# KC5000 - Further Computational Mathematics

## What will I learn on this module?

This module continues the numerical methods and computational mathematics thread established with KC4012: Computational Mathematics. The module aims to present an introduction to advanced numerical mathematics, in particular multivariable problems, and associated transferable skills. Numerical methods are applied to the solution of several classes of problems, including: systems of linear and nonlinear equations, eigensystems, optimisation, ordinary and partial differential equations. Theoretical aspects are illustrated and discussed at the lectures, and computational implementation developed at the computer-lab workshops, using appropriate software (e.g. MATLAB).

Topics may include (note this is indicative rather than prescriptive):
1. Vector and matrix spaces: normed spaces; vector norms; matrix norms; compatible norms; spectral radius; condition number.
2. Systems of linear equations: direct and iterative methods.
3. Matrix eigensystems: iterative methods for eigenvalues and eigenvectors.
4. Systems of nonlinear equations: multidimensional Newton method; fixed-point iterations method.
5. Numerical optimization: pattern search methods; descent methods.
6. Ordinary differential equations (ODEs): forward and backward Euler methods; Crank-Nicolson method; convergence, consistency and stability of a method; conditional stability; simple adaptive-step methods; Runge-Kutta methods; predictor-corrector methods; Heun method; systems of ODEs; stiff problems.
7. Numerical approximation of initial, boundary value problems (IBVP) for ordinary and partial differential equations (PDEs): finite difference method for the (Dirichlet) IBVP for the one- and two-dimensional Poisson equations; finite difference method for the (Dirichlet) IBVP for the one-dimensional heat equation; finite-difference method for the (Dirichlet) IBVP for the one-dimensional wave equation.

# How will I learn on this module?

You will learn through a combination of lectures, computer-lab workshops, and independent learning. Lectures give a formal introduction to theoretical aspects of numerical mathematics. Computer-lab workshops offer the opportunity to implement the numerical methods using appropriate programming software (e.g. MATLAB) and to apply them to the solution of problems of mathematical and applicative relevance, including problems coming from physics, biology, chemistry and engineering. Workshops will also be an opportunity to present you with open research problems, and will strengthen your transferable skills and employability. Northumbria’s computer labs and facilities are fully equipped with the latest industry-standard software.

Assessment is by a closed-book, computer-lab-based in-class test, worth 30%, and a formal closed-book, computer-lab-based examination, worth 70%. The test will assess your problem solving abilities, for instance by asking the implementation and use of a method that has not been directly presented in class, but whose understanding is readily achievable by means of the knowledge acquired in class. The examination will cover all aspects of the module and will assess your knowledge of the discipline, along with your problem solving abilities.

Exam feedback will be provided individually and also generically to indicate where the cohort has a strong or a weaker answer to examination questions. You will receive both written and oral feedback from the coursework assessment, as well as formative feedback throughout the course, in particular during the problem-solving/computer-based workshops.

Independent study is supported by further technology-enhanced resources provided via the e-learning portal, including codes for the programming software used (e.g. MATLAB), e-lecture notes and seminar sheets with answers and solution, and past paper questions.

# How will I be supported academically on this module?

Lectures and workshops will be the main point of academic contact, providing you with a formal teaching environment for core learning. In particular, workshops will provide you with opportunities for critical enquiry and exchanges. Outside formal scheduled teaching, you will be able to contact the module team (module tutor, year tutor, programme leader) either via email or the open door policy operated throughout the programme. Further academic support will be provided through technology-enhanced resources via the e-learning portal. You will also have the opportunity to give your feedback formally through periodic staff-student committees and directly to the module tutor at the end of the semester.

# What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk

# What will I be expected to achieve?

Knowledge & Understanding:
1. Understand theoretically and implement numerically methods and schemes to solve multivariable problems in numerical mathematics, including: systems of linear and nonlinear equations, eigensystems, optimisation, ordinary and partial differential equations (see KU1 from Programme Learning Outcomes).
2. Implement and apply numerical methods using a mathematical software / a programming language (e.g. MATLAB), understanding and discussing critically the results obtained (see KU2 and KU3 from Programme Learning Outcomes).

