research article the eh interpolation spline and its...

Research ArticleThe EH Interpolation Spline and Its Approximation

Jin Xie1 and Xiaoyan Liu2

1 Department of Mathematics and Physics Hefei University Hefei 230601 China2Department of Mathematics University of La Verne La Verne CA 91750 USA

Correspondence should be addressed to Jin Xie hfuuxiejin126com

Received 28 February 2014 Accepted 15 June 2014 Published 30 June 2014

Academic Editor Grzegorz Nowak

Copyright copy 2014 J Xie and X Liu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new interpolation spline with two parameters called EH interpolation spline is presented in this paper which is the extensionof the standard cubic Hermite interpolation spline and inherits the same properties of the standard cubic Hermite interpolationspline Given the fixed interpolation conditions the shape of the proposed splines can be adjusted by changing the values of theparameters Also the introduced spline could approximate to the interpolated function better than the standard cubic Hermiteinterpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm

1 Introduction

Spline interpolation is a useful and powerful tool for curvesand surfaces modeling Standard cubic Hermite spline isone of those interpolation functions But for the giveninterpolation condition the cubic Hermite interpolationspline is fixed that is to say the shape of the interpolationcurve or surface is fixed for the given interpolation data [1ndash6] Since the interpolation function is unique for the giveninterpolation data to modify the shape of the interpolationcurve to approximate the given curve seems to be impossibleand it is contradictory to the uniqueness of the interpolationfunction for the given interpolation data For the giveninterpolation condition how to improve the approximationaccuracy of the interpolation spline is an important problemin the computer aided geometric design In recent yearsmany authors have presented some new method to modifythe shape of the interpolation curve to satisfy the industrialproduct design with several kinds of new interpolationsplines with parameters [7ndash20]

These new splines all have similar properties of thestandard cubic Hermite spline For example for the giveninterpolation data if interpolation interval approaches zerotheoretically speaking these splines can approximate thegiven curve and surfacewell However there exists a problemin the process of the actual calculation the amount of

computation will increase dramatically if the length of theinterpolating intervals decreases On the other hand theapproximation accuracy of these new splines may not bebetter than the standard cubic Hermite spline

In [7ndash17] many rational form interpolation splines withmultiple parameters were presented For the given interpo-lation data the change of the parameters causes the changeof the interpolation curve Nevertheless the computation ofthe splines with multiple parameters is very complicatedSeveral kinds of rational splines with a single parameter werepresented in the papers [18 19] which is simple to computebut its approximation accuracy is not good for the givencurves and surfaces In general polynomial-form splinesare suitable to be used to design and compute In [20] apolynomial-form spline called quartic Hermite spline withsingle parameter is presented as the extension of the standardHermite spline The quartic spline has a simple form and itsapproximation rate to the given curves and surfaces is nothigh

In this paper a class of new quartic splines with twoparameters is developed which is the extension of thestandard cubic Hermite interpolation spline and inherits thesame properties of the standard cubic Hermite interpolationspline For the given interpolation conditions the shape ofthe proposed splines can be adjusted by changing the valuesof the parameters Furthermore the introduced splines could

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 745765 6 pageshttpdxdoiorg1011552014745765

2 Abstract and Applied Analysis

approximate to the interpolated functions better than thestandard cubic Hermite interpolation splines and the quarticHermite interpolation splines with single parameter

The remainder of the paper is organized as followsSection 2 introduces the standard cubic Hermite spline andsome of its properties A kind of interpolation spline with twoparameters is presented in Section 3 Section 4 discusses theapproximation of the introduced spline curve with numericalexamples In the end a summary and conclusions are given

2 The Standard Cubic Hermite Splineand Its Basis Functions

Generally for 119905 isin [0 1] the following four basis functions

1205720 (119905) = 1 minus 3119905

2+ 21199053

1205721(119905) = 3119905

2minus 21199053

1205730 (119905) = 119905 minus 2119905

2+ 1199053

1205731(119905) = minus 119905

2+ 1199053

(1)

are called the standard cubic Hermite basesThese bases satisfy

1205720(0) = 120572

1(1) = 1 120572

0(1) = 120572

1(0) = 0

1205721015840

1(0) = 120572

1015840

1(1) = 0 120572

1015840

0(0) = 120572

1015840

0(1) = 0

1205730(0) = 120573

0(1) = 0 120573

1(0) = 120573

1(1) = 0

1205731015840

0(0) = 120573

1015840

1(1) = 1 120573

1015840

0(1) = 120573

1015840

1(0) = 0

1205720(119905) + 120572

1(119905) = 1 120573

0(119905) + 120573

1(1 minus 119905) = 0

(2)

For given knots 119886 = 1199090lt 1199091lt sdot sdot sdot lt 119909

119899= 119887 and data

(119909119894 119910119894 119889119894) 119894 = 0 1 119899 where 119910

119894and 119889

119894are the values of

the function value and the first-order derivative value of thefunction being interpolated let ℎ

