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MTH307Sciences3 Unitsintermediate

Numerical Analysis Ii

This course, Numerical Analysis II, builds upon the foundational concepts of numerical methods. It delves into advanced topics such as polynomial approximations, orthogonal polynomials including Legendre and Chebyshev, and further interpolation techniques like cubic splines and Hermite approximations. The course also covers numerical integration methods and boundary value problems, equipping students with the tools to solve complex mathematical problems numerically.

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150h
Study Time
13
Weeks
12h
Per Week
intermediate
Math Level
Course Keywords
Numerical AnalysisApproximationInterpolationIntegrationBoundary Value Problems

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level
Builds on foundational knowledge
65%
intermediate
📊
Math Level
Moderate Math
🔬
Learning Type
Hands-on Practice

Course Topics

Key areas covered in this course

1

Polynomial Approximation

2

Orthogonal Polynomials

3

Legendre Polynomials

4

Chebyshev Polynomials

5

Cubic Spline Interpolation

6

Hermite Approximation

7

Numerical Integration

8

Trapezoidal Rule

9

Simpson's Rule

10

Newton-Cotes Formulas

11

Boundary Value Problems

12

Finite Difference Method

Total Topics12 topics

Requirements

Knowledge and skills recommended for success

MTH301: Real Analysis

MTH303: Ordinary Differential Equations

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

assignments

Comprehensive evaluation of course material understanding

Written Assessment

tutor-marked assessments

Comprehensive evaluation of course material understanding

Written Assessment

final examination

Comprehensive evaluation of course material understanding

Written Assessment

Career Opportunities

Explore the career paths this course opens up for you

Numerical Analyst

Apply your skills in this growing field

Data Scientist

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Computational Mathematician

Apply your skills in this growing field

Financial Modeler

Apply your skills in this growing field

Engineering Analyst

Apply your skills in this growing field

Industry Applications

Real-world sectors where you can apply your knowledge

FinanceEngineeringData ScienceScientific ResearchComputer Simulation

Study Schedule Beta

A structured 13-week journey through the course content

Week
1

Module 1: Approximations

9h

Unit 1: Polynomials

4 study hours
  • •Understand the definition of a polynomial and its degree.
  • •Distinguish between polynomial functions and polynomial equations.
  • •Express simple functions as polynomials using series expansion.
  • •Identify different types of function approximation methods.

Unit 2: Least Squares Approximation (Discrete Case)

5 study hours
  • •Understand the basic idea of least square approximation.
  • •Derive the least square formula for discrete data.
  • •Learn to fit a linear polynomial to a set of data points.
  • •Learn to fit a quadratic or parabolic polynomial to a set of data points.
Week
2

Module 1: Approximations

6h

Unit 3: Least Squares Approximation (Continuous Case)

6 study hours
  • •Distinguish between discrete data and continuous functions.
  • •Learn to fit polynomials to continuous functions using the least squares approach.
  • •Practice integration techniques required for continuous least squares approximation.
Week
3

Module 2: Orthogonal Polynomials

5h

Unit 1: Introduction to Orthogonal System

5 study hours
  • •Define orthogonal polynomials and understand their properties.
  • •Formulate orthogonal and orthonormal polynomials.
  • •Handle inner product of functions and understand its properties.
  • •Verify orthogonality of functions through integration.
Week
4

Module 2: Orthogonal Polynomials

5h

Unit 2: The Legendre Polynomials

5 study hours
  • •State the Rodrigues' formula for generating Legendre polynomials.
  • •Generate Legendre polynomials using Rodrigues' formula and recurrence relation.
  • •Understand the orthogonality property of Legendre polynomials.
  • •Solve problems using Legendre polynomials
Week
5

Module 2: Orthogonal Polynomials

5h

Unit 3: Least Squares Approximation by Legendre Polynomials

5 study hours
  • •Apply Legendre polynomials to least squares procedures.
  • •Obtain least square approximations using Legendre polynomials.
  • •Solve numerical problems using Legendre polynomial approximation.
Week
6

