Q1. Using the definition W=e^{i(2π/N)} , and the Euler identity e^{±iθ} = cos(θ) ± i sin(θ), the value of W^{(N/6)} can be computed as 0.866  0.5i 0.866 + 0.5i 0.5  0.866i 0.5  0.866i Q2. Using the definition W=e^{i(2π/N)}, and the Euler identity e^{±iθ} = cos(θ) ± i sin(θ), the value of W^{(6N)} can be computed as 1 + i 1  i 1 1 Q3. Given N=2, and . The first part of can be expressed as
The values for can be computed as
Q4. For N =2^{4 }=16, level L=2 and referring to the figure 1 shown at this link, the only terms of vector ƒ_{2}()which only need to compute are ƒ_{2}(47,1215) ƒ_{2}(03,811) ƒ_{2}(07) ƒ_{2}(815) Q5. For N =2^{4 }=16, level L=3 and referring to referring to the figure 1 shown at this link, the only companion nodes associated with ƒ_{3}(0) and ƒ_{3}(1) are ƒ_{3}(4) and ƒ_{3}(5) ƒ_{3}(6) and ƒ_{3}(7) ƒ_{3}(14) and ƒ_{3}(15) ƒ_{3}(2) and ƒ_{3}(3) Q6. Given N = 4, and .
Corresponding to level L = 1, one can compute ƒ_{3}(2) as 2  2i 4  6i 4  6i 4  4i

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Copyrights: University of South Florida, 4202 E Fowler Ave, Tampa, FL 336205350. All Rights Reserved. Questions, suggestions or comments, contact kaw@eng.usf.edu This material is based upon work supported by the National Science Foundation under Grant# 0126793, 0341468, 0717624, 0836981, 0836916, 0836805, 1322586. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Other sponsors include Maple, MathCAD, USF, FAMU and MSOE. Based on a work at http://mathforcollege.com/nm. Holistic Numerical Methods licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. 
