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    "## Simple iteration to solve a linear system\n",
    "\n",
    "We want to solve the equation $Ax = b$.  Recall that every root-finding problem has equivalent, associated fixed-point iterations.  If $x$ is a solution of $Ax = b$ then $x$ satisfies\n",
    "\n",
    "$$x = (I - A)x + b.$$\n",
    "\n",
    "and we want to find a fixed-point of  \n",
    "\n",
    "$$g(x) = (I - A)x + b.$$"
   ]
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       "array([[ -5.30764743e-09],\n",
       "       [ -2.58408872e-09],\n",
       "       [ -3.60924202e-09],\n",
       "       [ -3.46897711e-09]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "A = np.array([[1.1, .1, .3, .4], \n",
    "              [.2, .9, -.1, .33], \n",
    "              [-.3, .4, 1., .8], \n",
    "              [.1, .2, .3, 1]])\n",
    "b = np.array([[1], \n",
    "              [1], \n",
    "              [1], \n",
    "              [1]])\n",
    "T = np.eye(4) - A\n",
    "x = np.array([[0],\n",
    "              [0],\n",
    "              [0],\n",
    "              [0]])\n",
    "for i in range(40):\n",
    "    x = np.dot(T,x) + b\n",
    "np.dot(A,x) - b"
   ]
  }
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