Editing Numerical analysis (section)

===Computing values of functions===
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Interpolation: Observing that the temperature varies from 20 degrees Celsius at 1:00 to 14 degrees at 3:00, a linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm.

Extrapolation: If the [[gross domestic product]] of a country has been growing an average of 5% per year and was 100 billion last year, it might be extrapolated that it will be 105 billion this year.

[[Image:Linear-regression.svg|right|100px|A line through 20 points]]

Regression: In linear regression, given ''n'' points, a line is computed that passes as close as possible to those ''n'' points.

[[Image:LemonadeJuly2006.JPG|right|100px|How much for a glass of lemonade?]]

Optimization: Suppose lemonade is sold at a [[lemonade stand]], at $1.00 per glass, that 197 glasses of lemonade can be sold per day, and that for each increase of $0.01, one less glass of lemonade will be sold per day. If $1.485 could be charged, profit would be maximized, but due to the constraint of having to charge a whole-cent amount, charging $1.48 or $1.49 per glass will both yield the maximum income of $220.52 per day.

[[Image:Wind-particle.png|right|Wind direction in blue, true trajectory in black, Euler method in red]]

Differential equation: If 100 fans are set up to blow air from one end of the room to the other and then a feather is dropped into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the [[Euler method]] for solving an ordinary differential equation.
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One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the [[Horner scheme]], since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control [[round-off error]]s arising from the use of [[floating-point arithmetic]].