Book in Numerical analysis

Numerical Analysis by L Ridgeway Scott has a thorough discussion of systems of linear equations, interpolation and quadrature problems. It's halfway between a calculus textbook and Rudin's Principles of Real Analysis.

Here is a typical result, on the error term of the Simpson rule:

If $f(x)$ has four derivatives on $[a,b]$, let $x_k = \left( 1 - \frac{k}{2n}\right) a + \frac{k}{2n} \, b$: $$ \int_a^b f(t) \, dt = \frac{b-a}{6n} \sum_{k=1}^n [ f(x_{2k-2}) + 4 f(x_{2k-1}) + f(x_{2k})] - f^{(4)}(\xi) \frac{(b-a)^5}{180(2n)^4} $$ the error to Simpson's rule is proportional to $f^{(4)}$ and decays like $1/n^4$.

So we see that Simpson's rule is much faster convergence than Trapezoid rule.

Wikipedia has all these GREAT observations:

• Simpson's rule breaks up a curve into a bunch of parabolas of best fit and integrates those instead. Find the best fit curve: $$ \left|\begin{array}{cccc} 1 & a & a^2 & f(a) \\ 1 & \frac{a+b}{2} & (\frac{a+b}{2})^2 & f(\frac{a+b}{2}) \\ 1 & b & b^2 & f(b) \\ 1 & x & x^2 & P(x) \end{array} \right| \approx 0 $$ then integrate $$\int_0^a P(x) = \frac{b-a}{6} \left[ f(a) + f\left(\frac{a+b}{2}\right) + f(b)\right]$$
• Simpson rule is (2/3)Midpoint Rule + (1/3)Trapezoid Rule.

So this book is full of facts like these I wish I had learned in calculus class.