Introduction to Numerical Analysis, Study Guides, Projects, Research of Mathematical Analysis

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Numerical Analysis

Mathematics Informatics

Informatics Engineering

Gunadarma University

Overview of Numerical Analysis Numerical Analysis is a branch of mathematics that develops, analyzes, and applies algorithms for solving numerical problems. It involves approximating solutions to problems that may not be solvable by exact methods, often involving differential equations, integrals, and systems of equations. Key Concepts in Numerical Analysis

1. Error Analysis - Absolute Error : The difference between the exact value and the approximate value. Absolute Error = ∣x − x̃∣ - Relative Error : The absolute error divided by the exact value. Relative Error = |𝑥 − x̃| |𝑥| - Round-off Error : Errors that occur because of the limitations of representing numbers in a computer. - Truncation Error : Errors that result from approximating a mathematical procedure (e.g., using a finite number of terms of a series). 2. Solutions of Nonlinear Equations - Bisection Method : A root-finding method that repeatedly divides an interval in half and selects the subinterval in which the root lies. - Newton's Method : A root-finding method that uses function values and derivatives to approximate the root.

xn+1 = xn -

𝑓(xn) f′(xn)

5. Interpolation and Polynomial Approximation - Lagrange Interpolation : A method for constructing a polynomial that passes through a given set of points.

P(x) = ∑^ y

n

i = 0 i^ ∏^

x - xj xi - xj 0 ≤ j ≤ n j ≠ i

• Newton Interpolation : A method for constructing a polynomial using divided differences. P(x) = f[x 0 ] + f[x 0 , x 1 ](x – x 1 ) + f[x 0 , x 1 , x 2 ](x – x 0 )(x – x 1 ) + … **Example Problems

1. Bisection Method Problem** : Use the bisection method to find the root of f(x) = x^2 − 4 in the interval [1, 3]. Solution :
2. Compute the midpoint: c =

2. Evaluate f(c) = 22 −4 = 0.
3. Since f(c) = 0 , the root is c = 2. 2. Trapezoidal Rule Problem : Approximate the integral of f(x) = x^2 from 0 to 2 using the trapezoidal rule. Solution :
4. A = 0 , b = 2b, and f(x) = x^2.
5. ∫ x 2 0 (^2) dx ≈ 2 -^0 2 [f(0) + f(2)] = 1[0 + 4] = 4.

3. Gaussian Elimination Problem : Solve the system of equations using Gaussian elimination: { 2 𝑥 + 3y = 5 4x + 7y = 10 Solution : 1. Form the augmented matrix : (

2 3 |^5

2. Perform row operations to obtain an upper triangular form: o R2 ← R2 − 2R1: (

0 1 |^0

3. Solve the resulting system using back substitution: { y = 0 2 x + 3 ( 0 ) = 5 => x = 5 2 Solution: x = 2.5, y = 0. 4. Lagrange Interpolation Problem : Construct the Lagrange interpolating polynomial for the points (1, 1), (2, 4) and (3, 9). Solution :
4. Basis polynomials: L 0 (x) = (x - 2 )(x - 3 ) ( 1 - 2 )( 1 - 3 )

(x - 2 )(x - 3 ) 2

Practice Problems

1. Use Newton's method to approximate the root of f(x) = x^3 − x − 1 starting from x 0 = 1.
2. Approximate the integral of f(x) = ex^ from 0 to 1 using Simpson's rule.
3. Solve the system of equations using LU decomposition: { x + y + z = 6 2y + 5 z = - 4 2x + 5y = 27
4. Construct the Newton interpolating polynomial for the points (0, 1), (1, 3) and (2, 2 ).