Polynomials Content Evaluation Root finding Root Bracketing Interpolation

Polynomials

Content Evaluation Root finding Root Bracketing Interpolation Resultant 2

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

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

Details 5

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

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 pages Understand how to implement in program 7

Say the exponent k in base-2: a 2 a 1 a 0 (k>3) Except for the leading digit, replace 1 by [SX], 0 by [S] # of : 7 4 Each symbol, S or X, represents a multiplication 3 3 8

Implementation Left-to-right scan Right-to-left scan 2 2 2 z 1 0 1 1 x 16 x 8 x 4 x 2 23 11 5 2 1 0 1 1 x 9

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

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

Calculator City (ref) 12

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

Theorems for Polynomial Equations Sturm theorem: n 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. 14

Sturm Chain 15

Example 16

Sturm Theorem (cont) For roots with multiplicity: n n The theorem does not apply, but … The new equation : f(x)/gcd(f(x), f’(x)) w All roots are simple w All roots are same as f(x) 17

Sturm Chain by Maxima 18

Maxima (cont) 19

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 20

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 21

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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 -3 x 2+0. 91 x-3 23

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

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

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

Divided Difference 27

Divided Difference (cont) 28

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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. 30

Resultant (cont) with The above system has common zero: The equation Qz = 0 has nonzero solution IFF R=det(Q) vanishes. R is called the resultant of the equations. 31

Example 32

Sylvester Matrix and Resultant 33

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

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

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 36

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

Resultant in Maxima x as independent variable 38

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

Algebraic Curve Intersection 2 real roots; 2 intersection points 40

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 41

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