119894= 119909119894+1minus 119909119894 119905 = (119909 minus 119909

119894)ℎ119894

and then the standard cubic Hermite spline in the interval[119909119894 119909119894+1] can be defined as follows

119867119894(119909) = 120572

0(119905) 119910119894+ 1205721(119905) 119910119894+1+ 1205730(119905) ℎ119894119889119894+ 1205731(119905) ℎ119894119889119894+1

119894 = 0 1 119899 minus 1

(3)

Obviously we have119867119894(119909119894) = 119910119894 119867119894(119909119894+1) = 119910119894+1 1198671015840

119894(119909119894) =

119889119894 1198671015840

119894(119909119894+1) = 119889119894+1

The standard cubic Hermite spline is C1 continuous

However if interpolation data is given the shape and approx-imation of the spline cannot be changed

3 The EH Interpolation Spline

In order to overcome the disadvantage of the standard cubicHermite spline we extend its basis functions firstly

31 The Basis Functions of the EH Interpolation Spline

Definition 1 For arbitrary real number 120582119894 120583119894and 0 le 119905 le 1

the following four functions with parameters 120582119894 120583119894

1198901205720(119905) = 1 + (120582

119894minus 3) 1199052+ 2 (1 minus 120582

119894) 1199053+ 1205821198941199054

1198901205721(119905) = (3 minus 120582

119894) 1199052+ 2 (120582

119894minus 1) 1199053minus 1205821198941199054

1198901205730(119905) = 119905 + (120583

119894minus 2) 1199052+ (1 minus 2120583

119894) 1199053+ 1205831198941199054

1198901205731(119905) = minus (120583

119894+ 1) 1199052+ (1 + 2120583

119894) 1199053minus 1205831198941199054

(4)

are calledbasis functions of the EH interpolation splinebriefly EH bases

The EH bases are the extension of the standard cubicHermite bases When 120582

119894= 120583119894= 0 the bases are the standard

cubic Hermite bases The bases have the similar properties ofthe standard cubic Hermite bases

By computing directly we have 1198901205720(0) = 119890120572

1(1) =

1 1198901205720(1) = 119890120572

1(0) = 0 119890120572

1015840

1(0) = 119890120572

1015840

1(1) = 0 119890120572

1015840

0(0) =

1198901205721015840

0(1) = 0 119890120573

0(0) = 119890120573

0(1) = 0 119890120573

1(0) = 119890120573

1(1) =

0 11989012057310158400(0) = 119890120573

1015840

1(1) = 1 119890120573

1015840

1(1) = 119890120573

1015840

1(0) = 0 and 119890120572

0(119905) +

1198901205721(119905) = 1 119890120573

0(119905) + 119890120573

1(1 minus 119905) = 0

When 120582119894= 120583119894 the EH bases (4) are basis functions with

single parameter in [20]Figure 1 shows the four EHbases where the solid lines are

the standard Hermite bases the parameters 120582119894= 2 120583

119894= minus2

are for the dot-dash lines and 120582119894= minus2 120583

119894= 2 are for dashed

lineSo we may construct the Ferguson curve with two pa-

rameters based on the EH bases as follows

EH119894(119905) = 119890120572

0(119905) 119901119894+ 1198901205721(119905) 119901119894+1+ 11989012057301199011015840

119894+ 1198901205731(119905) 1199011015840

119894+1 (5)

where 119901119894 119901119894+1

and 1199011015840119894 1199011015840

119894+1are two interpolation points and

their tangent vectors respectivelyFor the given two interpolation points and tangent

vectors with the different parameters 120582119894 120583119894 we may obtain

different shape of the Ferguson curve with two parametersaccordingly

Figure 2 shows the Ferguson curves with different param-eters where the solid line is the standard Ferguson splinecurve with 120582

119894= 120583119894= 0 the parameters 120582

119894= 2 120583

119894= minus2

are for the dot-dash line and 120582119894= minus2 120583


line


Definition 2 Given a data set (119909119894 119910119894 119889119894) 119894 = 0 1 119899

where 119910119894and 119889

119894are the values of the function value and the

first-order derivative value of the function being interpolatedand 119886 = 119909

0lt 1199091lt sdot sdot sdot lt 119909

119899= 119887 is the knot spacing let

Abstract and Applied Analysis 3

00 02 04 06 08 1000

02

04

06

08

10

(a) 1198901205720(119905)

00 02 04 06 08 1000

02

04

06

08

10

(b) 1198901205721(119905)

00 02 04 06 08 10000

005

010

015

020

(c) 1198901205730(119905)

00 02 04 06 08 10

000

minus020

minus015

minus010

minus005

(d) 1198901205731(119905)

Figure 1 The graph of the four EH bases

00 02 04 06 08 1000

01

02

03

04

05

p998400i

p998400i+1

pi pi+1

Figure 2 The Ferguson curve with different parameters

ℎ119894= 119909119894+1

minus 119909119894 119905 = (119909 minus 119909

119894)ℎ119894 then the EH interpolation

spline in the interval [119909119894 119909119894+1] can be defined as follows

EH (119909)|[119909119894 119909119894+1]

