Numerical Methods · Unit 2
Root Finding Methods
Exam-focused notes for Root Finding Methods (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 8 solved past questions from this unit.
What this unit covers
- Bisection method derivation and application
- Newton Raphson method formula and convergence
- Secant method formula and derivation
- Comparison of root finding methods
- Horner's method for polynomial evaluation
- Algorithm and implementation of root finding methods
Comparison of root finding methods
Explain how bisection method differ from secant. Derive the formula for Newton Raphson. Use Newton Raphson method to solve the equation $f(x) = x^3 + 2x - 2$ correct upto three decimal places. [2+4+4]
- Equation: $f(x) = x^3 + 2x - 2 = 0$ - Required accuracy: three decimal places - Marks split: $2 + 4 + 4$ --- (a) Bisection Method vs Secant Method Feature Bisection Method Secant Method --------- Type Bracketing (closed) method Open method Initial points ...
Full solved answer →Algorithm and implementation of root finding methods
Write an algorithm and a C-Program to obtain roots of non-linear equation using Newton Raphson Method.[10]
The Newton-Raphson method is an iterative numerical technique used to find roots of a non-linear equation f(x) = 0. Starting from an initial guess x₀, it uses the tangent line at each point to converge to the root. Iterative Formula: $$x{n+1} = xn - \frac{f...
Full solved answer →Secant method formula and derivation
Use secant method to estimate the root of the equation $x^2-5x+6=0$, with initial estimate $x_1 = 4$ and $x_2 = 2$ (EPS=0.05). [5]
- Equation: $f(x) = x^2 - 5x + 6$ - Initial estimates: $x1 = 4$, $x2 = 2$ - Tolerance: $\text{EPS} = 0.05$ Secant formula: $$x{n+1} = xn - f(xn)\cdot\frac{xn - x{n-1}}{f(xn) - f(x{n-1})}$$ Stopping criterion: $$\varepsilon = \left\frac{x{n+1} - xn}{x{n+1}}\...
Full solved answer →How Secant methods differs from Newton Raphson method? Derive the formula for Secant Method. Solve the equation $\cos x + 2\sin x - x^2 = 0$ using Secant method. Assume error precision is 0.01. [10]
Feature Newton-Raphson Secant --------- Requires $f(x)$ and $f'(x)$ Only $f(x)$ Initial guesses One ($x0$) Two ($x0,x1$) Derivative Exact analytic Finite-difference approximation Convergence order Quadratic ($2$) Superlinear ($\approx 1.618$) Function evalu...
Full solved answer →Bisection method derivation and application
Question
Define true error and relative error. Derive the bisection method for solving non-linear equation and using this method solve $2x^3 - 2x - 5$ with initial $x_0 = 1$ and $x_1 = 2$. Calculate upto 10th iteration.[10]
True Error is the difference between the exact (true) value and the approximate value: $$Et = \text{True Value} - \text{Approximate Value}$$ Relative Error is the true error normalized by the true value (often as a percentage): $$\epsilonr = \frac{\text{Tru...
Full solved answer →Horner's method for polynomial evaluation
Write an algorithm for Honer's method. Evaluate the polynomial $f(x) = x^4 + 3x^3 + 5x^2 + 7^x + 9$ at x = 2 by using Honer's method. [5]
Polynomial: $f(x) = x^4 + 3x^3 + 5x^2 + 7x + 9$ (Note: the source text shows "$7^x$", but from the pattern of a standard 4th-degree polynomial this is clearly a typo for the linear term $7x$.) Coefficients (highest to lowest degree): Degree 4 3 2 1 0 ------...
Full solved answer →Define the terms true error and relative error? Write down algorithm for Horner' method to evaluate polynomial and use the method to evaluate the polynomial $2x^3 - 3x^2 + 5x - 2$ at x=3. [5]
- Polynomial: $P(x) = 2x^3 - 3x^2 + 5x - 2$ - Coefficients (high to low degree): $a3 = 2,\ a2 = -3,\ a1 = 5,\ a0 = -2$ - Evaluation point: $x = 3$ The true error is the difference between the exact (true) value and the approximate (computed) value. $$Et = X...
Full solved answer →Newton Raphson method formula and convergence
Show that the rate of convergence of Newtons Raphson method is quadratic. [5]
Let $x^$ be the exact root of $f(x) = 0$, and let $xn$ be the $n$-th approximation. Define the error at the $n$-th step as: $$en = xn - x^$$ The Newton-Raphson iteration formula is: $$x{n+1} = xn - \frac{f(xn)}{f'(xn)}$$ --- Expand $f(x^)$ about $xn$ using ...
Full solved answer →Make Unit 2 stick
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