Chapter 16-1 CHAPTER 16 INVESTMENTS Accounting Principles, Eighth Edition.
Chapter 16
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Transcript of Chapter 16
![Page 1: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/1.jpg)
Chapter 16Chapter 16
Curve Fitting:Curve Fitting:SplinesSplines
![Page 2: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/2.jpg)
Spline InterpolationSpline Interpolation
There are cases where polynomial interpolation is bad
Noisy dataSharp Corners (slope discontinuity)Humped or Flat data
Overshoot
Oscillations
![Page 3: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/3.jpg)
ExampleExample: : ff((xx) = ) = sqrtsqrt((absabs((xx))))
Interpolation at 4, 3, 2, 1, 0, 1, 2, 3, 4
![Page 4: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/4.jpg)
Cubic interpolation
5th-order
7th-order
Linear spline
Spline InterpolationSpline Interpolation
![Page 5: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/5.jpg)
Spline InterpolationSpline Interpolation
Idea behind splines
• Use lower order polynomials to connect subsets of data points
• Make connections between adjacent splines smooth
• Lower order polynomials avoid oscillations and overshoots
![Page 6: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/6.jpg)
Spline InterpolationSpline Interpolation
Use “piecewise” polynomials instead of a single polynomial
Spline -- a thin, flexible metal or wooden lathBent the lath around pegs at the required pointsSpline curves -- curves of minimum strain energyPiecewise Linear InterpolationPiecewise Quadratic InterpolationPiecewise Cubic Interpolation (cubic splines)
![Page 7: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/7.jpg)
Drafting SplineDrafting SplineContinuous function and derivatives
Zero curvatures at end points
![Page 8: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/8.jpg)
SplinesSplines
There are n-1 intervals and n data points
si(x) is a piecewise low-order polynomial
![Page 9: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/9.jpg)
Spline fits of a set of 4 points
![Page 10: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/10.jpg)
![Page 11: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/11.jpg)
Example: Ship LinesExample: Ship Lines
waterline
![Page 12: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/12.jpg)
Bow
Stern
![Page 13: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/13.jpg)
Original databaseOriginal database
Spline interpolation Spline interpolation for submerged hull for submerged hull (below waterline)(below waterline)
![Page 14: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/14.jpg)
1n1n1n1n
iiii
2222
1111
xxbaxs
xxbaxs
xxbaxs
xxbaxs
b is the slope between points
i1i
i1i
i1i
i1ii
ii
xx
ff
xx
xfxfb
fa
Linear SplinesLinear Splines Connect each two points with straight line
functions connecting each pair of points
![Page 15: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/15.jpg)
Linear SplinesLinear Splines
n1n1n322211
nn,33,22,11,
xxI xxI xxI interval
y(xy(xy(xy(x points data
,,,,,,:
),),),),:
x1 x2 x3 x4
si (x) = ai + bi (x xi)
![Page 16: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/16.jpg)
Linear SplinesLinear Splines
n1-nn
1nn
1n1n
n1n
n
32323
22
32
3
21212
11
21
2
i
xxx , xf xx
xxxf
xx
xx
xxx ,xf xx
xxxf
xx
xx
xxx , xf xx
xxxf
xx
xx
xs
)()(
)()(
)()(
)(
Identical to Lagrange interpolating polynomials
n1n1n322211
nn,33,22,11,
xxI xxI xxI interval
y(xy(xy(xy(x points data
,,,,,,:
),),),),:
![Page 17: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/17.jpg)
Connect each two points with straight line Functions connecting each pair of points are
slope i1i
i1ii xx
ffb
Linear splinesLinear splines
1 1 1 1 1 2
2 2 2 2 2 3
1
1 1 1 1 1
;
;
;
;
i i i i i i
n n n n n n
s x a b x x x x x
s x a b x x x x x
s x a b x x x x x
s x a b x x x x x
![Page 18: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/18.jpg)
Linear splines are exactly the same as linear interpolation!
