer software to solve your integrals, you have two choices:\n", | ||
| 699 | "\n", | ||
| 700 | "1. You can try to compute a primitive function exactly, and then (if you are computing a definite integral) substitute the endpoints of your integration interval to get the result. We can call this *symbolic integration*.\n", | ||
| 701 | "2. You can get an *approximated* result with a *numerical method*. This method always gives some kind of result, but it cannot be used to compute indefinite integrals.\n", | ||
| 702 | "\n", | ||
| 703 | "Sage can do both of these things, although people that work in numerical analysis and use often the second method tend to prefer other programs, such as Matlab (or its open-source clone Octave)." | ||
| 704 | ] | ||
| 705 | }, | ||
| 706 | { | ||
| 707 | "cell_type": "markdown", | ||
| 708 | "metadata": {}, | ||
| 709 | "source": [ | ||
| 710 | "### Symbolic integration\n", | ||
| 711 | "\n", | ||
| 712 | "Symbolic integrals work more or less like derivatives. You must specify an integration variable, but the endpoints of the integration interval are optional. If they are not given you get an indefinite integral." | ||
| 713 | ] | ||
| 714 | }, | ||
| 715 | { | ||
| 716 | "cell_type": "code", | ||
| 717 | "execution_count": 60, | ||
| 718 | "metadata": {}, | ||
| 719 | "outputs": [ | ||
| 720 | { | ||
| 721 | "name": "stdout", | ||
| 722 | "output_type": "stream", | ||
| 723 | "text": [ | ||
| 724 | "1/2*x^2 - cos(x)\n", | ||
| 725 | "0\n", | ||
| 726 | "-1/2*a^2 + 1/2*b^2 + cos(a) - cos(b)\n" | ||
| 727 | ] | ||
| 728 | } | ||
| 729 | ], | ||
| 730 | "source": [ | ||
| 731 | "var('a', 'b')\n", | ||
| 732 | "f = x + sin(x)\n", | ||
| 733 | "print( f.integral(x) ) # Alternative: integral(f, x)\n", | ||
| 734 | "print( f.integral(x, -10, 10) )\n", | ||
| 735 | "print( f.integral(x, a, b) )" | ||
| 736 | ] | ||
| 737 | }, | ||
| 738 | { | ||
| 739 | "cell_type": "markdown", | ||
| 740 | "metadata": {}, | ||
| 741 | "source": [ | ||
| 742 | "Your endpoints can also be $\\pm\\infty$:" | ||
| 743 | ] | ||
| 744 | }, | ||
| 745 | { | ||
| 746 | "cell_type": "code", | ||
| 747 | "execution_count": 61, | ||
| 748 | "metadata": {}, | ||
| 749 | "outputs": [ | ||
| 750 | { | ||
| 751 | "name": "stdout", | ||
| 752 | "output_type": "stream", | ||
| 753 | "text": [ | ||
| 754 | "1\n", | ||
| 755 | "sqrt(pi)\n" | ||
| 756 | ] | ||
| 757 | } | ||
| 758 | ], | ||
| 759 | "source": [ | ||
| 760 | "print( integral(e^(-x), x, 0, infinity) )\n", | ||
| 761 | "print( integral(e^(-x^2), x, -infinity, infinity) )" | ||
| 762 | ] | ||
| 763 | }, | ||
| 764 | { | ||
| 765 | "cell_type": "markdown", | ||
| 766 | "metadata": {}, | ||
| 767 | "source": [ | ||
| 768 | "The last function is also an example of an integral that perhaps you might want to compute numerically. In fact:" | ||
| 769 | ] | ||
| 770 | }, | ||
| 771 | { | ||
| 772 | "cell_type": "code", | ||
| 773 | "execution_count": 65, | ||
| 774 | "metadata": {}, | ||
| 775 | "outputs": [ | ||
| 776 | { | ||
| 777 | "name": "stdout", | ||
| 778 | "output_type": "stream", | ||
| 779 | "text": [ | ||
| 780 | "1/2*sqrt(pi)*erf(x)\n", | ||
| 781 | "1/2*sqrt(pi)*erf(2) - 1/2*sqrt(pi)*erf(1)\n" | ||
| 782 | ] | ||
| 783 | } | ||
| 784 | ], | ||
| 785 | "source": [ | ||
| 786 | "print( integral(e^(-x^2), x) )\n", | ||
| 787 | "print( integral(e^(-x^2), x, 1, 2) )" | ||
| 788 | ] | ||
| 789 | }, | ||
| 790 | { | ||
| 791 | "cell_type": "markdown", | ||
| 792 | "metadata": {}, | ||
| 793 | "source": [ | ||
| 794 | "Here `erf(x)` denotes the [error function](https://en.wikipedia.org/wiki/Error_function)." | ||
| 795 | ] | ||
| 796 | }, | ||
| 797 | { | ||
| 798 | "cell_type": "markdown", | ||
| 799 | "metadata": {}, | ||
| 800 | "source": [ | ||
| 801 | "### Numerical integration\n", | ||
| 802 | "\n", | ||
| 803 | "In order to get an explicit value for the computations above, we can use a *numerical* method.\n", | ||
| 804 | "\n", | ||
| 805 | "The word \"numerical\" does not have much to do with numbers, but it refers to the fact that we are trying to compute explicit results rather than symbolic or algebraic ones. [Numerical analysis](https://en.wikipedia.org/wiki/Numerical_analysis) is the branch of mathematics that studies methods to approximate computations over the real or complex numbers. With these methods there is usually a trade-off between speed and precision.\n", | ||
| 806 | "\n", | ||
| 807 | "The Sage function [`numerical_integral()`](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/integration.html#sage.calculus.integration.numerical_integral) takes as a parameter a real-valued one-variable function and the integration endpoints, and it returns both an approximate value for the integral and an error estimate." | ||
| 808 | ] | ||
| 809 | }, | ||
| 810 | { | ||
| 811 | "cell_type": "code", | ||
| 812 | "execution_count": 40, | ||
| 813 | "metadata": {}, | ||
| 814 | "outputs": [ | ||
| 815 | { | ||
| 816 | "data": { | ||
| 817 | "text/plain": [ | ||
| 818 | "(0.13525725794999466, 1.5016572202374808e-15)" | ||
| 819 | ] | ||
| 820 | }, | ||
| 821 | "execution_count": 40, | ||
| 822 | "metadata": {}, | ||
| 823 | "output_type": "execute_result" | ||
| 824 | } | ||
| 825 | ], | ||
| 826 | "source": [ | ||
| 827 | "numerical_integral(e^(-x^2), 1, 2)" | ||
| 828 | ] | ||
| 829 | }, | ||
| 830 | { | ||
| 831 | "cell_type": "markdown", | ||
| 832 | "metadata": {}, | ||
| 833 | "source": [ | ||
| 834 | "The result above means, in symbols\n", | ||
| 835 | "\\begin{align*}\n", | ||
| 836 | "\\int_1^2 e^{-x^2}\\mathrm dx = 0.13525725794999466 \\pm 1.5016572202374808\\times 10^{-15}\n", | ||
| 837 | "\\end{align*}\n", | ||
| 838 | "\n", | ||
| 839 | "There is also a [`monte_carlo_integral()`](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/integration.html#sage.calculus.integration.monte_carlo_integral) method for functions with more than one variable." | ||
| 840 | ] | ||
| 841 | }, | ||
| 842 | { | ||
| 843 | "cell_type": "markdown", | ||
| 844 | "metadata": {}, | ||
| 845 | "source": [ | ||
| 846 | "**Exercise.** Compute the area of the ellipse of equation $y^2+\\left(\\frac x3\\right)^2=1$." | ||
| 847 | ] | ||
| 848 | }, | ||
| 849 | { | ||
| 850 | "cell_type": "markdown", | ||
| 851 | "metadata": {}, | ||
| 852 | "source": [ | ||
| 853 | "## Differential equations\n", | ||
| 854 | "**Reference:** [[13](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/desolvers.html)]\n", | ||
| 855 | "\n", | ||
| 856 | "A [differential equation](https://en.wikipedia.org/wiki/Differential_equation) is an equation involving an unknwon function and its derivatives. They can be of two kinds: *ordinary* differential equations ([ODE](https://en.wikipedia.org/wiki/Ordinary_differential_equation)) and *partial* differential equations ([PDE](https://en.wikipedia.org/wiki/Partial_differential_equation)). The latter involve multivariate functions and their partial derivatives.\n", | ||
| 857 | "\n", | ||
| 858 | "Differential equations are in general hard to solve *exactly* (or *symbolically*): even a simple equation of the form $f'(x)=g(x)$, where $g(x)$ is someknown function, requires solving the integral $\\int g(x)\\mathrm{d}x$ in order to find $f$, which as we know is not always easy!\n", | ||