= 1199101198941198901205720 (119905) + 119910119894+11198901205721 (119905) + 119889119894ℎ1198941198901205730 (119905) + 119889119894+1ℎ1198941198901205731 (119905)

119894 = 0 1 119899 minus 1

(6)

where 1198901205720(119905) 119890120572

1(119905) 119890120573

0(119905) 119890120573

1(119905) are the EH bases

Obviously for the data set (119909119894 119910119894 119889119894) 119894 = 0 1 119899

EH(119909) satisfies

RH (119909119894) = 119910119894 RH1015840 (119909

119894) = 119889119894

119894 = 0 1 119899

(7)

If 120582119894= 120583119894= 0 it is just the standard cubic Hermite

spline It is of interest that for suitable selected parameters120582119894 120583119894 the piecewise interpolation function EH(119909) can be

1198622-continuous in the interval [119909

0 119909119899] In fact denote Δ

119894=

(119910119894+1minus 119910119894)ℎ119894 and let

EH10158401015840 (119909119894+) = EH10158401015840 (119909

119894minus) 119894 = 1 2 119899 minus 1 (8)

then the equations connecting the parameters 120582119894and 120583

119894

ℎ119894[Δ119894minus1(120582119894minus1+ 3) minus (119889

119894(2 minus 120583

119894minus1) + 119889119894minus1(120583119894minus1+ 1))]

= ℎ119894minus1[Δ119894(120582119894minus 3) + (119889

119894+1(1 + 120583

119894) + 119889119894(2 minus 120583

119894))]

119894 = 1 2 119899 minus 1

(9)

may be obtained If the successive parameters (120582119894minus1 120583119894minus1)

and (120582119894 120583119894) satisfy (9) at 119894 = 1 2 119899 minus 1 then EH(119909) isin

1198622(1199090 119909119899) Furthermore if the knots are equally spaced and

120582119894= 120583119894= 0 then (9) becomes the well-known tridiagonal

system for a cubic spline

119889119894minus1+ 4119889119894+ 119889119894+1

= 3 (Δ119894minus1+ Δ119894) 119894 = 1 2 119899 minus 1 (10)

Hence if given the parameter values 1205820 1205830in the interval

[1199090 1199091] by (9) we may obtain the 120582

1and 1205831and so onThus

we can construct a 1198622-continuous interpolation curve

4 The Approximation of the EHInterpolation Spline

According to the interpolation reminder of cubic Hermitespline when interpolation interval approaches zero the cubicHermite spline curve can approximate well to the functionbeing interpolated However for the EH interpolation splinewe constructed it can approximate well to the function being


Table 1 The parameters 120582119894and 120583

119894for EH interpolation spline and the max error

119909119894

119910119894

119889119894

120582119894

120583119894

119877119867120576119894

119867120576119894

00000 10000 10000 00421 00412 02569 times 10minus4 09062 times 10minus3

05000 12071 minus01107 00146 00129 02111 times 10minus4 02569 times 10minus3




25000 17929 21107

interpolated without interpolation interval approaching zeroand it can approximate to the interpolated functions betterthan the standard cubic Hermite interpolation spline

Firstly we give the definition of the ldquogood approximationrdquo

Definition 3 Let 119867119894(119909) be the standard cubic Hermite

spline EH119894(119909) be the EH interpolation spline and 119910(119909)

be the function being interpolated Denoting EH120576119894

=

max119909119894lt119909lt119909119894+1

|EH119894(119909) minus 119910(119909)| 119867120576

119894= max

119909119894lt119909lt119909119894+1|119867119894(119909) minus

119910(119909)| then if EH120576119894lt 119867120576

119894 we can call RH

119894(119909) has ldquogood

approximationrdquo to the interpolated function119910(119909) better than119867119894(119909)

According to the Definition 3 if EH120576119894lt 119867120576119894 we can get

the range of the parameters value 120582119894and 120583

119894 In the range

of the parameters value selecting the arbitrary values of theparameters 120582

119894and 120583

119894 we have a ldquogood approximationrdquo curve

Example 4 Given the function 119910(119909) = 119909 + cos((1205872)119909) andknots 119909

119894= (1198942) (119894 = 0 1 5) namely ℎ

119894= (1198942) (119894 =

0 4) According to the inequality EH120576119894lt 119867120576

119894 we may

get the range of the parameters 120582119894and 120583

119894 For the fixed

interpolation condition the max error and the parameters 120582119894

and 120583119894are given for every interval [119909

119894 119909119894+1] in Table 1 The

error curves of the EH(119909) and 119867(119909) to 119910(119909) are shown inFigure 3

By using the tensor productmethod we can construct theEH interpolation spline surfaces which has the similar EHinterpolation spline curve

Definition 5 Let Ω [119886 119887] times [119888 119889] be the plane regionand 119891(119909 119910) a bivariate function defined in the region Ω andlet 119886 = 119909