Example:
586
39
195
7
22
3
40
bxfx i
![Page 19: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/19.jpg)
Linear SplinesLinear Splines
Problem with linear splines -- not smooth at data points (or knots)
First derivative (slope) is not continuousUse higher-order splines to get continuous
derivativesEquate derivatives at neighboring splinesContinuous functional values and derivatives
![Page 20: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/20.jpg)
Quadratic splines - continuous first derivatives
May have discontinuous second and higher derivatives
Derive second order polynomial between each pair of points
For n points (i=1,…,n): (n-1) intervals & 3(n-1) unknown parameters (a’s, b’s, and c’s)
Need 3(n-1) equations
ii2
ii cxbxaxf
Quadratic SplinesQuadratic Splines
![Page 21: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/21.jpg)
Quadratic SplinesQuadratic Splines
x1 x2 x3 x4 x5
s 1 (x)
s2 (x)s 4
(x)
si(x) = ai + bi (xxi) + ci(xxi)2 3(n-1)3(n-1) unknownsunknowns
Interval 1 Interval 2 Interval 3 Interval 4
f(x1)
f(x2) f(x3)f(x4)
f(x5)
s3 (x)
n1n1n322211
nn,33,22,11,
xxI xxI xxI interval
y(xy(xy(xy(x points data
,,,,,,:
),),),),:
![Page 22: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/22.jpg)
Piecewise Quadratic SplinesPiecewise Quadratic Splines
244444454
233333343
222222232
211111121
xxcxxbaxs xx on
xxcxxbaxs xx on
xxcxxbaxs xx on
xxcxxbaxs xx on
)()()(:,
)()()(:,
)()()(:,
)()()(:,
1. The functional must pass through all the points xi (continuity condition)
2. The function values of adjacent polynomials must be equal at all nodes (identical to condition 1.) 2(n-2) + 2 = 2(n-1)
3. The 1st derivatives are continuous at interior nodes xi, (n-2) 4. Assume that the second derivatives is zero at the first points
2(n1) + (n2) + (1) = 3(n1) 3(n1) equations for 3(n1) unknowns
Example with n = 5
![Page 23: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/23.jpg)
Function must pass through all the points x = xi (2n 2 eqns)
This also ensure the same function values of adjacent polynomials at the interior knots (hi = xi+1 – xi)
Apply to every interval (4 intervals, 8 equations)
Piecewise Quadratic SplinesPiecewise Quadratic Splines
2iiiii
2i1iii1iii1i1ii
iiii
hchbfxxcxxbafxs
afxs
)()()(
)(
2444445
2333334
2222223
2111112
44
33
22
11
hchbff
hchbff
hchbff
hchbff
&
fa
fa
fa
fa
![Page 24: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/24.jpg)
Continuous slope at interior knots x = xi (n2 equations)
Apply to interior knots x2 , x3 and x4 (3 equations)
Zero curvatures at x = x1 (1 equation)
Piecewise Quadratic SplinesPiecewise Quadratic Splines
5444
4333
3222
bhc2b
bhc2b
bhc2b
0c 0c2xs 1111 )(
1iiii
1i1i1i1ii1iii
1i1i1ii
bhc2b
xxc2bxxc2b
xsxs
)()(
)()(
![Page 25: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/25.jpg)
Example: Quadratic SplineExample: Quadratic Spline
0
0
0
0
ff
f
ff
f
ff
f
ff
f
c
b
a
c
b
a
c
b
a
c
b
a
100
0102h10
0102h10
0102h10
hh0
001
hh0
001
hh0
001
hh0
001
45
4
34
3
23
2
12
1
4
4
4
3
3
3
2
2
2
1
1
1
3
2
1
244
233
222
211
![Page 26: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/26.jpg)
function [A, b] = quadratic(x, f)
% exact solution f(x)=x^3-5x^2+3x+4
% x=[-1 0 2 5 6]; f=[-5 4 -2 19 58];
h1=x(2)-x(1); h2=x(3)-x(2); h3=x(4)-x(3); h4=x(5)-x(4);
