index.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<div style=\"float:right\">\n",
    "<img src=\"https://www.ugent.be/++theme++ugent/static/images/logo_ugent_nl.svg\" style=\"height:150px\">\n",
    "<img src=\"https://opendreamkit.org/public/logos/odk-elected-logo.svg\" style=\"height:150px\">\n",
    "<img src=\"https://opendreamkit.org/public/logos/Flag_of_Europe.svg\" style=\"height:150px\">\n",
    "</div>\n",
    "\n",
    "# cypari2: Python bindings for PARI/GP\n",
    "\n",
    "### Jeroen Demeyer\n",
    "\n",
    "### Universiteit Gent / OpenDreamKit\n",
    "\n",
    "(joint work with Vincent Delecroix, Luca De Feo, Vincent Klein, …)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Quick examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.64493406684823"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from cypari2 import Pari\n",
    "pari = Pari()\n",
    "pari.zeta(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3, 10.3910994007158]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pari.ellinit([-112, 400]).ellanalyticrank()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^4 + (-y^3 + 2232*y^2 - 1069956*y + 36864000)*x^3 + (2232*y^3 + 2587918086*y^2 + 8900222976000*y + 452984832000000)*x^2 + (-1069956*y^3 + 8900222976000*y^2 - 770845966336000000*y + 1855425871872000000000)*x + (y^4 + 36864000*y^3 + 452984832000000*y^2 + 1855425871872000000000*y)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pari.polmodular(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Plotting (using SVG images)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
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291.95,56.85 292.36,57.75 292.77,59.52 293.17,62.17 293.58,65.65 293.99,69.93 294.40,74.96 294.81,80.69 295.22,87.05 295.63,93.98 296.04,101.40 296.45,109.22 296.86,117.37 297.27,125.75 297.68,134.27 298.09,142.83 298.50,151.36 298.91,159.74 299.32,167.90 299.73,175.75 300.13,183.20 300.54,190.18 300.95,196.61 301.36,202.43 301.77,207.59 302.18,212.04 302.59,215.73 303.00,218.65 303.41,220.76 303.82,222.07 304.23,222.57 304.64,222.27 305.05,221.21 305.46,219.41 305.87,216.91 306.28,213.77 306.69,210.04 307.09,205.80 307.50,201.12 307.91,196.08 308.32,190.77 308.73,185.27 309.14,179.68 309.55,174.09 309.96,168.61 310.37,163.31 310.78,158.30 311.19,153.66 311.60,149.48 312.01,145.83 312.42,142.78 312.83,140.41 313.24,138.76 313.65,137.89 314.05,137.81 314.46,138.58 314.87,140.18 315.28,142.64 315.69,145.94 316.10,150.05 316.51,154.96 316.92,160.61 317.33,166.96 317.74,173.94 318.15,181.48 318.56,189.50 318.97,197.90 319.38,206.61 319.79,215.52 320.20,224.52 320.61,233.51 321.01,242.38 321.42,251.03 321.83,259.35 322.24,267.24 322.65,274.60 323.06,281.34 323.47,287.37 323.88,292.62 324.29,297.01 324.70,300.48 325.11,303.00 325.52,304.51 325.93,305.00 326.34,304.45 326.75,302.87 327.16,300.26 327.57,296.66 327.97,292.09 328.38,286.61 328.79,280.29 329.20,273.18 329.61,265.38 330.02,256.96 330.43,248.04 330.84,238.71 331.25,229.08 331.66,219.26 332.07,209.37 332.48,199.53 332.89,189.83 333.30,180.41 333.71,171.35 334.12,162.77 334.53,154.76 334.93,147.41 335.34,140.79 335.75,134.97 336.16,130.01 336.57,125.95 336.98,122.82 337.39,120.64 337.80,119.42 338.21,119.15 338.62,119.81 339.03,121.36 339.44,123.75 339.85,126.94 340.26,130.84 340.67,135.39 341.08,140.49 341.49,146.04 341.89,151.95 342.30,158.10 342.71,164.40 343.12,170.72 343.53,176.95 