| 859 | "\n", | ||
| 860 | "Theoretical results on differential equations usually ensure the existence and/or uniquess of a solution under certain conditions, but in general they do not give a way to solve them. There exits many methods to find approximate solutions, and some of them are implemented in Sage as well (see [[13](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/desolvers.html)]). However we will focus on the simple ODEs that can be solved exactly.\n", | ||
| 861 | "\n", | ||
| 862 | "Let's start with a simple example. Let's find all functions $f(x)$ such that $f'(x)=f(x)$. In order to do so, we need to use the `function()` construct, which allows us to define an \"unknwon\" function inside Sage, like we define variables with `var()`." | ||
| 863 | ] | ||
| 864 | }, | ||
| 865 | { | ||
| 866 | "cell_type": "code", | ||
| 867 | "execution_count": 4, | ||
| 868 | "metadata": {}, | ||
| 869 | "outputs": [ | ||
| 870 | { | ||
| 871 | "data": { | ||
| 872 | "text/plain": [ | ||
| 873 | "_C*e^x" | ||
| 874 | ] | ||
| 875 | }, | ||
| 876 | "execution_count": 4, | ||
| 877 | "metadata": {}, | ||
| 878 | "output_type": "execute_result" | ||
| 879 | } | ||
| 880 | ], | ||
| 881 | "source": [ | ||
| 882 | "var('x')\n", | ||
| 883 | "function('f')\n", | ||
| 884 | "equation = derivative(f(x)) == f(x)\n", | ||
| 885 | "desolve(equation, f(x)) # f is the unknown function" | ||
| 886 | ] | ||
| 887 | }, | ||
| 888 | { | ||
| 889 | "cell_type": "markdown", | ||
| 890 | "metadata": {}, | ||
| 891 | "source": [ | ||
| 892 | "As you can expect, they are all the functions $Ce^x$ for some constant $C$. The constant $C$ plays the same role as the constant in the solution of an integral, but in this case Sage writes it explicitly.\n", | ||
| 893 | "\n", | ||
| 894 | "We can also specify *initial conditions* for our function. For example we can impose that $f(0)=3$ as follows:" | ||
| 895 | ] | ||
| 896 | }, | ||
| 897 | { | ||
| 898 | "cell_type": "code", | ||
| 899 | "execution_count": 5, | ||
| 900 | "metadata": {}, | ||
| 901 | "outputs": [ | ||
| 902 | { | ||
| 903 | "data": { | ||
| 904 | "text/plain": [ | ||
| 905 | "3*e^x" | ||
| 906 | ] | ||
| 907 | }, | ||
| 908 | "execution_count": 5, | ||
| 909 | "metadata": {}, | ||
| 910 | "output_type": "execute_result" | ||
| 911 | } | ||
| 912 | ], | ||
| 913 | "source": [ | ||
| 914 | "desolve(equation, f(x), (0,3))" | ||
| 915 | ] | ||
| 916 | }, | ||
| 917 | { | ||
| 918 | "cell_type": "markdown", | ||
| 919 | "metadata": {}, | ||
| 920 | "source": [ | ||
| 921 | "You can also solve *second order* equations, that is equations where the second derivative also appears. In this case if you want to specify an initial condition you should write the triple of values $(x_0, f(x_0), f'(x_0))$." | ||
| 922 | ] | ||
| 923 | }, | ||
| 924 | { | ||
| 925 | "cell_type": "code", | ||
| 926 | "execution_count": 6, | ||
| 927 | "metadata": {}, | ||
| 928 | "outputs": [ | ||
| 929 | { | ||
| 930 | "data": { | ||
| 931 | "text/plain": [ | ||
| 932 | "-1/2*I*sqrt(2)*sqrt(pi)*integrate(erf(1/2*I*sqrt(2)*x)*e^(-1/2*x^2), x)" | ||
| 933 | ] | ||
| 934 | }, | ||
| 935 | "execution_count": 6, | ||
| 936 | "metadata": {}, | ||
| 937 | "output_type": "execute_result" | ||
| 938 | } | ||
| 939 | ], | ||
| 940 | "source": [ | ||
| 941 | "equation = derivative(f(x), x, 2) + x*derivative(f(x)) == 1\n", | ||
| 942 | "desolve(equation, f(x), (0, 0, 0))" | ||
| 943 | ] | ||
| 944 | }, | ||
| 945 | { | ||
| 946 | "cell_type": "markdown", | ||
| 947 | "metadata": {}, | ||
| 948 | "source": [ | ||
| 949 | "**Exercise.** Use Sage to find out the functions $f(x)$ that satisfy\n", | ||
| 950 | "\\begin{align*}\n", | ||
| 951 | " \\begin{array}{rlcrl}\n", | ||
| 952 | " (A) &\n", | ||
| 953 | " \\begin{cases}\n", | ||
| 954 | " f(0) &= 1\\\\\n", | ||
| 955 | " f'(0) &= 0\\\\\n", | ||
| 956 | " f''(x) &= -f(x)\n", | ||
| 957 | " \\end{cases}\n", | ||
| 958 | " & \\qquad \\qquad &\n", | ||
| 959 | " (B) &\n", | ||
| 960 | " \\begin{cases}\n", | ||
| 961 | " f(0) &= 0\\\\\n", | ||
| 962 | " f'(0) &= 1\\\\\n", | ||
| 963 | " f''(x) &= -f(x)\n", | ||
| 964 | " \\end{cases}\n", | ||
| 965 | " \\end{array}\n", | ||
| 966 | "\\end{align*}" | ||
| 967 | ] | ||
| 968 | }, | ||
| 969 | { | ||
| 970 | "cell_type": "code", | ||
| 971 | "execution_count": null, | ||
| 972 | "metadata": {}, | ||
| 973 | "outputs": [], | ||
| 974 | "source": [] | ||
| 975 | }, | ||
| 976 | { | ||
| 977 | "cell_type": "markdown", | ||
| 978 | "metadata": {}, | ||
| 979 | "source": [ | ||
| 980 | "### A real-world example\n", | ||
| 981 | "\n", | ||
| 982 | "Differential equations have countless applications in Science, so it would be a shame not to see at least a simple one.\n", | ||
| 983 | "\n", | ||
| 984 | "Consider an object moving with constant acceleration $a$. Its velocity at time $t$ is described by the formula $v(t) = v(0) + at$. For example an object falling from the sky has acceleration $g\\sim 9.8 m/s^2$ towards the ground, so its velocity is $v(t) = -gt$.\n", | ||
| 985 | "\n", | ||
| 986 | "However in the real world you need to take into account the air's resistance, which depends (among other things) on the velocity of the object. In this case the acceleration $a(t)$ is not constant anymore, and it satisfies an equation of the form $a(t)=-g -kv(t)$, where $k$ is some constant that may depend on the shape and mass of the object (in practice it may be more complicated than this).\n", | ||
| 987 | "\n", | ||
| 988 | "Since the acceleration is the derivative of the velocity, we have a differential equation\n", | ||
| 989 | "\\begin{align*}\n", | ||
| 990 | " v'(t) = -g -kv(t)\n", | ||
| 991 | "\\end{align*}\n", | ||
| 992 | "and we can try to solve it with Sage!" | ||
| 993 | ] | ||
| 994 | }, | ||
| 995 | { | ||
| 996 | "cell_type": "code", | ||
| 997 | "execution_count": 7, | ||
| 998 | "metadata": {}, | ||
| 999 | "outputs": [ | ||
| 1000 | { | ||
| 1001 | "data": { | ||
| 1002 | "text/plain": [ | ||
| 1003 | "-98/15*(e^(3/2*t) - 1)*e^(-3/2*t)" | ||
| 1004 | ] | ||
| 1005 | }, | ||
| 1006 | "execution_count": 7, | ||
| 1007 | "metadata": {}, | ||
| 1008 | "output_type": "execute_result" | ||
| 1009 | } | ||
| 1010 | ], | ||
| 1011 | "source": [ | ||
| 1012 | "var('t')\n", | ||
| 1013 | "function('v')\n", | ||
| 1014 | "g = 9.8\n", | ||
| 1015 | "k = 1.5\n", | ||
| 1016 | "conditions = (0, 0) # Start with velocity 0\n", | ||
| 1017 | "desolve(derivative(v(t)) == -g -k*v(t), v(t), conditions)" | ||
| 1018 | ] | ||
| 1019 | }, | ||
| 1020 | { | ||
| 1021 | "cell_type": "markdown", | ||
| 1022 | "metadata": {}, | ||
| 1023 | "source": [ | ||
| 1024 | "If you want to solve this equation symbolically (that is, keeping $g$ and $k$ in symbols) you need to specify that $t$ is the *independent variable* of the equation:" | ||
| 1025 | ] | ||
| 1026 | }, | ||
| 1027 | { | ||
| 1028 | "cell_type": "code", | ||
| 1029 | "execution_count": 10, | ||
| 1030 | "metadata": {}, | ||
| 1031 | "outputs": [ | ||
| 1032 | { | ||
