MATH 174: Numerical Analysis I. Math Division, IMSP, UPLB 1 st Sem AY

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1 MATH 74: Numerical Analysis I Math Division, IMSP, UPLB st Sem AY 0809

2 Eample : Prepare a table or the unction e or in [0,]. The dierence between adjacent abscissas is h step size. What should be the step size or linear interpolation to give an absolute error o at most 0 6. ma ma ma! [0,] h j jh e h j jh e P j j j j i i Î

3 Using Etreme Value Theorem, Hence, 4 ma h h j h j jh h j h j jh j j ø ö ç è æ ø ö ç è æ ø ö ç è æ ø ö ç è æ 8 eh P

4 Thereore, eh P 0 8 Þ h Þ h» Since n must be an integer, we can choose: h0.006 or n65. Remember: we must start rom 0 then end with. 6 0 Þ n ³ 58.9 Þ n» 65 h We can also use n800 or h We can also use n000 or h0.00.

5 Eample : Determine the mathematical error bound ecluding the propagated roundo error or the interpolation using the ollowing data: [ 0, ] [ 0, ] ln ln ln.0986

6 . Using Cauchy Remainder Theorem éd ln P ê! ë d ù û where Î, P! Since / is monotonically decreasing at [,]: P!

7 . Using Cauchy Remainder Theorem P Eample: Let say.5,.5 P The interpolating polynomial is: Error bound 0. 5 P» ln.5 P.5» < 0.5

8 . Using Cauchy Remainder Theorem P Error bound Note: You can also get the Pero sa eam okay lang hanggang dito P ma Î [,] { } Now use your Math 6 technique in inding etrema

9 . Using Cauchy Remainder Theorem P ma Î[,] { } Graph o ma Î[,] { }

10 . Using Cauchy Remainder Theorem P æ ç è ö ø» Use round up since we are getting error bound.

11 . Using Corollary o Cauchy Remainder Theorem » ø ö ç è æ M M P [,], ln Î " M d d 4 ø ö ç è æ n n a b n M P M n Error bound

12 " Î[,]:ma error <

13 NOTE: Corollary o Cauchy Remainder Theorem 4 ø ö ç è æ n n h O n a b n M P h Interpret this big Oh!

14 . Using Net Term Rule [ 0, ] [ 0, ] [ 0,,] ln C B0.48/ ln B A / ln.0986 A ln/

15 . Using Net Term Rule P é ê ë B 0.48 ù û R Notice that we have propagated round o error. This is the disadvantage o using Net Term Rule. So i is given use Cauchy Remainder Theorem.

16 . Using Net Term Rule consider.5 B error éé êê êê êê êê ëë é ê ë R ln.5 lnù û.5.5 ù û ù 0.48 û.5.5.5

17 . Using Net Term Rule consider.5 error ln.5 ln R error R when.5 This error is the same as the absolute error. Eplain why? Absolute error : ln.5 P.5» 0.00

18 . Using Net Term Rule Generalizing the error bound or the given data points: R error û ù ê ê ê ê ë é û ù ê ê ê ê ë é û ù ê ë é ln B

19 . Using Net Term Rule Generalizing the error bound or the given data points: error ln R error ma{ln ln Î[,] 0.48 R}

20 . Using Net Term Rule Generalizing the error bound or the given data points: Using Numerical Optimization: error R The ma happens at».677 Some Numerical Optimization methods will be discussed in Math 75. But you can just use your Math 6 techniques in getting the maimum.

21 THE NEXT SLIDE WILL SHOW YOU ONE OF THE BIGGEST PROBLEMS IN POLYNOMIAL INTERPOLATION.

22 RUNGE PHENOMENON: one o the biggest disadvantages o adding more nodes Runge Function CauchyLorentz Function: POLYNOMIAL WIGGLES 5 lim n ma P n To minimize this problem, we can use Chebyshev nodes! To be discussed later.

Section 3.4: Concavity and the second Derivative Test. Find any points of inflection of the graph of a function.

Unit 3: Applications o Dierentiation Section 3.4: Concavity and the second Derivative Test Determine intervals on which a unction is concave upward or concave downward. Find any points o inlection o the