CS 205A class_15 notes

Scientific Computing

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CS205 – Class 15 Covered in class: 3, 4, 5 Readings: 8.7, 9.1, 9.2 1. Gaussian Quadrature – for each subinterval 1 [, ] ii x x + , use k specially spaced points to obtain a method that is exact on 2k-1 degree polynomials and thus has an order of accuracy of 2k a. 11 1 1 1 22 2 23 i i i i x xx x x x x x x x Gf f ++ + + + ⎛⎞ −+ + =− + + ⎜⎟ ⎝⎠ i. 2 points, piecewise cubic, exact for piecewise cubic functions, 4 th order accurate 2. Can extend quadrature to higher dimensions a. One dimension () b a f xdx - subdivide [,] ab into smaller intervals b. Two dimensions (, ) A f xydA ∫∫ - subdivide A into rectangles or triangles c. Three dimensions (,,) V f xyzdV ∫∫∫ - 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.
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CS 205A class_15 notes - CS205 Class 15 Covered in class:...

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