Thoughts on Computational Methods

Thoughts on Computational Methods

Assigned to students of Fall 2000 in EML 3041- Computational Methods course, students were asked to write a 300 word essay with the following lead sentence.

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nano-machines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development.

Read their collective writings. I gave an option whether they wanted to put either their names or their initials next to their contribution. Judging from how many students chose the latter option, I am keeping my eyes open at night as I do not know how many of the students in the class are running from the law!

Roberto Montane
Todd Rebol
JF

Cyntra Sonnylal
Greg Rosa

Giancarlo Battaglini

Jack Al-Kahwati

Jose Busquets
Sal Nsheiwat

Compiled by Roberto Montane, Fall 2000
From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nano-machines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development.

Throughout all walks of life computational methods (C.M.) is used to explain scientific development and discoveries by means of mathematical modeling, solving, and approximating equations. C.M can be defined as the art of converting real life problems into mathematical solutions; the mathematical equations derived by C.M allow us- the engineers to interpret the data accurately and systematically so that we may define and solve major real world issues.

To apply these mathematical concepts we use a variety of different means, some of which taught by Dr. Kaw include Roots of Equations, Linear Algebraic Equations, Interpolation, Numerical Differentiation and Integration, and Ordinary Differential Equations. These means can be combined for tough problems but in general serve very well in solving complex tasks.

Through the use of these concepts, C.M has been able to predict many variables from velocities and positions for satellite installations, and missile guidance to the determination of the most efficient setup for a home sprinkler system. Previously these computations were tedious and often not monetarily feasible, however through the development and continuous improvement of the personal computer C.M has been brought to the fore front. It is because of this that we can afford to concentrate on solutions and correct interpretations of problems by leaving the tedious computations to the computers.

For all the power and knowledge we have gained through computational Methods I would like to thank all those before me who have contributed to its success, and of course Dr. Kaw for the involvement he encourages in class.
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Compiled by Todd Rebol , Fall 2000
From the earliest times of approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nano-machines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development. Well if that first sentence left you scratching your head, do not feel bad because I did too.

Simply defined, computational methods are nothing more than different techniques to reformulate complicated mathematical problems so they can be solved by using simple arithmetic operations. Most different types of computational methods do have a couple things in common, many numbers and tedious calculations. Thus, enters an engineer's best friend, the computer. The lower cost and increased technology of computers have elevated the use and development of computational
methods.

The combination of the personal computer and numerical methods are invaluable items in an engineer's "toolbox." Together they can help solve complex mathematical models that may otherwise be extremely difficult or perhaps impossible to solve.

Besides problem solving, computational methods can also reinforce the understanding of mathematics in general. This is of course needed by all engineers. Computational methods breakdown higher mathematics into more basic arithmetic operations making several topics easier to comprehend. Some of the mathematical subjects that are involved in computational
methods are as follows:
1. Systems of linear algebraic equations
2. Curve fitting (regression and interpolation)
3. Integration
4. Ordinary and partial differential equations.

Numerical methods are often used to help solve mathematically modeled systems in many fields of research and development. Aircraft manufacturing, oil exploration, and gaining a better understanding of the circulation of the global oceans are just a few examples of specific topics that utilize the tools of numerical methods in the real world.

Combining numerical methods along with other areas of knowledge is crucial to continue to improve as an engineer. No matter which discipline of engineering, numerical methods will eventually cross your path.
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Compiled by J.F Fall , Fall 2000

Archimedes' methods were simpler than some of today's computational methods but not anymore important. He first showed that the perimeter of regular polygons circumscribed about the circle eventually becomes less than three and one-seventh the diameter as the number of their sides increases. Then he showed that the perimeters of the inscribed polygons eventually become greater than three and ten seventy-firsts. Therefore, he said that these became the upper and lower bounds of pi. This is a great example of using numbers and rounding to solve a difficult problem.

Numerical methods are used in everything. The computer would not be invented today without numerical methods. The calculator is an example of numerical methods, as it uses numerical methods to calculate answers. I can see why it would be important to learn numerical methods because you would be able to understand all of the procedures that computers are based upon. Any computer person would benefit to understand these procedures and to learn even more complicated ones.

