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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

T

beginning, 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,

  • n
  • n

  • {bi}i=0, and the weight, {ti}i=0, then
  • were published by Joseph-Louis Lagrange under his name.

However, the same formula had been produced by Waring sixteen years earlier.

  • n
  • i

  • ꢀ
  • ꢁ

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

  • i=0
  • j=0

where

f[ti, ti−1, .., tk+1] − f[ti−1, ti−2, .., tk]

  • f[ti, ti−1, .., tk] =
  • ,

ti − tk

(2) and

f[ti] = bi.

(3)

B. Lagrange Polynomial

Let L(t) be the Lagrange polynomial of control points,

  • n
  • n

{vi}i=0, and the weight, {ti}i=0, then

  • n
  • n

  • ꢀ
  • ꢁ

t − tj

L(t) =

vi

, tꢀ[t0, tn].

(4)

ti − tj

i=0

  • j=0,j
  • ꢀ=i

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

  • International Scholarly and Scientific Research & Innovation 13(8) 2019
  • 448
  • ISNI:0000000091950263

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

  • computed by the following definition.
  • Wang-Ball, DP or Dejdumrong curves.

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

n

points, {bi}i=0. Newton and Lagrange polynomial can be represented by using power basis as follows:

  • n
  • n

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

n

Wang-Ball control points, {pi}i=0, can be obtained from

where fi,j is Newton-Lagrange power basis defined by

n

Newton control points, {bi}i=0, and Lagrange control points,

nn

{vi}i=0, as follows:

F

i,n−j(0)

ꢂ

fi,j

=

ni=0,i

=j tj − ti

  • ꢄ
  • ꢄ

ꢀ

p0 p1 · · · pn

=

b0 b1 · · · bn · A−1

and

(12)

⎧⎨

and

p0 p1 · · · pn

1

,,

for j = 0,

n

F

i,n−j(u) =

  • ꢄ
  • ꢄ

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

  • ⎡
  • ⎤

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)
...
...
...
.
.

  • ⎡
  • ⎤

.

1

t

an,0 an,1 . . . an,n

⎢⎢⎢⎣
⎥⎥⎥⎦
ꢄ

N(t) = L(t) = b0 b1 · · · bn · M ·

(8)

...

and

tn

ꢅ ꢆ
⎧

⎪

n

2

(2tj)i(1 − tj)i+2

,,,,

for 0 ≤ i ≤

n

for i =

2

− 1

⎪⎪

where M is Newton-Lagrange monomial matrix defined as

⎪⎪⎪⎪
ꢅ ꢆ

in−i

follows:

⎪ (2tj) (1 − tj)

  • ⎡
  • ⎤

⎪⎨

f0,0 f0,1 . . . f0,n f1,0 f1,1 . . . f1,n

ani,j(t) =

ꢇ ꢈ
⎢⎢⎢⎣
⎥⎥⎥⎦

n

for i =

2

(2(1 − tj))n−i(tj)i

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

M =

.

(9)

  • .
  • .
  • .
  • .

  • .
  • .
  • .
  • .

.

ꢇ ꢈ

  • .
  • .
  • .

n

ann−i,j(1 − tj)

for
+ 1 ≤ i ≤ n

2

fn,0 fn,1 . . . fn,n

  • (15)
  • The matrix inverse of Newton-Lagrange monomial matrix

can be expressed as follows:

  • ⎡
  • ⎤

−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

==

...
...
...
.
.
.

n

DP control points, {qi}i=0, can be obtained from Newton

fn,0 fn,1 . . . fn,n

  • n
  • n

  • control points, {bi}i=0, and Lagrange control points, {vi}i=0
  • ,

(10)

  • ⎡
  • ⎤

as follows:

g0,0 g0,1 . . . g0,n g1,0 g1,1 . . . g1,n

⎢⎢⎢⎣
⎥⎥⎥⎦
ꢄꢄ
ꢄ

q0 q1 · · · qn

==

b0 b1 · · · bn ·C−1 (16)

,

...
...
...
.
.
.

gn,0 gn,1 . . . gn,n

and

ꢄ

where

⎧

q0 q1 · · · qn v0 v1 · · · vn ·C−1 (17)