| 1033 | "data": { | ||
| 1034 | "text/plain": [ | ||
| 1035 | "-(g*e^(k*t) - g)*e^(-k*t)/k" | ||
| 1036 | ] | ||
| 1037 | }, | ||
| 1038 | "execution_count": 10, | ||
| 1039 | "metadata": {}, | ||
| 1040 | "output_type": "execute_result" | ||
| 1041 | } | ||
| 1042 | ], | ||
| 1043 | "source": [ | ||
| 1044 | "var('t', 'g', 'k')\n", | ||
| 1045 | "function('v')\n", | ||
| 1046 | "conditions = (0, 0) # Start with velocity 0\n", | ||
| 1047 | "desolve(derivative(v(t)) == -g -k*v(t), v(t), conditions, ivar=t)" | ||
| 1048 | ] | ||
| 1049 | }, | ||
| 1050 | { | ||
| 1051 | "cell_type": "markdown", | ||
| 1052 | "metadata": {}, | ||
| 1053 | "source": [ | ||
| 1054 | "# Basic data analysis and visualization\n", | ||
| 1055 | "\n", | ||
| 1056 | "## Statistics\n", | ||
| 1057 | "**References:** [[14](https://doc.sagemath.org/html/en/reference/stats/sage/stats/basic_stats.html)]\n", | ||
| 1058 | "\n", | ||
| 1059 | "Sage includes the most basic functions for statistical analysis." | ||
| 1060 | ] | ||
| 1061 | }, | ||
| 1062 | { | ||
| 1063 | "cell_type": "code", | ||
| 1064 | "execution_count": 20, | ||
| 1065 | "metadata": {}, | ||
| 1066 | "outputs": [ | ||
| 1067 | { | ||
| 1068 | "name": "stdout", | ||
| 1069 | "output_type": "stream", | ||
| 1070 | "text": [ | ||
| 1071 | "Values:\t [1, 2, 3, 3, -6, -2, 4, -1, 0, 2, 3, -4, 0]\n", | ||
| 1072 | "Mean:\t\t\t 5/13\n", | ||
| 1073 | "Median:\t\t\t 1\n", | ||
| 1074 | "Mode:\t\t\t [3]\n", | ||
| 1075 | "Standard deviation:\t 2*sqrt(29/13)\n", | ||
| 1076 | "Variance:\t\t 116/13\n", | ||
| 1077 | "Moving average (5): [3/5, 0, 2/5, -2/5, -1, 3/5, 8/5, 0, 1/5]\n" | ||
| 1078 | ] | ||
| 1079 | } | ||
| 1080 | ], | ||
| 1081 | "source": [ | ||
| 1082 | "L = [1, 2, 3, 3, -6, -2, 4, -1, 0, 2, 3, -4, 0]\n", | ||
| 1083 | "\n", | ||
| 1084 | "print(\"Values:\\t\", L)\n", | ||
| 1085 | "\n", | ||
| 1086 | "print(\"Mean:\\t\\t\\t\", mean(L))\n", | ||
| 1087 | "print(\"Median:\\t\\t\\t\", median(L))\n", | ||
| 1088 | "print(\"Mode:\\t\\t\\t\", mode(L))\n", | ||
| 1089 | "\n", | ||
| 1090 | "print(\"Standard deviation:\\t\", std(L))\n", | ||
| 1091 | "print(\"Variance:\\t\\t\", variance(L))\n", | ||
| 1092 | "\n", | ||
| 1093 | "print(\"Moving average (5):\", moving_average(L,5))" | ||
| 1094 | ] | ||
| 1095 | }, | ||
| 1096 | { | ||
| 1097 | "cell_type": "markdown", | ||
| 1098 | "metadata": {}, | ||
| 1099 | "source": [ | ||
| 1100 | "You can also compare your data to a probability distribution, see [this page](https://doc.sagemath.org/html/en/reference/probability/sage/probability/probability_distribution.html). If you need to do more advanced statistics you should consider using [R](https://www.r-project.org/); you can also use it inside Sage." | ||
| 1101 | ] | ||
| 1102 | }, | ||
| 1103 | { | ||
| 1104 | "cell_type": "markdown", | ||
| 1105 | "metadata": {}, | ||
| 1106 | "source": [ | ||
| 1107 | "## Plotting\n", | ||
| 1108 | "**Reference:** [[15](https://doc.sagemath.org/html/en/reference/plotting/index.html)], more specifically the subsection [[16](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html)].\n", | ||
| 1109 | "\n", | ||
| 1110 | "Some Sage objects can be plotted:" | ||
| 1111 | ] | ||
| 1112 | }, | ||
| 1113 | { | ||
| 1114 | "cell_type": "code", | ||
| 1115 | "execution_count": 21, | ||
| 1116 | "metadata": {}, | ||
| 1117 | "outputs": [ | ||
| 1118 | { | ||
| 1119 | "data": { | ||
| 1120 | "image/png": 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\n", | ||
| 1121 | "text/plain": [ | ||
| 1122 | "Graphics object consisting of 1 graphics primitive" | ||
| 1123 | ] | ||
| 1124 | }, | ||
| 1125 | "execution_count": 21, | ||
| 1126 | "metadata": {}, | ||
| 1127 | "output_type": "execute_result" | ||
| 1128 | } | ||
| 1129 | ], | ||
| 1130 | "source": [ | ||
| 1131 | "f = sin(x)\n", | ||
| 1132 | "plot(f)" | ||
| 1133 | ] | ||
| 1134 | }, | ||
| 1135 | { | ||
| 1136 | "cell_type": "markdown", | ||
| 1137 | "metadata": {}, | ||
| 1138 | "source": [ | ||
| 1139 | "Sage's plotting functions are based on Python's [matplotlib](https://matplotlib.org/).\n", | ||
| 1140 | "\n", | ||
| 1141 | "You can give a number of options to adjust the aspect of your plot, see [here](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/plot.html#sage.plot.plot.plot). Let's see some of them:" | ||
| 1142 | ] | ||
| 1143 | }, | ||
| 1144 | { | ||
| 1145 | "cell_type": "code", | ||
| 1146 | "execution_count": 67, | ||
| 1147 | "metadata": {}, | ||
| 1148 | "outputs": [ | ||
| 1149 | { | ||
| 1150 | "data": { | ||
| 1151 | "image/png": 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\n", | ||
| 1152 | "text/plain": [ | ||
| 1153 | "Graphics object consisting of 1 graphics primitive" | ||
| 1154 | ] | ||
| 1155 | }, | ||
| 1156 | "execution_count": 67, | ||
| 1157 | "metadata": {}, | ||
| 1158 | "output_type": "execute_result" | ||
| 1159 | } | ||
| 1160 | ], | ||
| 1161 | "source": [ | ||
| 1162 | "f = sin(x)\n", | ||
| 1163 | "plot(f,\n", | ||
| 1164 | " -2*pi, 2*pi, # bounds for x\n", | ||
| 1165 | " ymin = -0.7, ymax = 0.7, # bounds for y\n", | ||
| 1166 | " color = \"red\",\n", | ||
| 1167 | " title = \"The sin function\",\n", | ||
| 1168 | " )" | ||
| 1169 | ] | ||
| 1170 | }, | ||
| 1171 | { | ||
| 1172 | "cell_type": "markdown", | ||
| 1173 | "metadata": {}, | ||
| 1174 | "source": [ | ||
| 1175 | "Some of the options are not described precisely in Sage's documentation, but you can find them on [matplotlib's documentation](https://matplotlib.org/stable/contents.html). You can find many examples online for adjusting your plot as you like!" | ||
| 1176 | ] | ||
| 1177 | }, | ||
| 1178 | { | ||
| 1179 | "cell_type": "markdown", | ||
| 1180 | "metadata": {}, | ||
| 1181 | "source": [ | ||
| 1182 | "If you need to plot more than one object at the time, you can sum two plots and show them together with `show()`:" | ||
| 1183 | ] | ||
| 1184 | }, | ||
| 1185 | { | ||
| 1186 | "cell_type": "code", | ||
| 1187 | "execution_count": 36, | ||
| 1188 | "metadata": {}, | ||
| 1189 | "outputs": [ | ||
| 1190 | { | ||
| 1191 | "data": { | ||
| 1192 | "image/png": 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\n", | ||
| 1193 | "text/plain": [ | ||
| 1194 | "Graphics object consisting of 2 graphics primitives" | ||
| 1195 | ] | ||
| 1196 | }, | ||
| 1197 | "metadata": {}, | ||
| 1198 | "output_type": "display_data" | ||
| 1199 | } | ||
| 1200 | ], | ||
| 1201 | "source": [ | ||
| 1202 | "cosine = plot(cos(x), (x,-pi/2,pi/2), color=\"red\")\n", | ||
| 1203 | "exponential = plot(exp(x), (x,-2,0.5))\n", | ||
| 1204 | "\n", | ||
| 1205 | "show(cosine + exponential)" | ||
| 1206 | ] | ||
| 1207 | }, | ||
| 1208 | { | ||
| 1209 | "cell_type": "markdown", | ||
| 1210 | "metadata": {}, | ||
| 1211 | "source": [ | ||
| 1212 | "Finally, there are other types of plots that you can use, like [scatter plots](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/scatter_plot.html#sage.plot.scatter_plot.scatter_plot) and [bar charts](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/bar_chart.html#sage.plot.bar_chart.bar_chart). You can also add [text](https://doc.sagemath.org/html/en/reference/plotting/sage/plot/text.html#sage.plot.text.text) to your plot:" | ||
