Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform

World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:13, No:8, 2019 Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves Taweechai Nuntawisuttiwong, Natasha Dejdumrong Abstract—Newton-Lagrange Interpolations are widely used in quadratic computation time, which is slower than the others. numerical analysis. However, it requires a quadratic computational However, there are some curves in CAGD, Wang-Ball, DP time for their constructions. In computer aided geometric design and Dejdumrong curves with linear complexity. In order to (CAGD), there are some polynomial curves: Wang-Ball, DP and reduce their computational time, Be´zier curve is converted into any of Wang-Ball, DP and Dejdumrong curves. Hence, Dejdumrong curves, which have linear time complexity algorithms. Thus, the computational time for Newton-Lagrange Interpolations can be reduced by applying the algorithms of Wang-Ball, DP and the computational time for Newton-Lagrange interpolations Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong can be reduced by converting them into Wang-Ball, DP and algorithms, first, it is necessary to convert Newton-Lagrange Dejdumrong algorithms. Thus, it is necessary to investigate polynomials into Wang-Ball, DP or Dejdumrong polynomials. In the conversion from Newton-Lagrange interpolation into the curves with linear complexity, Wang-Ball, DP and this work, the algorithms for converting from both uniform and non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and Dejdumrong polynomials are investigated. Thus, the computational Dejdumrong curves. An application of this work is to modify time for representing Newton-Lagrange polynomials can be reduced sketched image in CAD application with the computational into linear complexity. In addition, the other utilizations of using time reduction for plotting Newton-Lagrange curves. CAGD curves to modify the Newton-Lagrange curves can be taken. Keywords—Newton interpolation, Lagrange interpolation, linear II. NEWTON & LAGRANGE POLYNOMIALS complexity. Although Newton and Lagrange polynomials are different representations, their simple forms are the same polynomials. I. INTRODUCTION Newton applies Divided Difference method to represent its HERE are many interpolation methods used in numerical polynomial while Lagrange uses simpler formula. At the analysis. The popular one is Newton’s divided-difference Tbeginning, it is important to introduce both methods. Newton interpolating polynomial. The Newton’s method was given by and Lagrange polynomials can be shown as follows: Isaac Newton written in Lemma 5 of Book III of his Principia Mathematica of 1687 [1] but it was known to him before A. Newton Polynomial from the letter he wrote to Oldenburg [2]. The simple forms Let N(t) be the Newton polynomial of control points, of Newton interpolating polynomials, Lagrange interpolations, <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul><ul style="display: flex;"><li style="flex:1">{bi}i=0, and the weight, {ti}i=0, then </li><li style="flex:1">were published by Joseph-Louis Lagrange under his name. </li></ul>However, the same formula had been produced by Waring sixteen years earlier. <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">i</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>N(t) = f[ti, ti−1, .., t0] (t − tj), tꢀ[t0, tn] (1) Newton-Lagrange interpolations can be beneficial to estimate intermediate data between precise data points or to predict the future data. In addition, Newton-Lagrange interpolations can be used to vectorize a raster-image into a vector image. However, a quadratic computational time, O(n2), must be taken to construct a Newton-Lagrange interpolation. In computer aided geometric design (CAGD), there are many polynomial curves, e.g. Be´zier [3], Said-Ball [4], Wang-Ball [5], DP [6], NB1 [7], [8] and Dejdumrong [9] curves. The difference between Newton-Lagrange and CAGD curves is the utilization of CAGD curves that is for CAD/CAM modeling. In CAGD curves, Be´zier curves are widely used in CAD software because they are better in shape preserving and possess simple forms. Unfortunately, Be´zier curve has a <ul style="display: flex;"><li style="flex:1">i=0 </li><li style="flex:1">j=0 </li></ul>where f[ti, ti−1, .., tk+1] − f[ti−1, ti−2, .., tk] <ul style="display: flex;"><li style="flex:1">f[ti, ti−1, .., tk] = </li><li style="flex:1">,</li></ul>ti − tk (2) and f[ti] = bi. (3) B. Lagrange Polynomial Let L(t) be the Lagrange polynomial of control points, <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>{vi}i=0, and the weight, {ti}i=0, then <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>t − tj L(t) = vi , tꢀ[t0, tn]. (4) ti − tj i=0 <ul style="display: flex;"><li style="flex:1">j=0,j </li><li style="flex:1">ꢀ=i </li></ul>T. Nuntawisuttiwong and N. Dejdumrong are with the Department of Computer Engineering, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand (e-mail: [email protected], [email protected]). In CAGD curves, there exists the power basis form for representing curves. In this work, the power basis <ul style="display: flex;"><li style="flex:1">International Scholarly and Scientific Research & Innovation 13(8) 2019 </li><li style="flex:1">448 </li><li style="flex:1">ISNI:0000000091950263 </li></ul>World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:13, No:8, 2019 representations for Newton and Lagrange polynomial are O(n). Thus, the computational time can also be reduced by investigated. Thus, Newton and Lagrange polynomial can be using the conversion from Newton-Lagrange polynomials into <ul style="display: flex;"><li style="flex:1">computed by the following definition. </li><li style="flex:1">Wang-Ball, DP or Dejdumrong curves. </li></ul>Definition 1. Newton-Lagrange Power Basis: Let N(t) and L(t) be Newton and Lagrange polynomial with their control III. CONVERSION FROM UNIFORM NEWTON-LAGRANGE POLYNOMIALS INTO CAGD CURVE npoints, {bi}i=0. Newton and Lagrange polynomial can be represented by using power basis as follows: <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>A. Conversion from Uniform Newton-Lagrange Curve into Wang-Ball Curve ꢀ ꢀ N(t) = L(t) = bi · fi,j · ti, (5) (6) i=0 j=0 nWang-Ball control points, {pi}i=0, can be obtained from where fi,j is Newton-Lagrange power basis defined by nNewton control points, {bi}i=0, and Lagrange control points, nn{vi}i=0, as follows: Fi,n−j(0) ꢂfi,j =ni=0,i =j tj − ti <ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢄ</li></ul>ꢀp0 p1 · · · pn =b0 b1 · · · bn · A−1 and (12) ⎧⎨and p0 p1 · · · pn 1,,for j = 0, nFi,n−j(u) = <ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢄ</li></ul>ꢃ=v0 v1 · · · vn ·A−1 (13) ⎩n−j+1 n=i Fi,j−1(k + 1)tk for j > 0. (7) k=u,k ꢀwhere <ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>Definition 2. Newton-Lagrange Monomial Matrices: Newton and Lagrange polynomials can be represented by using monomial matrix forms as follows: a0,0 a0,1 . . . a0,n a1,0 a1,1 . . . a1,n ⎢⎢⎢⎣ ⎥⎥⎥⎦A = (14) ... ... ... . .<ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>.1tan,0 an,1 . . . an,n ⎢⎢⎢⎣ ⎥⎥⎥⎦ ꢄN(t) = L(t) = b0 b1 · · · bn · M · (8) ...and tn ꢅ ꢆ ⎧⎪n2(2tj)i(1 − tj)i+2 ,,,,for 0 ≤ i ≤ nfor i = 2− 1 ⎪⎪where M is Newton-Lagrange monomial matrix defined as ⎪⎪⎪⎪ ꢅ ꢆ in−i follows: ⎪ (2tj) (1 − tj) <ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>⎪⎨f0,0 f0,1 . . . f0,n f1,0 f1,1 . . . f1,n ani,j(t) = ꢇ ꢈ ⎢⎢⎢⎣ ⎥⎥⎥⎦nfor i = 2(2(1 − tj))n−i(tj)i ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩M = .(9) <ul style="display: flex;"><li style="flex:1">.</li><li style="flex:1">.</li><li style="flex:1">.</li><li style="flex:1">.</li></ul><ul style="display: flex;"><li style="flex:1">.</li><li style="flex:1">.</li><li style="flex:1">.