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Unformatted text preview: 15.2 The points can be ordered so that they are close to and centered around the lull(noun. A
divideddifference table can then be developed as x for} First Second Third Fourth 3 5.25 T25 2 [1.25 {I
5 1925 5.25 2.?5 0.25 2 4 B 1.?5 6 35 6.25 ‘l 4.?5 Note that the fact that the founh divided difference is. zero means that the data was. generated with
a thirdorder polynomial. First order: I; [—1) = 5.25 + T.25(4— 5) =12.5
fzt4j=115 + [:4 — 5)(4 — 5E =1D.5 Third older: f3 (4) = “3.5 + [4—3)(4 — 5}(4— 30.25 =10
Fourth order: f3(4}=10.5 + (—1— 3)[—1— 5)(4 — 2}(4 — 6}D=1{) 15.3 Lagrange polynomial. First order:
4—5 4—3
4 =—5.25+ 19.?5=12.5
fl( ) 3—5 5—3 Second order: f3 {4:}:ng35 +W19I75 +W4=105
' T3  5X3 — 3) (5  3H5 — 3) (  3X2  5) Third order: = [4—5}(4—2)[4—6} 525+ (4—3}(4—2}(4—6}19ITS
(3—5)t3—2)<3—6)' (S—BHS—EDCS—ﬁ) ' . (Lam—53mm“ (43)(45}(42}
(2—3)t2—5)(—6) (mars—5x64) 15.5 MATLAB can be used To generate a cubic polynomial Through The ﬁrst 4 points in the Table. f3 (4} 36:10 3*} X = [l 2 3 ‘9];
3*} fx = [3.6 1.8 1.2 0.9];
3*} p = polyfitllxr fx,3}
p :
U.15C'U 1.5000 —5.25C'0 7.5000 Therefore. The roots problem to be solved is 1.6 =—a.15x3+1.5x3—5.25x + as 01‘
f(.r} = —a.15x3+1.5x3—5.25x + 5.9 = a Biseetion can be employed the root of This polynomial. Using initial guesses ofx: = .. and In = 3. a
value of 2.2156 is obtained with £3 = 0.0006991} after 16 iterations. 15.7 (:1) Because they bracket the tuilqtown. the two last points are used for linear interpolation. .? — .5 5
flail18)=5.5453+M(0.113—a11144)=5.643t 0.1254T—0.lll44 (b) The quadratic interpolation can be implemented easily in MATLAB. >> v = [0.103?7 0.1144 0.1254T];
>> s = [6.414? 6.5453 6.166%];
5) p = polyfit{v,s,2}
p: 354.2358 —Eé.99?6 9.3450
?} polyvalip,0.ll$)
ans = 6.601? Therefore. to the level of signiﬁcance reported in the table the estimated entropy is 6.6077 (c) The inverse interpolation can be implemented in MATLAB. First. as in part u—‘e can ﬁt a
quadratic polynomial to the data to yield. P [0 H
[H 4.2353 6é.99?€ 9.3450
“We must therefore find the root of 6.45 = 354.2358):2 — 64.9976x + 9.3450
or 6.45 = 354.2358):2 — 54.99%] + 2.8950 In hiATLAB. we can generate this polynomial by subtracting 6.45 from the constant coefﬁcient
of the polynomial >5 p(3)=p(3)—6.1’:5
[3 LL'I 
til 4.2358 —Eé.99?6 2. ro
'sD
in
CI Then. we can use the roots function to determine the solution. >> root5{p] ans =
0.1074
0.0761 Thus. the value of the specific volume corresponding to an entropy of 6.45 is 0.1074 15.10 Errata: In the first printing, the problem statement asked for a fifth—order interpolating polynomial which cannot be obtained with the 5 given points. Subsequent printings only ask
for third—order and fourthorder polynomials. Thirdorder case: The MATLAB polyﬁt function can be used to generate the cubic polynomial
and perform the interpolation. }} x = [1.8 2 2.2 2.4];
}} J = [0.5615 0.5?67 0 5560 0 520:],
>> p = polyfit{x,J,3}
p =
0.016? —0.2988 0 9306 —0.2229
}} Jpred=polyval(p,2.1}
Jpred =
0.5683 The builtin function besselj can be employed to determine the true value which can then be
used to determine the percent relative error >> Jtrue=besselj(1,2.1}
Jtrue =
0.5683
}} ea=ab5i(Ctrue—Jpred}thrue]*100
a =
3.15T39—004 L'D Fourthorder case: >> x = [1.3 2 2.2 2 4 2.6],
>> J = 0.5815 0.5?67 0.5560 0.520: 0.4?08];
}} p = polyfit{x,J,4}
p =
0.0182 —0.1365 0.1818 0.2630 0.123?