119898= 119887 and 119888 = 119910


119899=

119889 be the knot sequences Denote ℎ119894= 119909119894+1minus119909119894 ℎ119895= 119910119895+1minus119910119895

119906 = (119909 minus 119909119894)ℎ119894 V = (119910 minus 119910


spline surface on the region [119909119894 119909119894+1]times[119910119894 119910119894+1] can be defined

as follows

EH (119909 119910)1003816100381610038161003816

[119910119894 119910119894+1]

[119909119894 119909119894+1]

= (1198901205720 (119906) 1198901205721 (119906) 1198901205730 (119906) 1198901205731 (119906))119872(

1198901205720(V)

1198901205721(V)

1198901205730(V)

1198901205731(V)

)

(11)where

119872 =(

119891(119909119894 119910119894) 119891 (119909

119894 119910119894+1) ℎ

1198951198911015840

V (119909119894 119910119894) ℎ1198951198911015840

V (119909119894 119910119894+1)

119891 (119909119894+1 119910119894) 119891 (119909

119894+1 119910119894+1) ℎ

1198951198911015840

V (119909119894+1 119910119894) ℎ1198951198911015840

V (119909119894+1 119910119894+1)

ℎ1198941198911015840

119906(119909119894 119910119894) ℎ

1198941198911015840

119906(119909119894 119910119894+1) ℎ

119894ℎ11989511989110158401015840

119906V (119909119894 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894 119910119894+1)

ℎ1198941198911015840

119906(119909119894+1 119910119894) ℎ1198941198911015840

119906(119909119894+1 119910119894+1) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894+1)

) (12)

Given the end-points the first order partial derivative andthe second-order blending partial derivative of the functioninterpolated with proper parameters the EH interpolationspline surfaces could approximate to the bivariate functionsbeing interpolated better than the standard cubic Hermitespline surfaces

Example 6 Given the bivariate function being interpolated119891(119909 119910) = sin(1205872)119909 cos(1205872)119910 let 119886 = 0 lt 1 lt 2 = 119887 and 119888 =minus1 lt 0 lt 1 = 119889 be the knot sequences Denote ℎ

119894= 119909119894+1minus 119909119894

ℎ119895= 119910119895+1minus119910119895 119906 = (119909minus119909

119894)ℎ119894 and V = (119910minus119910

119895)ℎ119895 By selecting

1205820= 1205821= 03208 120583

0= 1205831= 06995 we can work out that

the max error of the EH(119909 119910) minus119891(119909 119910) equals 05069 times 10minus3but themax error of the119867(119909 119910)minus119891(119909 119910) equals 01061times 10minus1

Figure 4 shows the error surface of the EH(119909 119910)minus119891(119909 119910)Figure 5 shows the error surface of the 119867(119909 119910) minus 119891(119909 119910)

5 Conclusion

This paper introduced a kind of EH interpolation splinewhich is the extension of the standard cubic Hermite inter-polation spline The shape of the proposed splines can beadjusted by changing the values of the parameters for the


00 05 10 15 20 25

00000

00005

00010

H(x) minus y(x)

minus00005

EH(x) minus y(x)

Figure 3 The error curves of the EH(119909) minus 119910(119909) and119867(119909) minus 119910(119909)

0005

1015

20

00

0510

00000

00005

minus05

minus10

minus00005

Figure 4 The error surface of the EH(119909 119910) minus 119891(119909 119910)

0005

1015

20

00

0510

0000

0005

0010

0015

minus05

minus10

Figure 5 The error surface of the119867(119909 119910) minus 119891(119909 119910)

fixed interpolation conditions Also the introduced splinecould approximate to the interpolated function better thanthe standard cubic Hermite interpolation spline

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work was funded by the Natural Science Foundationof Anhui Province of China under Grant no 1208085MA15the Key Project Foundation of Scientific Research Edu-cation Department of Anhui Province under Grant noKJ2014ZD30 and the Key Construction Disciplines Founda-tion of Hefei University under Grant no 2014XK08

References

[1] S Butt and KW Brodlie ldquoPreserving positivity using piecewisecubic interpolationrdquoComputersampGraphics vol 17 no 1 pp 55ndash64 1993

[2] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquo Computers and Graphics vol 15 no 1 pp15ndash23 1991

[3] R L Dougherty A S Edelman and J M Hyman ldquoNon-negativity monotonicity or convexity-preserving cubic andquintic Hermite interpolationrdquo Mathematics of Computationvol 52 no 186 pp 471ndash494 1989

[4] R E Carlson and F N Fritsch ldquoMonotone piecewise cubicinterpolationrdquo SIAM Journal on Numerical Analysis vol 17 no2 pp 238ndash246 1980

[5] F N Fritsch and J Butland ldquoA method for constructinglocal monotone piecewise cubic interpolantsrdquo SIAM Journal onScientific and Statistical Computation vol 5 no 2 pp 300ndash3041984

[6] L L Schumaker ldquoOn shape preserving quadratic spline inter-polationrdquo SIAM Journal on Numerical Analysis vol 20 no 4pp 854ndash864 1983