f1=f(1); f2=f(2); f3=f(3); f4=f(4); f5=f(5);
A=[1 0 0 0 0 0 0 0 0 0 0 0
0 h1 h1^2 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 h2 h2^2 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 h3 h3^2 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 h4 h4^2
0 1 2*h1 0 -1 0 0 0 0 0 0 0
0 0 0 0 1 2*h2 0 -1 0 0 0 0
0 0 0 0 0 0 0 1 2*h3 0 -1 0
0 0 1 0 0 0 0 0 0 0 0 0];
b=[f1; f2-f1; f2; f3-f2; f3; f4-f3; f4; f5-f4; 0; 0; 0; 0];
Quadratic Spline InterpolationQuadratic Spline Interpolation
![Page 27: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/27.jpg)
>> x=[-1 0 2 5 6]; f=[-5 4 -2 19 58];>> [A,b]=quadratic(x,f)p = A\b -5.0000 9.0000 0 4.0000 9.0000 -6.0000 -2.0000 -15.0000 7.3333 19.0000 29.0000 10.0000
dx = 0.1; z1=x(1):dx:x(2); s1=p(1)+p(2)*(z1-x(1))+p(3)*(z1-x(1)).^2;z2=x(2):dx:x(3); s2=p(4)+p(5)*(z2-x(2))+p(6)*(z2-x(2)).^2;z3=x(3):dx:x(4); s3=p(7)+p(8)*(z3-x(3))+p(9)*(z3-x(3)).^2;z4=x(4):dx:x(5); s4=p(10)+p(11)*(z4-x(4))+p(12)*(z4-x(4)).^2;H=plot(z1,s1,'r',z2,s2,'b',z3,s3,'m',z4,s4,'g');xx=-1:0.1:6; fexact=xx.^3-5*xx.^2+3*xx+4;hold on; G=plot(xx,fexact,'c',x,f,'ro');set(H,'LineWidth',3); set(G,'LineWidth',3,'MarkerSize',8);H1=xlabel('x'); set(H1,'FontSize',15);H2=ylabel('f(x)'); set(H2,'FontSize',15);H3=title('f(x)=x^3-5x^2+3x+4'); set(H3,'FontSize',15);
Quadratic Spline InterpolationQuadratic Spline Interpolation
4
4
4
3
3
3
2
2
2
1
1
1
c
b
a
c
b
a
c
b
a
c
b
a
p
![Page 28: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/28.jpg)
s1(x)
s2(x)
s3(x)
s4(x)Quadratic Spline
Exact function
![Page 29: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/29.jpg)
Cubic splines avoid the straight line and the over-swing
3ii
2iiiiii xxdxxcxxbaf )()()(
Can develop method like we did for quadratic – 4(n–1) unknowns – 4(n–1) equations
interior knot equality end point fixed interior knot first derivative equality assume derivative value if needed
Cubic SplinesCubic Splines
![Page 30: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/30.jpg)
Piecewise Cubic SplinesPiecewise Cubic Splines
x1 x2 x3 xn
s 1 (x
)
s2 (x)s n-1
(x)
44((nn11)) unknownsunknowns
Continuous Continuous slopes and slopes and curvaturescurvatures
xn-1
n1n1n322211
nn,33,22,11,
xxI xxI xxI interval
y(xy(xy(xy(x points data
,,,,,,:
),),),),:
3ii
2iiiiii xxdxxcxxbaf )()()(
![Page 31: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/31.jpg)
Piecewise Cubic SplinesPiecewise Cubic Splines
x0 x1 x2 xn
ssi i (x)(x) - piecewise cubic polynomials - piecewise cubic polynomials
ssii’(x)’(x) - piecewise quadratic polynomials (slope) - piecewise quadratic polynomials (slope)
ssii”(x) ”(x) - piecewise linear polynomials (curvatures)- piecewise linear polynomials (curvatures)
xn-1
s 1” (x)
s2”(x)
s3 ”(x)
S n-1” (x
)
Reduce to (n-1) unknowns and (n-1) equations for si’’
n1n1n322211
nn,33,22,11,
xxI xxI xxI interval
y(xy(xy(xy(x points data
,,,,,,:
),),),),:
![Page 32: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/32.jpg)
Cubic SplinesCubic Splines Piecewise cubic polynomial with continuous
derivatives up to order 2
1. The function must pass through all the data points
gives 2(n-1) equations
2 3
2 31 1 1 1 1 1
: ( ) ( ) ( ) ( )
: ( ) ( ) ( ) ( )
i i i i i i i i i i i i i i i
i i i i i i i i i i i i i i
x x s x f a b x x c x x d x x a