343.94,182.99 344.35,188.73 344.76,194.07 345.17,198.91 345.58,203.16 345.99,206.75 346.40,209.61 346.81,211.68 347.22,212.91 347.63,213.28 348.04,212.76 348.45,211.36 348.85,209.07 349.26,205.94 349.67,201.98 350.08,197.26 350.49,191.83 350.90,185.78 351.31,179.18 351.72,172.13 352.13,164.74 352.54,157.12 352.95,149.37 353.36,141.63 353.77,134.00 354.18,126.62 354.59,119.59 355.00,113.04 355.41,107.07 355.81,101.79 356.22,97.29 356.63,93.66 357.04,90.96 357.45,89.27 357.86,88.62 358.27,89.05 358.68,90.58 359.09,93.21 359.50,96.92 359.91,101.69 360.32,107.48 360.73,114.23 361.14,121.87 361.55,130.30 361.96,139.45 362.37,149.19 362.77,159.41 363.18,169.99 363.59,180.80 364.00,191.71 364.41,202.57 364.82,213.26 365.23,223.65 365.64,233.59 366.05,242.98 366.46,251.68 366.87,259.60 367.28,266.64 367.69,272.71 368.10,277.74 368.51,281.68 368.92,284.48 369.33,286.12 369.73,286.58 370.14,285.88 370.55,284.04 370.96,281.10 371.37,277.10 371.78,272.13 372.19,266.26 372.60,259.59 373.01,252.23 373.42,244.28 373.83,235.87 374.24,227.13 374.65,218.18 375.06,209.17 375.47,200.22 375.88,191.46 376.29,183.02 376.69,175.01 377.10,167.55 377.51,160.74 377.92,154.66 378.33,149.38 378.74,144.98 379.15,141.49 379.56,138.95 379.97,137.37 380.38,136.74 380.79,137.05 381.20,138.26 381.61,140.32 382.02,143.16 382.43,146.70 382.84,150.85 383.25,155.51 383.65,160.57 384.06,165.89 384.47,171.37 384.88,176.86 385.29,182.24 385.70,187.38 386.11,192.15 386.52,196.43 386.93,200.11 387.34,203.08 387.75,205.24 388.16,206.52 388.57,206.86 388.98,206.19 389.39,204.48 389.80,201.72 390.21,197.90 390.61,193.04 391.02,187.18 391.43,180.36 391.84,172.66 392.25,164.16 392.66,154.96 393.07,145.17 393.48,134.91 393.89,124.31 394.30,113.53 394.71,102.69 395.12,91.95 395.53,81.46 395.94,71.37 396.35,61.81 396.76,52.92 397.17,44.84 397.57,37.68 397.98,31.55 398.39,26.54 398.80,22.71 399.21,20.14 399.62,18.85 400.03,18.88 400.44,20.21 400.85,22.85 401.26,26.74 401.67,31.83 402.08,38.06 402.49,45.32 402.90,53.52 403.31,62.53 403.72,72.22 404.13,82.45 404.53,93.08 404.94,103.94 405.35,114.88 405.76,125.74 406.17,136.38 406.58,146.62 406.99,156.34 407.40,165.39 407.81,173.66 408.22,181.03 408.63,187.40 409.04,192.71 409.45,196.88 409.86,199.89 410.27,201.70 410.68,202.31 411.08,201.74 411.49,200.03 411.90,197.22 412.31,193.41 412.72,188.67 413.13,183.11 413.54,176.84 413.95,170.00 414.36,162.72 414.77,155.14 415.18,147.41 415.59,139.67 416.00,132.09 416.41,124.79 416.82,117.91 417.23,111.58 417.64,105.91 418.04,101.01 418.45,96.97 418.86,93.86 419.27,91.73 419.68,90.61 420.09,90.52 420.50,91.45 420.91,93.38 421.32,96.27 421.73,100.05 422.14,104.65 422.55,109.95 422.96,115.86 423.37,122.25 423.78,128.99 424.19,135.93 424.60,142.92 425.00,149.82 425.41,156.47 425.82,162.73 426.23,168.45 426.64,173.50 427.05,177.75 427.46,181.09 427.87,183.43 428.28,184.68 428.69,184.78 429.10,183.69 429.51,181.39 429.92,177.89 430.33,173.19 430.74,167.35 431.15,160.43 431.56,152.51 431.96,143.69 432.37,134.10 432.78,123.86 433.19,113.13 433.60,102.07 434.01,90.84 434.42,79.61 434.83,68.55 435.24,57.85 435.65,47.67 436.06,38.17 436.47,29.51 436.88,21.83 437.29,15.26 437.70,9.91 438.11,5.87 438.52,3.22 438.92,2.00 439.33,2.25 439.74,3.97 440.15,7.14 440.56,11.72 440.97,17.66 441.38,24.86 441.79,33.22 442.20,42.62 442.61,52.92 443.02,63.98 443.43,75.62 443.84,87.68 444.25,99.98 444.66,112.34 445.07,124.59 445.48,136.55 445.88,148.06 