Introduction to computational science and mathematics by Charles F. Van Loan

By Charles F. Van Loan

Arithmetic for desktop Scientists

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Figure 1. The Research Triangle But interesting ideas can flow in the other direction as well. A physical experiment may be restricted in scope for reasons of budget or safety, so the scene shirts to computer simulation. The act of writing the program to perform the simulation will most likely have a clarifying influence, prompting some new mathematical pursuit. Innovative models are discovered, leading to a modification of the initial set of experiments, and so forth. A parable will serve to clarify these interactions.

The reason is related to the precision of the computer's arithmetic. 3. 2. The surface area of an oblate spheroid such as the Earth is given by A = 4pr1r2 where r1 is the equatorial radius and r2 is the polar radius. Write a program that reads in these two radii and computes the difference between 4pr1r2 and 4p((r1+r2)/2)2. Use the Earth data r1 = 3963, r2 = 3957. 2 Sines and Cosines Let us continue the discussion of formulas and programs using as examples some well-known trigonometric identities.

Want the surface area of a sphere? Use Have the cosine of some angle qÎ[0, p/2] and want cos(q/2)? Use Want the minimum value µ of the quadratic function q(x) = x2 + bx + c on the interval [L, R]? Use Want to know if year y is a leap year? Use the rule that y is a leap year if it is a century year divisible by 400 or a non-century year divisible by 4. Sometimes the application of a formula involves a simple substitution. Thus, the surface area of a one-inch ball bearing is 4p(1/2)2 = p square inches.

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