Topics:

Gaussian Elimination & Back Substitution•LU Factorization•Operation Count and Complexity for each method•Sources of error: Condition Number, Swamping•Partial Pivoting•

Gaussian Elimination & Back Substitution

Example. Solve using Gaussian Method

Example. Solve the same problem using LU factorization

Start by using matrix notation instead of a tableau:

March 19 2012, Gaussian Method & LU FactorizationAlexandros Sopasakis & Claus Fuhrer

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Operation Count and Complexity for each method

Assume that we wish to solve an nxn size system AX=B.Gaussian Elimination requires n^3/3+n^2/2-5n/6 operationsBack Substitution (after Gaussian Elimination) requires n^2/2+n/2 operations

Overall therefore the Gaussian Method requires approximately n^3/3+n^2 -5n/6 operations to solve a problem. In general we say that Gaussian Method is an O(n^3) order method.

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problem. In general we say that Gaussian Method is an O(n^3) order method.•

Overall LU factorization & Back Substitution requires approximately n^3/3 + n^2 operations. In general we say that LU is an O(n^3) order method.

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Question: is there a best method to use and if so under which circumstances is one method better than the other?

Sources of Error.

Ill-conditioning=sensitivity of solution to the input data (not much we can do about it)•Swamping = arithmetic errors due to large numerical discrepancy in parameters (fixable)•

Ill-conditioning & Condition Number

First we need to define the concept of norm of a vector and norm of a matrix. As usual we wish to solve the problem Ax = B. Assume that x* is an approximate solution to that problem.

Definition. Infinity vector norm (maximum vector norm):

Definition. Infinity matrix norm (maximum matrix norm):

Definition. Residual:

Definition. Backward Error:

Definition. Forward Error:

Definition. Condition Number: For a square matrix A, cond(A) is the maximum possible error magnification factor for solving Ax = B, over all possible right hand sides B.

Interpretation of Condition Number: The larger the condition number is then the hardest will be to solve the system Ax = B.

For example if cond(A)=10^k then we should be prepared to lose k digits of accuracy in computing the solution x. In double precision computer arithmetic we may have up to 16 digits accurate. In this case however the accuracy will now be reduced to 16-k digits.

Example. Calculate the condition number for the following matrix A:

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