# Lecture 10 handout.pdf - Math 472 Lecture 10 Chapter 2...

• 12

This preview shows page 1 - 5 out of 12 pages.

The preview shows page 3 - 5 out of 12 pages.
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.
2.swamping: A deficiency of the naive Gaussian Elimination.
2 / 12
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 askrkwherer=b-Axais called theresidual)Therelative backward erroriskb-Axak/kbkTheforward erroriskx-xakTherelative forward erroriskx-xak/kxkSolving 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= 2.00014 / 12

Course Hero member to access this document