[7] Q Duan Y Zhang L Wang and E H Twizell ldquoRegion controland approximation of a weighted rational interpolating curvesrdquoCommunications in Numerical Methods in Engineering vol 22no 1 pp 41ndash53 2006

[8] Q I Duan K DjidjeliW G Price and EH Twizell ldquoA rationalcubic spline based on function valuesrdquoComputers andGraphicsvol 22 no 4 pp 479ndash486 1998

[9] Q Duan K Djidjeli W G Price and E H Twizell ldquoTheapproximation properties of some rational cubic splinesrdquo Inter-national Journal of Computer Mathematics vol 72 no 2 pp155ndash166 1999

[10] M Sarfraz ldquoCubic spline curves with shape controlrdquoComputersand Graphics vol 18 no 5 pp 707ndash713 1994

[11] QDuan A K Liu and FH Cheng ldquoConstrained interpolationusing rational cubic spline with linear denominatorsrdquo TheKorean Journal of Computational amp Applied Mathematics AnInternational Journal vol 6 no 1 pp 203ndash215 1999

[12] M Z Hussain and M Sarfraz ldquoPositivity-preserving interpo-lation of positive data by rational cubicsrdquo Journal of Compu-tational and Applied Mathematics vol 218 no 2 pp 446ndash4582008

[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012


[14] F IbraheemM Hussain M Z Hussain and A A Bhatti ldquoPos-itive data visualization using trigonometric functionrdquo Journalof Applied Mathematics vol 2012 Article ID 247120 19 pages2012

[15] Q Duan H Zhang Y Zhang and E H Twizell ldquoErrorestimation of a kind of rational splinerdquo Journal of Computationaland Applied Mathematics vol 200 no 1 pp 1ndash11 2007

[16] M Tian and H L Geng ldquoError analysis of a rational interpola-tion splinerdquo International Journal of Mathematical Analysis vol5 no 25ndash28 pp 1287ndash1294 2011

[17] F Bao Q Sun J Pan and Q Duan ldquoPoint control of rationalinterpolating curves using parametersrdquoMathematical andCom-puter Modelling vol 52 no 1-2 pp 143ndash151 2010

[18] J Xie J Q Tan and S F Li ldquoRational cubic Hermite interpo-lating spline and its approximation propertiesrdquo Chinese Journalof Engineering Mathematics vol 28 no 3 pp 385ndash392 2010

[19] J Xie J Q Tan and S F Li ldquoA kind of rational cubic spline andits applicationsrdquo Acta Mathematicae Applicatae Sinica vol 23no 35 pp 847ndash855 2010

[20] C-Y Liu L Yang and J-C Li ldquoQuartic Hermite interpolatingsplines with parametersrdquo Journal of Computer Applications vol32 no 7 pp 1868ndash1870 2012

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Stochastic AnalysisInternational Journal of


approximate to the interpolated functions better than thestandard cubic Hermite interpolation splines and the quarticHermite interpolation splines with single parameter

The remainder of the paper is organized as followsSection 2 introduces the standard cubic Hermite spline andsome of its properties A kind of interpolation spline with twoparameters is presented in Section 3 Section 4 discusses theapproximation of the introduced spline curve with numericalexamples In the end a summary and conclusions are given

2 The Standard Cubic Hermite Splineand Its Basis Functions

Generally for 119905 isin [0 1] the following four basis functions

1205720 (119905) = 1 minus 3119905

2+ 21199053

1205721(119905) = 3119905

2minus 21199053

1205730 (119905) = 119905 minus 2119905

2+ 1199053

1205731(119905) = minus 119905

2+ 1199053

(1)

are called the standard cubic Hermite basesThese bases satisfy

1205720(0) = 120572

1(1) = 1 120572

0(1) = 120572

1(0) = 0

1205721015840

1(0) = 120572

1015840

1(1) = 0 120572

1015840

0(0) = 120572

1015840

0(1) = 0

1205730(0) = 120573

0(1) = 0 120573

1(0) = 120573

1(1) = 0

1205731015840

0(0) = 120573

1015840

1(1) = 1 120573

1015840

0(1) = 120573

1015840

1(0) = 0

1205720(119905) + 120572

1(119905) = 1 120573

0(119905) + 120573

1(1 minus 119905) = 0

(2)

For given knots 119886 = 1199090lt 1199091lt sdot sdot sdot lt 119909

119899= 119887 and data

(119909119894 119910119894 119889119894) 119894 = 0 1 119899 where 119910

119894and 119889

119894are the values of

the function value and the first-order derivative value of thefunction being interpolated let ℎ

119894= 119909119894+1minus 119909119894 119905 = (119909 minus 119909

119894)ℎ119894

and then the standard cubic Hermite spline in the interval[119909119894 119909119894+1] can be defined as follows

119867119894(119909) = 120572

0(119905) 119910119894+ 1205721(119905) 119910119894+1+ 1205730(119905) ℎ119894119889119894+ 1205731(119905) ℎ119894119889119894+1

