class_15

class_15 - CS205 Class 15 Covered in class: 3, 4, 5...

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CS205 – Class 15 Covered in class: 3, 4, 5 Readings: 8.7, 9.1, 9.2 1. Can extend quadrature to higher dimensions a. One dimension ( ) b a f x dx - subdivide [ , ] a b into smaller intervals b. Two dimensions ( , ) A f x y dA  - subdivide A into rectangles or triangles c. Three dimensions ( , , ) V f x y z dV  - subdivide V into boxes or tetrahedral d. Monte Carlo methods – usually used in higher dimensions i. Random or pseudo random numbers are used to generate sample points that are averaged and multiplied by the element “size” (e.g. length, area, volume) ii. Error decreases like 1/2 n where n is the number of sample points 1. 100 times more points are needed to gain one more digit of accuracy 2. Slow convergence, but independent of the number of dimensions 3. Not competitive for lower dimensional problems, but the only alternative for higher dimensional problems 2. Richardson extrapolation eliminate the leading order error term using 2 calculations. a. Start an integration scheme with some step size
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class_15 - CS205 Class 15 Covered in class: 3, 4, 5...

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