CE 601: Numerical Methods Lecture 7. Course Coordinator: Dr. Suresh A. Kartha, Associate Professor, Department of Civil Engineering, IIT Guwahati.

Transcription

1 CE 60: Numerical Methods Lecture 7 Course Coordinator: Dr. Suresh A. Kartha, Associate Professor, Department of Civil Engineering, IIT Guwahati.

2 Drawback in Elimination Methods There are various drawbacks while using elimination methods using computers o Theoretically, the elimination methods should give actual solution. o However in computers we declare or assign variables with only certain precision (say single precision, double precision, etc.) Due to this there are drawbacks like: o Accumulation of round-off errors. o Failure in solving ill-conditioned systems.

3 Round-off Errors o Round-off errors are generated by approimating precision numbers by finite precision numbers. o Usually in computer you have single precision numbers (7 significant digits) and double precision numbers (4 significant digits). o The case of effect of round-off errors in elimination method is demonstrated in net slide.

4 Using Gauss elimination method, ( R2 R2 (/ ) R ) o But while using computer, you need to use finite precision numbers. Accuracy of significant Digits Precision X decimals decimals decimals decimals

5 o You can see from the above table, how based on the significant digits and precision the round-off errors creep. o You can reduce round-off errors by doing partial pivoting. Ill-conditioned System Consider the system or, It's solution is.0 and.0. 2

6 o if we give a slight perturbation on the coefficient of 2 in the above equation, i. e., or, Now the solutions are and 3. 2 o Huge difference in solution for a very slight changes in the system. o This is because the system is ill-conditioned.

7 An ill-conditioned system is one where small changes in the numerical values of coefficient matric [A] or right side vector{b} cause large changes in the solution vector {}. A well-conditioned system is one which produce only small change in the solution vector for small changes in [A] or {b}. Eample of an ill-conditioned system has already been discussed in our previous class. Ill-conditioning of a system is determined by its condition number.

8 Norms o Measure of magnitude of a matri, vector, etc. For [A], {b}, {} etc, their norms are given as : A, b, etc. The norm of a matri or vector shall be always greater than zero. i.e., A 0, 0 If A 0, then [ A] 0. If k is a scalar quantity, then k[a]=[ka], but ka k A,(norm of a scalar is its absolute value)

9 For [ A] and [ B], A B A B AB A B Norm of a vector { } You have, 2 e 2 2 i i (sum of magnitudes) (Euclidean Norm) ma (Maimum magnitude Norm) i n i

10 For an n n matri [ A], A A ma a (Maimum of column sums) j n i n n ma a (Maimum of row sums) i n j ij ij A e i n j 2 Condition Number n a 2 ij (Euclidean norm) o Measure of sensitivity of the system to small changes in any of its elements.

11 Recall, [ A]{ } { b} Now it is clear, b A (from properties of norm) Small change { b} in { b} will cause change in { },say by { }. So, new system will be, [ A] { } { } { b} { b} [ A]{ } { b} ( [ A]{ } { b}) { A b } [ ] { } A b b A A b b A ( ) A A b b where c( A) A A is called condition number. i.e., ca ( ) b b

12 Smaller values of c(a) -> well conditioning Larger values of c(a) -> ill conditioning Eample Find condition number of [ A] A [ A] 0000 A e e c A ( ) A A ca ( ) Very large value Ill-conditioned system.