Math 472 Lecture 10Chapter 2Systems of EquationsRuoyu WuU Michigan1 / 12
2.3 Sources of errorThere are two major sources of error for the Gaussian Elimination Algorithm:1.ill-conditioning: The sensitivity of the solution to the input data (Aorb),as illustrated in Example 2.12 in the textbook.
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2.swamping: A deficiency of the naive Gaussian Elimination.
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2.3 Sources of errorDefinition 2.4 (Backward and Forward Error)Consider the systemAx=b, and letxabe an approximate solution.•Thebackward erroriskb-Axak∞(also written askrk∞wherer=b-Axais called theresidual)•Therelative backward erroriskb-Axak∞/kbk∞•Theforward erroriskx-xak∞•Therelative forward erroriskx-xak∞/kxk∞•Solving the systemAx=bis equivalent to solving the equation(s)f(y) = 0, wheref(y) =Ay-b.•The solution (root) isx.•The backward error iskf(xa)k∞=kAxa-bk∞.•The forward error iskxa-xk∞.3 / 12
2.3 Sources of errorExample 2.11Find the forward, backward, relative forward, and relative backward errors forthe approximate solution(-1,3.0001)of the systemx1+x2= 21.0001x1+x2= 2.0001☼4 / 12