119894 = 0 1 119899 minus 1

(3)

Obviously we have119867119894(119909119894) = 119910119894 119867119894(119909119894+1) = 119910119894+1 1198671015840

119894(119909119894) =

119889119894 1198671015840

119894(119909119894+1) = 119889119894+1

The standard cubic Hermite spline is C1 continuous

However if interpolation data is given the shape and approx-imation of the spline cannot be changed


In order to overcome the disadvantage of the standard cubicHermite spline we extend its basis functions firstly

31 The Basis Functions of the EH Interpolation Spline

Definition 1 For arbitrary real number 120582119894 120583119894and 0 le 119905 le 1

the following four functions with parameters 120582119894 120583119894

1198901205720(119905) = 1 + (120582

119894minus 3) 1199052+ 2 (1 minus 120582

119894) 1199053+ 1205821198941199054

1198901205721(119905) = (3 minus 120582

119894) 1199052+ 2 (120582

119894minus 1) 1199053minus 1205821198941199054

1198901205730(119905) = 119905 + (120583

119894minus 2) 1199052+ (1 minus 2120583

119894) 1199053+ 1205831198941199054

1198901205731(119905) = minus (120583

119894+ 1) 1199052+ (1 + 2120583

119894) 1199053minus 1205831198941199054

(4)

are calledbasis functions of the EH interpolation splinebriefly EH bases

The EH bases are the extension of the standard cubicHermite bases When 120582

119894= 120583119894= 0 the bases are the standard

cubic Hermite bases The bases have the similar properties ofthe standard cubic Hermite bases

By computing directly we have 1198901205720(0) = 119890120572

1(1) =

1 1198901205720(1) = 119890120572

1(0) = 0 119890120572

1015840

1(0) = 119890120572

1015840

1(1) = 0 119890120572

1015840

0(0) =

1198901205721015840

0(1) = 0 119890120573

0(0) = 119890120573

0(1) = 0 119890120573

1(0) = 119890120573

1(1) =

0 11989012057310158400(0) = 119890120573

1015840

1(1) = 1 119890120573

1015840

1(1) = 119890120573

1015840

1(0) = 0 and 119890120572

0(119905) +

1198901205721(119905) = 1 119890120573

0(119905) + 119890120573

1(1 minus 119905) = 0

When 120582119894= 120583119894 the EH bases (4) are basis functions with

single parameter in [20]Figure 1 shows the four EHbases where the solid lines are

the standard Hermite bases the parameters 120582119894= 2 120583

119894= minus2

are for the dot-dash lines and 120582119894= minus2 120583


lineSo we may construct the Ferguson curve with two pa-

rameters based on the EH bases as follows

EH119894(119905) = 119890120572

0(119905) 119901119894+ 1198901205721(119905) 119901119894+1+ 11989012057301199011015840

119894+ 1198901205731(119905) 1199011015840

119894+1 (5)

where 119901119894 119901119894+1

and 1199011015840119894 1199011015840

119894+1are two interpolation points and

their tangent vectors respectivelyFor the given two interpolation points and tangent

vectors with the different parameters 120582119894 120583119894 we may obtain

different shape of the Ferguson curve with two parametersaccordingly

Figure 2 shows the Ferguson curves with different param-eters where the solid line is the standard Ferguson splinecurve with 120582

119894= 120583119894= 0 the parameters 120582

119894= 2 120583

119894= minus2

are for the dot-dash line and 120582119894= minus2 120583


line


Definition 2 Given a data set (119909119894 119910119894 119889119894) 119894 = 0 1 119899

where 119910119894and 119889

119894are the values of the function value and the

first-order derivative value of the function being interpolatedand 119886 = 119909


119899= 119887 is the knot spacing let


00 02 04 06 08 1000

02

04

06

08

10

(a) 1198901205720(119905)

00 02 04 06 08 1000

02

04

06

08

10

(b) 1198901205721(119905)

00 02 04 06 08 10000

005

010

015

020

(c) 1198901205730(119905)

00 02 04 06 08 10

000

minus020

minus015

minus010

minus005

(d) 1198901205731(119905)


00 02 04 06 08 1000

01

02

03

04

05

p998400i

p998400i+1

pi pi+1


ℎ119894= 119909119894+1

minus 119909119894 119905 = (119909 minus 119909



EH (119909)|[119909119894 119909119894+1]

= 1199101198941198901205720 (119905) + 119910119894+11198901205721 (119905) + 119889119894ℎ1198941198901205730 (119905) + 119889119894+1ℎ1198941198901205731 (119905)

119894 = 0 1 119899 minus 1

(6)

where 1198901205720(119905) 119890120572

1(119905) 119890120573

0(119905) 119890120573




RH (119909119894) = 119910119894 RH1015840 (119909

119894) = 119889119894

119894 = 0 1 119899

(7)





119894=

(119910119894+1minus 119910119894)ℎ119894 and let

EH10158401015840 (119909119894+) = EH10158401015840 (119909

119894minus) 119894 = 1 2 119899 minus 1 (8)