</li><li style="flex:1">.</li></ul>.ꢇ ꢈ <ul style="display: flex;"><li style="flex:1">.</li><li style="flex:1">.</li><li style="flex:1">.</li></ul>nann−i,j(1 − tj) for + 1 ≤ i ≤ n 2fn,0 fn,1 . . . fn,n <ul style="display: flex;"><li style="flex:1">(15) </li><li style="flex:1">The matrix inverse of Newton-Lagrange monomial matrix </li></ul>can be expressed as follows: <ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>−1 f0,0 f0,1 . . . f0,n f1,0 f1,1 . . . f1,n B. Conversion from Uniform Newton-Lagrange Curve into DP Curve ⎢⎢⎢⎣ ⎥⎥⎥⎦M−1 ==... ... ... . . .nDP control points, {qi}i=0, can be obtained from Newton fn,0 fn,1 . . . fn,n <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul><ul style="display: flex;"><li style="flex:1">control points, {bi}i=0, and Lagrange control points, {vi}i=0 </li><li style="flex:1">,</li></ul>(10) <ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>as follows: g0,0 g0,1 . . . g0,n g1,0 g1,1 . . . g1,n ⎢⎢⎢⎣ ⎥⎥⎥⎦ ꢄꢄ ꢄq0 q1 · · · qn ==b0 b1 · · · bn ·C−1 (16) ,... ... ... . . .gn,0 gn,1 . . . gn,n and ꢄwhere ⎧q0 q1 · · · qn v0 v1 · · · vn ·C−1 (17) 1,,for i = 0, ⎨⎩gi,j =(11) tji for i > 0. where <ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>c0,0 c0,1 . . . c0,n c1,0 c1,1 . . . c1,n ... Newton and Lagrange polynomial are very useful for polynomial interpolation. However, they take a lot of computational time, O(n2). In CAGD curves, Wang-Ball, DP and Dejdumrong curves consume linear computational time, ⎢⎢⎢⎣ ⎥⎥⎥⎦C = (18) ... ... . . .cn,0 cn,1 . . . cn,n <ul style="display: flex;"><li style="flex:1">International Scholarly and Scientific Research & Innovation 13(8) 2019 </li><li style="flex:1">449 </li><li style="flex:1">ISNI:0000000091950263 </li></ul>World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:13, No:8, 2019 and IV. CONVERSION FROM NONUNIFORM NEWTON-LAGRANGE INTO CAGD CURVES ⎧(−1)n−i(1 − tj)n (−1)n−i(1 − tj)n−itj , for i = 0 ⎪⎪⎪⎪A. Conversion from Nonuniform Newton-Lagrange curve into Nonuniform Wang-Ball curve ⎪⎪⎪⎪⎪⎪, for 0 ꢅ ꢆ n⎪⎪⎪⎪− 1 2nWang-Ball control points, {pi}i=0, can be transformed from ⎪⎪⎪⎪nNewton control points, {bi}i=0, and Lagrange control points, ⎪ (−1)n−i(n − 2i)(1 − tj)i+1tj+ ⎪⎪n⎪n2{vi}i=0, as follows: <ul style="display: flex;"><li style="flex:1">⎪</li><li style="flex:1">ꢅ ꢆ </li></ul>n2n2+1 ꢂ ꢃ ⎪⎪⎪⎪⎪⎪⎪⎨−n1n, for i = 2ꢀ ꢁ ꢂ ꢃ <ul style="display: flex;"><li style="flex:1">(2 ) </li><li style="flex:1">(1 − tj </li><li style="flex:1">+</li></ul>ꢄꢄ ꢄb0 b1 · · · bn ·A−1 (24) n+1 <ul style="display: flex;"><li style="flex:1">ꢂ ꢃ </li><li style="flex:1">ꢂ ꢃ </li></ul>p0 p1 · · · pn ==<ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><ul style="display: flex;"><li style="flex:1">(−1) </li><li style="flex:1">(1 − tj) </li><li style="flex:1">)</li></ul>cni,j(t) = and ⎪tj −a <ul style="display: flex;"><li style="flex:1">(n − 2i)(tb−−ab )( </li><li style="flex:1">)</li><li style="flex:1">+</li></ul>+1 jn−i+1 ⎪⎪⎪⎪⎪⎪⎪⎪b−a ꢄ ꢇ ꢈ n2<ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>2tj −a v0 v1 · · · vn ·A−1 (25) −1n, for i = 2<ul style="display: flex;"><li style="flex:1">ꢀ ꢁ ꢂ ꢃ </li><li style="flex:1">ꢂ ꢃ </li></ul>2p0 p1 · · · pn (2 ) (1 − ( b−a <ul style="display: flex;"><li style="flex:1">)</li><li style="flex:1">+</li></ul>⎪⎪⎪⎪<ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>+1 ꢂ ꢃ 2ꢂ ꢃ 2<ul style="display: flex;"><li style="flex:1">(−1) </li><li style="flex:1">(1 − tj) </li><li style="flex:1">)</li></ul>⎪⎪⎪where <ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul>⎪⎪⎪ ꢇ ꢈ a0,0 a0,1 . . . a0,n a1,0 a1,1 . . . a1,n ...n2<ul style="display: flex;"><li style="flex:1">(−1)(1 − tj)tij </li><li style="flex:1">, for </li><li style="flex:1">+ 1 </li></ul>⎪⎪<ul style="display: flex;"><li style="flex:1">⎪</li><li style="flex:1">⎢</li></ul>⎢⎢⎣ ⎥⎥⎥⎦ ⎪⎪⎪⎪⎪⎪⎪

Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform

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