A Problem involving Simpson’s Rule

Since I’m teaching a course on Computational Physics here in Maynooth and have just been doing methods of numerical integration (i.e. quadrature) I thought I’d add this little item to the Cute Problems folder. You might answer it by writing a short bit of code, but it’s easy enough to do with a calculator and a piece of paper if you prefer.

Use the above expression, displayed using my high-tech mathematical visualization software, to obtain an approximate value for π/4 (= 0.78539816339…) by estimating the integral on the left hand side using Simpson’s Rule at ordinates x =0, 0.25, 0.5, 0.75 and 1.

Comment on the accuracy of your result. Solutions and comments through the box please.

HINT 1: Note that the calculation just involves two applications of the usual three-point Simpson’s Rule with weights (1/3, 4/3, 1/3). Alternatively you could do it in one go using weights (1/3, 4/3, 2/3, 4/3, 1/3).

HINT 2: If you’ve written a bit of code to do this, you could try increasing the number of ordinates and see how the result changes…

P.S. Incidentally I learn that, in Germany, Simpson’s Rule is sometimes called called Kepler’s rule, or Keplersche Fassregel after Johannes Kepler, who used something very similar about a century before Simpson…


3 Responses to “A Problem involving Simpson’s Rule”

  1. There are some Great Ideas, ideas which are obvious in retrospect but at the time were far from obvious, and which are very efficient. One is FM synthesis (as opposed to additive or subtractive), I believe first introduced by Yamaha’s DX7. Another is the Bulisch-Stoer integration method. Be sure to teach it!

  2. Very interesting. Yet, there are many ways to calculate π/4.
    In my view, the issue goes way beyond of how to calculate it but about what the MEANING of this π/4 carries.
    We have always taken the ‘integration from zero to infinity’ for granted in math.

    In fact, there are some fundamental issues here.
    One, there are at least two infinities in math (discounting the continuum hypothesis) while there is seemingly without any infinity in the physical universe as a physical REALITY.

    Two, are the two zeros equal?
    Zero (c) = 1/ (countable infinity)
    Zero (u) = 1/ (uncountable infinity)
    There is no reason for these two zeros to be equal if the two infinities are not equal.

    Three, there is no way to transform infinity to finite in math and vice versa. But, can infinity (of math) transform into concrete OBJECT (of physics)?

    If the answer is NO, then there is no direct link between math and physics, although math is a very useful tool and language for physics.

    If the answer is YES, then physics and math can be unified at a rock bottom level.

    I would like to show that Yes is the right answer with some PHYSICS evidences.

    First, the COUNTABLE infinity is embedded in TRISECTING an angle: it takes COUNTABLE steps to trisect any angle by evenly dividing it, with the following steps.
    1/3 = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/256 + 1/512 – 1/1024 + 1/2048 -… +…
    = .33349 – … + … = .3333333333333…..
    Of course, a trisected angle is definitely a concrete object. If humans do not have the time to complete this task, nature will definitely have a better chance of doing it.

    Can uncountable infinity manifest into a concrete object? Of course, it must, for the rising of a physical universe from a singular point (zeros).
    In 1/3 = .3333333….., it has countable digits (taking countable steps). Yet, for the number pi (3.14159…), it has uncountable digits as it is a “normal” number.

    {(Pi / 4) = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 – … + … (with “countable” infinite steps)}

    That is, pi (having uncountable digits) is reached with the above equation. Yet, there is a concrete object (a circle) which always associates with the number pi. Thus, the uncountable infinity MANIFESTS as a circle which is a concrete object.

    This COUNTABLE transformation demands a trisected universe: three quark colors, three generations, etc.

    The UNCOUNTABLE transformation demands a universe-lock (the Alpha, electron fine structure constant), which also leads to the Planck CMB data.

    In math, these two equations {Eq(1/3) and Eq(Pi / 4)} show that ODD number can only reached by EVEN numbers and vice versa. And, this leads to the phenomena of “unreachable numbers”, show up in the:
    Fermat’s last theory, cannot be reached by nature numbers.
    All kind of PRIME numbers issues: cannot be reached by multiplication-operations, etc.

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