Question

1. Fit cubic splines to the following data

4 2 9.0 7.0 5.0 2.5 1.0 Xi 2.5 0.5 2.5 1.0 fi 2,5 er

(1) using natural end condition and estimate f(4.0) with the cubic spline.

(2) using clamped end condition with f1’=-2.5, f5’=3.0 and estimate f(4.0) with the cubic spline.

(3) using “not-a-knot” end condition and estimate f(4.0) with the cubic spline.

Answer 1

4- l 0.S 4gs4me tuat C)=a,(2-1) b, (2-1)()td,, 1s2 sr S) 303.)fd-25 2 T-setCrDS +)'g 's()'s -25 S2(3)(5)a (0.5)3 b2(2)(2.)t-f2 9 Sz (m4)-ST)s(2t b )t2r 14- q tto(s- ()ts - 2.9 0.S 3.375a, t225b,t1, d 8a4+464 s(a) = 392(x-2-5)2ら2 (1ース5)+ C

2 + Since -G ( 6.75a,421tG-2 11 2 Now, S)%) 94,+26/-21 12 14 Natuoal euel Condhtiso IS 16 SAA tate GE.4518946 a,-0.2568 42106, b041S18446e-0 0842a526 b.015232, 22-0.221a52 at02035288 a=0.21S26316 a4--0.167105 232, b4,00 26315, d.d, d2d 033 68 4211 A Sjhee 4'e[s, fC4)= S a-3- 3.49605232

clampedt endl est:f f 3.0 1204-t 464t= 3 zet and - On Solvi ug - a,-0.11676975P a2-0.201796275 20.2 5654216 bF 7519312 1O.8638G77 b4 0:67S3 0 625D C19.5, C2 0.22720703/2 2984375 C- -o.675390625, 에e2s, d2키,dg-20, d4-05 fi4)=a-b-e, tde3.40161119 not-a-knot Conditun ,) (2 S(24)S4(4 (19 a4 121 111 A Tmes 6a,-642 94 -1 Ou sobint D-t et LA a,a2-0.18S127b = 1.4-214997 a0.2178511421-4 b0,5871 42857 ,d,- a40.21 78571429 ,d,2.5 b30.8071428S11, da2 b4 20.29 64285714 Di2714285, C40.71428 S71 Ca f(4.0)a-b33t de33464-28 571