Roots of Equations - Examples & Applications

This section explores methods for finding the roots of equations, including bracketing methods (Bisection, False Position), open methods (Newton-Raphson, Secant), handling multiple roots, and solving systems of nonlinear equations.

Bracketing Methods: Bisection

The Bisection method repeatedly halves the interval that contains the root.

Basic: Bisection Method

Use the bisection method to find the root of the function f(x)=e−x−xf(x) = e^{-x} - x. Perform three iterations using an initial bracket of xl=0x_l = 0 and xu=1x_u = 1. Calculate the approximate relative error for each iteration.

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Intermediate: Drag Coefficient with Bisection

The velocity of a falling parachutist is given by v(t)=gmc(1−e−(c/m)t)v(t) = \frac{gm}{c} \left(1 - e^{-(c/m)t}\right). Given v=40 m/sv = 40 \text{ m/s}, m=68.1 kgm = 68.1 \text{ kg}, t=10 st = 10 \text{ s}, and g=9.8 m/s2g = 9.8 \text{ m/s}^2, use the bisection method to determine the drag coefficient cc to a level of ϵa≤5%\epsilon_a \leq 5\%. The initial guesses are cl=12c_l = 12 and cu=16c_u = 16.

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Bracketing Methods: False Position

The False Position method connects the endpoints of the interval with a straight line and estimates the root at the intersection of the line with the x-axis.

Intermediate: False Position Method

Use the false position method to determine the root of f(x)=−0.5x2+2.5x+4.5f(x) = -0.5x^2 + 2.5x + 4.5. The initial interval is [5,10][5, 10]. Perform two iterations.

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Open Methods: Newton-Raphson Method

The Newton-Raphson method is a fast-converging open method that utilizes the derivative of the function to extrapolate to the root.

Advanced: Newton-Raphson Method

Use the Newton-Raphson method to estimate the root of f(x)=e−x−xf(x) = e^{-x} - x. Employ an initial guess of x0=0x_0 = 0. Perform three iterations.

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Open Methods: Secant Method

Intermediate: Secant Method

Use the Secant method to estimate the root of f(x)=e−x−xf(x) = e^{-x} - x. Start with initial estimates of x−1=0x_{-1} = 0 and x0=1x_0 = 1. Perform two iterations.

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Multiple Roots

The Modified Newton-Raphson method improves convergence for multiple roots.

Advanced: Modified Newton-Raphson

Find the root of f(x)=(x−3)(x−1)(x−1)=x3−5x2+7x−3f(x) = (x - 3)(x - 1)(x - 1) = x^3 - 5x^2 + 7x - 3, which has a double root at x=1x=1. Use the Modified Newton-Raphson method with an initial guess x0=0x_0 = 0. Perform one iteration.

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Systems of Nonlinear Equations

The Newton-Raphson method can be extended to systems of nonlinear equations using partial derivatives (Jacobian matrix).

Advanced: System of Nonlinear Equations

Solve the following system using the multivariable Newton-Raphson method with an initial guess of x0=1.5,y0=2.0x_0 = 1.5, y_0 = 2.0. Perform one iteration.
f1(x,y)=x2+xy−10=0f_1(x, y) = x^2 + xy - 10 = 0
f2(x,y)=y+3xy2−57=0f_2(x, y) = y + 3xy^2 - 57 = 0

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Roots of Polynomials

Polynomials can have complex roots. Methods like Bairstow's method extract quadratic factors from the polynomial to find complex conjugate root pairs without requiring complex arithmetic.

Advanced: Finding Complex Roots with the Quadratic Formula

Find the roots of the quadratic polynomial extracted from Bairstow's method: x2−2x+5=0x^2 - 2x + 5 = 0.

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