Polynomials

2

Content

EvaluationRoot findingRoot BracketingInterpolationResultant

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IntroductionBest understood and most applied functions

Taylor’s expansion Basis of parametric curve/surface Data fitting

Basic Theorem: Weierstrass Approximation Theorem Fundamental Theorem of Algebra

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Evaluation – Horner’s Method

Compare number of multiplications and additions

01321)( axaaxaxaxaxp nnnn

Evaluate p(t)

p(t) = b0

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Details

3210

33

2210

)(

)(

taatatatp

xaxaxaaxp

100

211

322

33

)(

tbabtp

tbab

tbab

ab

6

Details (cont)

)()(

)()()(

)(

)(

0

122

30

10212

323

3

012

23

3

tqtp

xqtxxqxp

xqtxb

btxxbtxxbtxb

tbbxtbbxtbbxb

axaxaxaxp

122

3)( bxbxbxq

100

211

322

33

)(

tbabtp

tbab

tbab

ab

Evaluate p(t) and p’(t)

p(t) = b0

p’(t) = c1

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Evaluating xk Efficiently

Instead of using pow(x,k), or any iterative/recursive subroutines, think again!The S-and-X method: S(square) X(multiply-by-x)See how it works in the next pagesUnderstand how to implement in program

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Say the exponent k in base-2: a2a1a0 (k>3)

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12121

22

]][[

7111

x

x

xxx

SXSX

6

02121

22

]][[

6110

x

x

xx

SSX

5

12021

22

]][[

5101

x

x

xx

SXS

Except for the leading digit, replace 1 by [SX], 0 by [S]

Except for the leading digit, replace 1 by [SX], 0 by [S]

# of :

Each symbol, S or X, represents a multiplication

23

12121221

2222

]][][][[

2310111

x

x

xxxx

SXSXSXS

7 4 3 3

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ImplementationLeft-to-right scan

23

121212021

2222

]][][][[

2310111

x

x

xxxx

SXSXSXS

11101x2x4x16 x8 xz

Right-to-left scan

2321125222

120

1

1

1

0

1

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[Multiplication, division]

FFT…GCD of polynomials (Euler algorithm)

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Roots of Polynomials

Low order polynomials (for degree 4)

Quartics: see notes

Hi-precision formula for quadratics

Quadratics Cubics

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Calculator City (ref)

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Root Counting

P(x) has n complex roots, counting multiplicitiesIf ai’s are all real, then the complex roots occur in conjugate pairs.Descarte’s rules of sign Sturm’s sequenceBounds on roots[Deflation]

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Theorems for Polynomial Equations

Sturm theorem: The number of real roots of an

algebraic equation with real coefficients whose real roots are simple over an interval, the endpoints of which are not roots, is equal to the difference between the number of sign changes of the Sturm chains formed for the interval ends.

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Sturm Chain

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Example

38879.1 ,32836.10802951.0 ,334734.0 ,21465.1

13)( 5

ix

xxxf

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Sturm Theorem (cont)

For roots with multiplicity: The theorem does not apply, but … The new equation : f(x)/gcd(f(x),f’(x))

All roots are simple All roots are same as f(x)

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Sturm Chain by Maxima

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Maxima (cont)

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Descarte’s Sign RuleA method of determining the maximum number of positive and negative real roots of a polynomial. For positive roots, start with the sign of the coefficient of the lowest power. Count the number of sign changes n as you proceed from the lowest to the highest power (ignoring powers which do not appear). Then n is the maximum number of positive roots.

For negative roots, starting with a polynomial f(x), write a new polynomial f(-x) with the signs of all odd powers reversed, while leaving the signs of the even powers unchanged. Then proceed as before to count the number of sign changes n. Then n is the maximum number of

negative roots.

