Today’s class

Today’s class• Roots of Equations• Polynomials

Numerical Methods, Lecture 6 1 Prof. Jinbo Bi

CSE, UConn

• Locate the roots of a set of simultaneous equations

• Use extensions of the various open methods

• Fixed-point iteration is not practical for nonlinear systems

Numerical Methods, Lecture 6 2 Prof. Jinbo Bi

CSE, UConn

Systems of nonlinear equations

• Using Newton-Raphson method• First order Taylor approximation

• Solve for xi+1 and yi+1

Systems of nonlinear equations

yf

yyxf

xxyxfyxf iii

iiiii

,1

1,1

1,11,1 )()(),(),(

yf

yyxf

xxyxfyxf iii

iiiii

,2

1,2

1,21,2 )()(),(),(

Numerical Methods, Lecture 6 3 Prof. Jinbo Bi

CSE, UConn

Numerical Methods, Lecture 6 4 Prof. Jinbo Bi

CSE, UConn

Systems of nonlinear equations

• Newton-Raphson method can still diverge as with the single equation method

• Can be generalized to more than two equations - requires solving larger linear systems

Numerical Methods, Lecture 6 5 Prof. Jinbo Bi

CSE, UConn

Systems of nonlinear equations

Polynomials• A polynomial is of the form:

• The roots of a polynomial follow these rules:• There will be n roots to an n-th order

polynomial. The roots may be real or complex and need not be distinct

• If n is odd, there is at least one real root• If complex roots exist, they exist in conjugate

pairs (λ+μi and λ-μi)Numerical Methods, Lecture 6 6 Prof. Jinbo Bi

CSE, UConn

Polynomials• Uses in electrical engineering

• Solving for poles and zeros in transfer functions

• Solving characteristic equations in linear ordinary differential equations

Numerical Methods, Lecture 6 7 Prof. Jinbo Bi

CSE, UConn

Computing with Polynomials

Numerical Methods, Lecture 6 8 Prof. Jinbo Bi

CSE, UConn

Computing with Polynomials

Numerical Methods, Lecture 6 9 Prof. Jinbo Bi

CSE, UConn

Polynomial Deflation• Given a polynomial and a single known

root, deflating the polynomial can reduce the order of the polynomial and remove possible redundant roots

• Example:

• Root x=4 is known

Numerical Methods, Lecture 6 10 Prof. Jinbo Bi

CSE, UConn

Polynomial Deflation

Numerical Methods, Lecture 6 11 Prof. Jinbo Bi

CSE, UConn

• Synthetic Division

Polynomial Deflation

Numerical Methods, Lecture 6 12 Prof. Jinbo Bi

CSE, UConn

Roots of polynomials• Previous methods (Newton-Raphson,

bracketing, etc.) have a few problems when applied to polynomials• Must determine an initial guess of the root• Can not find complex roots• May not converge

• Polynomial-specific methods• Müller’s method• Bairstow‘s method

Numerical Methods, Lecture 6 13 Prof. Jinbo Bi

CSE, UConn

• Similar idea to secant method• Instead of projecting a straight line

through two points to estimate the root, project a parabola through three points to estimate the root

Muller’s method

Numerical Methods, Lecture 6 14 Prof. Jinbo Bi

CSE, UConn

Muller’s method

Numerical Methods, Lecture 6 15 Prof. Jinbo Bi

CSE, UConn

• Find a parabola such that it intersects the function at three points x0, x1, and x2

Muller’s method

Numerical Methods, Lecture 6 16 Prof. Jinbo Bi

CSE, UConn

• Solve for c

• Substitute back into set of equations

• Define new set of variables

Muller’s method

Numerical Methods, Lecture 6 17 Prof. Jinbo Bi

CSE, UConn

• Substitute new variables back into equations

• Solve for a and b

Muller’s method

Numerical Methods, Lecture 6 18 Prof. Jinbo Bi

CSE, UConn

• Now that we have found the coefficients of the parabola, find where it intersects the x-axis to get the next root estimate

• The intersection point with the x-axis is simply the root of the parabola which we can find using the quadratic formula