119894


119894(2 minus 120583


= ℎ119894minus1[Δ119894(120582119894minus 3) + (119889

119894+1(1 + 120583

119894) + 119889119894(2 minus 120583

119894))]

119894 = 1 2 119899 minus 1

(9)






119889119894minus1+ 4119889119894+ 119889119894+1

= 3 (Δ119894minus1+ Δ119894) 119894 = 1 2 119899 minus 1 (10)










119909119894

119910119894

119889119894

120582119894

120583119894

119877119867120576119894

119867120576119894






25000 17929 21107






=

max119909119894lt119909lt119909119894+1

|EH119894(119909) minus 119910(119909)| 119867120576

119894= max

119909119894lt119909lt119909119894+1|119867119894(119909) minus

119910(119909)| then if EH120576119894lt 119867120576






119894 In the range


119894and 120583



119894= (1198942) (119894 = 0 1 5) namely ℎ

119894= (1198942) (119894 =


119894 we may





119894 119909119894+1] in Table 1 The





119898= 119887 and 119888 = 119910


119899=


119906 = (119909 minus 119909119894)ℎ119894 V = (119910 minus 119910



as follows

EH (119909 119910)1003816100381610038161003816

[119910119894 119910119894+1]

[119909119894 119909119894+1]

= (1198901205720 (119906) 1198901205721 (119906) 1198901205730 (119906) 1198901205731 (119906))119872(

1198901205720(V)

1198901205721(V)

1198901205730(V)

1198901205731(V)

)

(11)where

119872 =(

119891(119909119894 119910119894) 119891 (119909

119894 119910119894+1) ℎ

1198951198911015840

V (119909119894 119910119894) ℎ1198951198911015840

V (119909119894 119910119894+1)

119891 (119909119894+1 119910119894) 119891 (119909

119894+1 119910119894+1) ℎ

1198951198911015840

V (119909119894+1 119910119894) ℎ1198951198911015840

V (119909119894+1 119910119894+1)

ℎ1198941198911015840

119906(119909119894 119910119894) ℎ

1198941198911015840

119906(119909119894 119910119894+1) ℎ

119894ℎ11989511989110158401015840

119906V (119909119894 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894 119910119894+1)

ℎ1198941198911015840

119906(119909119894+1 119910119894) ℎ1198941198911015840

119906(119909119894+1 119910119894+1) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894+1)

) (12)



119894= 119909119894+1minus 119909119894

ℎ119895= 119910119895+1minus119910119895 119906 = (119909minus119909

119894)ℎ119894 and V = (119910minus119910


1205820= 1205821= 03208 120583




5 Conclusion



00 05 10 15 20 25

00000

00005

00010

H(x) minus y(x)

minus00005

EH(x) minus y(x)


0005

1015

20

00

0510

00000

00005

minus05

minus10

minus00005


0005

1015

20

00

0510

0000

0005

0010

0015

minus05

minus10





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 1000

02

04

06

08

10

(a) 1198901205720(119905)

00 02 04 06 08 1000

02

04

06

08

10

(b) 1198901205721(119905)

00 02 04 06 08 10000

005

010

015

020

(c) 1198901205730(119905)

00 02 04 06 08 10

000

minus020

minus015

minus010

minus005

(d) 1198901205731(119905)


00 02 04 06 08 1000

01

02

03

04

05

p998400i

p998400i+1

pi pi+1


ℎ119894= 119909119894+1

minus 119909119894 119905 = (119909 minus 119909



EH (119909)|[119909119894 119909119894+1]

= 1199101198941198901205720 (119905) + 119910119894+11198901205721 (119905) + 119889119894ℎ1198941198901205730 (119905) + 119889119894+1ℎ1198941198901205731 (119905)

119894 = 0 1 119899 minus 1

(6)

where 1198901205720(119905) 119890120572

1(119905) 119890120573

0(119905) 119890120573




RH (119909119894) = 119910119894 RH1015840 (119909

119894) = 119889119894

119894 = 0 1 119899

(7)





119894=

(119910119894+1minus 119910119894)ℎ119894 and let

EH10158401015840 (119909119894+) = EH10158401015840 (119909

119894minus) 119894 = 1 2 119899 minus 1 (8)


119894


119894(2 minus 120583


= ℎ119894minus1[Δ119894(120582119894minus 3) + (119889

119894+1(1 + 120583

119894) + 119889119894(2 minus 120583

119894))]

119894 = 1 2 119899 minus 1

(9)






119889119894minus1+ 4119889119894+ 119889119894+1

= 3 (Δ119894minus1+ Δ119894) 119894 = 1 2 119899 minus 1 (10)