>> Jpred=polyval{p,2.1}
Jpred =
0.5683
>> ea=abat(Ctrue—Jpred}thrue]*100
ea = 0.0021 16.1 (a) The simultaneous equations for the natural spline can be set up as 1 cl' 0 '
1 3 0.5. 1:2 0
0.5 2 0.5 c3, _ —6
0.5 3 1 c4 _ —24
1 4 1 ('5 1s
1 es 0 These equations can be solved for the (3‘s and then Eqs. (16.21) and (16.13) can be used to solve
for the [1's and the d‘s. The coefﬁcients for the intervals can be summarized as interval a b C d
1 1 3.9T9954 9 9.829948
2 5 4.858991 8.88713? 8.48884
3 T 3.849249 8.52282 43.31535
4 8 —1.41989 9.99585 5.414938
5 2 —5.‘8598 8.248983 2.88299 These can be used to generate the following plot of the natural spline: DINB01030 Notice how the notaknot version exhibits much more curvature. partielﬂaﬂy between the laat
points. [13) The piecewiae eubie Hermite polynomial and ita plot can be generated with MATLAB as 3>) X [l C 4 5]; 3;; y [l E 1]; >> xx = linspaceﬂl,5}: >> yy = interpl {erJXXr 'E'ChiF'JF
>3: plct{x,y, 'C'TrxerY} ..J b.)
m in
l' 3 LL! 16.3 (a) The llﬂtﬂkllOT ﬁt can be r:.e’r up in MﬁJ'LAB as >? x = linspace{ﬂ,l,ll}; >> y = l.f{[x—O.3].“C+O.Ulj+l.f{[x—D.9).“C+U.04}—E; >> xx = linapaceiﬂ,l}; >§ yy 5p1ine{x,y,xx}: >> yh l.f[[xx—0.3).“Z+U.Dl}+1.fﬁ{xx—D.9}.A2+D.Oé]—E;
>§ plct{x,y,'c*,xx,yg,xx,yh,T——'j 10D ED ED 40 20 D ﬂ.2 0.4 0.6 (LEI 1 (b) The piecewise cubic Hermite polynomial ﬁt can be rget up in [VIATLAB as >> = linspace{ﬂ,l,ll}; >> — l.f{[x—O.3].“C+O.Ulj+l.f{[x—O.9).“C+U.Dé}—E; >§ XX = linspace(ﬂ,l}: >> yy = interpl{x,y,xx,'pchip']; >} yh = l.f[[xx—0.3).“:+U.Dl}+1.f[{xx—D.9}.*:+D.Dé]E;
>> plct{x,y,'GT,xx,yy,Xx,yh,T——'j H1 N
 16.5 (a) The llotakno‘r ﬁt can be SET up in NIATLAB as
D? x = [1 D.E U.2 0.2 0.6 1];
>:»y=[000111]: >> xx = linspacei—l,l}: >) yy 2 spline§xryJXX}F
>> 1::1C‘t:{:>s‘..r y, 'cT . xx; YE} 12
1
DE
DE 04 31 05 D D5 1 (b) The clamped spline with zero end alopea can be set up in MQLTLAB as >§ x = [—1 —D.E —U.2 0.2 0.6 1];
>> y = [O U D l l l]:
>§ ya = [U y D]; }} xx = linspacei—l,l}:
}} yy = 5pline{x,ys,xx};
>> E'lct {Kr Y: '6‘ .XX: FY} 12 11 —D5 U 05 1 {e} The piecewise eubie Hermite polynomial ﬁt can he get up in NIATLAB as 3} x = [—1 —D.6 —u.2 02 0.6 1];
3} y = [e a D l 1 i]: §§ xx = linspaceil:1}F }} yy = interpl{er;KKr'PChiF'}F
>3 plct{xry:'GT:XX:YY} 16.10 (:1) Here is a MATLAB seaaion to develop The 5p1i11e and plot it along with The dam: >> x=[D C 4 T 1012];
>> y=[20 El] 12 “J 'E E];
>3> xx=linspace{0,12):
>> yy=splinetx,y,xxj;
>3> plet{x,y, 'cT,xx,yy}l 25 ‘15 1C! Here is the command to make the prediction: >> yp=splinetx,y,1.5} (b) To prescribe zero first derivatives at the end knots. the y vector is modiﬁed so that the first
and last elements are set to the desired values of zero. The plot and the prediction both indicate
that there is less overshoot between the first two points because of the prescribed zero slopes. )} yd=[0 y D]; }) yy=splinetx,yd,xx};
>3 plottmy; 'cHXXIYY}
>> yy=splinetx,yd,l.5} 20.5?01 ...
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 Spring '09
 RAPHAELHAFTKA
 Polynomials, Numerical Analysis, Hermite polynomial ﬁt, First Second Third Fourth, Jpred

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