## Posts Tagged 'Numerical Analysis'

### Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein

In last week’s post we looked at aspects of puzzles of the form “What is the next number”. We are presented with a short list of numbers, for example ${1, 3, 5, 7, 9}$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as the next number.

Borwein integrals evaluated by Mathematica. The first seven integrals are all equal to ${\pi}$. The eighth is a tiny bit less than this.

In this article we consider a sequence of seven ones: ${1, 1, 1, 1, 1, 1 ,1}$. Most people would agree that the next number in the sequence is ${1}$. We will show how the number ${1 - 1.47\times10^{-11} \approx 0.999\,999\,999\,985}$ could be the “correct” answer. Continue reading ‘Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein’

### What’s the Next Number?

We are all familiar with simple mathematical puzzles that give a short sequence and ask “What is the next number in the sequence”. Simple examples would be

$\displaystyle \begin{array}{rcl} && 1, 3, 5, 7, 9, 11, \dots \\ && 1, 4, 9, 16, 25, \dots \\ && 1, 1, 2, 3, 5, 8, \dots \,, \end{array}$

the sequence of odd numbers, the sequence of squares and the Fibonacci sequence.

### The Power of the 2-gon: Extrapolation to Evaluate Pi

Lewis Fry Richardson

Richardson’s extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of ${\pi}$, and show how the estimates improve dramatically with higher order extrapolation.

[This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).]

### Following the Money around the Eurozone

Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican City use euro coins so, from 2015, there have been 23 countries, each with its own national coin designs. For the €1 and €2 coins, there are 23 distinct national patterns; for the smaller denominations, there are many more. Thus, there is a wide variety of designs in circulation.

National designs of Finland, France, Germany, Ireland and Netherlands.

### Real Derivatives from Imaginary Increments

The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.

Rounding error and formula error as functions of step size ${h}$ [Image from Wikimedia Commons].

For finite difference approximations, the choice of step size ${h}$ is crucial: if ${h}$ is too large, the estimate of the derivative is poor, due to truncation error; if ${h}$ is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if ${h}$ is very small, they produce zero values.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing ${f^\prime(x)}$.

### Can You Believe Your Eyes?

Scene from John Ford’s Stagecoach (1939).

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

### Does Numerical Integration Reflect the Truth?

Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The big question is:

Does the numerical solution resemble the true solution of the equation?

There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.

Smooth curve: True solution. Black dots: stable solution. Red dots: unstable solution (time step too large).

### Dimension Reduction by PCA

We live in the age of “big data”. Voluminous data collections are mined for information using mathematical techniques. Problems in high dimensions are hard to solve — this is called “the curse of dimensionality”. Dimension reduction is essential in big data science. Many sophisticated techniques have been developed to reduce dimensions and reveal the information buried in mountains of data.

### The Monte-Carlo Method

Learning calculus at school, we soon find out that while differentiation is relatively easy, at least for simple functions, integration is hard. So hard indeed that, in many cases, it is impossible to find a nice function that is the integral (or anti-derivative) of a given one. Thus, given ${f(x)}$ we can usually find ${d f /d x}$, whereas we may not be able to find ${\int f(x)\,d x}$.

### The Rambling Roots of Wilkinson’s Polynomial

Finding the roots of polynomials has occupied mathematicians for many centuries. For equations up to fourth order, there are algebraic expressions for the roots. For higher order equations, many excellent numerical methods are available, but the results are not always reliable.

A 10th-order polynomial (blue) and a slightly perturbed version, with the coefficient of ${x^9}$ changed by one part in a million.

### Bouncing Billiard Balls Produce Pi

There are many ways of evaluating ${\pi}$, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method.

### Staying Put or Going with the Flow

The atmospheric temperature at a fixed spot may change in two ways. First, heat sources or sinks may increase or decrease the thermal energy; for example, sunshine may warm the air or radiation at night may cool it. Second, warmer or cooler air may be transported to the spot by the air flow in a process called advection. Normally, the two mechanisms act together, sometimes negating and sometimes reinforcing each other. What is true for temperature is also true for other quantities: pressure, density, humidity and even the flow velocity itself. This last effect may be described by saying that “the wind blows the wind” [TM132 or search for “thatsmaths” at irishtimes.com].

Hurricane Ophelia approaching Ireland, 16 October 2017, 1200Z. Image from https://earth.nullschool.net/

### Marvellous Merchiston’s Logarithms

Log tables, invaluable in science, industry and commerce for 350 years, have been consigned to the scrap heap. But logarithms remain at the core of science, as a wide range of physical phenomena follow logarithmic laws  [TM103 or search for “thatsmaths” at irishtimes.com].

Android app RealCalc with natural and common log buttons indicated.

### Simulating the Future Climate

The Earth’s climate is changing, and the consequences may be very grave. This week, That’s Maths in The Irish Times ( TM040  ) is about computer models for simulating and predicting the future climate.

### French Curves and Bézier Splines

A French curve is a template, normally plastic, used for manually drawing smooth curves. These simple drafting instruments provided innocent if puerile merriment to generations of engineering students, but they have now been rendered obsolete by computer aided design (CAD) packages, which enable us to construct complicated curves and surfaces using mathematical functions called Bézier splines.

### Singularly Valuable SVD

In many fields of mathematics there is a result of central importance, called the “Fundamental Theorem” of that field. Thus, the fundamental theorem of arithmetic is the unique prime factorization theorem, stating that any integer greater than 1 is either prime itself or is the product of prime numbers, unique apart from their order.

The fundamental theorem of algebra states that every non-constant polynomial has at least one (complex) root. And the fundamental theorem of calculus shows that integration and differentiation are inverse operations, uniting differential and integral calculus.

The Fundamental Theorem of Linear Algebra

### The Lambert W-Function

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$
It would seem that when ${x>1}$ this must blow up. Surprisingly, it has finite values for a range of x>1. Continue reading ‘The Lambert W-Function’