Find the first five diagonal Pade approximants \([1/1], ..., [5/5]\) to \(e^x\) around the
origin. Remember that the numerator and denominator can be multiplied by a
constant to make the numbers as convenient as possible. Evaluate the approximations
at \(x = 1\) and compare with the correct value of \(e = 2.718281828459045\). How
is the error improving with the order? How does that compare to the polynomial
error?
Take as a data set \(x = \{-10,-9, ... , 9, 10\}\), and \(y(x) = 0\) if \(x \leq 0\) and \(y(x) = 1\) if \(x > 0\).