3 positive roots

4 negative roots

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Computing Roots Numerically

Newton is the main method for root findingCan be implemented efficiently using Horner’s methodQuadratic convergence except for multiple rootUse deflation to resolve root multiplicity; but can accumulate error; polynomials are sensitive to coefficient variationsNewton-Maehly’s method: roots converges quadratically even if previous roots are inaccurate

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Experiment

P(x)=(x-1)(x-2)(x-3)(x2+1)Assume two imprecise roots have been found, 1.1 & 1.9The deflated (cubic) polynomial is then

x3-3x2+0.91x-3

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Maehly procedureFast convergence; accurate solution

Deflated cubicFaster convergence, but solution was plagued with the propagated error

Original quinticConverged to 3in many steps (5 was not a good guess for 3)

9.11

1.111 )()(

)(

kk xxkk

kkk xpxp

xpxx

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Polynomial Interpolation

Given (n+1) pairs of points (xi,f(xi)) find a nth-degree polynomial Pn(x) to pass through these pointsCompute the function value not listed in the table by evaluating the interpolating polynomial

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Lagrange Polynomial

High computational cost; cannot reuse pointsDivided difference: a better alternative

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Divided Difference

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Divided Difference (cont)

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Resultant (of two polynomials)

An expression involving polynomial coefficients such that the vanishing of the expression is necessary and sufficient for these polynomials to have a common zero.

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Resultant (cont)

The equation Qz = 0 has nonzero solution IFF R=det(Q) vanishes.R is called the resultant of the equations.

with

The above system has common zero:

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Example

0

341

341

211

211

034

022

2

R

xx

xx

8

341

341

211

211

034

022

2

R

xx

xx

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Sylvester Matrix and Resultant

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Applications of Resultant

0)(

0)(

xq

xp

0),(

0),(

yxq

yxp

0)(

0)(

curve same for the 0),(

equation ingcorrespond Find

)(),()( :curve

ytq

xtp

yxF

implicit

tqtptcParamateri

R(x) = 0 IFF intersection exists

II. Algebraic curve intersection

I. Common zero

III. Implicitization

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Find Common Zero

The system has simultaneous zero iff the resultant is zero

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Result from Linear Algebra

has non-trivial solution if det(A) = 0

A reduced system (remove row n) gives the ratio:

Ai: remove column i

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Sylvester Resultant

To find the common zero, consider the reduced system

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Resultant in Maxima

x as independent variable

Byy

Byy

Ay

R

yxyB

Ayx

2

2

2

2

0

0

101

0

01

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Algebraic Curve Intersection

01),(

01),( 22

xyyxq

yxyxp

No real roots; no intersection (circle & hyperbola)

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Algebraic Curve Intersection

01),(

011),( 22

xyyxq

yxyxp

2 real roots; 2 intersection points

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Implicitization

The above system in t has a common zero whenever (x,y) is a point on the curve.

… a parabola

Application: parametric curve intersectionFind the corresponding parameter of (x,y) on the curve

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Parametric Curve Intersection

Cubic Bezier curve and its control points

Bernstein polynomial

implicitize f(x,y)=0 )(),()(:1 tytxtpc

)(),()(:2 sysxsqc

F(s)=f(x(s),y(s))=0

Find roots in [0,1]

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0012

210

0012

210

221

20

221

20

2

2

2

1

12

0

0

222

222

121

121

121)(

)()(

ytyytyyy

xtxxtxxx

tyttyty

txttxtx

ty

xtt

y

xt

y

x

ty

txtp

12

2)(:

1

2,

1

1,

1

0:)(

23

2)(:

1

2,

1

1,

0

0:)(

222

211

s

sscsc

tt

ttctc

29.0,69.0

04820

01242423

0443),(

2

22

21

s

ss

sss

yxxyxf

Quadratic Bezier Curve Intersection

Sketch using de Casteljau algorithm

[1, 2]

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De Casteljau Algorithm [ref]

A cubic Bezier curve with 4 control points

p(t), t[0,1] is defined.

Locate p(0.5)

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Inversion

41

419

21

,,

yx

t

2

1

05

10

05

/

01

15

01

0

1015

150

015

:rowfourth theDiscarding

0

115

0115

15

0315

43

47

47

43

47

47

2

3

2

3

43

47

47

2

3

43

41

47

419

t

tt

t

t

t

t

t

t

Example

Given (x,y) on the curve, find the corresponding parameter t