Numerical analysis and methods are applied to a very wide range of subject areas. Indeed, whenever a mathematical problem is encountered in science or engineering, which cannot readily or rapidly be solved by a traditional mathematical analysis, then a numerical method is usually sought and a numerical analysis carried out.
This one-week course aims to give a broad understanding of the theoretical basis for numerical analysis, including an awareness of the nature and types of numerical errors that occur in computer calculations, and extensive "hands-on" experience in the application of numerical methods. In this context, a wide variety of elementary but important problems will be considered, such as collocation and interpolation, differentiation, integration, initial value problems for differential equations, non-linear algebraic equations, and systems of linear algebraic equations.
This course is specifically intended for practising scientists and engineers of all disciplines who need to understand the fundamental principles of modern numerical analysis and key numerical methods for the computer solution of basic scientific problems. There are no rigid entry requirements, but it is expected that students will be educated to degree level in a scientific discipline.
The course includes:
Nature of numerical analysis:
Introduction to
computational methods and the requirement for a numerical analysis. The
floating-point representation of a number in the computer, and analysis of
floating point arithmetic. The existence of rounding error, significance error
and truncation error. Consideration of the posedness and conditioning of a
problem and the stability of a numerical method used to solve it.
Perturbation analysis for systems of
linear equations:
Introduction to vector and matrix norms as a measure of
magnitude. The properties of vector and matrix norms, and their application in
perturbation analysis for systems of linear equations. Definition of the
Condition Number of a matrix, and its effect upon the upper bound for the
relative error of the solution arising from small perturbations in the data of
the problem. The importance of a forward and backward error analysis when
solving a system of linear equations by a numerical method.
Non-linear algebraic equations:
The uni-variate
non-linear equation f(x) = 0 and the existence and uniqueness of a
solution in an interval (a,b). Algorithms for the Bisection method, the Secant
method, the Newton-Raphson method and the method of Fixed-Point iteration are
developed. Aitken's Acceleration technique for linearly convergent iteration
methods. For each of these methods, the rate of convergence and a full error
analysis is provided.
Matrices and systems of linear equations:
Review of matrices and fundamental matrix algebra. Systems of linear
equations and particular structures which frequently occur in
practice. Existence
and uniqueness of the solution.
Decomposition methods for systems of
linear equations:
The method of Gaussian Elimination with Back-substitution,
and the requirement for partial pivoting and row scaling in practical
application. The LU decomposition algorithm with Forward- and
Back-substitution. The Cholesky Decomposition algorithm for symmetric,
positive-definite systems of linear equations. Each of these algorithms is
derived by considering a full matrix but amendments for particular matrix
structures are discussed.
Iteration methods for systems of linear equations:
The iteration algorithms of Jacobi and
Gauss-Seidel, and the method of Successive-over-Relaxation, for systems of
linear equations with a sparse matrix structure. Conditions for convergence of
each technique and estimation of the optimum acceleration parameter in the SOR
method for large systems. The Conjugate Gradient method is introduced as a
method for solving a system of linear equations, and the requirement for
pre-conditioning is considered.
Collocation and interpolation:
Definition of the collocation problem. Collocation
to n+1 data points by a polynomial of degree n. The Lagrange
representation of the collocating polynomial is developed to demonstrate the
existence of a solution, and the Uniqueness Theorem for polynomial collocation
is discussed. An error analysis for the Lagrange representation is also given.
Hermite collocation to function values and derivatives. Orthogonal polynomials
are introduced and the advantages of collocating at the zeros of an orthogonal
polynomial are demonstrated.
Finite difference operators:
Definition of the finite difference operators, and derivation
of the relationships that exist between them. The application of finite
difference operators is demonstrated by deriving Newton's Forward
Difference and Backward Difference Interpolation formulae, and the associated
formation of Difference Tables.
Numerical differentiation:
Derivation of
difference formulae for approximating derivatives of functions from discrete
data, eg Forward and Backward Difference formulae and Mean Central Difference
formulae with a discussion of the truncation error in each case.
Numerical integration:
Introduction to
open and closed Newton-Cotes quadrature formulae for the approximation of
definite integrals. Derivation of the Trapezium Rule and Simpson's Rule, plus
composite forms, and Romberg's extrapolation method. Derivation of Gaussian
quadrature rules, with the Gauss-Legendre rule being used for
demonstration. Adaptive
techniques for numerical integration with an example based on Simpson's rule
being used for demonstration.
Initial value ordinary differential equations:
Introduction to numerical methods for solving first
order initial value ordinary differential equations. Taylor's series
derivation of Euler's method and the Euler-Trapezium method; the
Taylor's series method
is also discussed. Runge-Kutta algorithms are considered, without derivation,
and a comparison between Runge-Kutta and predictor-corrector methods is
presented. Application of these numerical methods to higher order initial value
problems is also considered.
Practicals:
Each of the
major topics is supported by a practical session.
The course lectures will be given by the teaching and research staff of the Applied Mathematics and Operational Research Group under the direction of Dr Stephen Shaw with the assistance of other colleagues.