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
NA-FMN050 Page 1
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.
•
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:
NA-FMN050 Page 3