Intellectual / Professional skills & abilities:
3. Construct rigorous mathematical arguments to build, implement and apply, using a mathematical programming language, numerical methods allied to computational techniques, understanding their effectiveness and range of applicability (see IPSA1 and IPSA2 from Programme Learning Outcomes).
4. Understand, compare, select and critically evaluate a range of numerical methods, their effectiveness and the resulting numerical output, typically from a mathematical software (see IPSA3 from Programme Learning Outcomes).

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Demonstrate critical enquiry and the ability to learn independently and to manage time and resources (see PVA1 and PVA4 from Programme Learning Outcomes).

# How will I be assessed?

SUMMATIVE
1. Lab-based test (30%) – 1, 2, 3, 4, 5
2. Lab-based exam (70%) – 1, 2, 3, 4, 5

FORMATIVE
Problem-solving/computer-based workshops
– 1, 2, 3, 4, 5

Feedback will take several forms, including verbal feedback during the computer-lab workshops; individual verbal and written comments on the coursework assessment delivered in class and via blackboard; written feedback on the exam.

None

None

# Module abstract

Further Computational Mathematics is designed to introduce you to advanced numerical mathematics, in particular multivariable problems and associated transferable skills, extending the thread established with MS0402: Computational Mathematics. Numerical methods are rigorously derived, compared, and applied, using appropriate software (e.g. MATLAB), to the solution of several classes of problems, spanning from systems of linear and nonlinear equations, to ordinary and partial differential equations.

You will learn through a combination of lectures and problem-solving/computer-based workshops. Lectures give a formal introduction to theoretical aspects of numerical mathematics. Computer-lab workshops offer the opportunity to implement and apply the numerical methods to the solution of problems coming from mathematics, physics, biology, chemistry, engineering, and finance. Workshops will also be an opportunity to present you with open research problems, and will strengthen your transferable skills and employability.

You will be assessed by a closed-book, computer-lab-based in-class test, worth 30%, and a formal closed-book, computer-lab-based examination, worth 70%.

# What will I learn on this module?

This module continues the numerical methods and computational mathematics thread established with KC4012: Computational Mathematics. The module aims to present an introduction to advanced numerical mathematics, in particular multivariable problems, and associated transferable skills. Numerical methods are applied to the solution of several classes of problems, including: systems of linear and nonlinear equations, eigensystems, optimisation, ordinary and partial differential equations. Theoretical aspects are illustrated and discussed at the lectures, and computational implementation developed at the computer-lab workshops, using appropriate software (e.g. MATLAB).

Topics may include (note this is indicative rather than prescriptive):
1. Vector and matrix spaces: normed spaces; vector norms; matrix norms; compatible norms; spectral radius; condition number.
2. Systems of linear equations: direct and iterative methods.
3. Matrix eigensystems: iterative methods for eigenvalues and eigenvectors.
4. Systems of nonlinear equations: multidimensional Newton method; fixed-point iterations method.
5. Numerical optimization: pattern search methods; descent methods.
6. Ordinary differential equations (ODEs): forward and backward Euler methods; Crank-Nicolson method; convergence, consistency and stability of a method; conditional stability; simple adaptive-step methods; Runge-Kutta methods; predictor-corrector methods; Heun method; systems of ODEs; stiff problems.
7. Numerical approximation of initial, boundary value problems (IBVP) for ordinary and partial differential equations (PDEs): finite difference method for the (Dirichlet) IBVP for the one- and two-dimensional Poisson equations; finite difference method for the (Dirichlet) IBVP for the one-dimensional heat equation; finite-difference method for the (Dirichlet) IBVP for the one-dimensional wave equation.

### Course info

UCAS Code G101

Credits 20

Mode of Study 4 years full-time or 5 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2020

## Mathematics MMath (Hons)

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