446.29,158.96 446.70,169.10 447.11,178.36 447.52,186.62 447.93,193.80 448.34,199.81 448.75,204.61 449.16,208.17 449.57,210.49 449.98,211.59 450.39,211.50 450.80,210.28 451.21,208.03 451.62,204.83 452.03,200.82 452.44,196.11 452.84,190.86 453.25,185.22 453.66,179.34 454.07,173.39 454.48,167.53 454.89,161.91 455.30,156.70 455.71,152.02 456.12,148.01 456.53,144.79 456.94,142.44 457.35,141.06 457.76,140.71 458.17,141.40 458.58,143.17 458.99,146.00 459.40,149.86 459.80,154.69 460.21,160.42 460.62,166.96 461.03,174.18 461.44,181.95 461.85,190.14 462.26,198.58 462.67,207.11 463.08,215.57 463.49,223.77 463.90,231.55 464.31,238.75 464.72,245.20 465.13,250.76 465.54,255.30 465.95,258.71 466.36,260.88 466.76,261.74 467.17,261.24 467.58,259.35 467.99,256.07 468.40,251.42 468.81,245.45 469.22,238.21 469.63,229.83 470.04,220.39 470.45,210.05 470.86,198.96 471.27,187.28 471.68,175.19 472.09,162.89 472.50,150.57 472.91,138.43 473.32,126.65 473.72,115.44 474.13,104.97 474.54,95.42 474.95,86.93 475.36,79.64 475.77,73.67 476.18,69.11 476.59,66.04 477.00,64.49\" style=\"fill:none;stroke:#ff0000;\"/></svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "[0.E-307, 100.000000000000, -7.26626527057968, 7.80311802062338]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pari(\"my(Z=lfuninit(1, [100])); ploth(x=0, 100, lfunhardy(Z,x))\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Python interaction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-3, -4, -7, -8, -11, -19, -43, -67, -163]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "discs = set(pari.quaddisc(-1-n) for n in range(200))\n",
    "sorted([D for D in discs if D.qfbclassno() == 1], reverse=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 8, 27, 64, 125, 216, 343, 512, 729]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def cube(x):\n",
    "    return x**3\n",
    "\n",
    "pari.apply(cube, range(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "ename": "PariError",
     "evalue": "impossible inverse in Fp_inv: Mod(2, 6)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mPariError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-27-817f380fe1ec>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;36m1\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mpari\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mMod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m6\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32mcypari2/gen.pyx\u001b[0m in \u001b[0;36mcypari2.gen.Gen.__div__\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mcypari2/handle_error.pyx\u001b[0m in \u001b[0;36mcypari2.handle_error._pari_err_handle\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;31mPariError\u001b[0m: impossible inverse in Fp_inv: Mod(2, 6)"
     ]
    }
   ],
   "source": [
    "1 / pari.Mod(4, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Implementation\n",
    "\n",
    "`cypari2` is written in Cython, a Python-like language compiling to C (like `gp2c` but for Python)\n",
    "\n",
    "PARI objects (`GEN`) are wrapped in a Python object `Gen` (independent of the PARI type):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<type 'cypari2.gen.Gen'>\n",
      "t_REAL\n"
     ]
    }
   ],
   "source": [
    "x = pari.pi()\n",
    "print(type(x))\n",
    "print(x.type())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Auto-generation\n",
    "\n",
