Numerical Linear Algebra

- Computer arithmetic. Fixed point computations, floating point computations, rounding errors in computations, numerically effective algorithms. - Error analysis. Laws of floating point arithmetic, Addition, multiplication and inner product of n-floating point numbers, floating point matrix operations, stability of algorithms and conditioning of problems. - Gaussian elimination and LU factorisation. LU factorisation using Gaussian elimination, partial and complete pivoting, Gauss-Jordan transformations, computation of the inverse of a matrix, Stability of Gaussian elimination. Cholesky factorisation. - Numerical Solutions of Linear Systems. Direct methods: Solution of upper and lower triangular linear systems, solution of a system using LU factorisation and their stability, solving linear systems with multiple right-hand side. Sensitivity analysis of linear systems. Iterative methods: sparse matrices, stationary methods, conjugate gradients, preconditioning.

Objectives

NA

Prerequisites

The course has no prerequisites

Syllabus

Computer arithmetic. Fixed point computations, floating point computations, rounding errors in computations, numerically effective algorithms. Error analysis. Laws of floating point arithmetic, Addition, multiplication and inner product of n-floating point numbers, floating point matrix operations, stability of algorithms and conditioning of problems. Gaussian elimination and LU factorisation. LU factorisation using Gaussian elimination, partial and complete pivoting, Gauss-Jordan transformations, computation of the inverse of a matrix, Stability of Gaussian elimination. Cholesky factorisation. Numerical Solutions of Linear Systems. Direct methods: Solution of upper and lower triangular linear systems, solution of a system using LU factorisation and their stability, solving linear systems with multiple right-hand side. Sensitivity analysis of linear systems. Iterative methods: sparse matrices, stationary methods, conjugate gradients, preconditioning.