Norms and CG added

Author: atcemgil

AR1-Model-Octave.ipynb

@@ -44,17 +44,17 @@
     "here \n",
     "\n",
     "\\begin{eqnarray}\n",
-    "\\|e\\|_2 & = & \\left(e_1^2 + e_2^2 + \\dots + e_N\\right)^{\\frac{1}{2}}\n",
+    "\\|e\\|_2 & = & \\left(e_1^2 + e_2^2 + \\dots + e_N^2\\right)^{\\frac{1}{2}}\n",
     "\\end{eqnarray}\n",
     "\n",
     "Another possibility is measuring the error with other norms, such as the absolute error\n",
     "\\begin{eqnarray}\n",
-    "\\|e\\|_1 & = &  \\left|e_1\\right| + \\left|e_2\\right| + \\dots + \\left|e_2\\right|\n",
+    "\\|e\\|_1 & = &  \\left|e_1\\right| + \\left|e_2\\right| + \\dots + \\left|e_N\\right|\n",
     "\\end{eqnarray}\n",
     "\n",
     "and more general $p$ norms\n",
     "\\begin{eqnarray}\n",
-    "\\|e\\|_p & = & \\left( \\left|e_1\\right|^p + \\left|e_2\\right|^p + \\dots + \\left|e_2\\right|^p\\right)^{\\frac{1}{p}}\n",
+    "\\|e\\|_p & = & \\left( \\left|e_1\\right|^p + \\left|e_2\\right|^p + \\dots + \\left|e_N\\right|^p\\right)^{\\frac{1}{p}}\n",
     "\\end{eqnarray}\n",
     "\n",
     "\n",
@@ -827,28 +827,6 @@
     "plt.show()"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Drawing norm-balls in 2-D\n",
-    "\n",
-    "Calculate the points in polar coordinates. \n",
-    "\n",
-    "In $2-D$, take rays of the form \n",
-    "$$x = \\left( \\begin{array}{c} x_1 \\\\ x_2 \\end{array} \\right) = \n",
-    "\\left( \\begin{array}{c} \\alpha \\cos(\\theta) \\\\ \\alpha \\sin(\\theta) \\end{array} \\right)\n",
-    "$$\n",
-    "where $\\alpha>0$ and $0\\leq \\theta \\leq 2\\pi$.\n",
-    "\n",
-    "The $p$-norm of a vector $x$ with $\\|x\\|_p = 1 $ satisfies \n",
-    "\n",
-    "$\\left|x_1\\right|^p + \\left|x_2\\right|^p = \\alpha^p (\\left|\\cos(\\theta)\\right|^p + \\left|\\sin(\\theta)\\right|^p) = 1 $\n",
-    "\n",
-    "Solve for $\\alpha(\\theta)$:\n",
-    "$$\\alpha = \\left(\\frac{1}{\\left|\\cos(\\theta)\\right|^p + \\left|\\sin(\\theta)\\right|^p}\\right)^{1/p} $$"
-   ]
-  },
   {
    "cell_type": "code",
    "execution_count": 11,
@@ -1009,18 +987,6 @@
     "plt.xlim((-5,N))\n",
     "plt.show()"
    ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The Matrix norm\n",
-    "\n",
-    "\\begin{eqnarray}\n",
-    "\\|A\\|_{(p,q)} = \\sup_{\\|x\\|\\neq 0} \\frac{\\|A x\\|_{(p)}}{\\|x\\|_{(q)}} = \\sup_{\\|x\\|_{(q)} = 1} {\\|A x\\|_{(p)}}\n",
-    "\\end{eqnarray}\n",
-    "\n"
-   ]
   }
  ],
  "metadata": {