Muller’s method

Numerical Methods, Lecture 6 19 Prof. Jinbo Bi

CSE, UConn

• Which of the two quadratic roots do you choose?• Choose the sign that agrees with the sign of

b, making the denominator the largest• That will bring xr closer to x2

Muller’s method

Numerical Methods, Lecture 6 20 Prof. Jinbo Bi

CSE, UConn

• How do you pick the three points for the next iteration?• If there are only real roots, choose xr and the

closer two of the orignal three roots to xr.• If there are complex roots, use a sequential

approach: use xr, x2, and x1 to replace x2, x1, and x0

Muller’s method

Numerical Methods, Lecture 6 21 Prof. Jinbo Bi

CSE, UConn

• Find roots of f(x)=x3-13x-12• Use x0=4.5, x1=5.5, and x2=5 as the

initial guesses

Example

Numerical Methods, Lecture 6 22 Prof. Jinbo Bi

CSE, UConn

Example

Numerical Methods, Lecture 6 23 Prof. Jinbo Bi

CSE, UConn

Example

Numerical Methods, Lecture 6 24 Prof. Jinbo Bi

CSE, UConn

• Based on dividing the polynomial by (x-t) where t is the root estimate

• If there is no remainder, we found the root

• If not, we adjust the guess

Bairstow’s method

Numerical Methods, Lecture 6 25 Prof. Jinbo Bi

CSE, UConn

• Factor by x-t

• Factor by x2-rx-s

Bairstow’s method

Numerical Methods, Lecture 6 26 Prof. Jinbo Bi

CSE, UConn

• We have to find values of r and s such that the remainder is 0

• The remainder is 0 if b1=b0=0

• b1 and b0 are both functions of r and s, so we can give a Taylor series expansion of both

Bairstow’s method

Numerical Methods, Lecture 6 27 Prof. Jinbo Bi

CSE, UConn

Bairstow’s method

Numerical Methods, Lecture 6 28 Prof. Jinbo Bi

CSE, UConn

• Solve for Δr and Δs• Use the results to adjust r and s and iterate• Stop when the approximate error in r and s is

sufficient

Bairstow’s method

Numerical Methods, Lecture 6 29 Prof. Jinbo Bi

CSE, UConn

• Find roots of f(x)= x5 - 3.5x4 + 2.75x3 + 2.125x5 - 3.875x + 1.25

• Use s= -1, r= -1 as the initial guesses

Example

Numerical Methods, Lecture 6 30 Prof. Jinbo Bi

CSE, UConn

• Solve linear system

• New s and r

Example

Numerical Methods, Lecture 6 31 Prof. Jinbo Bi

CSE, UConn

• Use s= 0.1381, r= -0.6442

Example

Numerical Methods, Lecture 6 32 Prof. Jinbo Bi

CSE, UConn

• Solve linear system

• New s and r

Example

Numerical Methods, Lecture 6 33 Prof. Jinbo Bi

CSE, UConn

• Keep going until s= 0.5 and r= -0.5• Then solve the following:

Example

Numerical Methods, Lecture 6 34 Prof. Jinbo Bi

CSE, UConn

• The quotient coefficients are as follows

• Now we have to solve:

Example

Numerical Methods, Lecture 6 35 Prof. Jinbo Bi

CSE, UConn

• If no s or r are known, start with s=0, r=0• Converges fast• May diverge

Bairstow’s method

Numerical Methods, Lecture 6 36 Prof. Jinbo Bi

CSE, UConn

• Bisection• 2 initial guesses, slow convergence, will not

diverge• False-position

• 2 initial guesses, slow-medium convergence, will not diverge

Roots of equations

Numerical Methods, Lecture 6 37 Prof. Jinbo Bi

CSE, UConn

• Newton-Raphson• 1 initial guess, fast convergence, may

diverge• Secant

• 2 initial guesses, medium-fast convergence, may diverge

Roots of equations

Numerical Methods, Lecture 6 38 Prof. Jinbo Bi

CSE, UConn

• Muller• 3 initial guesses, medium-fast convergence,

may diverge, only polynomials, complex roots

• Bairstow• 2 initial guesses, medium-fast convergence,

may diverge, only polynomials, complex roots

Roots of equations

Numerical Methods, Lecture 6 39 Prof. Jinbo Bi

CSE, UConn

Next class• Linear algebraic equations• Read Chapter 9• HW3, due 09/24

Numerical Methods, Lecture 6 40 Prof. Jinbo Bi

CSE, UConn