| 1213 | ] | ||
| 1214 | }, | ||
| 1215 | { | ||
| 1216 | "cell_type": "code", | ||
| 1217 | "execution_count": 53, | ||
| 1218 | "metadata": {}, | ||
| 1219 | "outputs": [ | ||
| 1220 | { | ||
| 1221 | "data": { | ||
| 1222 | "image/png": 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\n", | ||
| 1223 | "text/plain": [ | ||
| 1224 | "Graphics object consisting of 3 graphics primitives" | ||
| 1225 | ] | ||
| 1226 | }, | ||
| 1227 | "metadata": {}, | ||
| 1228 | "output_type": "display_data" | ||
| 1229 | } | ||
| 1230 | ], | ||
| 1231 | "source": [ | ||
| 1232 | "b = bar_chart(range(1,10))\n", | ||
| 1233 | "s = scatter_plot([(1,5), (4,2), (8,8), (4,7)],\n", | ||
| 1234 | " marker = \"*\", # symbol\n", | ||
| 1235 | " markersize = 100,\n", | ||
| 1236 | " edgecolor = \"black\",\n", | ||
| 1237 | " facecolor = \"red\"\n", | ||
| 1238 | " )\n", | ||
| 1239 | "t = text(\"wow, such plot!\", (1, 8), color=\"black\", fontsize=20)\n", | ||
| 1240 | "show(b + s + t)" | ||
| 1241 | ] | ||
| 1242 | }, | ||
| 1243 | { | ||
| 1244 | "cell_type": "markdown", | ||
| 1245 | "metadata": {}, | ||
| 1246 | "source": [ | ||
| 1247 | "## Interpolation\n", | ||
| 1248 | "**References:** [[17](https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_ring.html#sage.rings.polynomial.polynomial_ring.PolynomialRing_field.lagrange_polynomial)] and [[18](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/interpolation.html)].\n", | ||
| 1249 | "\n", | ||
| 1250 | "When you need to work with a discrete set of data, like measurements of real-world quantities, it can be useful to visualize a \"smoothed out\" version of this data, for example by plotting a function that approximates it.\n", | ||
| 1251 | "\n", | ||
| 1252 | "One way to do so is finding the lowest-degree polynomial that passes through all your points. This is called [Lagrange Polynomial](https://en.wikipedia.org/wiki/Lagrange_polynomial)." | ||
| 1253 | ] | ||
| 1254 | }, | ||
| 1255 | { | ||
| 1256 | "cell_type": "code", | ||
| 1257 | "execution_count": 65, | ||
| 1258 | "metadata": {}, | ||
| 1259 | "outputs": [ | ||
| 1260 | { | ||
| 1261 | "data": { | ||
| 1262 | "image/png": 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\n", | ||
| 1263 | "text/plain": [ | ||
| 1264 | "Graphics object consisting of 3 graphics primitives" | ||
| 1265 | ] | ||
| 1266 | }, | ||
| 1267 | "metadata": {}, | ||
| 1268 | "output_type": "display_data" | ||
| 1269 | } | ||
| 1270 | ], | ||
| 1271 | "source": [ | ||
| 1272 | "points = [ (0,1), (1,2), (1.5,0), (2,4), (3,5) ]\n", | ||
| 1273 | "polring.<x> = QQ[] # you need to specify a polynomial ring\n", | ||
| 1274 | "lp = polring.lagrange_polynomial(points)\n", | ||
| 1275 | "show(scatter_plot(points, facecolor=\"red\")\n", | ||
| 1276 | " + plot(lp, 0, 3) # slightly different notation for polynomials\n", | ||
| 1277 | " + text(lp, (1,8), color=\"black\")\n", | ||
| 1278 | " )" | ||
| 1279 | ] | ||
| 1280 | }, | ||
| 1281 | { | ||
| 1282 | "cell_type": "markdown", | ||
| 1283 | "metadata": {}, | ||
| 1284 | "source": [ | ||
| 1285 | "One can compute the Lagrange Polynomial over any base ring, and it has the advantage that it is a very \"nice\" function (continuous and differentiable as much as you like, with easily computable derivatives and primitives).\n", | ||
| 1286 | "\n", | ||
| 1287 | "However, it does not always give you good approximation of your data:" | ||
| 1288 | ] | ||
| 1289 | }, | ||
| 1290 | { | ||
| 1291 | "cell_type": "code", | ||
| 1292 | "execution_count": 2, | ||
| 1293 | "metadata": {}, | ||
| 1294 | "outputs": [ | ||
| 1295 | { | ||
| 1296 | "data": { | ||
| 1297 | "image/png": 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\n", | ||
| 1298 | "text/plain": [ | ||
| 1299 | "Graphics object consisting of 2 graphics primitives" | ||
| 1300 | ] | ||
| 1301 | }, | ||
| 1302 | "metadata": {}, | ||
| 1303 | "output_type": "display_data" | ||
| 1304 | } | ||
| 1305 | ], | ||
| 1306 | "source": [ | ||
| 1307 | "R = [x/10 for x in range(-10,10)]\n", | ||
| 1308 | "L = [1/(1+25*x^2) for x in R]\n", | ||
| 1309 | "points = [(R[i], L[i]) for i in range(len(L))]\n", | ||
| 1310 | "polring.<x> = RR[]\n", | ||
| 1311 | "lp = polring.lagrange_polynomial(points)\n", | ||
| 1312 | "\n", | ||
| 1313 | "show(plot(lp, -0.82, 0.72) + scatter_plot(points))" | ||
| 1314 | ] | ||
| 1315 | }, | ||
| 1316 | { | ||
| 1317 | "cell_type": "markdown", | ||
| 1318 | "metadata": {}, | ||
| 1319 | "source": [ | ||
| 1320 | "This particular example is called [Runge's phenomenon](https://en.wikipedia.org/wiki/Runge%27s_phenomenon). For a better approximation you can use a [spline](https://en.wikipedia.org/wiki/Spline_(mathematics)), which is a *piecewise* polynomial function:" | ||
| 1321 | ] | ||
| 1322 | }, | ||
| 1323 | { | ||
| 1324 | "cell_type": "code", | ||
| 1325 | "execution_count": 90, | ||
| 1326 | "metadata": {}, | ||
| 1327 | "outputs": [ | ||
| 1328 | { | ||
| 1329 | "data": { | ||
| 1330 | "image/png": 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jJdx220xcrkV4vV5cLhfp6WPJzMxq1rmJghWYAEpKSpg1ayb5+eY9WrgYM2Ysv/lt875HEQkYBSYRcdZtt8H8+bBxo4fvv3cTHx8flPE8wQxMlTweD+vWuenfP55XX41l9OigHFZEmk4TV4qIswoLIS0Njj02lvbt20f04OfY2FguuKA9XbvGUlTkdDUiEmgKTCLSLLZtg48/hmHDnK4keCzLBEQFJpHIo8AkIs1ixQoTIC65xOlKgistDTZuhC++cLoSEQkkBSYRaRaFhdCnD5x4otOVBNeQIeByqZVJJNIoMIlIwHm9poUpmrrjKrVpA337KjCJRBoFJhEJuPffh2++ic7ABKZb7vXX4aefnK5ERAJFgUlEAq6wEOLiTEtLNEpLg+++g/fec7oSEQkUBSYRCbiiIrj4YjjmGKcrcUZKCiQkmOAoIpFBgUlEAsrthtWro7c7DqBFC3N3oMYxiUQOBSYRCag33jBjd9LSgn/s7OxskpOTSUlJCf7Bj5KWBu++a7rmRCT8aWkUEQmoKVPMHXKffeZcDU4sjXK0zZuhWzdYuBDS0x0pQUT8o6VRRCT4CgujuzuuUteucNppGsckEikUmEQkYD7/HMrKnOmOC0XDhpnA1PSGfBFxmgKTiARMUZEZ8Dx4sNOVhIa0NLNEyqZNTlciIk2lwCQiAVNYCP37mzmYBC66yARI3S0nEv4UmEQkIA4cMHfIafzSYXFxJkBqHJNI+FNgEpGAWL0a9uxRYDrasGEmSB444HQlItIUCkwiEhCFhdC2LZx7rtOVhJa0NBMk//1vpysRkaZQYBKRgCgqgqFDIUZXlSrOPRdOPFHjmETCnS5tItJku3fD2rXqjquJy2WWSdE4JpHwpsAkIk22YoWZa2joUKcrCU3DhsF778HXXztdiYg0lgKTiDRZURH07AkdOzpdSWgaOtQEytdfd7oSEWksBSYRaRLbNoFJ3XG1O/lkSE7WOCaRcKbAJCJN8tFH8NVXWg6lPmlpWiZFJJwpMIlIkxQVQWwsDBzodCWhbdgw2LYNPvnE6UpEpDEUmESkSQoL4cILoXVrpysJbYMGQcuW6pYTCVcKTCLSaB4PvP126HTHZWdnk5ycTEpKitOlVHPssaYVToFJJDwpMIlIo739NuzbFzoDvqdOncqGDRtYs2aN06XUKC0N3nrL/JuJSHhRYBKRRissNHeA9ejhdCXhYdgw2LsXSkqcrkREGkqBSUQarajItJpYltOVhIeePaF9e3XLiYQjBSYRaZRt2+Djj0OnOy4cxMSYSSwVmETCjwKTiDRKUZFpWbrkEqcrCS9pafDBB7Brl9OViEhDKDCJSKMUFUFKCpxwgtOVhJfK9fZWrHC2DhFpGAUmEWkwr9d84as7ruESE+Hss9UtJxJuFJhEpMHefx+++SZ05l8KN2lpJjBpmRSR8KHAJCINVlgI8fHQt6/TlYSntDTYscOswyci4UGBSUQarLAQLr4YjjnG6UrC04ABZv29wkKnKxERfykwiUiDuN2werW645qidWuz/p7GMYmEDwUmEWmQN94wg7414Ltp0tLM0jIej9OViIg/FJhEpEEKC+G006BrV6crCW9paWZNuVWrnK5ERPyhwCQiDVK5HIo0TXIyJCVpHJNIuFBgEhG/ff45lJWpOy4QLOvw9AIiEvoUmETEb4WF5s64iy5yupLIkJZmphb46iunKxGR+igwiYjfCgrMLfFxcU5XUrPs7GySk5NJSUlxuhS/XHKJaWnSMikioc+ymz7VrOaqFYkC+/aZdePuuQd+9zunq6mb2+0mISGBiooK4uPjnS6nTikpZhD9Cy84XYlI1LL82UktTCLil1WrYO9eGDHC6UoiS1qaaWHy+ZyuRETqosAkIn4pKICOHeGss5yuJLKkpcHu3bBundOViEhdFJhExC8FBTB8uBlzI4HTrx8cd5zulhMJdQpMIlKvrVvh449NYJLAatkSBg/WfEwioU6BSUTqVVAAMTHmri4JvLQ0KCmBPXucrkREaqPAJCL1KiiACy6A4493upLIlJYGBw7AypVOVyIitVFgEpE6HTgA//qX7o5rTqedBl26aByTSChTYBKROv373+B2a/xSc7Iss9yMxjGJhC4FJhGp0/Ll0LYtnHee05VEtrQ0+PRT+PJLpysRkZooMIlInQoKTOtHjK4WzWrIEPNvrGVSREKTLoEiUqsdO+CDD5reHZeTk0PXrl1p3bo1vXv3ZtWqVXXuv2/fPu666y66dOlCq1at6N69O0899VTTighxxx8P55+vcUwioaqF0wWISOgqKjo8vqax5s+fT2ZmJjk5OaSmpjJ79mxGjBjBhg0b6Ny5c43Pufzyy9m5cydPPvkkp556Krt27eKnn35qfBFhYtgweOwx8HrB5XK6GhE5khbfFZFa/fKX8PnnsGZN41+jb9++nHfeeeTm5h56rEePHowdO5YZM2ZU27+goIArr7ySsrIyTjjhhEYdM5wW3z3SO+9AaiqsXm2mcRCRoNDiuyLSeF6vaWFqSnfc/v37Wbt2LWlpaVUeT0tL45133qnxOa+++ip9+vThwQcfJCkpidNPP53f/OY3eDyeWo+zb98+3G53lS0cnX++6ZpbtszpSkTkaApMIlKj996Db75pWmDavXs3Xq+X9u3bV3m8ffv27Nixo8bnlJWVUVxczEcffUR+fj6zZs1iwYIFTJ06tdbjzJgxg4SEhENbp06dGl+0g1q0MP/eS5c6XYmIHE2BSURqVFAAbdpA375Nfy3rqBV7bduu9lgln8+HZVk8//zznH/++YwcOZJHHnmEZ555ptZWpjvuuIOKiopD29atW5tetENGjYL334ft252uRESOpMAkIjVavhyGDjWtHo3Vtm1bXC5XtdakXbt2VWt1qtShQweSkpJISEg49FiPHj2wbZtt27bV+JxWrVoRHx9fZQtXw4aZgfbqlhMJLQpMIlLN7t3wn/80fTqBli1b0rt3b1YcNbnQihUr6N+/f43PSU1N5auvvmLPESvRbty4kZiYGE4++eSmFRQG2rY1A77VLScSWhSYRKSa5cvBtmHkyKa/1vTp05k7dy5PPfUUpaWlZGVlsWXLFiZPngyY7rQJEyYc2v+qq67ixBNP5LrrrmPDhg28/fbb/Pa3v+XXv/41sbGxTS8oDIwaZSaw3L/f6UpEpJICk4hU89prkJICiYlNf60rrriCWbNmcf/993POOefw9ttvs2zZMrp06QLA9u3b2bJly6H9jzvuOFasWMF3331Hnz59uPrqqxk9ejSPPfZY04sJE6NGwZ49UM/8niISRJqHSUSqOHDAdAvddhvcfbfT1TROuM7DVMm2oVMnuPxyeOQRp6sRiXiah0lEGq64GNxuuPRSpyuJXpZlukM1jkkkdCgwiUgVr70GHTvCuec6XUl0GzUKNm40M62LiPMUmESkitdeM1/WtUyTJEFy8cXQsqVamURChQKTiByycaPZ1B3nvOOOg4suUmASCRUKTCJyyNKl0KqVad0Q540aBStXmjvmRMRZCkwicshrr8GQIfCznzldiYAJTPv3w7/+5XQlIqLAJCIAVFTA22+rOy6UdO8OZ5yhbjmRUKDAJCIAFBXBTz+ZVg0JHSNHmnXlmj5lnog0hQKTiACmO65nTzg4AbeEiNGj4auvYO1apysRiW4KTCKC12taMdQdF3oGDoTjj4fFi52uRCS6KTCJCP/5D+zeHf6BKTs7m+TkZFJSUpwuJWBatDDdpApMIs7SWnIiwp13wpw5sHMnuFxOV9N04b6W3NEWLIDx46GsDLp2dboakYijteRExD/5+WasTCSEpUg0bJiZ9VutTCLOUWASiXKffGK29HSnK5HaxMWZyUQVmESco8AkEuXy881ElUOHOl2J1GXMGFi1Cr75xulKRKKTApNIlMvPhxEjIDbW6UqkLqNHm7sZNYmliDMUmESi2LZtsGaNuuPCQceOcP756pYTcYoCk0gUW7QIjjlGs3uHizFjoKAAfvzR6UpEoo8Ck0gUW7jQLLabkOB0JeKP9HT44QdYscLpSkSijwKTSJT6+muz2K6648JHjx5mW7DA6UpEoo8Ck0iUWrIEfD7TzSPhY9w4ePVV2L/f6UpEoosCk0iU8Xg87Ny5kwULPPTrB4mJTlckDZGRAd99B8uXm8/R4/E4XZJIVFBgEokSxcXFjMvIIC4ujsTERJYtjWPP9xmUlJQ4XZo0gNtdTNxxGfwi3XyOcXFxjMvQ5yjS3BSYRKJAbm4ugwYNonTdWv4+bRKLH7yXh2+dxIE9axk4cCB5eXlOlyh+yM3N5cILB5F04loeusV8jn+fNonSdfocRZqbFt8ViXDFxcUMGjSIaeMvY+atNxETc/j3JJ/PR+asPLIXLGHVqlWkpqY6WGngRNriuxCdn6NIkPi1+K4Ck0iEG5eRQem6tax/NrfKl2wln89Hr2tuJvm8PrzySmTcfhWJgSkaP0eRIPErMKlLTiSCeTweFi1ezKTRw2v8kgWIiYlh0ujh5Ocv0gDiEKXPUcR5CkwiEcztduP1eume1KHO/bolJeL1enG73UGqrHlkZ2eTnJxMSkqK06UEVLR9jiKhSIFJJILFx8fjcrnYVL69zv3KynfgcrnCvvtq6tSpbNiwgTVr1jhdSkBF2+coEooUmEQiWGxsLGPHjGHukgJ8Pl+N+/h8PuYuKSA9fSyxsbFBrlD8oc9RxHkKTCIRLjMri9LNW8h6dHa1L9vKu6tKN28hMzPLoQrFH/ocRZylu+REokBeXh5TpkzhtJM7M/kXw+mWlEhZ+Q7mLimgdPMWcnJymDx5stNlBkwk3iUHhz/HHl07M2n04c9x9qICNm6JvM9RJEg0rYCIHPb735fw4IMziYlZhNfrxeVykZ4+lszMrIibtydSAxNASUkJs2bNJD/ffI4xMS5iYsZSWJjFkCGR9TmKBIkCk4gcNngwtGgBr77qwe12Ex8fH7FjXSI5MFXyeMznuGtXPL16xfLyyzB+vNNViYQlzcMkIsbWrfDWW/CrX5kBxO3bt4/YsBQtKj/Hnj1j6dMHXnzR6YpEIpsCk0gUePFFaN0a0tOdrkSaw1VXwdKl8N13TlciErkUmESiwHPPwZgxEKG9U1HviivgwAFYuNDpSkQilwKTSIRbvx7++1/THSeRqWNHuOgieOEFpysRiVwKTCIR7rnn4MQTYdgwpyuR5nT11fDGG7Btm9OViEQmBSaRCOb1mlaHK6+EY45xuhppTuPHm3Fqzz7rdCUikUmBSSSCrVwJ5eXqjosG8fGQkQFPPw1Nny1GRI6mwCQSwZ57Drp3h759na5EguG66+Czz2D1aqcrEYk8CkwiEer77+Hll2HCBLD8mpZNwt1FF0GXLvDMM05XIhJ5FJhEItTLL8PevXDttU5XEjzZ2dkkJyeTkpLidCmOiImBiRPhpZfMZy8igaOlUUQiVL9+0KYNLF/udCXBFw1Lo9Rm0yY49VTTHXv11U5XIxIWtDSKSLT6+GP497/h+uudrkSCrXt3GDTIDP4WkcBRYBKJQE8+CW3bwmWXOV2JOOG668ycTF9+6XQlIpFDgUkkwuzbZ+bimTABWrZ0uhpxwrhxcNxxMHeu05WIRA4FJpEI8+qrsHu3uuOi2XHHwTXXmJbGAwecrkYkMigwiUSYJ580A76Tk52uRJx0002wfTssWeJ0JSKRQYFJJIJ8+SUUFal1SaBXLxOc8/KcrkQkMigwiUSQ2bMhLg6uuMLpSiQUTJ4MK1bA5587XYlI+FNgEokQ+/aZQb4TJ5oxLCLjx8Pxx8OcOU5XIhL+FJhEIsSCBfC//8GUKU5XIqEiNtbM9P7UUyZQi0jjKTCJRIicHBgyBH7+c6crkVBy003w9dfwz386XYlIeFNgEokA69bBO++EbutSTk4OXbt2pXXr1vTu3ZtVq1b59bySkhJatGjBOeec07wFRrAzzoDBgyE31+lKRMKbApNIBMjJgY4dYcwYpyupbv78+WRmZnLXXXfxwQcfMHDgQEaMGMGWLVvqfF5FRQUTJkzg4osvDlKlkevmm6G42ARrEWkcLb4rEua++w6SkuD22+Huu52uprq+ffty3nnnkXtEE0ePHj0YO3YsM2bMqPV5V155Jaeddhoul4tFixaxrgHf9tG8+G5NfvoJunWDiy/WGnMiNdDiuyLR4B//MF+IN97odCXV7d+/n7Vr15KWllbl8bS0NN55551an/f000+zadMm7rnnnuYuMSq0aAHTpsELL8CuXU5XIxKeFJhEwtiBA/DYY3DVVZCY6HQ11e3evRuv10v79u2rPN6+fXt27NhR43M+++wzfv/73/P888/TokULv46zb98+3G53lU2qmjQJXC4zV5eINJwCk0gY++c/Yds2yMx0upK6WVbVFm/btqs9BuD1ernqqqu47777OP300/1+/RkzZpCQkHBo69SpU5NrjjQnnGAWZM7Jgf37na5GJPxoDJNImLJt6NvXzOz9+utOV1Oz/fv3c+yxx/LKK6+Qnp5+6PFbb72VdevWsXLlyir7f/fddxx//PG4XK5Dj/l8PmzbxuVyUVRUxJAhQ6odZ9++few7YqIht9tNp06dNIbpKBs2wJlnwrPPwq9+5XQ1IiHDrzFM/rV3i0jIeecdWLMmtBdXbdmyJb1792bFihVVAtOKFSsYU8MtffHx8fz3v/+t8lhOTg5vvPEGCxYsoGvXrjUep1WrVrRq1SqwxUeg5GQYNgweegiuvhpqaOQTkVooMImEqZkz4fTTYeRIpyup2/Tp07nmmmvo06cP/fr1Y86cOWzZsoXJkycDcMcdd1BeXs68efOIiYnhrLPOqvL8k046idatW1d7XBrnd78zd8sVFZnwJCL+UWASCUMbN8LChWY8SkyIj0S84oor+Prrr7n//vvZvn07Z511FsuWLaNLly4AbN++vd45mSRwBg+GPn3gwQcVmEQaQmOYRMLQ9dfDsmWweTO0bu10NaFH8zDVbcECszDvmjUmPIlEOc3DJBKJtmyBefPgttsUlqRx0tPh1FPhgQecrkQkfCgwiYSZhx82d8bddJPTlUi4crngN78x01J89pnT1YiEBwUmkTCya5eZ2fuWW0xoEmmsiRPhpJPMHXMiUj8FJpEw8uijZpD3Lbc4XYmEu9at4dZb4Zln4KuvnK5GJPQpMImEiYoKeOIJs/L8CSc4XY1EgilT4NhjzR1zIlI3BSaRMJGTAz/+CNOnO12JRIqEBMjKMuvLbd/udDUioU2BSSQM7N1rJqr89a+hQwenq5FIcuutpntOd8yJ1E2BSSQMPPEEfPedmaVZJJASEkyrZV6exjKJ1EWBSSTEVVTA3/4GkyZBLUupiTTJLbdAbKxamUTqosAkEuIeeQQ8HvjDH5yuRCJVZSvT7NlqZRKpjQKTSAjbvdsEpmnToGNHp6sJfdnZ2SQnJ5OSkuJ0KWHnllvMHXMzZjhdiUho0lpyIiHsN7+BOXOgrAzatnW6mvChteQaZ8YMuOceKC2F7t2drkYkaLSWnEg4Ky+H7GzTVaKwJMFw661m9u+77nK6EpHQo8AkEqL+/GfTRaJ5lyRYjj0W7rsP5s+HNWucrkYktCgwiYSgsjKYOxd+/3tQj5IE08SJcOaZZgqLpo/YEIkcCkwiIej2203XyNSpTlci0aZFCzONxVtvwfLlTlcjEjo06FskxKxaBYMGwbx5cM01TlcTnjTou2lsGy66CL75BtatA5fL6YpEmpVfg74VmERCiM8H558PlgXvvgsxagNuFAWmpvvPf6BvX3j6abj2WqerEWlWuktOJBx4PB527tyJx+Ph2Wdh7VqzbpzCkjjp/PNh/HgzYereveaxI89VkWijS7KIQ4qLixmXkUFcXByJiYnExcVx880ZDBlSwoABTlcnYuZl+t//YOrU6ufquIwMSkpKnC5RJGjUJSfigNzcXKZOnUqPrp2ZNHo43ZM6sKl8O3kLC/hs2xZycnKYPHmy02WGLXXJBc7w4bkUFppz9YbLDp+rc5cUULpZ56pEBI1hEglFxcXFDBo0iGnjL2PmrTcRc0Tfm8/nI3NWHtkLlrBq1SpSU1MdrDR8KTAFhs5ViRIKTCKhaFxGBqXr1rL+2dwqX0CVfD4fva65meTz+vDKKwscqDD8KTAFhs5ViRIa9C0SajweD4sWL2bS6OE1fgEBxMTEMGn0cPLzF2lwrThG56pIVQpMIkHkdrvxer10T+pQ537dkhLxer243e4gVSZSlc5VkaoUmESCKD4+HpfLxaby7XXuV1a+A5fLpe4kcYzOVZGqFJhEgig2NpaxY8Ywd0kBPp+vxn18Ph9zlxSQnj6W2NjYIFcoYuhcFalKgUkkyDKzsijdvIWsR2dX+yKqvPOodPMWMjOzHKowfGVnZ5OcnExKSorTpUQEnasih+kuOREHTJiQx7PPTjk0t023pETKyndobpsA0V1ygZOXl8eUKVMOzRlWea7m5Rfw2VadqxIRNK2ASCjauROSkyElpYS4uJnk5y/C6/XicrlITx9LZmaW5rRpIgWmwCopKWHWrKrnakzMWNLSsnjtNZ2rEvYUmERCjW3D2LGwejV8/DG0a2du33a73cTHx2scSIAoMDWPI8/VZ5+N5aab4PXXYcgQpysTaRIFJpFQM28eTJwI+fkmOEnzUGBqfj6fCUpffgnr10NcnNMViTSaJq4UCSVbt8Itt8A11ygsSfiLiYGnnjKL8/7mN05XI9L8FJhEgsC2YdIkOO44ePRRp6sRCYxu3eChh2DOHCgsdLoakealwCQSBHPmQFERPPkkHH+809WIBM5NN8HQoXD99fDNN05XI9J8FJhEmllZGdx2m/liGTbM6WpEAsuyzC8Ce/eaVtSmD4sVCU0KTCLN6Kef4Npr4aST4O9/d7oakebRqZMJTfn5MHu209WINA8FJpFm9Kc/QUkJ/N//6S4iiWzp6XDzzZCVBR995HQ1IoGnaQVEmskbb8All8D998Mf/uB0NdFF0wo4w+OBvn3B64U1a+DYY52uSMQvmlZAxCm7dsHVV8PgwXDHHU5XIxIcsbHw0kuweTNMn+50NSKBpcAkEmA+H0yYYP587jlwuZyuSCR4kpNh1iwzlumf/3S6GpHAUWASCbAHHjBz0jz7LHTo4HQ1IsF3ww0wbpyZaqCszOlqRAJDgUkkgAoK4K67zJiltDSnqxFxhmXBP/4BbduaWe1/+