1

,,

for i = 0,

⎨⎩

gi,j

=

(11)

tji

for i > 0.

where

  • ⎡
  • ⎤

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

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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 < i ≤

ꢅ ꢆ

n

⎪⎪⎪⎪

− 1

2

n

Wang-Ball control points, {pi}i=0, can be transformed from

⎪⎪⎪⎪

n

Newton control points, {bi}i=0, and Lagrange control points,

⎪ (−1)n−i(n − 2i)(1 − tj)i+1tj+

⎪⎪

n

⎪

n

2

{vi}i=0, as follows:

  • ⎪
  • ꢅ ꢆ

n

2

n

2

+1

ꢂ ꢃ

⎪⎪⎪⎪⎪⎪⎪⎨

−

n

1

n

, for i =

2

ꢀ ꢁ ꢂ ꢃ

  • (2 )
  • (1 − tj
  • +

ꢄꢄ
ꢄ

b0 b1 · · · bn ·A−1 (24)

n

+1

  • ꢂ ꢃ
  • ꢂ ꢃ

p0 p1 · · · pn

==

  • 2
  • 2

  • (−1)
  • (1 − tj)
  • )

cni,j(t) =

and

⎪

tj −a

  • (n − 2i)(tb−−ab )(
  • )
  • +

+1

j

n−i+1

⎪⎪⎪⎪⎪⎪⎪⎪

b−a

ꢄ
ꢇ ꢈ

n

2

  • n
  • n

2

tj −a

v0 v1 · · · vn ·A−1 (25)

−

1

n

, for i =

2

  • ꢀ ꢁ ꢂ ꢃ
  • ꢂ ꢃ

2

p0 p1 · · · pn

(2 )
(1 − ( b−a

  • )
  • +

⎪⎪⎪⎪

  • n
  • n

+1

ꢂ ꢃ

2

ꢂ ꢃ

2

  • (−1)
  • (1 − tj)
  • )

⎪⎪⎪

where

  • ⎡
  • ⎤

⎪⎪⎪
ꢇ ꢈ

a0,0 a0,1 . . . a0,n a1,0 a1,1 . . . a1,n

...

n

2

  • (−1)(1 − tj)tij
  • , for
  • + 1

⎪⎪

  • ⎪
  • ⎢

⎢⎢⎣
⎥⎥⎥⎦
⎪⎪⎪⎪⎪⎪⎪

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    Mathematics for IT Engineers Pál Burai Copyright c 2019 Pál Burai UNIVERSITY OF DEBRECEN WWW.UNIDEB.HU Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (the “License”). You may not use this file except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. This work was supported by the construction EFOP-3.4.3-16-2016-00021. The project was sup- ported by the European Union, co-financed by the European Social Fund. Contents 1 Complex numbers .............................................5 1.1 A possible motivation introducing complex numbers5 1.2 Operations with complex numbers in algebraic form6 1.2.1 Algebraic form, real part, imaginary part..............................6 1.2.2 Graphical representation of complex numbers on the complex plane.......7 1.2.3 Addition of complex numbers in algebraic form........................8 1.2.4 Multiplication of complex numbers in algebraic form.....................9 1.2.5 Conjugate of complex numbers.................................... 12 1.3 Polar form and exponential form of complex numbers 13 1.3.1 Length and angle of complex numbers.............................. 13 1.3.2 Calculation with complex numbers in polar form....................... 18 1.3.3 Euler’s formula.................................................. 26 1.3.4 Calculation with complex numbers in exponential form................
  • Math 375: Lecture Notes

