A Beginner's Guide to Numerical Methods in MATLAB

How do you store numbers using only 0s and 1s?

How computers actually store numbers. A discussion of mantissa, exponent and bias.

We talk about the notion of the machine epsilon and the fact that it's not really that useful when specifying calculation tolerances.

The two types of errors that reduce the accuracy of numerical methods.

What makes a function continuous?

The claim that a differentiable function that has equal states at two distinct point has a stationary point somewhere in between.

Another theorem you need to be aware of.

If a<b and f(a)f(b)<0 then the root of f(x) = 0 lies between a and b.

What are one-point iterative methods and what are they used for?

The notion of convergence (and divergence, too).

A look at Aitken's Δ² process and Steffensen's method.

What is the order of convergence and why do we care?

An extremely efficient and popular root finding method. Quadratic convergence, woo-hoo!

Newton's method applied in many dimensions. Useful for solving systems of non-linear equations!

What if you cannot get the Jacobian matrix in analytic form? Use finite differences! (Note: finite differences are actually discussed in a later section, so you can come back to this clip later.)

A way of speeding up polynomial evaluations.

A very simple method that leverages Bolzano's theorem.

Similar to the Bisection method, Regula Falsi can, in most cases, provide faster convergence than the Bisection method.

Yet another single-point iteration method.

An introduction to the concept of interpolation, with a simple example.

A better way of defining the interpolating polynomial.

Did you think the Newton basis was cool? With divided difference, you don't even have to solve the triangular set of equations!

The derivations of divided differences took too much time, so the examples get their own separate lesson.

What kind of simplicifications can be made to divided differences if we assume the points are equally spaced?

An interpolation formula for Lagrange polynomial.

An improvement of Lagrangian interpolation.

Finite difference approximations of derivatives - forward, backward and central differences.

Now a formula for the 2nd derivative approximation.

A look that the error terms in first and second derivatives that arise from using finite difference methods.

A method of combining approximations for improving accuracy.

Why would we want to integrate things numerically?

The simplest way of estimating the value of an integral.

Subdivide an integral into several strips, evaluate functions as midpoints, treat strips as rectangles. Profit!

A way of numerically calculating a specific type of integral.

Another numeric procedure for a very specific integral. Usable for calculating the Gamma function!