MA2715 (and MA2895 material)

Summary of what was done each week in the lectures

The first lecture started with an overview of all the topics in MA2715. The printed notes is likely to be in 5 chapters. The first chapter was handed out at the first lecture and the first exercise sheet was handed out at the Thursday lecture. Comments were made about the numerical projects module which has Matlab labs on Thursday.

The lecture started with mostly revision which is the start of chapter 1. The underlined notation for column vectors and upper case letters for matrices was introduced and also the letter I was given for the identity matrix with e_1, ..., e_n for the standard base vectors. The inner product and the outer product of vectors was given. Linear independence and linear dependence of vectors was defined. The product A*x was written as a combination of the columns of A. The definition of eigenvalues and eigenvectors was given and that eigenvalues are solutions of the characteristic equation.

Spectrum and spectral radius were defined and the computation of the eigenvalues and eigenvectors of a 2-by-2 matrix was done.

The vector norm axioms and the 2-norm, infinity-norm and 1-norm were given.

Definition of matrix norms induced by vector norms. The property norm(AB)<= norm(A)*norm(B) was proved.

The spectral radius of A is less than or equal to any induced matrix norm was shown.

The expression for the infinity matrix norm nd the 1-matrix norm with comments as to a vector x of size 1 in the norm such that norm(Ax)=norm(A). The matrix 2-norm in terms of the square root of the spectral radius of A^T*A.

A 3-by-3 example with the matrix norms computed was done.

A brief mention of the matrix condition number with a representation in terms of the eigenvalues of the matrix A in the real symmetric case.

Chapter 2 material on direct methods was started. An example with an upper triangular system solved by back substitution and a lower triangular system solved by forward substitution was done. The back substitution algorithm in the general n-by-n case was given and the number of operation involved was determined. Comments about forward substitution for lower triangular systems in the general n=by-n case. Solving A*x=b when A=L*U in 2 stages, forward substitution for L*y=b followed by back substitution for U*x=y and the operation count.

The structure of basic Gauss elimination was mentioned.

Basic Gauss elimination to get to upper triangular form. A matrix description using Gauss transformation matrices M_k. The formula for M_k^{-1} and the product M_1^{-1}...M_k^{-1} has a simple representation as a unit lower triangular matrix. Thus A=LU when basic Gauss elimination runs to completion.

For a nxn matrix the number of operations grows like 2n^3/3 where an operation mostly means *, + or - (there are a smaller number of divisions).

A=LU implies that we immediately have the corresponding factorization of all principal sub-matrices.

The determinant of A is the product of the diagonal entries of U.

Basic Gauss elimination is possible when we always have non-zero pivot elements and this is when all principal sub-matrices are non-singular.

A=LU implies that we immediately have the corresponding factorization of all principal sub-matrices and to get L and U we can do the computations in a different order to what is done with basic Gauss elimination. On the OHPs a 3-by-3 example indicating getting the entries with growing principal sub-matrices.

The need for (partial) pivoting which involves re-arranging the equations. Solving A*x=b when we have PA=LU.

An example with a badly conditioned unit lower triangular matrix.

Getting a column of the inverse of A when we know A=LU.

Brief mention of complete pivoting.

The start of chapter 3 and a reminder about solving u'=au with u(0)=u0. Generalization to systems of linear differential equations with a constant matrix A. The method involves getting the eigenvalues and the eigenvectors of A. If V denotes the matrix with the columns as the eigenvectors then we can get the solution satisfying a given initial condition when V is non-singular by solving a system of equations with V as the matrix. In this case the general solution is a linear combination of terms exp(\lambda_i x)v_i where \lambda_i is the eigenvalue and v_i is the eigenvector.

Three examples of systems of linear differential equations with n=2. In one cases the matrix had a complex conjugate pair of eigenvalues and eigenvectors.

The exponential matrix and a representation of the solution as u(x)=exp(xA)*u0 for all cases of A, i.e. deficient matrices as well. When A=V*D*inv(V) the representation is exp(xA)=V*exp(x*D)*inv(V).

u(x) tends to 0 as x tends to infinity when the real part of all the eigenvalues is negative.

Converting a higher order differential equation to a system of first order equations with the characteristic equation to find the eigenvalues being the same as the auxiliary equation in methods that has been covered in earlier modules.

The start of the finite difference method for the 2-point BVP which is chapter 4. A uniform mesh of an interval [a, b] and Taylor expansions about x_i evaluated at x_i+h and at x_i-h. The central difference finite difference approximations of the first and second derivative at x_i using values at x_i-h, x_i and x_i+h. Taylor's expansion with a remainder term is briefly mentioned.

The local truncation error.

The tri-diagonal structure of the linear system.

A short example with N=4 in setting up the 3-by-3 linear system to solve.

The tri-diagonal linear system involved in the general cases. Discussion of how this can be done efficiently in Matlab using spdiags to set-up the system. Matlab code as to how to do everything when u''=r.

Using U_0, U_1 and U_2 to approximate a derivative boundary condition at x_0. Similarly using U_N, U_{N-1} and U_{N-2} to approximate a derivative boundary condition at x_N.

A very brief look at numerical methods for the initial value problem. Taylor series methods and the growing complexity of the expressions for the derivatives in the general case. Runge Kutta methods which are methods which only use function values. Statement of Heun's order 2 method and statement of the popular Runge Kutta order 4 method.

The start of Fourier series which is chapter 5. The case of 2\pi periodic functions. Using the relation f(x+2\pi)=f(x) to sketch over more than 1 period. Statement of what is the Fourier series for a 2\pi periodic function f(x). Steps towards justifying the expressions starting with the definition of orthogonal functions on (-\pi, \pi) and trig-addition formulas to give expressions for products of cos(nx), cos(mx), sin(nx) and sin(mx).

Getting the series for f_1(x)=1 in [0, \pi] and 0 in (-\pi, 0).

Getting the series for f_2(x)=|x| in (-\pi, \pi].

Getting the series for f_3(x)=x in (-\pi, \pi].

Getting the series for f_4(x)=(x+|x|)/2 in (-\pi, \pi].

Left and right limits of a function f(x) at a point x. Statement of sufficient conditions for the series to converge to (f(x+)+f(x-))/2. This is f(x) at a point of continuity.

The of sum series obtained by evaluating the Fourier series at a point of continuity of f(x).

The half range cosine series and the half range sine series. The example of two representations of x on (0, \pi).

Face-to-face lectures were stopped across the university due to cornoavirus.

Face-to-face lectures were stopped across the university due to cornoavirus.

The university revision week.

Miscellaneous links