Least Square Plane and Leastsquare Quadric Surface Approximation by Using Modified Lagrange’s Method

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 77-79 www.iosrjournals.org

Least Square Plane and Leastsquare Quadric Surface Approximation by Using Modified Lagrange’s Method C.V.Rao, GS Department Assistant professor, JubailIndustrial College, Jubail, KSA.

Abstract: Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.

Keywords: Least square, quadric surface , Normal equations

I. Introduction Least square principle method is one the best approximation methodin numerical analysis for line and curve fitting, this method was invented by Lagrange's. A function đ?‘Ś = đ?‘“(đ?‘Ľ)may be given in discretedata đ?‘Ľđ?‘˜ , đ?‘Śđ?‘˜ .The best approximation in the least square is defined as that for which the constantsđ?‘?đ?‘– , đ?‘– = 0,1,2,3 ‌ đ?‘›are determined so that the aggregate of đ?‘¤ đ?‘Ľ đ??¸2over given domain D is as small as possible, where đ?‘¤ đ?‘Ľ > 0 $ is the weight function for the function whose values are given at đ?‘ + 1 pointsđ?‘Ľ0 , đ?‘Ľ1 ‌ đ?‘Ľđ?‘ .

We have đ??ź đ?‘?0 , đ?‘?1 ‌ đ?‘?đ?‘ =

đ?‘ đ?‘˜=đ?‘œ

đ?‘› đ?‘–=0 ∅đ?‘–

đ?‘¤( đ?‘Ľđ?‘˜ ) đ?‘“ đ?‘Ľđ?‘˜ −

đ?‘Ľđ?‘˜

2

= �������------------(1)

Where ∅đ?‘– đ?‘Ľ = đ?‘Ľ đ?‘– , đ?‘– = 0,1,2,3 ‌ đ?‘› and đ?‘¤ đ?‘Ľ = 1 The necessary conditions for (1) to have a minimum value is that đ?œ•đ??ź = 0 , đ?‘– = 0,1,2,3 ‌ đ?‘› . đ?œ•đ?‘? đ?‘–

This gives a system of đ?‘› + 1linear equations in đ?‘› + 1constants. These equations are called normal equations.

Then we get approximated nth degree polynomial function of đ?‘Ľ.

1.Least square plane: Given a discrete data đ?‘Ľđ?‘˜ , đ?‘Śđ?‘˜ , đ?‘˜ = 0,1, ‌ đ?‘ , Consider∅ đ?‘Ľ, đ?‘Ś = đ?‘?0 + đ?‘?1 đ?‘Ľ + đ?‘?2 đ?‘Ś Such that đ??ź đ?‘?0 , đ?‘?1 , đ?‘?2 = đ?‘ đ?‘˜=đ?‘œ đ?‘§đ?‘˜ − đ?‘?0 + đ?‘?1 đ?‘Ľđ?‘˜ + đ?‘?2 đ?‘Śđ?‘˜

Then Normal equations

đ?œ•đ??ź đ?œ•đ?‘? đ?‘–

2

đ?‘ ≼ 2.

= �������

= 0 , đ?‘– = 0,1,2 .

ie., đ?‘?0 đ?‘ + 1 + đ?‘?1 đ?‘?0 đ?‘?0

đ?‘Śđ?‘˜ + đ?‘?1

đ?‘Ľđ?‘˜ đ?‘Śđ?‘˜ + đ?‘?2

đ?‘Śđ?‘˜ 2 =

đ?‘Ľđ?‘˜ + đ?‘?1

đ?‘Ľđ?‘˜ + đ?‘?2 đ?‘Ľđ?‘˜ 2 + đ?‘?2

đ?‘Śđ?‘˜ = đ?‘Ľđ?‘˜ đ?‘Śđ?‘˜ =

đ?‘§đ?‘˜ đ?‘Ľđ?‘˜ đ?‘§đ?‘˜

đ?‘Śđ?‘˜ đ?‘§đ?‘˜

Solving this system of equations we getđ?‘?0 , đ?‘?1 đ?‘Žđ?‘›đ?‘‘ đ?‘?2 .

Example 1: Suppose that given data points −1, 1, −2 , 1, 2 , 3 , 1, 1 , 2 that lie on đ?‘§ = đ?‘“(đ?‘Ľ , đ?‘Ś).These points define uniquely a linear function in two variables, sođ?‘§đ?‘˜ = đ?‘?0 + đ?‘?1 đ?‘Ľđ?‘˜ + đ?‘?2 đ?‘Śđ?‘˜ , đ?‘˜ = 0,1,2 DOI: 10.9790/5728-1301047779

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Least Square Plane and Leastsquare Quadric Surface Approximation by Using Modified Lagrange’s The coefficients satisfy the normal equations 3𝑐0 + 1𝑐1 + 4𝑐2 = 3 𝑐0 + 3 𝑐1 + 2𝑐2 = 7 4𝑐0 + 2𝑐1 + 6𝑐2 = 6 Solving these equations we get𝑐0 = −1 , 𝑐1 = 2 , 𝑐2 = 1 Thus 𝑧 = −1 + 2𝑥 + 𝑦 .