    "Cython code wrapping most GP functions is auto-generated from `pari.desc`. Example:\n",
    "```\n",
    "Function: cos\n",
    "Class: basic\n",
    "Section: transcendental\n",
    "C-Name: gcos\n",
    "Prototype: Gp\n",
    "Help: cos(x): cosine of x.\n",
    "Doc: cosine of $x$.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This becomes\n",
    "```cython\n",
    "    def cos(self, x, long precision=0):\n",
    "        r'''\n",
    "        Cosine of :math:`x`.\n",
    "        '''\n",
    "        x = objtogen(x)\n",
    "        sig_on()\n",
    "        cdef GEN _x = (<Gen>x).g\n",
    "        precision = prec_bits_to_words(precision)\n",
    "        cdef GEN _ret = gcos(_x, precision)\n",
    "        return new_gen(_ret)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Performance\n",
    "\n",
    "`cypari2` is very fast:\n",
    "\n",
    "- Cython allows direct C calls to the PARI library\n",
    "- Objects remain on the PARI stack if possible, no needless copying\n",
    "  (several unsafe operations require cloning anyway: resizing PARI stack, getting/setting entry of vector/matrix)\n",
    "\n",
    "Only overhead comes from dealing with Python objects and methods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PARI stack size set to 268435456 bytes, maximum size set to 268435456\n"
     ]
    }
   ],
   "source": [
    "pari.allocatemem(2**28)\n",
    "def hilberttrace(n):\n",
    "    m = pari.mathilbert(n)\n",
    "    m.trace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 5 ms, sys: 13 ms, total: 18 ms\n",
      "Wall time: 18.7 ms\n"
     ]
    }
   ],
   "source": [
    "%time hilberttrace(1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time = \u001b[0;32m61 ms\u001b[0m.\n"
     ]
    }
   ],
   "source": [
    "%%script gp -q --default timer=1\n",
    "allocatemem(2^28)\n",
    "hilberttrace(n) = {my(m=mathilbert(n)); trace(m); return;}\n",
    "hilberttrace(1000);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## History (a.k.a. why is it called `cypari2`?)\n",
    "\n",
    "- ≤ 2006: Very early versions of SageMath using Pyrex/Cython wrappers of PARI\n",
    "- 2012: Marc Culler and Nathan Dunfield forked these into a separate Python package `cypari` (to use it with their `SnapPy` topology package)\n",
    "- 2015: auto-generation of wrappers in SageMath\n",
    "- 2016: these were made into a separate package `cypari2`\n",
    "- 2018: `cypari2` version 2 released, keeping `GEN`s on the PARI stack instead of always copying\n",
    "- `cypari` and `cypari2` are still separate packages because of build/packaging issues and Windows compatibility of `cysignals`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## TODO\n",
    "\n",
    "- better support for functions\n",
    "- support iterators (problem: PARI/GP iterators are not easily usable from C)\n",
    "- support attributes like `.clgp` (problem: `omega()` versus `.omega` incompatibility)\n",
    "- slicing (`M[,1]` for a matrix `M`) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Try it!\n",
    "\n",
    "Assuming Python ≥ 2.7 and PARI/GP version ≥ 2.9.4 (installed in a location where Python will find it):\n",
    "```\n",
    "pip install cypari2\n",
    "```\n",
    "\n",
    "...or use SageMath.\n",
    "\n",
    "Sources: https://github.com/sagemath/cypari2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": []
  }
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