1. Polynomials

2. 2 Content Evaluation Root finding Root Bracketing Interpolation Resultant

3. 3 Introduction Best understood and most applied functions Taylor’s expansion Basis of parametric curve/surface Data fitting Basic Theorem: Weierstrass Approximation Theorem Fundamental Theorem of Algebra

4. 4 Evaluation – Horner’s Method Compare number of multiplications and additions Evaluate p(t) p(t) = b 0

5. 5 Details

6. 6 Details (cont) Evaluate p(t) and p’(t) p(t) = b 0 p’(t) = c 1

7. 7 Evaluating x k 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 pages Understand how to implement in program

8. 8 Say the exponent k in base-2: a2a1a0 (k>3) Except for the leading digit, replace 1 by [SX], 0 by [S] # of  : Each symbol, S or X, represents a multiplication 7433

9. 9 Implementation Left-to-right scan 11101 x2x2 x4x4 x 16 x8x8 x z Right-to-left scan 23 2 11 2 5 2 2 2 1 2 0 1 1 1 0 1

10. 10 [Multiplication, division] FFT… GCD of polynomials (Euler algorithm)

11. 11 Roots of Polynomials Low order polynomials (for degree  4) Quartics: see notes Hi-precision formula for quadratics Quadratics Cubics

12. 12 Calculator City (ref)ref

13. 13 Root Counting P(x) has n complex roots, counting multiplicities If a i ’s are all real, then the complex roots occur in conjugate pairs. Descarte’s rules of sign Sturm’s sequence Bounds on roots [Deflation]

14. 14 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.

15. 15 Sturm Chain

16. 16 Example

17. 17 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)

18. 18 Sturm Chain by Maxima

19. 19 Maxima (cont)

20. 20 Descarte’s Sign Rule A 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

21. 21 Computing Roots Numerically Newton is the main method for root finding Can be implemented efficiently using Horner’s method Quadratic convergence except for multiple root Use deflation to resolve root multiplicity; but can accumulate error; polynomials are sensitive to coefficient variations Newton-Maehly’s method: roots converges quadratically even if previous roots are inaccurate

22. 22

23. 23 Experiment P(x)=(x-1)(x-2)(x-3)(x 2 +1) Assume two imprecise roots have been found, 1.1 & 1.9 The deflated (cubic) polynomial is then x 3 -3x 2 +0.91x-3

24. 24 Maehly procedure Fast convergence; accurate solution Deflated cubic Faster convergence, but solution was plagued with the propagated error Original quintic Converged to 3 in many steps (5 was not a good guess for 3)

25. 25 Polynomial Interpolation Given (n+1) pairs of points (x i,f(x i )) find a nth-degree polynomial P n (x) to pass through these points Compute the function value not listed in the table by evaluating the interpolating polynomial

26. 26 Lagrange Polynomial High computational cost; cannot reuse points Divided difference: a better alternative

27. 27 Divided Difference

28. 28 Divided Difference (cont)

29. 29

30. 30 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.

31. 31 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:

32. 32 Example

33. 33 Sylvester Matrix and Resultant

34. 34 Applications of Resultant R(x) = 0 IFF intersection exists II. Algebraic curve intersection I. Common zero III. Implicitization

35. 35 Find Common Zero The system has simultaneous zero iff the resultant is zero

36. 36 Result from Linear Algebra has non-trivial solution if det(A) = 0 A reduced system (remove row n) gives the ratio: A i : remove column i

37. 37 Sylvester Resultant To find the common zero, consider the reduced system

38. 38 Resultant in Maxima x as independent variable

39. 39 Algebraic Curve Intersection No real roots; no intersection (circle & hyperbola)

40. 40 Algebraic Curve Intersection 2 real roots; 2 intersection points

41. 41 Implicitization The above system in t has a common zero whenever (x,y) is a point on the curve. … a parabola Application: parametric curve intersection Find the corresponding parameter of (x,y) on the curve

42. 42 Parametric Curve Intersection Cubic Bezier curve and its control points Bernstein polynomial implicitize f(x,y)=0 F(s)=f(x(s),y(s))=0 Find roots in [0,1]

43. 43 Quadratic Bezier Curve Intersection Sketch using de Casteljau algorithm [1, 2]12

44. 44 De Casteljau Algorithm [ref]ref A cubic Bezier curve with 4 control points p(t), t  [0,1] is defined. Locate p(0.5)

45. 45 Inversion Example Given (x,y) on the curve, find the corresponding parameter t