Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

The need for multiple integrations, integrating the results of integration, occurs in content problems that we have in our space of three physical dimensions. The need also occurs in theoretical investigations that may involve a great many dimensions, see next.

*
Schlafli
, 1814 - 1895, a Swiss mathematician is credited with developing the Euclidean geometry
of n dimensions. He introduced the notion of a group of n coordinate values
representing a point in an n space. He introduced polytopes as higher dimensional
analogues of polygons and polyhedra. *

*In a modern computer or communications context,
all of the 2^64 possible 64-bit binary words can be described as the coordinates of
the vertices of a cube in 64-dimensions.*

*While
Euclid
~325 BC -~265 BC, proved that there are exactly five regular solids in a three-dimensional
space, Schlafli proved that there are exactly six in four dimensions and exactly
3 in five and higher dimensions. *

*Schlafli made an important contribution
to non-Euclidean geometry by proposing that spherical 3-dimensional space could
be regarded as the surface of a hyper sphere in Euclidian 4-dimensional space.*

As numerical integration provides only an approximation to a true integration, one can expect the result of successive such integrations to worsen in comparison with the true result. To test this expectation the true result of successive integrations must be accurately known. Since the integration of sinusoids results in sinusoids the sinusoid was chosen as the integrand. A simple numerical integration process was chosen for study.

It was decided first to employ the 4-point Gaussian Integrator to integrate sin(x). This emulates the use of a very precise DA Input Table. Then, next, to follow with a sequence of integrators each employing rectangular quadrature.Each integration employed 1000 equal steps in the range x = 0 to x = 2 * π. The "Input Table" section is shown next.

The expressions contained in the outlined cells, foregoing, can be seen by clicking their addresses next.

. | . |

B1 | I2 | F7 | G7 | H7 | I7 |

B1 contains the step size and I2 contains the maximum integration error. F7 contains the weighted evaluations of sin(x) at the 4 Gaussian points multiplied by the step size.

G7 shows a step of the accumulation given the initial value of -1 in G6. This is expected to be, closely, the negative cosine of x, A7.

The negative cosines of x are calculated in column H and can be seen to agree quite closely with the accumulator values, the results of 4-point Gaussian Quadrature.

The results for some of the calculation steps of the first two of 61 successive rectangular quadrature integrators are shown next. The maximum errors are shown in cells M2 and G2.

The expressions contained in the outlined cells, foregoing, are available next.

. | . |

J7 | N7 |

The essential difference from the Input Table that receives its input from the sine function is that the first of these integrators receives its input, Dy, from the accumulator of the Input Table and the second of these integrators receives its input from the accumulator of the first integrator.

Each of the remaining 59 integrators is coupled to its predecessor in the same manner.

Note that the maximum error at the accumulator of these integrators seems to increase as the number of integration stages increases.

The final two integrators of the sequence are shown next.

Note that the maximum errors have increased but remain quite tolerable. This may be better grasped from the chart following:

The eye would not be able distinguish the foregoing plots from -cos(x) and -sin(x).

The behaviour of the maximum error versus the stage of integration can be of interest. The maximum errors have been collected into an array with four columns so that all integration outputs that are nominally the same function occur in the same column. That array follows:

Note that the maximum errors seem to settle to values that appear to be identical for all subsequent integrations that result in the same function. This may not be true. The foregoing figure shows 16 decimal digits and after stage 7 there are four leading zeros. With Excel's arithmetic there can be no more than 16 significant digits. This suggests that 4 more decimal digits could be shown with likely little significance to be attached to the last digit. That array follows:

It is now seen that after stage 30 of integration the maximum errors remain constant and, as one might expect, do not depend on the sign of the function. The constancy achieved by the maximum errors is a remarkable result!

What is the effect of reducing step size? With 1/2 or 1/4 of the step size, the values of the functions will still cover the range 0 to 1. See the effect on maximum error of half the step size next.

The maximum rectangular quadrature error in the stabilized zone has been reduced by a factor of about 10.

Next, the step size is again halved.

The rectangular integration maximum errors in the stabilized zone have again decreased by approximately an order of magnitude.

Notice that the reduction of step size has resulted in earlier stage stabilization of the maximum errors.

The same is true for an electronic integrator such as an Op Amp in that electrical transmission time, albeit very small, is required before its output can be applied to its input.

There is a similar effect, a very small delay in the mechanical transmission of force, when the output of a mechanical integrator is returned to its input.

In numerical integration the size of the first step can be made quite small relative to the size of the steps that follow should a smaller initial lag be desired.

Feeding the output of stage 60 back as the input to stage 1 is accomplished by changing cell J8 from =(G7+G8)/2*$B$1 to =(IM6+IM7)/2*$B$1, and changing subsequent cells in column J accordingly. Cell J7 was set to 0 and is to be considered later in this topic. The outputs of the 60th and 61st integrators are shown next.

Surprise! As far as the eye can tell, both are sinusoids.

The maximum errors are shown next.

This change has reduced the worst error, stage 1, by a factor of about 4.

The worst of these occurs with the first three integrators. Perhaps this could be ameliorated by selecting a value for cell J7, a cell that had been set to 0.

Now employ

The new value for cell J7 is:

The 58th stage will be fed back to illustrate. Also, the value for J7 will again be chosen to minimize the maximum error of the first three integrators. The result is seen next.

This case required exactly the same value for cell J7 as did the previous case and provides identical maximum errors.

Although the foregoing adjustments of J7 did lessen maximum error, the process, although interesting, does not seem very general in its application, although it could possibly be of use at times. Accordingly, the value 0 is restored to cell J7

A general procedure to lessen error is to employ more steps and make them small enough to satisfy ones own precision needs.

Maximum error was then calculated for each integrator output for 4 equal ranges of x, 0 to π, π to 2π, 2π to 3π, and 3π to 4π, every half cycle of the sinusoids. These errors are shown next.

The sinusoids from the first two integrators are plotted next.

Should one desire more precision, the step size can be reduced further and more steps employed.

The viewer might wonder whether variable step size might be used to improve precision. Don't know? Give it a try!

The spreadsheet has been shown to be an excellent tool for the exploration of fed back and non-fed back systems of integrators and their associated parameters, albeit in a very particular situation.

Subsequent topics will delve further into aspects of applying the spreadsheet integrator.

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