MATH 174: Numerical Analysis. Lecturer: Jomar F. Rabajante 1 st Sem AY

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1 MATH 74: Numeric Aysis Lecturer: Jomr F. Rbjte st Sem AY -

2 INTERPOLATION THEORY We wt to seect fuctio p from give css of fuctios i such wy tht the grph of y=p psses through fiite set of give dt poits odes. We re goig to strt with POLYNOMIAL INTERPOLATION.

3 POLYNOMIAL INTERPOLATION Give set of + dt poits i,y i where o two i re the sme, we wt to fid poyomi p of degree t most with the property p i =y i, i=,,,

4 POLYNOMIAL INTERPOLATION UNISOLVENCE THEOREM Eistece d Uiqueess Theorem Poyomi iterpotio defies ier bijectio where { i,y i, i=,,,} ϵ K +. П is the vector spce of poyomis defied o y iterv cotiig the odes of degree t most.

5 POLYNOMIAL INTERPOLATION UNISOLVENCE THEOREM Eistece d Uiqueess Theorem I other words, Give + iterpotio odes i,y i, where i s re distict, there is poyomi p of degree ess th or equ to tht iterpotes y i t i, i=,,,. This poyomi p is uique mog the set of poyomis of degree t most.

6 POLYNOMIAL INTERPOLATION Proof of eistece d uiqueess of the iterpotig poyomi: We c write p s There re m+ coefficiets. By hypothesis, there re + coditios o p, set m=. We wt to fid the vues of the coefficiets: This is system of + ier equtios i + ukows. Sovig it is equivet to sovig the poyomi iterpotio probem. m m p... y y y

7 POLYNOMIAL INTERPOLATION Proof of eistece d uiqueess of the iterpotig poyomi We c write the system of ier equtios i mtri form: Vdermode mtri

8 POLYNOMIAL INTERPOLATION Proof of eistece d uiqueess of the iterpotig poyomi Deote the Vdermode mtri s X. It c be show usig cofctor epsio tht det X Sice the poits i re distict, ji det X i j Thus, X is osigur d the system of ier equtio hs uique soutio for i s. This proves the eistece d uiqueess of iterpotig poyomi of degree ess th or equ to.

9 POLYNOMIAL INTERPOLATION SOME WAYS OF CONSTRUCTING THE INTERPOLATING POLYNOMIAL UNIVARIATE Geer Cse c be used for uequy spced dt poits Lgrge Iterpotio Newto s Divided Differeces Nevie s Method Speci Cse for equy spced dt poits Forwrd Differeces Bckwrd Differeces Cetered Differeces MULTIVARIATE OSCULATING POLYNOMIALS The outputs here re the sme usig sme set of odes.

10 LAGRANGE INTERPOLATION Give the fiite dt poits beow, the Lgrge Formu is s foows: P, beig i ier i y i combitio of the y poyomis i, is itsef poyomi. y The formu is... ivrit uder - - y differet order of the - y dt set. p i where i ji j y i i i j j

11 LAGRANGE INTERPOLATION Empe : Fuctio Iterpotio Represetig cotiuous fuctios by iterpotig poyomis Iterpote usig equy-spced poits Assume tht the give poits re rouded off up to 6 decim pces.986 y y p

12 LAGRANGE INTERPOLATION Empe : y y p.986 The iterpotig poyomi This is ced ier iterpotio. Try y- =.986- /

13 LAGRANGE INTERPOLATION Does this poyomi rey psses through the give poits? Assume j is oe of the give bscisss. The i fuctios obey the Kroecker det equtio: Try it here: j i if j i if { ij i j.986

14 LAGRANGE INTERPOLATION Empe : Dt Iterpotio Iterpotig discrete dt sets The viscosity of wter hs bee eperimety determied t differet tempertures, s idicted i the foowig tbe: Temp deg 5 5 Viscosity From this tbe, how c we estimte resobe vue for the viscosity t temperture 8 degrees? Use Poyomi Iterpotio.