CS 205A class_12 notes

Scientific Computing

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CS 205 – Class 12 Readings : Same as last Covered in class : All 1. Convergence and the Error a. The error looks like 0 jj j ea s = where the j s are the search directions and j a are numerical coefficients b. 0 kkj j j k j k k k sA e sA a s a s a s ⋅= = ∑∑ since the search directions are orthogonal in A space i. thus 1 0 1 0 () k kj j j k k kk e s e a s s α = ⋅+ == ⋅⋅ where the summation can be added since it is identically zero when multiplied by k ii. now 11 1 2 2 21 1 k kkk k ee s e s s αα −− − − − − − =+ + = 1. i.e. 1 0 1 k j j s = iii. thus k e a s = and (from above) a = − c. so the error is 0 j es =− and 1 1 k j ess = + i. after n steps the second term is equal to the first term and the error is zero d. multiplying this error equation by i gives 1 1 k ik j ij j i j e s s = i. for i < k, there is exactly one nonzero term in each sum, and these terms cancel ii. thus for i < k, 0 e and thus 0 sr = (we’ll use this below) iii.
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

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CS 205A class_12 - CS 205 Class 12 Readings Same as last Covered in class All 1 Convergence and the Error a The error looks like e0 = j a j s j

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