Numerical Method · Unit 1 · 8 hrs
Solution of Nonlinear Equations
Exam-focused notes for Solution of Nonlinear Equations (Numerical Method, CSC212): what the TU syllabus asks and how it has actually been tested, with 13 solved past questions from this unit.
What this unit covers
- Errors in Numerical Calculations
- Sources of Errors
- Propagation of Errors
- Review of Taylor's Theorem
- Solving Non-linear Equations by Trial and Error method
- Half-Interval method and Convergence
- Newton's method and Convergence
- Secant method and Convergence
- Fixed point iteration and its convergence
- Newton's method for calculating multiple roots
- Horner's method
Newton's method and Convergence
What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve the following equation up to 3 decimal places: $x^2 - 5x + 6 = 0$ [10]
Inherent errors are errors that already exist in a problem before any numerical computation is carried out. They are caused by the data or by the mathematical model, not by the numerical technique used. Main sources: - Data errors: Values obtained from phys...
Full solved answer →Derive the formula for Newton Raphson Method. Solve the equation using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method. $x^2 + 4x - 9 = 0$ [10]
- Equation: $f(x) = x^2 + 4x - 9 = 0$ - Error precision: $E = 0.01$ - Initial guess: not specified in the problem, so I take $x0 = 4$ The exact positive root is $x = -2 + \sqrt{13} = 1.60555...$ --- Expand $f(x)$ by Taylor series about $xn$: $$f(xn + h) = f...
Full solved answer →Calculate a real negative root of following equation using Newton’s method for polynomial. $x^4 + 2x^3 + 3x^2 + 4x = 5$. [5]
Equation: $$x^4 + 2x^3 + 3x^2 + 4x = 5$$ Rearranged: $$f(x) = x^4 + 2x^3 + 3x^2 + 4x - 5 = 0$$ $$x{n+1} = xn - \frac{f(xn)}{f'(xn)}, \qquad f'(x) = 4x^3 + 6x^2 + 6x + 4$$ $$f(-1) = 1 - 2 + 3 - 4 - 5 = -7 \;(<0)$$ $$f(-2) = 16 - 16 + 12 - 8 - 5 = -1 \;(<0)$$...
Full solved answer →Secant method and Convergence
How secant methods differs from Newton Raphson method? Derive the formula for Secant Method. Solve the equation using Secant method. Assume error precision as 0.01. Discuss the drawbacks of the Newton Raphson method.cosx+2sinx−x2=0\cos x + 2\sin x - x^2 = 0cosx+2sinx−x2=0[10]
- Equation: $f(x) = \cos x + 2\sin x - x^2 = 0$ - Error precision (tolerance): $E = 0.01$ - Initial guesses: not specified in question. I choose $x0 = 1$, $x1 = 2$ (bracketing the root, angles in radians). Feature Newton-Raphson Secant --------- Requires $f...
Full solved answer →How secant method can approximate the root of a non-linear equation? Explain with necessary derivation. Estimate a real root of following equation using secant method. Assume error precision of 0.01. $x^3 + 2x - \cos(x) = 4$ [10]
- Equation: $x^3 + 2x - \cos(x) = 4$, i.e. $f(x) = x^3 + 2x - \cos(x) - 4 = 0$ - Error precision: $E = 0.01$ - Method: Secant method - Initial guesses: not specified (to be chosen from sign change). All angles in radians. Newton-Raphson formula: $$x{n+1} = ...
Full solved answer →What is non-linear equation? Derive the required expression to calculate the root of non-linear equation using secant method. Using this expression find a root of following equation.
$$x^2 + \cos(x) - e^{-x} - 2 = 0$$
[10]
A non-linear equation is an equation $f(x)=0$ in which the unknown appears in a degree higher than one, or inside transcendental functions (trigonometric, exponential, logarithmic). Its graph is not a straight line, and it generally cannot be solved by dire...
Full solved answer →Half-Interval method and Convergence
Define the terms approximate error and relative approximate error? Discuss the working of Half Interval method for finding the roots of non-linear equation. [5]
--- Approximate Error is the difference between the current approximation and the previous approximation of a root. It gives an estimate of how much the solution has changed between two successive iterations. $$Ea = x{new} - x{old}$$ Since the true value is...
Full solved answer →Calculate a real root of the following function using bisection method correct upto 3 significant figures. $x^2 - e^x = 3$ [5]
Equation: $x^2 - e^x = 3$ Rearranged as: $f(x) = x^2 - e^x - 3 = 0$ Required accuracy: 3 significant figures. --- $x$ $f(x) = x^2 - e^x - 3$ Sign --------- $-1$ $1 - 0.3679 - 3 = -2.368$ $-$ $-2$ $4 - 0.1353 - 3 = +0.865$ $+$ Sign change between $-2$ and $-...
Full solved answer →How the half-interval method can be estimate a root of a non-linear equation? Find a real root of the following equation using the half-interval method to correct up to two decimal places. $x^2 - e^{-x} - x = 1$ [5]
- Equation: $x^2 - e^{-x} - x = 1$ - Rearranged: $f(x) = x^2 - e^{-x} - x - 1 = 0$ - Required accuracy: two decimal places The half-interval (bisection) method is based on the Intermediate Value Theorem: if $f(x)$ is continuous on $[a,b]$ and $f(a)\cdot f(b...
Full solved answer →Fixed point iteration and its convergence
What is fixed point iteration method? How can it converge to the root of a non-linear equation? Also explain the diverging cases with suitable examples. [5]
In the Fixed Point Iteration Method, we rearrange the non-linear equation f(x) = 0 such that x is isolated on the left-hand side of the equation. The rearranged form is expressed as: $$x = g(x) \quad \cdots (1)$$ Since f(x) = 0 and x = g(x) are equivalent f...
Full solved answer →Calculate the real root of the given equation using fixed point iteration correct up to 3 significant figures. $2x^3 - 2x = 5$. [5]
- Equation: $2x^3 - 2x = 5$, i.e. $f(x) = 2x^3 - 2x - 5 = 0$ - Method: Fixed point iteration - Required accuracy: 3 significant figures No initial guess given; must be determined by root location. Rearrange to $x = g(x)$: $$2x^3 = 5 + 2x \implies x = \left(...
Full solved answer →Horner's method
How can Horner’s rule be used to evaluate the f(x) and f(x) of a polynomial at a given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner’s rule.[10]
Horner's rule is an efficient scheme for evaluating a polynomial and its derivative at a point by rewriting the polynomial in nested form. It is closely related to synthetic division and the Remainder Theorem. For a polynomial of degree $n$: $$f(x) = a0 x^n...
Full solved answer →Define the terms true error and relative error? Use Horner' method to evaluate polynomial at x = 3 and write down its algorithm. $2x^3 - 3x^2 + 5x - 2$ [5]
- Polynomial: $P(x) = 2x^3 - 3x^2 + 5x - 2$ - Evaluation point: $x = 3$ - Coefficients: $a3 = 2,\ a2 = -3,\ a1 = 5,\ a0 = -2$ --- True error is the difference between the exact (true) value of a quantity and its approximate (computed) value. $$Et = \text{Tr...
Full solved answer →Make Unit 1 stick
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