MHpikSaToFJJEA+/xx++UsYMQLuvdfpakSc1aYNLFpkWph+/WvNzyThT4FJJAD27DG/SbdrB88/r3FLTsnOziY5OZmUlBSnSxHgrLPMlBovv6x5yCT8aVoBkSaybRg/3oxbevddM+hVnKVpBULLXXfB3/4Gy5erq1pCkqYVEAmGGTPM3UDz5iksidTk/vvNskBXXgmbNjldjUjjKDCJNMH8+ea35z/+0cx0LCLVuVzwwgtmEPill8K33zpdkUjDKTCJNNLbb5v5lq6+WoO8RerTpg0sXQr/+58Z77dvn9MViTSMApNII5SWwpgxkJoKTz0FMfpJEqnXaafBq6+asX7XXmsmdxUJF7rMizTQ9u1m6oCTT4aFC6FlS6crEgkf/fubO0kru7NFwoUCk0gDfP89jBoFP/0Ey5aZbgYRaZiMDHjoIXPnXF6e09WI+KeF0wWIhIv9++Hyy80ElcXF0KmT0xWJhK+sLNiyBaZMgYQEM+mrSChTYBLxw08/mcHdb7xhWpZ69XK6IpHwZlnwyCPmjrkJEyAuztxBJxKq1CUnchSPx8POnTvxeDyAGZj6619Dfj688gpcfLHDBYpEiJgYePJJGD3aTP761ltV//7on0URJykwiRxUXFzMuIwM4uLiSExMJC4ujnEZGfziFyU89xw89xxcdpnTVYpElhYt4MUXYeBA08K0alXtP4slJSVOlytRTIFJBMjNzWXQoEGUrlvL36dNYvGD9/L3aZP4aO1aFi8eyNVX53HllU5XGb5ycnLo2rUrrVu3pnfv3qxatarWfRcuXMjQoUNp164d8fHx9OvXj8LCwiBWK8HWqpVpwe3bFy6+uOafxdJ1axk4cCB5GiUuDtFachL1iouLGTRoENPGX8bMW28i5ohJlXw+H5mz8shesIRVq1aRmprqYKXhaf78+VxzzTXk5OSQmprK7NmzmTt3Lhs2bKBz587V9s/MzKRjx44MHjyYNm3a8PTTT/PQQw/x7rvvcu655/p1TK0lF57+9a9i0oYOYtrl+lmUoPJrLTkFJol64zIyKF23lvXP5la5QFfy+Xz0uuZmks/rwyuvLHCgwvDWt29fzjvvPHJzcw891qNHD8aOHcuMGTP8eo0zzzyTK664grvvvtuv/RWYwtO4jAxKP1jL+uf0syhBpcV3Rerj8XhYtHgxk0YPr/ECDRATE8Ok0cPJz1+kwacNtH//ftauXUvaUUvUp6Wl8c477/j1Gj6fj++//54TTjih1n327duH2+2uskl4OfSzeJl+FiU0KTBJVHO73Xi9Xrondahzv25JiXi9Xn0RN9Du3bvxer20b9++yuPt27dnx44dfr3Gww8/zA8//MDll19e6z4zZswgISHh0NZJk2SFHf0sSqhTYJKoFh8fj8vlYlP59jr3KyvfgcvlUvdOI1lW1RZv27arPVaTF198kXvvvZf58+dz0kkn1brfHXfcQUVFxaFt69atTa5Zgks/ixLqFJgkqsXGxjL0kjHk5Rfgq2UlUJ/Px9wlBaSnjyU2NjbIFYa3tm3b4nK5qrUm7dq1q1qr09Hmz5/P9ddfz8svv8wll1xS576tWrUiPj6+yibhJTY2lrFjxjB3iX4WJTQpMElU++ADePc/WWzcsoWsR2dXu1BX3plTunkLmZlZDlUZvlq2bEnv3r1ZsWJFlcdXrFhB//79a33eiy++yLXXXssLL7zAqFGjmrtMCRGZWVmUbq7nZ7FsC5Mn62dRgk9Lo0jUys+HX/0KkpMHcPnlOdx++xRef28dk0YPp1tSImXlO5i7pIDSzVsO3RIvDTd9+nSuueYa+vTpQ79+/ZgzZw5btmxh8uTJgOlOKy8vZ968eYAJSxMmTODRRx/lggsuONQ6FRsbS0JCgmPvQ5rfgAEDyMnJYcqU2n8WWxyTwx/+kEqvXtCundMVS1Sxbbupm0hY8fls+y9/sW2w7fHjbfuHH8zjxcXF9rhxGbbL5bIB2+Vy2ePGZdjFxcXOFhwBsrOz7S5dutgtW7a0zzvvPHvlypWH/m7ixIn2hRdeeOj/L7zwQhszXUmVbeLEiX4fr6KiwgbsioqKAL4LCZa6fhbXrLHtk06y7VNOse1165yuVCKEX3lH8zBJVPF44MYbzTIn99wDd99t1rOquo8Ht9tNfHy8xkmEKc3DFBlq+1ncsgXGjoVPP4VnnjHr0Ik0gSauFDlSWRlkZMAnn8DTT6OlTiKYAlPk27sXrr8eXnoJ7rwT7r8fXC6nq5IwpYkrRSotXw59+oDbDf/+t8KSSLg79lh44QV44AGYMQNGjICdO52uSiKZApNEtJ9+gj/8AUaNgtRUeO89OPtsp6sSkUCwLPjd76CoCNavNz/b//qX01VJpFJgkoj1xRcwaBD87W/w5z/D4sVw/PFOVyUigXbJJbBuHfTsCWlp8Mc/ml+WRAJJgUki0oIFcM458NVXsGqVGeNQy/JUIhIBEhOhsND8cjRjBgwZAtu2OV2VRBJ9hUhE+fZbmDjR3DWTlmZ+6+zXz+mqRCQYYmLML0crV5oW5rPPhuefh6bf2ySiwCQRZPlyOOss0/X29NMwfz60aeN0VSISbKmpZhb/4cPN5LSXXQbl5U5XJeFOgUnCntsNkybByJFmDMNHH8G115oBoSISnU480bQuLV4Ma9dCcjLMnavWJmk8BSYJaR6Ph507d+LxeGr8+8WLTavS/PkwZ45pZTr55CAXKSIh67LLYMMGGDcObrjBdNV/8UXt+9d3zZHopcAkIam4uJhxGRnExcWRmJhIXFwc4zIyKCkpAeDLL2HMGDPb71lnwX//ay6GalUSkaO1aQNPPmkGhW/caK4ZTzwBXu/hfeq75ohopm8JObm5uUydOpUeXTszafRwuid1YFP59kOLb44dm0Nh4WSOPx4efRR+8QsFJTGys7PJzs7G6/WyceNGzfQt1bjd8PvfQ26umcz2iSfg/ffrvubk5OQcWixaIpKWRpHwU1xczKBBg5g2/jJm3noTMUfMBeDz+ciclccTryzh8stXMXduKnFxDhYrIUtLo0h9Skpg2jRYt64Yy6r7mpO9YAmrVq0iNTXVwYqlGSkwSfgZl5FB6bq1rH82t8qFq5LP56Pnr27mzN59eOWVBQ5UKOFAgUn84fVCSkoGP3y9lg0v1n7N6XXNzSSfp2tOBNNachJePB4PixYvZtLo4TVeuABiYmK44bLh5Ocv0qBMEWmS/fs9rF+/mMnpdV9zJo3WNUcUmCSEuN1uvF4v3ZM61Llft6REvF4vbrc7SJWJSCTSNUcaQoFJQsYxx8QTE+NiU/n2OvcrK9+By+VSV4uINEl8fDwul6454h8FJnHcvn2QnQ1nnRULjGH2ogJ8Pl+N+/p8PuYuKSA9fSyxsbHBLVREIkpsbCxjx4xh7pK6rzl5Cws4qd1YPvxQ15xopsAkjtmzB2bNgtNOg1tugaFD4YUXsti4ZQtZj86udgGrvGOldPMWMjOznClaRCJKZlYWpZvrvuZ8tm0LrVpn0a+fuU6tXOlQseKoFk4XINFn1y54/HHTqvT993DVVWZelB49AAbw7bc5TJkyhdffW8ek0cPplpRIWfmOKnOi6PZeEQmEAQMGkJNT/zXnxhtTWbgQ/vxnuOgiGDgQ/vAHE6A0D1x00LQCEjRlZfDQQ2ZhXJfLzMydlQWdO1fft6SkhFmzZpKfvwiv14vL5SI9fSyZmVkKS1IvTSsgDeXvNce24bXX4E9/gjVroFcvmD4drrwSWrVy8A1IU2geJnGebZsJ4h5/HBYsgBNOMN1vU6ea/66Px+PB7XYTHx+vMUviNwUmaSx/rzm2DW+9BQ8/DEuXQocOZiLMG26Adu2CV68EhAKTBJ6/F5M9e+CFF0y32/r1cOqpkJkJ110Hxx4bvHolOikwSTCVlsLMmTBvnglS48bB5MkwYED93XX6pTAkaOJKCRx/F6b89FO49VZISoKbb4ZTToGCAvP41KkKSyISeXr0gDlzYNs2+OtfTVfdoEFw5pnw2GPw7bfVn6PFfsOPWpikXvUthjtzZg4JCZN56ilYtco0R0+aBDfdBF26OF29RCO1MImTfD7TXZeXB/n5cMwxcMUVptXp/PMhL0+L/YYYdclJ0/m7GC6sYujQVK67Dn7xCw1+FGcpMEmo2LEDnnrKtEB9+SWcckoxX36pxX5DjAKTNJ0/i+GeddXNdEvuw9JlWphSQoMCk4QarxdWrIAbb8wglrVseEmL/YYQjWGSpvF3Mdwbxw6nsEgLU4qI1Mblggsv9PDVV4uZ/Ast9huOFJikRqWl8Kc/aWFKCS/Z2dkkJyeTkpLidCki1TR0sd+vvtI1NZQoMAlgboVdswbuvBN+/nNIToZHH40nxtLClBI+pk6dyoYNG1izZo3TpYhU05DFfi1cJCfHc+mlMHs2lJcHqUiplQJTGPN4POzcubPRzbbffWcmk7z+ejj5ZHP3xpw50L8/LFkCX38dS3p6/QtTajFcEZH6+bvY79wlBYwYMZa//jWWPXvMlCwnnwy9e8N998H775tfchujqd8b0UyBKQw1dv4O24YPPjDzhAwaBG3bwvjxsHq1mdb/jTcO39Fx6aXQurV/C1NqMVwREf/4e029864sbrvNTE+waxc895xZqPyRR0xw6tTJ/LL78svw9df1H1fzPjWd7pILM/XNiXT0/B07d8Kbb0JhoZlAcscOOO44uPhiGDEChg+vf66kvLw8pkyZcuiYNS1MqTlDJJToLjkJZU25pu7fb+a7W7rUXNc3bDCziaekwLBhcMkl0Ldv1aldGvq9EYU0rUCk8WdOpOwFS/jTn1axfXsqb75pfpjAzDg7YoTZBgyAli0bdmwthivhRIFJQl2grqlbt5rpCgoLzZ/ffmt6B/r3h4sughNPLGbaNM37VA8Fpkjjz5xIyVfezMatfTj11AUMHgyDB5sfmg5135ThN617JOFAgUnCRSCvqV4vfPih6cZ76y14+21wV2RwWifN+1QPBaZga84w4fF4iIuL4+/TJpF5ZXqt+816KZ/fPjGX77//XoFGopYCkwjs2eOhTRvnvzfC4BdtTVwZLM0xmM62YdMmmD8ffvtbuPhizYkkIiL+++GHhn1vjBvn5uGH4Z13YO/eph8/0gaat3C6gHB35GC6v0+bVGUw3cCBA/0aTGfbZpXrNWvgvfcOb5UrXHfpAuecE09MjOZEEhER/zRk3qeYGBdudzx//CN4PBATY+bj690bzjvP/HnOOfCzn/l37EB8N4Yadck1gb+DsI8cTLdvn5lFe/36w9uHH5rbRgE6doQ+fapu7dqZv/NnDJP6okXUJSdSqaHfGwcOwH//C2vXHt7Wrzd358XEmImNe/eGc8+FXr3MVvkdVakx340Oi+4xTMHoM/XnROx59c20iOvDWWctYP16+OQTMzAPoFu3wydc794mHHXsWPvxwvAkFHGEApOIEYjvjQMH4OOPq4eoH380f9++vfke69nTbM89m8H2zWtZ/5wzv9w34vs/OgNTcXExs2bOZNHixYdu1Rw7ZgxZ06cHNEQ0ZBD2bY/NpV+/7zn33NhDAenMMyEuruHH1ZxIIvVTYBI5rDm+N7xeM852/XrTIlXZY1JW5sEijodvDf5A8yZ8//sVmCJqDFNz9plWVMCnn5oWok8+gQ8+8H8wnW17WbjQTfv2TT8pJk+eTM+ePZk1aya/fWJulfk75sx7Xi1LIiJSRXN8b7hccPrpZhs37vDjmza5OfVU/wea33efm969YznjDNPrctxxDS4FCM6YqaAFpubuIis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| ||
| 1331 | "text/plain": [ | ||
| 1332 | "Graphics object consisting of 2 graphics primitives" | ||
| 1333 | ] | ||
| 1334 | }, | ||
| 1335 | "metadata": {}, | ||
| 1336 | "output_type": "display_data" | ||
| 1337 | } | ||
| 1338 | ], | ||
| 1339 | "source": [ | ||
| 1340 | "show(plot(spline(points), -1, 1) + scatter_plot(points))" | ||
| 1341 | ] | ||
| 1342 | }, | ||
| 1343 | { | ||
| 1344 | "cell_type": "markdown", | ||
| 1345 | "metadata": {}, | ||
| 1346 | "source": [ | ||
| 1347 | "A detailed explanation of splines is a good topic for a course of numerical analysis. For this course it is enough that you know that they exist and they can be plotted." | ||
| 1348 | ] | ||
| 1349 | } | ||
| 1350 | ], | ||
| 1351 | "metadata": { | ||
| 1352 | "kernelspec": { | ||
| 1353 | "display_name": "SageMath 9.2", | ||
| 1354 | "language": "sage", | ||
| 1355 | "name": "sagemath" | ||
| 1356 | }, | ||
| 1357 | "language_info": { | ||
| 1358 | "codemirror_mode": { | ||
| 1359 | "name": "ipython", | ||
| 1360 | "version": 3 | ||
| 1361 | }, | ||
| 1362 | "file_extension": ".py", | ||
| 1363 | "mimetype": "text/x-python", | ||
| 1364 | "name": "python", | ||
| 1365 | "nbconvert_exporter": "python", | ||
| 1366 | "pygments_lexer": "ipython3", | ||
| 1367 | "version": "3.8.5" | ||
| 1368 | } | ||
| 1369 | }, | ||
