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What books are good an introductory course in Numerical analysis? I look for a book with many applications, especially in biology

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I don't know this book but a Google search found Numerical Methods for the Life Scientist. – lhf Jan 24 '13 at 11:50
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One of the main books of the subject of numerical methods is Trefethen's Numerical Linear Algebra, which covers numerical methods for linear algebra. I'm not sure how well this relates to life sciences and biology though. A classic text on the subject of numerical methods for partial differential equations is that of Quateroni and Valli. I can personally vouch for both of these books, but not necessarily with respect to biology.

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Numerical Analysis by L Ridgewood 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.

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Numerical Recipes in C by William H. Press etal has a lot of pages... But which language or system you are going to use? Some of them have a lot of introduction material, models and data that you can try and which definitely help you to learn.

If you use, say R, you might find very nice introductions from different univ cources, also related to biology. R-system has also some introduction in the system that are related to biology. See, e.g.

For a book, see, eg. (I haven't read)

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