notebooks/sage-1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolic Expressions \n",
    "\n",
    "In this notebook we explore things that can be done with expressions. We need to inform sage that we are using certain symbols as variables. To start with, let's use x and y as such symbols."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x,y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can assign an expression to a variable which will be of the appropriate type. Unlike in numerical computation tools, you don't need to specify the value of x yet. The expression is retained in its original form unless you want it to be simplified or expanded etc.,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 3\\right)}^{3}</script></html>"
      ],
      "text/plain": [
       "(x + 3)^3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly1 = (x + 3)^3\n",
    "poly1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.symbolic.expression.Expression'>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(poly1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can ask sage to expand the expression and show it. Change the power from 3 to say 99 and run the cells again and see how sage has implemented arbitrary precision arithmetic to keep all the digits intact. Also change the expression for poly1 to use a floating number inside and see how the expanded form looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} + 9 \\, x^{2} + 27 \\, x + 27</script></html>"
      ],
      "text/plain": [
       "x^3 + 9*x^2 + 27*x + 27"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly1.expand().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can factorize a polynomial expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} + 2 \\, x - 3</script></html>"
      ],
      "text/plain": [
       "x^2 + 2*x - 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly2 = x^2+2*x-3\n",
    "poly2.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 3\\right)} {\\left(x - 1\\right)}</script></html>"
      ],
      "text/plain": [
       "(x + 3)*(x - 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "poly2.factor().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equations as expressions\n",
    "\n",
    "Equations are also expression. Notice the use of == sign to indicate the equality of left hand side and right hand side of the equation. Sage understands the two sides and can show those to you."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y - 3}{x^{2} + 1} = x^{3} - 5</script></html>"
      ],
      "text/plain": [
       "(y - 3)/(x^2 + 1) == x^3 - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr1 = (y-3)/(x^2+1)==x^3-5\n",
    "expr1.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{y - 3}{x^{2} + 1}</script></html>"
      ],
      "text/plain": [
       "(y - 3)/(x^2 + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr1.lhs().show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} - 5</script></html>"
      ],
      "text/plain": [
       "x^3 - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr1.rhs().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can multiple both sides of the equation with another expression, like you do with a pen and a paper."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y - 3 = {\\left(x^{3} - 5\\right)} {\\left(x^{2} + 1\\right)}</script></html>"
      ],
      "text/plain": [
       "y - 3 == (x^3 - 5)*(x^2 + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr2 = expr1.multiply_both_sides(x^2+1)\n",
    "expr2.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can add / substract another expression to both sides. In this case, we are adding a number 3 to both sides."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y = {\\left(x^{3} - 5\\right)} {\\left(x^{2} + 1\\right)} + 3</script></html>"
      ],
      "text/plain": [
       "y == (x^3 - 5)*(x^2 + 1) + 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr3 = expr2.add_to_both_sides(3)\n",
    "expr3.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expanded form of the equation shows the solution for y."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y = x^{5} + x^{3} - 5 \\, x^{2} - 2</script></html>"
      ],
      "text/plain": [
       "y == x^5 + x^3 - 5*x^2 - 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr3a = expr3.expand().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analytical solutions\n",
    "\n",
    "In the above case, we solved for y using steps. We can ask sage to do all that by itself and give us the solution directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[y == x^5 + x^3 - 5*x^2 - 2]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = solve(expr1,y)\n",
    "sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results of the solution is a sequence (list) of rules. We can use the individual elements of the sequence to extract the rhs as solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{5} + x^{3} - 5 \\, x^{2} - 2</script></html>"
      ],
      "text/plain": [
       "x^5 + x^3 - 5*x^2 - 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ysol = sol[0].rhs()\n",
    "ysol.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If there are multiple solutions to a polynomial equation, those will be shown as elements of a sequence of rules. Some of these solutions could be complex numbers too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{5} - x^{4} + x^{3} - x^{2} + x - 1 = 0</script></html>"
      ],
      "text/plain": [
       "x^5 - x^4 + x^3 - x^2 + x - 1 == 0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr4 = x^5-x^4 + x^3-x^2+x-1 == 0\n",
    "expr4.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x == -1/2*I*sqrt(3) + 1/2, x == 1/2*I*sqrt(3) + 1/2, x == 1, x == -1/2*I*sqrt(3) - 1/2, x == 1/2*I*sqrt(3) - 1/2]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol4 = solve(expr4,x)\n",
    "sol4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can give the expression right inside the solve function and seek roots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x == -1/2*sqrt(21) - 9/2, x == 1/2*sqrt(21) - 9/2]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x^2+9*x+15==0,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also use the solve function to seek roots of expressions with unknown coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The first root is:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{b + \\sqrt{b^{2} - 4 \\, a c}}{2 \\, a}</script></html>"
      ],
      "text/plain": [
       "-1/2*(b + sqrt(b^2 - 4*a*c))/a"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The second root is:\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{b - \\sqrt{b^{2} - 4 \\, a c}}{2 \\, a}</script></html>"
      ],
      "text/plain": [
       "-1/2*(b - sqrt(b^2 - 4*a*c))/a"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('a,b,c')\n",
    "sol3 = solve(a*x^2+b*x+c==0,x)\n",
    "print(\"The first root is:\")\n",
    "sol3[0].rhs().show()\n",
    "print(\"The second root is:\")\n",
    "sol3[1].rhs().show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulation of polynomials\n",
    "\n",
    "You can find the greatest common divisors of two polynomial expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} - 8 \\, x + 15</script></html>"
      ],
      "text/plain": [
       "x^2 - 8*x + 15"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} - 5 \\, x + 6</script></html>"
      ],
      "text/plain": [
       "x^2 - 5*x + 6"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gcdpart1 = x^2-8*x+15\n",
    "gcdpart1.show()\n",
    "gcdpart2 = x^2-5*x+6\n",
    "gcdpart2.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x - 3\\right)} {\\left(x - 5\\right)}</script></html>"
      ],
      "text/plain": [
       "(x - 3)*(x - 5)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x - 2\\right)} {\\left(x - 3\\right)}</script></html>"
      ],
      "text/plain": [
       "(x - 2)*(x - 3)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "x - 3"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(gcdpart1).show()\n",
    "factor(gcdpart2).show()\n",
    "gcd(gcdpart1, gcdpart2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Substituting values\n",
    "\n",
    "You can substitute values for the variables in an expression easily."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 3\\right)}^{3}</script></html>"
      ],
      "text/plain": [
       "(x + 3)^3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "64"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poly1.show()\n",
    "value1 = poly1.subs(x=1)\n",
    "value1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes the value is reported using symbols. You can ask for a numerical value specifically. The alias for the function numerical_approx is n."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(pi + 3)^3"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value2 = poly1.subs(x=pi)\n",
    "value2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "231.655717937028"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value2.numerical_approx()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "231.655717937028"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value2.n()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can ask for numerical value to a given precision. Change the number of digits to, say, 100 and see."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "231.655717937028465183478105415"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "value2.n(digits=30)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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