| 1370 | "nbformat": 4, | ||
| 1371 | "nbformat_minor": 4 | ||
| 1372 | } | ||
diff --git a/Lectures/8-SageCalculus.pdf b/Lectures/8-SageCalculus.pdf new file mode 100644 index 0000000..5b533b6 --- /dev/null +++ b/Lectures/8-SageCalculus.pdf | |||
| Binary files differ | |||
diff --git a/Lectures/9-SageLatex.ipynb b/Lectures/9-SageLatex.ipynb new file mode 100644 index 0000000..63ea8c6 --- /dev/null +++ b/Lectures/9-SageLatex.ipynb | |||
| @@ -0,0 +1,346 @@ | |||
| 1 | { | ||
| 2 | "cells": [ | ||
| 3 | { | ||
| 4 | "cell_type": "markdown", | ||
| 5 | "metadata": {}, | ||
| 6 | "source": [ | ||
| 7 | "It can happen that you need to include the results of your Sage computations and/or Sage code inside a LaTeX document. Luckily Sage provides some functions to translate its objects into LaTeX, and the listings package for LaTeX can be used to include any code (Sage, Python or any other language) in a LaTeX document.\n", | ||
| 8 | "\n", | ||
| 9 | "In this document we will describe some of these interactions between LaTeX and Sage." | ||
| 10 | ] | ||
| 11 | }, | ||
| 12 | { | ||
| 13 | "cell_type": "markdown", | ||
| 14 | "metadata": {}, | ||
| 15 | "source": [ | ||
| 16 | "# The `show()` command\n", | ||
| 17 | "**Reference:** [[1](https://doc.sagemath.org/html/en/reference/repl/sage/repl/display/pretty_print.html)] (`show()` is just an alternative name for `pretty_print()`).\n", | ||
| 18 | "\n", | ||
| 19 | "With this command Sage will generate a picture displaying the object. The result depends on the object itself: most of them will be typeset in Latex, but for example graphics primitives (such as plots) will be displayed as pictures.\n", | ||
| 20 | "\n", | ||
| 21 | "You can see it as an alternative to `print()`." | ||
| 22 | ] | ||
| 23 | }, | ||
| 24 | { | ||
| 25 | "cell_type": "code", | ||
| 26 | "execution_count": 4, | ||
| 27 | "metadata": {}, | ||
| 28 | "outputs": [ | ||
| 29 | { | ||
| 30 | "name": "stdout", | ||
| 31 | "output_type": "stream", | ||
| 32 | "text": [ | ||
| 33 | "1 + 1*x + 1/2*x^2 + 1/6*x^3 + Order(x^4)\n" | ||
| 34 | ] | ||
| 35 | }, | ||
| 36 | { | ||
| 37 | "data": { | ||
| 38 | "text/html": [ | ||
| 39 | "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1 + 1 x + \\frac{1}{2} x^{2} + \\frac{1}{6} x^{3} + \\mathcal{O}\\left(x^{4}\\right)</script></html>" | ||
| 40 | ], | ||
| 41 | "text/latex": [ | ||
| 42 | "\\begin{math}\n", | ||
| 43 | "\\newcommand{\\Bold}[1]{\\mathbf{#1}}1 + 1 x + \\frac{1}{2} x^{2} + \\frac{1}{6} x^{3} + \\mathcal{O}\\left(x^{4}\\right)\n", | ||
| 44 | "\\end{math}" | ||
| 45 | ], | ||
| 46 | "text/plain": [ | ||
| 47 | "1 + 1*x + 1/2*x^2 + 1/6*x^3 + Order(x^4)" | ||
| 48 | ] | ||
| 49 | }, | ||
| 50 | "metadata": {}, | ||
| 51 | "output_type": "display_data" | ||
| 52 | }, | ||
| 53 | { | ||
| 54 | "name": "stdout", | ||
| 55 | "output_type": "stream", | ||
| 56 | "text": [ | ||
| 57 | "[ 1 2 3]\n", | ||
| 58 | "[ 4 5 6]\n", | ||
| 59 | "[ 8 9 10]\n" | ||
| 60 | ] | ||
| 61 | }, | ||
| 62 | { | ||
| 63 | "data": { | ||
| 64 | "text/html": [ | ||
| 65 | "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | ||
| 66 | "1 & 2 & 3 \\\\\n", | ||
| 67 | "4 & 5 & 6 \\\\\n", | ||
| 68 | "8 & 9 & 10\n", | ||
| 69 | "\\end{array}\\right)</script></html>" | ||
| 70 | ], | ||
| 71 | "text/latex": [ | ||
| 72 | "\\begin{math}\n", | ||
| 73 | "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n", | ||
| 74 | "1 & 2 & 3 \\\\\n", | ||
| 75 | "4 & 5 & 6 \\\\\n", | ||
| 76 | "8 & 9 & 10\n", | ||
| 77 | "\\end{array}\\right)\n", | ||
| 78 | "\\end{math}" | ||
| 79 | ], | ||
| 80 | "text/plain": [ | ||
| 81 | "[ 1 2 3]\n", | ||
| 82 | "[ 4 5 6]\n", | ||
| 83 | "[ 8 9 10]" | ||
| 84 | ] | ||
| 85 | }, | ||
| 86 | "metadata": {}, | ||
| 87 | "output_type": "display_data" | ||
| 88 | }, | ||
| 89 | { | ||
| 90 | "name": "stdout", | ||
| 91 | "output_type": "stream", | ||
| 92 | "text": [ | ||
| 93 | "pi\n" | ||
| 94 | ] | ||
| 95 | }, | ||
| 96 | { | ||
| 97 | "data": { | ||
| 98 | "text/html": [ | ||
| 99 | "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\pi</script></html>" | ||
| 100 | ], | ||
| 101 | "text/latex": [ | ||
| 102 | "\\begin{math}\n", | ||
| 103 | "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\pi\n", | ||
| 104 | "\\end{math}" | ||
| 105 | ], | ||
| 106 | "text/plain": [ | ||
| 107 | "pi" | ||
| 108 | ] | ||
| 109 | }, | ||
| 110 | "metadata": {}, | ||
| 111 | "output_type": "display_data" | ||
| 112 | } | ||
| 113 | ], | ||
| 114 | "source": [ | ||
| 115 | "s = (e^x).series(x==0, 4)\n", | ||
| 116 | "M = matrix([[1,2,3],[4,5,6],[8,9,10]])\n", | ||
| 117 | "print(s)\n", | ||
| 118 | "show(s)\n", | ||
| 119 | "print(M)\n", | ||
| 120 | "show(M)\n", | ||
| 121 | "print(pi)\n", | ||
| 122 | "show(pi)" | ||
| 123 | ] | ||
| 124 | }, | ||
| 125 | { | ||
| 126 | "cell_type": "markdown", | ||
| 127 | "metadata": {}, | ||
| 128 | "source": [ | ||
| 129 | "In a Jupyter notebook, the results above are displayed using [MathJax](https://www.mathjax.org/).\n", | ||
| 130 | "\n", | ||
| 131 | "If you are running this code in an interactive console (terminal) instead of a Jupyter notebook, you will get the Latex source code for those objects. You can force this behavior by using the `latex()` command." | ||
| 132 | ] | ||
| 133 | }, | ||
| 134 | { | ||
| 135 | "cell_type": "markdown", | ||
| 136 | "metadata": {}, | ||
| 137 | "source": [ | ||
| 138 | "# The `latex()` command\n", | ||
| 139 | "**Reference:** [[2](https://doc.sagemath.org/html/en/reference/misc/sage/misc/latex.html)]\n", | ||
| 140 | "\n", | ||
| 141 | "This command is potentially very useful if you need to include the results of Sage computations in a Latex file, especially with complex objects like matrices or very large polynomials.\n", | ||
| 142 | "\n", | ||
| 143 | "Technically, this is a function that returns a string, so you need to `print()` it to see the result." | ||
| 144 | ] | ||
| 145 | }, | ||
| 146 | { | ||
| 147 | "cell_type": "code", | ||
| 148 | "execution_count": 5, | ||
| 149 | "metadata": {}, | ||
| 150 | "outputs": [ | ||
| 151 | { | ||
| 152 | "name": "stdout", | ||
| 153 | "output_type": "stream", | ||
| 154 | "text": [ | ||
| 155 | "1 + 1 x + \\frac{1}{2} x^{2} + \\frac{1}{6} x^{3} + \\mathcal{O}\\left(x^{4}\\right)\n", | ||
| 156 | "\n", | ||
| 157 | "\n", | ||
| 158 | "\\left(\\begin{array}{rrr}\n", | ||
| 159 | "1 & 2 & 3 \\\\\n", | ||
| 160 | "4 & 5 & 6 \\\\\n", | ||
| 161 | "8 & 9 & 10\n", | ||
| 162 | "\\end{array}\\right)\n" | ||
| 163 | ] | ||
| 164 | } | ||
| 165 | ], | ||
| 166 | "source": [ | ||
| 167 | "print(latex(s))\n", | ||
| 168 | "print(\"\\n\")\n", | ||
| 169 | "print(latex(M))" | ||
| 170 | ] | ||
| 171 | }, | ||
| 172 | { | ||
| 173 | "cell_type": "markdown", | ||
| 174 | "metadata": {}, | ||
| 175 | "source": [ | ||
| 176 | "Interestingly, Sage can use matplotlib's PGF backend to generate Latex code for a plot. (PGF is the graphics language underlying TikZ, like TeX is the language underlying Latex)." | ||
| 177 | ] | ||
| 178 | }, | ||
| 179 | { | ||
| 180 | "cell_type": "code", | ||
| 181 | "execution_count": 15, | ||
| 182 | "metadata": {}, | ||