Module 2: Orthogonal Polynomials

5h

Unit 4: The Chebyshev Polynomials

5 study hours
  • •State the necessary formulae for generating Chebyshev polynomials.
  • •Obtain Chebyshev polynomials Tn(x) up to n = 10 using recurrence formula.
  • •Classify Chebyshev polynomials as a family of orthogonal series.
  • •Understand the properties of Chebyshev polynomials.
Week
7

Module 2: Orthogonal Polynomials

5h

Unit 5: Series of Chebyshev Polynomials

5 study hours
  • •Identify the form of functions suitable for Chebyshev polynomial approximation.
  • •Apply Chebyshev polynomials to fit a cubic approximation to a function f(x).
  • •Evaluate the accuracy of Chebyshev polynomial approximations.
Week
8

Module 2: Orthogonal Polynomials

5h

Unit 6: Chebyshev Interpolation

5 study hours
  • •Use Lagrange's formula for interpolation.
  • •Interpolate using Chebyshev polynomials.
  • •Compute the error table from the approximation.
  • •Apply Chebyshev interpolation to approximate functions.
Week
9

Module 3: Further Interpolation Techniques

6h

Unit 1: Cubic Spline Interpolation

6 study hours
  • •Define a cubic spline and understand its properties.
  • •Derive a method of fitting a cubic spline to a set of data points.
  • •Fit a cubic spline to a set of data points.
  • •Interpolate a function from the fitted cubic spline.
Week
10

Module 3: Further Interpolation Techniques

5h

Unit 2: Hermite Approximations

5 study hours
  • •Distinguish between cubic spline and Hermite polynomial.
  • •Figure out the Hermite approximation formula.
  • •Fit polynomial by Hermite approximation technique.
  • •Find an estimate using Hermite approximation.
Week
11

Module 4: Numerical Integration

9h

Unit 1: Introduction to Numerical Integration

4 study hours
  • •Understand the concept of numerical integration.
  • •Differentiate between analytical and numerical approaches to integration.
  • •List various known methods for numerical quadrature.
  • •Understand the role of polynomial approximation in numerical integration.

Unit 2: Trapezoidal Rule

5 study hours
  • •Derive the Trapezoidal rule geometrically.
  • •Use the Newton-Gregory formula to derive the Trapezoidal rule.
  • •Implement the Trapezoidal rule to evaluate a definite integral.
  • •Estimate the error in the Trapezoidal rule.
Week
12

Module 4: Numerical Integration

10h

Unit 3: Simpson's Rules

5 study hours
  • •Derive Simpson's 1/3 rule using Newton Forward formula.
  • •Distinguish between Simpson's rule and Trapezoidal rule.
  • •Apply Simpson's 1/3 rule to evaluate definite integrals.
  • •Understand the conditions for applying Simpson's rule.

Unit 4: Newton-Cotes Formulas

5 study hours
  • •Derive the Simpson's 3/8 rule.
  • •Understand the structure of Newton-Cotes formulas.
  • •Apply Newton-Cotes formulas to evaluate definite integrals.
  • •Compare the accuracy of different Newton-Cotes formulas.
Week
13

Module 5: Boundary Value Problems

12h

Unit 1: Introduction to BVP

6 study hours
  • •Distinguish between Initial Value Problems and Boundary Value Problems.
  • •Derive finite difference schemes for solving BVPs.
  • •Solve BVPs using finite difference schemes.
  • •Understand the application of Taylor series in deriving finite difference approximations.

Unit 2: BVP involving Partial Differential Equation

6 study hours
  • •Define a second-order PDE and a BVP involving a PDE.
  • •Classify various types of PDEs (parabolic, elliptic, hyperbolic).
  • •Classify types of boundary conditions for PDEs (Dirichlet, Neumann, Mixed).
  • •Derive finite difference schemes for PDEs.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

Read the complete course material as provided by NOUN.

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Review all examples and exercises in each unit, focusing on the application of formulas.

2

Practice deriving key formulas like Trapezoidal and Simpson's rules to understand their origins.

3

Create concept maps linking different approximation and interpolation techniques.

4

Focus on understanding the error terms for each numerical method to assess accuracy.

5

Practice solving boundary value problems using finite difference methods with varying step sizes.

6

Review past TMAs and identify areas needing further clarification.

7

Allocate sufficient time for practicing numerical problems, as this is a calculation-intensive course.

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