CHAPTER 6b. NUMERICAL INTERPOLATION
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1 CHAPTER 6. NUMERICAL INTERPOLATION A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering y Dr. Irhim A. Asskk Spring ENCE - Computtion s in Civil Engineering II Deprtment o Civil nd Environmentl Engineering University o Mrylnd, College Prk o Undetermined Coeicients A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 6 Develop ourth-order interpoltion polynomil or the ollowing set o dt, or which we know their originl unction, tht is, (). () 7 6 ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No.
2 o Undetermined Coeicients A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 6 (cont d) The generl orm o the interpoltion polynomil is given y Eq. s n () L n In our cse it is () ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. o Undetermined Coeicients A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 6 (cont d) We need to ind the coeicients,,, nd. We notice tht hve ive unknowns,(n ), tht require solving 5 simultneous liner equtions using the pir, i nd ( i ), o the given dt s ollows: ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 5
3 Asskk Slide No. 6 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION o Undetermined Coeicients Emple 6 (cont d) () () () () () () () () () () () () () () () () () () () () () 6 7 () () Asskk Slide No. 7 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION o Undetermined Coeicients Emple 6 (cont d) OR
4 o Undetermined Coeicients A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 6 (cont d) The solution o this set o equtions yields () Thereore, the interpoltion polynomil is () ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. o Undetermined Coeicients A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 6 (cont d) We notice in this emple tht s the degree o the interpoltion polynomil goes higher, the computtionl eort ecomes comple. When the polynomil is o the sme order s the unction used to derive tht dt, the polynomil will provide errorless interpolted vlues, nd will e ectly the sme s the originl unction. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 9
5 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering The previous emple shows tht the ith-order interpoltion polynomil gives errorless interpolted vlues, nd there ws signiicnt improvement in ccurcy. Thus, it is usully tempting to try n interpoltion polynomil o higher order to improve the ccurcy. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 5 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering However, it should e recognized tht the solution procedure ecomes more comple s the order increses. The solution in this cse would require more simultneous equtions to e solved or the coeicients. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 5 5
6 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering The Gregory-Newton provides mens o developing n nth-order interpoltion polynomil without requiring the solution o set o simultneous liner equtions. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 5 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Gregory-Newton Formul ( ) ( ) ( )( ) ( )( )( ) 5( )( )( )( ) L ( )( ) K( ) ( )( ) K( ) n n Where independent vrile i n known vlues or i,,, n i n unknown coeicients or i,,, (n ) n (7) n ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 5 6
7 7 Asskk Slide No. 5 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Gregory-Newton Formul The (n ) coeicients cn developed rom set o (n ) vlues o the depended vriles s given in tulr orm s ( n ) ( ) ( ) ( ) () n n i Asskk Slide No. 55 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Gregory-Newton Formul I, then () ( ) or this pir o vlues, every term in Eq. 7 ecept is zero, tht is ( ) ( ) ( )( ) ( )( )( ) ( )( )( )( ) ( ) ( ) ( )( ) ( )( )( ) ( )( )( )( ) L L 5 5
8 Asskk Slide No. 56 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Gregory-Newton Formul Thereore, ( ) () Asskk Slide No. 57 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Gregory-Newton Formul I, then () ( ) or this pir o vlues, every term in Eq. 7 ecept nd re zero, tht is ( ) ( ) ( )( ) ( )( )( ) ( )( )( )( ) ( ) ( ) ( )( ) ( )( )( ) ( )( )( )( ) L L 5 5
9 9 Asskk Slide No. 5 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Gregory-Newton Formul Thereore, ( ) ( ) ( ) ( ) ( ) ( ) OR (9) Asskk Slide No. 59 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Gregory-Newton Formul I, then () ( ) or this pir o vlues, every term in Eq. 7 ecept, nd re zero, tht is ( ) ( ) ( )( ) ( )( )( ) ( )( )( )( ) ( ) ( ) ( )( ) ( )( )( ) ( )( )( )( ) L L 5 5
10 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Gregory-Newton Formul Thereore, ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) OR ( ) ( ) ( ) ( ) ( ) ( )( ) () ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 6 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Gregory-Newton Formul The process cn e repeted to solve the remining coeicients. At ech step new pir o points is used with the vlues o i computed in previous step. Note tht the Gregory-Newton method yields the sme polynomil s the solution o the simultneous equtions used or the method o undetermined coeicients. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 6
11 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Gregory-Newton Formul Thus, it will give the sme interpolted vlue nd hve the sme ccurcy, The dvntge o the Gregory-Newton is tht it does not require the solution o n simultneous equtions with n unknowns. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 6 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 7: G-N Fit qudrtic polynomil or the ollowing set o dt using the Gregory-Newton interpoltion method. Using this polynomil pproimte the unction () when is equl to. nd.. () 7 ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 6
12 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 7 (cont d): G-N I, then () I, then () ( ) ( ) ( )( ) 7 ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 6 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 7 (cont d): G-N I, then () 7 ( ) ( )( ) ( )( )( ) 7() Hence, the polynomil is given y ( ) 7( ) 6( )( ) ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 65
13 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 7 (cont d): G-N For., ( ) 7( ) 6( )( ) (.) 7(. ) 6(. )(. ). For. ( ) 7( ) 6( )( ) (.) 7(. ) 6(. )(. ). ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 66 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple 7 (cont d): G-N Comprison With true vlues: () Approimtion True %Error ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 67
14 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple : G-N Interpoltion Show tht the qudrtic polynomil otined in Emple 7 could hve een otined using the method o undetermined coeicients tht will result in the sme interpolted vlues or equl. nd.. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 6 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple (cont d): G-N Interpoltion A qudrtic polynomil using the method o undetermined coeicients is given y () The coeicients will e determined s ollows () 7 ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 69
15 5 Asskk Slide No. 7 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Emple (cont d): G-N Interpoltion 7 () 7 () () () () () () () 7 Asskk Slide No. 7 A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering ENCE CHAPTER 6. NUMERICAL INTERPOLATION Gregory-Newton Interpoltion Emple (cont d): G-N Interpoltion () () () () () ()
16 Gregory-Newton Interpoltion A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering Emple (cont d): G-N Interpoltion Thereore, () OR () 6 6 (.) 6 (.) 6(.). (.) 6 (.) 6(.). These re the sme vlues otined y Gregory-Newton. ENCE CHAPTER 6. NUMERICAL INTERPOLATION Asskk Slide No. 7 6
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