x x s x f a b x x c x x d x x
2 31
i i
i i i i i i i i
a f
f b h c h d h f
i = 1,2,…, n-1
![Page 33: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/33.jpg)
Cubic SplinesCubic Splines2. First derivatives at the interior nodes must be equal:
(n-2) equations
3. Second derivatives at the interior nodes must be equal:
(n-2) equations
1i2iiiii
1i1i1ii
2iiiiii
bhd3hc2b
xsxs
xxd3xxc2bxs
)()(
)()()(
1iiii
1i1i1ii
iiii
chd3c
xsxs
xxd6c2xs
)()(
)()(
![Page 34: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/34.jpg)
Cubic SplinesCubic Splines
4. Two additional conditions are needed (arbitrary)
The last two equations
Total equations: 2(n-1) + (n-2) + (n-2) +2 = 4(n-1)
1n1n1nn1n
111
hd6c20xs
c20xs
)(
)(
0hd3cc
0c
0xs
0xs
1n1n1nn
1
n1n
11
)(
)(
![Page 35: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/35.jpg)
Cubic SplinesCubic Splines
3ii
2iiiiii xxdxxcxxbaxs )()()()(
Solve for (ai, bi, ci, di) – see textbook for details
Tridiagonal system with boundary conditions c1 = cn= 0
]),[],[(
)(
1iii1i
1i
1ii
i
i1i1iiii1i1i1i
xxfxxf3
h
ff3
h
ff3chchh2ch
1iii
i
i1ii
i
i1iiii cc2
3
h
h
ffb
h3
ccd fa
,,
![Page 36: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/36.jpg)
Cubic SplinesCubic Splines
0
xxfxxf3
xxfxxf3
xxfxxf3
0
c
c
c
c
c
1
hhh2h
hhh2h
hhh2h
1
2n1n1nn
2334
1223
n
1n
3
2
1
1n1n2n2n
3322
2211
],[],[
],[],[
],[],[
)(
)(
)(
Tridiagonal matrix
![Page 37: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/37.jpg)
Hand CalculationsHand Calculations
f(4)estimate 581924)x(f
6520x
0f(4) ,4x3x5x f(x) :solution Exact 23
391
1958
h
ffxxf
73
219
h
ffxxf
32
42
h
ffxxf
58f 19f 2f 4f
156xxh 325xxh 202xxh
3
3434
2
2323
1
1212
4321
343232121
],[
)(],[
],[
,,,
,,
![Page 38: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/38.jpg)
Hand CalculationsHand Calculations
0
96
30
0
0
7393
373
0
c
c
c
c
1
183
3102
1
0
xxfxxf3
xxfxxf3
0
c
c
c
c
1
hhh2h
hhh2h
1
4
3
2
1
2334
1223
4
3
2
1
3322
2211
)(
],[],[
],[],[
)(
)(
can be further simplified since c1 = c4 = 0 (natural spline)
25352112
6760560
c
c
96
30
c
c
83
310
3
2
3
2
.
.
![Page 39: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/39.jpg)
Cubic Spline InterpolationCubic Spline Interpolation
1iii
i
i1ii
i
i1iiii
3ii
2iiiiii
cc23
h
h
ffb
h3
ccd fa
xxdxxcxxbaxs
,,
)()()()(
08309863cc23
h
h
ffb
9014083cc23
h
h
ffb
5492962cc23
h
h
ffb
08450704h3
ccd
43661971h3
ccd
1126760h3
ccd
0c
25352112c
6760560c
0c
58fa
19fa
2fa
4fa
433
3
343
322
2
232
211
1
121
3
343
2
232
1
121
4
3
2
1
44
33
22
11
.)(
.)(
.)(
.
.
.
.
.
![Page 40: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/40.jpg)
Cubic SplinesCubic Splines
1iii
i
i1ii
i
i1iiii
3ii
2iiiiii
cc23
h
h
ffb
h3
ccd fa
xxdxxcxxbaxs
,,
)()()()(
32
333
2333333
32
322
2222222
3
311
2111111
5x084507045x253521125x8309863019
xxdxxcxxbaxs
2x4366197212x67605602x90140832
xxdxxcxxbaxs
x1126760x54929624
xxdxxcxxbaxs
)(.)(.)(.
)()()()(
)(.)(.)(.
)()()()(
..
)()()()(
Piecewise cubic splines (cubic polynomials)
![Page 41: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/41.jpg)
Cubic Spline InterpolationCubic Spline Interpolation
The exact solution is a cubic function Why cubic spline interpolation does not give the exact
solution for a cubic polynomial?
Because the conditions on the end knots are different! In general, f (x0) 0 and f (xn) 0 !!