| 183 | "outputs": [], | ||
| 184 | "source": [ | ||
| 185 | "#latex(plot(x^2)) # The output is more than 20 pages long" | ||
| 186 | ] | ||
| 187 | }, | ||
| 188 | { | ||
| 189 | "cell_type": "markdown", | ||
| 190 | "metadata": {}, | ||
| 191 | "source": [ | ||
| 192 | "It is probably easier to just generate the picture and include that in your Latex document with `\\includegraphics`." | ||
| 193 | ] | ||
| 194 | }, | ||
| 195 | { | ||
| 196 | "cell_type": "markdown", | ||
| 197 | "metadata": {}, | ||
| 198 | "source": [ | ||
| 199 | "## A Latex name for your variables\n", | ||
| 200 | "**Reference:** [[3](https://doc.sagemath.org/html/en/reference/calculus/sage/calculus/var.html)]\n", | ||
| 201 | "\n", | ||
| 202 | "Sometimes you might want to use variables and functions that have, for example, a Greek letter as a name. You can tell Sage that you want them displayed this way when you declare them:" | ||
| 203 | ] | ||
| 204 | }, | ||
| 205 | { | ||
| 206 | "cell_type": "code", | ||
| 207 | "execution_count": 16, | ||
| 208 | "metadata": {}, | ||
| 209 | "outputs": [ | ||
| 210 | { | ||
| 211 | "name": "stdout", | ||
| 212 | "output_type": "stream", | ||
| 213 | "text": [ | ||
| 214 | "phi1(epsilon)\n" | ||
| 215 | ] | ||
| 216 | }, | ||
| 217 | { | ||
| 218 | "data": { | ||
| 219 | "text/html": [ | ||
| 220 | "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e^{{\\varepsilon}} + \\phi_1\\left({\\varepsilon}\\right)</script></html>" | ||
| 221 | ], | ||
| 222 | "text/latex": [ | ||
| 223 | "\\begin{math}\n", | ||
| 224 | "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e^{{\\varepsilon}} + \\phi_1\\left({\\varepsilon}\\right)\n", | ||
| 225 | "\\end{math}" | ||
| 226 | ], | ||
| 227 | "text/plain": [ | ||
| 228 | "e^epsilon + phi1(epsilon)" | ||
| 229 | ] | ||
| 230 | }, | ||
| 231 | "metadata": {}, | ||
| 232 | "output_type": "display_data" | ||
| 233 | }, | ||
| 234 | { | ||
| 235 | "data": { | ||
| 236 | "text/plain": [ | ||
| 237 | "e^{{\\varepsilon}} + \\phi_1\\left({\\varepsilon}\\right)" | ||
| 238 | ] | ||
| 239 | }, | ||
| 240 | "execution_count": 16, | ||
| 241 | "metadata": {}, | ||
| 242 | "output_type": "execute_result" | ||
| 243 | } | ||
| 244 | ], | ||
| 245 | "source": [ | ||
| 246 | "var('epsilon', latex_name=\"\\\\varepsilon\")\n", | ||
| 247 | "function('phi1', latex_name=\"\\\\phi_1\")\n", | ||
| 248 | "\n", | ||
| 249 | "print(phi1(epsilon))\n", | ||
| 250 | "show(phi1(epsilon) + e^epsilon)\n", | ||
| 251 | "latex(phi1(epsilon) + e^epsilon)" | ||
| 252 | ] | ||
| 253 | }, | ||
| 254 | { | ||
| 255 | "cell_type": "markdown", | ||
| 256 | "metadata": {}, | ||
| 257 | "source": [ | ||
| 258 | "**Warning:** You need to use two backspaces `\\\\`. The reason is that in Python (like in many other programming languages) the backslash symbol inside a string is used to print special characters, such as a newline `\\n`." | ||
| 259 | ] | ||
| 260 | }, | ||
| 261 | { | ||
| 262 | "cell_type": "markdown", | ||
| 263 | "metadata": {}, | ||
| 264 | "source": [ | ||
| 265 | "# From Jupyter to Latex\n", | ||
| 266 | "**Reference:** [[4](https://nbconvert.readthedocs.io/en/latest/)]\n", | ||
| 267 | "\n", | ||
| 268 | "From the Jupyter menu `File > Download as` you can choose to download your work in many formats, among which there are also Latex and pdf. Personally I prefer downloading the .tex file, so then I can change the title, add an author name and make any other change I like before compiling it into a pdf file.\n", | ||
| 269 | "\n", | ||
| 270 | "If you choose to download the pdf file, you might need to install some extra packages. For example I had to install [`pandoc`](https://pandoc.org/), `texlive-XeTeX` and `texlive-Xdvi`, but this depends on your operating system and Latex distribution." | ||
| 271 | ] | ||
| 272 | }, | ||
| 273 | { | ||
| 274 | "cell_type": "markdown", | ||
| 275 | "metadata": {}, | ||
| 276 | "source": [ | ||
| 277 | "# SageTex\n", | ||
| 278 | "**Reference:** [[5](https://doc.sagemath.org/html/en/tutorial/sagetex.html)]\n", | ||
| 279 | "\n", | ||
| 280 | "With SageTex it is possible to run Sage commands directly inside Latex, using the `\\sage{}` command. In this way you don't need to run your Sage code first and then copy the results in Latex. It can be useful especially for short Sage commands.\n", | ||
| 281 | "\n", | ||
| 282 | "You might need to take some extra steps to make this work on your system, see the link above." | ||
| 283 | ] | ||
| 284 | }, | ||
| 285 | { | ||
| 286 | "cell_type": "markdown", | ||
| 287 | "metadata": {}, | ||
| 288 | "source": [ | ||
| 289 | "# The Latex `listings` package\n", | ||
| 290 | "**References:** [[6](https://en.wikibooks.org/wiki/LaTeX/Source_Code_Listings)] and [[7](https://ftp.snt.utwente.nl/pub/software/tex/macros/latex/contrib/listings/listings.pdf)]\n", | ||
| 291 | "\n", | ||
| 292 | "If you want to include some code (Sage, Python or anything else) in a Latex document you can use the listings package.\n", | ||
| 293 | "\n", | ||
| 294 | "```\n", | ||
| 295 | "\\usepackage{listings}\n", | ||
| 296 | "\n", | ||
| 297 | "...\n", | ||
| 298 | "\n", | ||
| 299 | "\\begin{lstlisting}[language=Python]\n", | ||
| 300 | "for i in range(0,100):\n", | ||
| 301 | " if i%5 == 0:\n", | ||
| 302 | " print(\"Multiple of 5!\")\n", | ||
| 303 | "\\end{lstlisting}\n", | ||
| 304 | "```\n", | ||
| 305 | "\n", | ||
| 306 | "You need to specify the language you are using with the `language=` option. This option can also be set at the beginning of the document using the `\\lstset{language=Python}` command.\n", | ||
| 307 | "\n", | ||
| 308 | "As an alternative, you can include a file directly without copying the code into the tex file, like you would do for a picture:\n", | ||
| 309 | "\n", | ||
| 310 | "```\n", | ||
| 311 | "\\lstinputlisting[language=Python]{file.py}\n", | ||
| 312 | "```\n", | ||
| 313 | "\n", | ||
| 314 | "It is technically possible to include Latex listings in a markdown cell of the Jupyter notebook using [this package](https://jupyter-contrib-nbextensions.readthedocs.io/en/latest/nbextensions/latex_envs/README.html), but it does not make much sense. So we will move to a Latex editor for the examples." | ||
| 315 | ] | ||
| 316 | }, | ||
| 317 | { | ||
| 318 | "cell_type": "code", | ||
| 319 | "execution_count": null, | ||
| 320 | "metadata": {}, | ||
| 321 | "outputs": [], | ||
| 322 | "source": [] | ||
| 323 | } | ||
| 324 | ], | ||
| 325 | "metadata": { | ||
| 326 | "kernelspec": { | ||
| 327 | "display_name": "SageMath 9.2", | ||
| 328 | "language": "sage", | ||
| 329 | "name": "sagemath" | ||
| 330 | }, | ||
| 331 | "language_info": { | ||
| 332 | "codemirror_mode": { | ||
| 333 | "name": "ipython", | ||
| 334 | "version": 3 | ||
| 335 | }, | ||
| 336 | "file_extension": ".py", | ||
| 337 | "mimetype": "text/x-python", | ||
| 338 | "name": "python", | ||
| 339 | "nbconvert_exporter": "python", | ||
| 340 | "pygments_lexer": "ipython3", | ||
| 341 | "version": "3.8.5" | ||
| 342 | } | ||
| 343 | }, | ||
| 344 | "nbformat": 4, | ||
| 345 | "nbformat_minor": 4 | ||
| 346 | } | ||
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