119909119894

119910119894

119889119894

120582119894

120583119894

119877119867120576119894

119867120576119894






25000 17929 21107






=

max119909119894lt119909lt119909119894+1

|EH119894(119909) minus 119910(119909)| 119867120576

119894= max

119909119894lt119909lt119909119894+1|119867119894(119909) minus

119910(119909)| then if EH120576119894lt 119867120576






119894 In the range


119894and 120583



119894= (1198942) (119894 = 0 1 5) namely ℎ

119894= (1198942) (119894 =


119894 we may





119894 119909119894+1] in Table 1 The





119898= 119887 and 119888 = 119910


119899=


119906 = (119909 minus 119909119894)ℎ119894 V = (119910 minus 119910



as follows

EH (119909 119910)1003816100381610038161003816

[119910119894 119910119894+1]

[119909119894 119909119894+1]

= (1198901205720 (119906) 1198901205721 (119906) 1198901205730 (119906) 1198901205731 (119906))119872(

1198901205720(V)

1198901205721(V)

1198901205730(V)

1198901205731(V)

)

(11)where

119872 =(

119891(119909119894 119910119894) 119891 (119909

119894 119910119894+1) ℎ

1198951198911015840

V (119909119894 119910119894) ℎ1198951198911015840

V (119909119894 119910119894+1)

119891 (119909119894+1 119910119894) 119891 (119909

119894+1 119910119894+1) ℎ

1198951198911015840

V (119909119894+1 119910119894) ℎ1198951198911015840

V (119909119894+1 119910119894+1)

ℎ1198941198911015840

119906(119909119894 119910119894) ℎ

1198941198911015840

119906(119909119894 119910119894+1) ℎ

119894ℎ11989511989110158401015840

119906V (119909119894 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894 119910119894+1)

ℎ1198941198911015840

119906(119909119894+1 119910119894) ℎ1198941198911015840

119906(119909119894+1 119910119894+1) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894+1)

) (12)



119894= 119909119894+1minus 119909119894

ℎ119895= 119910119895+1minus119910119895 119906 = (119909minus119909

119894)ℎ119894 and V = (119910minus119910


1205820= 1205821= 03208 120583




5 Conclusion



00 05 10 15 20 25

00000

00005

00010

H(x) minus y(x)

minus00005

EH(x) minus y(x)


0005

1015

20

00

0510

00000

00005

minus05

minus10

minus00005


0005

1015

20

00

0510

0000

0005

0010

0015

minus05

minus10





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909119894

119910119894

119889119894

120582119894

120583119894

119877119867120576119894

119867120576119894






25000 17929 21107






=

max119909119894lt119909lt119909119894+1

|EH119894(119909) minus 119910(119909)| 119867120576

119894= max

119909119894lt119909lt119909119894+1|119867119894(119909) minus

119910(119909)| then if EH120576119894lt 119867120576






119894 In the range


119894and 120583



119894= (1198942) (119894 = 0 1 5) namely ℎ

119894= (1198942) (119894 =


119894 we may





119894 119909119894+1] in Table 1 The





119898= 119887 and 119888 = 119910


119899=


119906 = (119909 minus 119909119894)ℎ119894 V = (119910 minus 119910



as follows

EH (119909 119910)1003816100381610038161003816

[119910119894 119910119894+1]

[119909119894 119909119894+1]

= (1198901205720 (119906) 1198901205721 (119906) 1198901205730 (119906) 1198901205731 (119906))119872(

1198901205720(V)

1198901205721(V)

1198901205730(V)

1198901205731(V)

)

(11)where

119872 =(

119891(119909119894 119910119894) 119891 (119909

119894 119910119894+1) ℎ

1198951198911015840

V (119909119894 119910119894) ℎ1198951198911015840

V (119909119894 119910119894+1)

119891 (119909119894+1 119910119894) 119891 (119909

119894+1 119910119894+1) ℎ

1198951198911015840

V (119909119894+1 119910119894) ℎ1198951198911015840

V (119909119894+1 119910119894+1)

ℎ1198941198911015840

119906(119909119894 119910119894) ℎ

1198941198911015840

119906(119909119894 119910119894+1) ℎ

119894ℎ11989511989110158401015840

119906V (119909119894 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894 119910119894+1)

ℎ1198941198911015840

119906(119909119894+1 119910119894) ℎ1198941198911015840

119906(119909119894+1 119910119894+1) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894) ℎ119894ℎ11989511989110158401015840

119906V (119909119894+1 119910119894+1)

) (12)



119894= 119909119894+1minus 119909119894

ℎ119895= 119910119895+1minus119910119895 119906 = (119909minus119909

119894)ℎ119894 and V = (119910minus119910


1205820= 1205821= 03208 120583




5 Conclusion



00 05 10 15 20 25

00000

00005

00010

H(x) minus y(x)

minus00005

EH(x) minus y(x)


0005

1015

20

00

0510

00000

00005

minus05

minus10

minus00005


0005

1015

20

00

0510

0000

0005

0010

0015

minus05

minus10





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 05 10 15 20 25

00000

00005

00010

H(x) minus y(x)

minus00005

EH(x) minus y(x)


0005

1015

20

00

0510

00000

00005

minus05

minus10

minus00005


0005

1015

20

00

0510

0000

0005

0010

0015

minus05

minus10





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the eh interpolation spline and its...

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