II. Least square quadric surface Given a discrete data 𝑥𝑘 , 𝑦𝑘 , 𝑘 = 0,1, … 𝑁 , 𝑁 ≥ 5 Consider ∅ 𝑥, 𝑦 = 𝑐0 + 𝑐1 𝑥 + 𝑐2 𝑦+ 𝑐3 𝑥 2 + 𝑐4 𝑦 2 + 𝑐5 𝑥𝑦 Such that 𝑁

𝑧𝑘 − 𝑐0 + 𝑐1 𝑥𝑘 + 𝑐2 𝑦𝑘 + 𝑐3 𝑥𝑘 2 + 𝑐4 𝑦𝑘 2 + 𝑐5 𝑥𝑘 𝑦𝑘

𝐼 𝑐0 , 𝑐1 , 𝑐2 , 𝑐3 , 𝑐4 , 𝑐5 =

2

= 𝑚𝑖𝑛𝑖𝑚𝑢𝑚

𝑘=𝑜

The coefficients satisfy normal equations 𝑐0 𝑁 + 1 + 𝑐1

𝜕𝐼 𝜕𝑐 𝑖

𝑥𝑘 + 𝑐2

= 0 , 𝑖 = 0,1,2,3,4,5 .

𝑐0

𝑥𝑘 + 𝑐1

𝑥𝑘 2 + 𝑐2

𝑐0

𝑦𝑘 + 𝑐1

𝑥𝑘 𝑦𝑘 + 𝑐2

𝑦𝑘 2 + 𝑐3

𝑥𝑘 2 𝑦𝑘 + 𝑐4

𝑐0

𝑥𝑘 2 + 𝑐1

𝑥𝑘 3 + 𝑐2

𝑥𝑘 2 𝑦𝑘 + 𝑐3

𝑥𝑘 4 + 𝑐4

𝑐0

𝑦𝑘 2 + 𝑐1

𝑥𝑘 𝑦𝑘 2 + 𝑐2

𝑐0

𝑥𝑘 𝑦𝑘 + 𝑐1

𝑥𝑘 𝑦𝑘 + 𝑐3

𝑦𝑘 3 + 𝑐3

𝑥𝑘 2 𝑦𝑘 + 𝑐2

𝑥𝑘 3 + 𝑐4

2

𝑥𝑘 𝑦𝑘 2 + 𝑐5

2

𝑥𝑘 2 𝑦𝑘 + 𝑐5 𝑦𝑘 4 + 𝑐5

𝑥𝑘 3 𝑦𝑘 + 𝑐4

𝑦𝑘 2 + 𝑐3

𝑥𝑘 2 =

𝑦𝑘 3 + 𝑐5

𝑥𝑘 2 𝑦𝑘 + 𝑐4

𝑥𝑘 𝑦𝑘 2 + 𝑐3

𝑥𝑘 2 + 𝑐4

𝑦𝑘 + 𝑐3

𝑧𝑘

𝑥𝑘 𝑧𝑘

𝑥𝑘 𝑦𝑘 2 =

𝑦𝑘 𝑧𝑘

𝑥𝑘 3 𝑦𝑘 = 𝑥𝑘 𝑦𝑘 3 =

𝑥𝑘 𝑦𝑘 3 + 𝑐5

𝑥𝑘 𝑦𝑘 =

𝑥𝑘 2 𝑧𝑘 𝑦𝑘 2 𝑧𝑘 2

𝑥𝑘 2 𝑦𝑘 =

𝑥𝑘 𝑦𝑘 𝑧𝑘

Solve these equations we get 𝑐0 , 𝑐1 , 𝑐2 , 𝑐3 , 𝑐4 , 𝑐5 .

Example 2: Suppose that given data points 0 , 0, 0 , 0 , 1 , −4 , 1, −1 , 1 DOI: 10.9790/5728-1301047779

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Least Square Plane and Leastsquare Quadric Surface Approximation by Using Modified Lagrange’s 1, 2, −2 , 2, 1 , 4 , 1, 3 , −3 those lie on đ?‘§ = đ?‘“(đ?‘Ľ , đ?‘Ś) .These data points satisfy uniquely a degree of two variable function. đ?‘§đ?‘˜ = đ?‘?0 + đ?‘?1 đ?‘Ľđ?‘˜ + đ?‘?2 đ?‘Ľđ?‘˜ + đ?‘?3 đ?‘Ľđ?‘˜ 2 + đ?‘?4 đ?‘Śđ?‘˜ 2 + đ?‘?5 đ?‘Ľđ?‘˜ đ?‘Śđ?‘˜ , đ?‘˜ = 0,1,2,3,4,5 The coefficients đ?‘?0 , đ?‘?1 , đ?‘?2 , đ?‘?3 , đ?‘?4 , đ?‘?5 satisfy the normal equations 6đ?‘?0 + 5đ?‘?1 + 6đ?‘?2 + 7đ?‘?3 + 16đ?‘?4 + 6đ?‘?5 = −4 5đ?‘?0 + 7đ?‘?1 + 6đ?‘?2 + 11đ?‘?3 + 16đ?‘?4 + 8đ?‘?5 = 4 6đ?‘?0 + 6đ?‘?1 + 16đ?‘?2 + 8đ?‘?3 + 36đ?‘?4 + 16đ?‘?5 = −14 7đ?‘?0 + 11đ?‘?1 + 8đ?‘?2 + 19đ?‘?3 + 18đ?‘?4 + 12đ?‘?5 = 12 16đ?‘?0 + 16đ?‘?1 + 36đ?‘?2 + 18đ?‘?3 + 100đ?‘?4 + 36đ?‘?5 = −34 6đ?‘?0 + 8đ?‘?1 + 16đ?‘?2 + 12đ?‘?3 + 36đ?‘?4 + 18đ?‘?5 = −6 Solving this system of equations we get đ?‘?0 = 0 , đ?‘?1 = −1 , đ?‘?2 = −4 , đ?‘?3 = 1 , đ?‘?4 = 0 , đ?‘?5 = 3 Thus đ?‘§ = − đ?‘Ľ − 4đ?‘Ś+ đ?‘Ľ 2 + 3đ?‘Ľđ?‘Ś

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DOI: 10.9790/5728-1301047779

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