    Math 375: Lecture Notes

    Math 375: Lecture notes Professor Monika Nitsche September 21, 2011 Contents 1 MATLAB Basics 8 1.1 Example:Plottingafunction . ... 8 1.2 Scripts....................................... 9 1.3 Settingupvectors................................ 10 1.4 VectorandMatrixoperations . 11 1.5 Plotting ...................................... 14 1.6 Printingdata ................................... 15 1.7 Forloops...................................... 15 1.8 Whileloops .................................... 16 1.9 Timingcode.................................... 17 1.10Functions ..................................... 17 2 Computing Fundamentals 20 2.1 Vectorizing, timing, operation counts, memory allocation........... 20 2.1.1 Vectorizing for legibility and speed . 20 1 2.1.2 Memoryallocation ............................ 21 2.1.3 Counting operations: Horner’s algorithm . 21 2.1.4 Counting operations: evaluating series . 22 2.2 Machine Representation of real numbers, Roundoff errors . ......... 23 2.2.1 Decimal and binary representation of reals . ..... 23 2.2.2 Floating point representation of reals . ..... 24 2.2.3 Machine precision, IEEE rounding, roundoff error . ...... 25 2.2.4 Loss of significant digits during subtraction . 27 2.3 Approximating derivatives, Taylor’s Theorem, plotting y = hp ........ 29 2.4 Big-ONotation .................................. 30 3 Solving nonlinear equations f(x) = 0 31 3.1 Bisection method to solve f(x)=0( 1.1).................... 32 § 3.2 Fixed point iteration to solve x = g(x) (FPI, 1.2)............... 34 § 3.2.1 Examples ................................. 34 3.2.2 TheFPI.................................. 34 3.2.3 ImplementingFPIinMatlab. 35 3.2.4 TheoreticalResults ............................ 35 3.2.5 Definitions................................. 37 3.2.6 Stoppingcriterion. 37 3.3 Newton’s method to solve f(x)=0( 1.4) ................... 38 § 3.3.1 Thealgorithm............................... 38 3.3.2 Matlabimplementation. 38 2 3.3.3 Theoreticalresults ............................ 39 3.4 Secant method ( 1.5)..............................
  • Piecewise Polynomial Taylor Expansions – the Generalization of Fa`A Di Bruno's Formula

    Piecewise Polynomial Taylor Expansions – the Generalization of Fa`A Di Bruno's Formula

    Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany TOM STREUBEL1,CAREN TISCHENDORF,ANDREAS GRIEWANK2 Piecewise Polynomial Taylor Expansions – The Generalization of Faa` di Bruno’s Formula 1 0000-0003-4917-7977 2 0000-0001-9839-1473 ZIB Report 18-24 (May 2018) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Piecewise Polynomial Taylor Expansions – The Generalization of Faa` di Bruno’s Formula Tom Streubel and Caren Tischendorf and Andreas Griewank Abstract We present an extension of Taylor’s theorem towards nonsmooth evalua- tion procedures incorporating absolute value operaions. Evaluations procedures are computer programs of mathematical functions in closed form expression and al- low a different treatment of smooth operations and calls to the absolute value value function. The well known classical Theorem of Taylor defines polynomial approx- imation of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential or differential algebraic equations, for the construction of solvers for the continuous nonlinear op- timization of finite dimensional objective functions and for root solving of nonlinear systems of equations. The herein provided proof is construtive and allow efficiently designed algorithms for the execution and computation of generalized piecewise polynomial expansions.
  • A Conic Sections

    A Conic Sections

    A Conic Sections The ellipse, hyperbola, and parabola (and also the circle, which is a special case of the ellipse) are called the conic section curves (or the conic sections or just conics), since they can be obtained by cutting a cone with a plane (i.e., they are the intersections of a cone and a plane). The conics are easy to calculate and to display, so they are commonly used in applications where they can approximate the shape of other, more complex, geometric figures. Many natural motions occur along an ellipse, parabola, or hyperbola, making these curves especially useful. Planets move in ellipses; many comets move along a hyperbola (as do many colliding charged particles); objects thrown in a gravitational field follow a parabolic path. There are several ways to define and represent these curves and this section uses asimplegeometric definition that leads naturally to the parametric and the implicit representations of the conics. y F P D P D F x (a) (b) Figure A.1: Definition of Conic Sections. Definition: A conic is the locus of all the points P that satisfy the following: The distance of P from a fixed point F (the focus of the conic, Figure A.1a) is proportional 364 A. Conic Sections to its distance from a fixed line D (the directrix). Using set notation, we can write Conic = {P|PF = ePD}, where e is the eccentricity of the conic. It is easy to classify conics by means of their eccentricity: =1, parabola, e = < 1, ellipse (the circle is the special case e = 0), > 1, hyperbola.