014114s4f :ioninterpolatspline Cubic
0f(4) 4x3x5x f(x) :solution Exact
2
23
.)()(
,
![Page 42: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/42.jpg)
Cubic Spline InterpolationCubic Spline Interpolation
![Page 43: Chapter 16](https://reader036.fdocuments.us/reader036/viewer/2022081519/56812d1b550346895d92120b/html5/thumbnails/43.jpg)
>> xx=[-1 0 2 5 6]; f = [-5 4 -2 19 58];>> Cubic_spline(xx,f);
Resulting piecewise function:s1 =(-5.000000)+(11.081218)*(x-(-1.000000))s2 =(0.000000)*(x-(-1.000000)).^2+(-2.081218)*(x-(-1.000000)).^3
Resulting piecewise function:s1 =(4.000000)+(4.837563)*(x-(0.000000))s2 =(-6.243655)*(x-(0.000000)).^2+(1.162437)*(x-(0.000000)).^3
Resulting piecewise function:s1 =(-2.000000)+(-6.187817)*(x-(2.000000))s2 =(0.730964)*(x-(2.000000)).^2+(1.221658)*(x-(2.000000)).^3
Resulting piecewise function:s1 =(19.000000)+(31.182741)*(x-(5.000000))s2 =(11.725888)*(x-(5.000000)).^2+(-3.908629)*(x-(5.000000)).^3
(i = 1)
(i = 2)
(i = 3)
(i = 4)
3ii
2iiiiii xxdxxcxxbaxs )()()()(
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Cubic Spline InterpolationCubic Spline Interpolation
Piecewise cubic Continuous slope Continuous curvature
zero curvatures at end knots
exact solution
06f )(
01f )(
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Natural spline : zero second derivative (curvature) at end points Clamped spline: prescribed first derivative (clamped) at end points Not-a-Knot spline: continuous third derivative at x2 and xn-1
End Conditions of SplinesEnd Conditions of Splines
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Comparison of Runge’s function (dashed red line) with a 9-point not-a-knot spline fit generated with MATLAB (solid green line)
>> x = linspace(-1,1,9); y = 1./(1+25*x.^2);>> xx = linspace(-1,1,100); yy = spline(x,y,xx);>> yr=1./(1+25*xx.^2);>> H=plot(x,y,'o',xx,yy,xx,yr,'--');
Continuous third derivatives
at x2 and xn-1
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Note that first derivatives of 1 and 4 are specified at the left and right boundaries, respectively.
>> yc = [1 y –4];% 1 and -4 are the 1st-order derivatives (or slopes)at 1st & last point, respectively.>> yyc = spline(x,yc,xx);>> >> H=plot(x,y,'o',xx,yyc,xx,yr,'--');
Clamped End Spline - Use 11 values including slopes at end pointsClamped End Spline - Use 11 values including slopes at end points
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MATLAB FunctionsMATLAB Functions
One-dimensional interpolationsyy = spline (x, y, xx)
yi = interp1 (x, y, xi, ‘method’)yi = interp1 (x, y, xi, ‘linear’) yi = interp1 (x, y, xi, ‘spline’) – not-a-knot splineyi = interp1 (x, y,, xi, ‘pchip’) – cubic Hermite yi = interp1 (x, y,, xi, ‘cubic’) – cubic Hermiteyi = interp1 (x, y,, xi, ‘nearest’) – nearest neighbor
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Piecewise polynomial interpolation on velocity time series for an automobile
MATLAB FunctionMATLAB Function: : interp1interp1
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>> x=[1 2 4 7 9 10]
x =
1 2 4 7 9 10
>> y=[3 -2 5 4 9 2]
y =
3 -2 5 4 9 2
>> xi=1:0.1:10;
>> y1=interp1(x,y,xi,'linear');
>> y2=interp1(x,y,xi,'spline');
>> y3=interp1(x,y,xi,'cubic');
>> plot(x,y,'ro',xi,y1,xi,y2,xi,y3,'LineWidth',2,'MarkerSize',12)
>> print -djpeg spline00.jpg
Spline InterpolationsSpline Interpolations
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Spline InterpolationsSpline Interpolations
Linear
Not-a-knot
Cubic
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MATLAB’s FunctionsMATLAB’s Functions
Two-dimensional interpolationszi = interp2 (x, y, z, xi, yi)
))(),((),(
),(,),,(),,(
),(,),,(),,(
),(,),,(),,(
jyixfjiz
yxyxyx
yxyxyx
yxyxyx
mn2n1n
m22212
m12111
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MATLAB FunctionMATLAB Function: : interp2interp2
>> [x,y,z]=peaks(100); [xi,yi]=meshgrid(-3:0.1:3,-3:0.1:3);>> zi = interp2(x,y,z,xi,yi); surf(xi,yi,zi)>> print -djpeg075 peaks1.jpg>> [x,y,z]=peaks(10); [xi,yi]=meshgrid(-3:0.1:3,-3:0.1:3);>> zi = interp2(x,y,z,xi,yi); surf(xi,yi,zi)>> print -djpeg075 peaks2.jpg
100 100 data base 10 10 data base
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CVEN 302-501CVEN 302-501Homework No. 11Homework No. 11
Chapter 16 Problem 16.1(a) (30) – Hand Calculation
Problem 16.1(b) (20)– MATLAB program not-a-knot spline: yy = spline(x, y, xx)
Problem 16.1(c)(20) – MATLAB program
Chapter 19 Problem 19.2 (20)– Hand Calculation
Problem 19.4 (20)– Hand Calculation
Due on Wed. 11/05/08 at the beginning of the periodDue on Wed. 11/05/08 at the beginning of the period