CHAPTER 6c. NUMERICAL INTERPOLATION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig y Dr. Irahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil ad Evirometal Egieerig Uiversity o Marylad, College Park Method o Udetermied Coeiciets Eample 6 Develop a ourth-order iterpolatio polyomial or the ollowig set o data, or which we kow their origial uctio, that is, (). 4 () 8 7 64 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 7
Fiite Dierece Iterpolatio I the values o the idepedet variales are equally spaced, a iite dierece scheme ca e used to develop a iterpolatio polyomial. I is the icremet o which the values o the idepedet variale are recorded, the the irst iite dierece o the depedet variale y () is ( ) () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 74 Fiite Dierece Iterpolatio The secod iite dierece ca e epressed as [ ] [ ( ) ( ) ] -[ ( ) ] I geeral, the iite dierece is [ ] () () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 75
Fiite Dierece Iterpolatio Rule The ith iite dierece, where i is less tha or equal to, o a th-degree polyomial is a polyomial o degree ( ). This rule implies that the th dierece would e a costat. Assume the ollowig th-order polyomial: () L (4) does ot equal zero. ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 76 Fiite Dierece Iterpolatio For ( ), the polyomial give y Eq. 4 ecomes ( ) ( ) ( ) ( ) L ( ) This polyomial ca e rearrage ito ( ) ( ) L (5) (6) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 77
Fiite Dierece Iterpolatio The irst iite dierece ca e costructed y sutractig Eq. 4 rom Eq.6 as ollows: () L ( ) ( ) L Or ( ) ( ) L (7) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 78 Fiite Dierece Iterpolatio For ( ), Eq. 5 ecomes ( ) ( ) ( ) L (8) Equatios 8, 6, ad 4 ca e sustituted ito Equatio to compute a secod iite dierece, with the ollowig result: ( )( ) L (9) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 79 4
Fiite Dierece Iterpolatio Equatio 9 is a polyomial o order (-). By iductio,the ith iite dierece is give y i i ( )( ) L ( i ) L i () Equatio is a polyomial o order ( ). The th iite dierece equals the ollowig costat: L! () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 8 Fiite Dierece Iterpolatio Eample Suppose that, the Eq. 4 ecomes Or () () () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 8 5
6 Slide No. 8 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Fiite Dierece Iterpolatio Eample (cot d): For ( ), ( ) ca e deied as ollows: () Slide No. 8 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Fiite Dierece Iterpolatio Eample (cot d): For ( ), ( ) ca e deied as ollows: 4 4 4 (4) 4
Fiite Dierece Iterpolatio Eample (cot d): Recall Eq.,, ad ( ) () ( ) ( ) ( ) The irst dierece is computed as ollows: ( ) (5) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 84 Fiite Dierece Iterpolatio Eample (cot d): Similarly, recall Eq.,,, ad 4 [ ] [ ( ) ( ) ] [ ( ) ] - () ( ) ( 4 ) ( ) ( ) The secod iite dierece is computed as ollows: ( ) ( ) ( ) (6) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 85 7
Fiite Dierece Iterpolatio First ad Secod Fiite Dierece The a quadratic polyomial o the orm () The irst ad secod iite dierece are give as ad ( ) ( ) (8) (7) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 86 Fiite Dierece Iterpolatio Eample Develop a iterpolatio polyomial or the ollowig data usig the iite dierece approach. Estimate the () or.7 () 5 8 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 87 8
Fiite Dierece Iterpolatio Eample (cot d): () 5 8 () ( ) 5 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 88 Fiite Dierece Iterpolatio Eample (cot d): The irst iite dierece ca e otaied usig Eq. as ( ) 5 This value ca e equated to Eq. 7 as ollows: ( ) ()() () () (9) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 89 9
Fiite Dierece Iterpolatio Eample (cot d): The secod iite dierece ca e otaied usig Eq. as [ ( ) ( ) ]-[ ( ) ] [ 8 5] [ 5 ] This value ca e equated to Eq. 8 as ollows: () () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9 Fiite Dierece Iterpolatio Eample (cot d): Solvig Eq., gives.5 Sustitutig this result ito Eq. 9, gives Or.5 (.5) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9
Fiite Dierece Iterpolatio Eample (cot d): Usig these values o ad ad ay oe o the three poits, a value o ca e calculated rom Eq. as ollows: () () 8.5().5() ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9 Fiite Dierece Iterpolatio Eample (cot d): Thus, the iterpolatio polyomial is () ().5.5 (.7) ca e estimated: (.7).5(.7).5(.7) 6. 995 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9
Fiite Dierece Iterpolatio Fiite-dierece Tales We see i the precedig eample that the iite dierece approach ca e used to costruct a iterpolatio polyomial. However, the procedure ca e tedious ad requires a lot o computatios. Thereore, we eed some sort o scheme to orgaize the procedure usig iitedierece tales. ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 94 Fiite Dierece Iterpolatio Fiite-dierece Tale The iite-diereces ca e orgaized ito a tale as show i the et viewgraph. The iite-dierece tale ca e used to derive a iterpolatio polyomial. As we will see, iite-dierece tales ca e used with Newto ormula to derive iterpolatio polyomials. ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 95
Fiite-dierece Tale () () () ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) : : : : ( -) [ ( -)] [ ( -)] [ ( -)] ( -) [ ( -)] [ ( -)] [ ( -)] () ( ) ( ) : [ ( -)] ::: () ( ) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 96 Fiite Dierece Iterpolatio Eample Costruct a iite-dierece tale or the ollowig set o data: 4 () 78 97 744 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 97
Fiite Dierece Iterpolatio Eample (cot d): () 4 66 97 6 78 7 469 6 97 78 547 4 744 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 98 4