Lab 4-28.ipynb

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    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.sparse import spdiags\n",
    "from scipy.sparse.linalg import spsolve\n",
    "%matplotlib inline"
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    "**Problem 1**: Write a function which generates the $N\\times N$ sparse matrix\n",
    "\n",
    "$$\n",
    "D_{L} = \\begin{pmatrix} 0 & 0 & 0 & \\cdots & 0 \\\\ 1 & 0 & 0 & \\cdots & 0\\\\ 0 & 1 & 0 & \\cdots & 0\\\\ \\vdots & \\vdots & \\ddots & \\ddots & 0\\\\ 0 & 0 & \\cdots & 1 & 0\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "Test your code on two different examples.  Don't forget about the `.todense()` method.  "
   ]
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   "source": [
    "def DL(N):\n",
    "    # write your code here"
   ]
  },
  {
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    "# place your examples here.  "
   ]
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    "**Problem 2**: Write a function which generates the $N\\times N$ sparse matrix\n",
    "\n",
    "$$\n",
    "D_{U} = \\begin{pmatrix} 0 & 1 & 0 & \\cdots & 0 \\\\ 0 & 0 & 1 & \\cdots & 0\\\\ 0 & 0 & 0 & \\ddots & 0\\\\ \\vdots & \\vdots & \\ddots & \\ddots & 1\\\\ 0 & 0 & \\cdots & 0 & 0\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "Test your code on two different examples.  Don't forget about the `.todense()` method.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "def DU(N):\n",
    "    # write your code here"
   ]
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   "source": [
    "# place your examples here."
   ]
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    "**Problem 3**: Given the interval $a\\leq x \\leq b$ and some function $f(x)$, for an $N+1$ point mesh $x_{j}=a + j\\delta x$, $\\delta x = (b-a)/N$, write a function which generates the left-sided derivative approximation defined by the equations:\n",
    "\n",
    "$$\n",
    "f'(x_{j}) \\approx -\\frac{3}{2\\delta x} f_{j} +\\frac{2}{\\delta x}f_{j+1} - \\frac{1}{2\\delta x}f_{j+2}, ~ 0\\leq j \\leq N-2.  \n",
    "$$\n",
    "\n",
    "Test your implementation on the function $f(x)=\\sin(x)$ for $0\\leq x \\leq 2\\pi$.  Use the notes from Lecture Nine as a guide.  "
   ]
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