# A7: Numerical Analysis - Material for the year 2019-2020

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16 lectures

Scientific computing pervades our lives: modern buildings and structures are designed using it, medical images are reconstructed for doctors using it, the cars and planes we travel on are designed with it, the pricing of "Instruments'' in the financial market is done using it, tomorrow's weather is predicted with it. The derivation and study of the core, underpinning algorithms for this vast range of applications defines the subject of Numerical Analysis. This course gives an introduction to that subject.

Through studying the material of this course students should gain an understanding of numerical methods, their derivation, analysis and applicability. They should be able to solve certain mathematically posed problems using numerical algorithms. This course is designed to introduce numerical methods - i.e. techniques which lead to the (approximate) solution of mathematical problems which are usually implemented on computers. The course covers derivation of useful methods and analysis of their accuracy and applicability.

The course begins with a study of methods and errors associated with computation of functions which are described by data values (interpolation or data fitting). Following this we turn to numerical methods of linear algebra, which form the basis of a large part of computational mathematics, science, and engineering. Key ideas here include algorithms for linear equations, least squares, and eigenvalues built on LU and QR matrix factorizations. The course will also include the simple and computationally convenient approximation of curves: this includes the use of splines to provide a smooth representation of complicated curves, such as arise in computer aided design. Use of such representations leads to approximate methods of integration. Techniques for improving accuracy through extrapolation will also be described. The course requires elementary knowledge of functions and calculus and of linear algebra.

Although there are no assessed practicals for this course, the tutorial work involves a mix of written work and computational experiments. Knowledge of Matlab is required, but many examples will be provided.

At the end of the course the student will know how to:

Find the solution of linear systems of equations.

Compute eigenvalues and eigenvectors of matrices.

Approximate functions of one variable by polynomials and piecewise polynomials (splines).

Compute good approximations to one-dimensional integrals.

Increase the accuracy of numerical approximations by extrapolation.

Use computing to achieve these goals.

Lagrange interpolation [1 lecture]

Newton-Cotes quadrature [2 lectures]

Gaussian elimination and LU factorization [2 lectures]

QR factorization [1 lecture]

Eigenvalues: Gershgorin's Theorem, symmetric QR algorithm, polynomial rootfinding via eigenvalues [3 lectures]

Best approximation in inner product spaces, least squares, orthogonal polynomials [4 lectures]

Piecewise polynomials, splines [2 lectures]

Richardson Extrapolation. [1 lecture].

You can find the material for this course in many introductory books on Numerical Analysis such as

1) A. Quarteroni, R Sacco and F Saleri, *Numerical Mathematics* (Springer, 2000).

2) K. E. Atkinson, *An Introduction to Numerical Analysis* (2nd Edition, Wiley, 1989).

3) S. D. Conte and C. de Boor, *Elementary Numerical Analysis* (3rd Edition, Graw-Hill, 1980).

4) G. M. Phillips and P. J. Taylor, *Theory and Applications of Numerical Analysis* (2nd Edition, Academic Press, 1996).

5) W. Gautschi, *Numerical Analysis: An Introduction* (Birkhauser, 1977).

6) H. R. Schwarz, *Numerical Analysis: A Comprehensive Introduction* (Wiley, 1989).

But the main recommended book for this course is:

1) E. Süli and D. F. Mayers, *An Introduction to Numerical Analysis* (CUP, 2003). Of which the relevant chapters are: 6, 7, 2, 5, 9, 11.