I am sure Archimedes would be pleased to see the technologies that we have today and realized that they are related to the methods he was using back in 250 BC. He was in the early stages of mathematics when geometry was the main subject. Mathematics has shaped our world into the computer world we have today. All the web and internet companies would not be possible without the numerical methods that were started so long ago. It is amazing to me because I have never thought of it that way.
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Compiled by Tomasz Talaljaj , Fall 2000

In today's world, most developed nations would not be able to function properly or function at all as they heavily rely upon technology based on or derived from computational methods. Nearly every aspect of daily life is dependent upon computers and their software. Whether going to the bank, grocery shopping or making a phone call; there is certainly some form of contact and interaction with the computers or computer controlled devices. Of course, people in general are still dependent on one another as they are the ones that operate and control the computers and develop the software.

The computational methods have certainly been the key to technological development and nowhere is it more evident than in the past 100 years. Throughout that time there were more advances in technology than in the previous 500 years combined, by fair comparison. The recent technological developments had much greater impact on our lives as well. Radio, telephone, television, automobile, aero plane, satellite, and computers etc. have completely changed the way people live. The world has become much smaller as most places and people are quickly reachable in one way or another. Much of the recent technological developments (in the past 40 years) can be largely credited to computers, as they have become more powerful, user friendly and easily accessible. Computers have taken the task of sorting data, solving mathematical problems and storing vast repositories of information in the manner of seconds. All of this is done with the basic skeleton of computational methods. Had it not been for computational methods, it would be fair to suggest that this report would have been written by hand or with the use of a typewriter. This would have been a definite step back in time for an engineering student.
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Compiled by J.D. , Fall 2000
From the earliest times of approximating 'pi' by the Greek mathematician Archimedes (in 250 B.C.) to finite elements methods used in developing nanomachines ( in 2000 A.D.), computational methods (also called numerical methods), have been the key to technological development. Archimedes' calculation of pi involved inscribing and circumscribing polygons about a circle. The differences in the inscribed and circumscribed polygons' perimeters led to more accurate approximations of pi. He used polygons with96 sides. He would have liked to have used up to 384 sides, but the arithmetic with Greek symbols was too difficult. Archimedes' mathematical and mechanical skills led him to develop several inventions, such as a boat launcher, Archimedean screw, and a water pump (Burton 215). Throughout history, just like Archimedes, those with minds that could handle logical reasoning and mathematical manipulations have developed technology that has led to the incredible devices available today.

    In the 1800s, George Boole created an algebra of logic. Boolean operators and Boolean data types are named after him. Charles Babbage designed a mechanical machine that had units for arithmetic and logic. Though it was never actually built, it helped bring about the development of modern computers. Herman Hollerith was another whose logical reasoning and mathematical skills helped advance technology. He made a computer that helped with the U.S. Census of 1890. He also later was the founder of IBM (Schneider 25).

     In the 1940s, Howard Aiken actually built a computer similar to the one Babbage envisioned, but by the time Aiken completed it, technology had already surpassed it. Grace Hopper created a method used to determine sin x that was the first subroutine written for a computer. Another development at this time was the ENIAC, which was a huge computer that could perform arithmetic operations. It had thousands of vacuum tubes, weighed 30 tons, and was the size of a small house. Later, physicists at Bell Labs created a device to replace the vacuum tubes, which led to great changes in computer design. This new device was the transistor (Schneider 26).

     In the 1950s, the logic language Fortran was developed. This allowed a compiler to translate an entire program into machine language. The computer disk drive was invented by Reynold Johnson, but it was not like the small unit used today; it weighed a ton. A step-by-step procedure for organizing lists of data was published by Donald Shell. This kind of sorting is an important part of a computer's function (Schneider 27).

     From the 1960s to the 1990s, scientists and mathematicians continued to develop more effective computer languages and more efficient, compact computers. Paul Allen and Bill Gates created an operating system known as MS-DOS. Robert Barnaby developed a program used by word processors called WordStar. Visual Basic is an easy to use language that was developed by Tom Button, and it has made the work of programmers easier (Schneider 28-29).

     All of these technological developments were possible because of mathematics and logic. Equations, symbols, flow charts, and computations area must in the technology available today. Computational methods have been the key to technological development in the past, and so, they will more than likely continue to be the key of future developments.

Works Cited: Burton, David. The History of Mathematics. Boston: Allyn and Bacon, Inc.,
1985. Schneider, David. Visual Basic 6.0. Upper Saddle River, New Jersey: Prentice Hall, 1999.

Compiled by Jim Hall, Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development. This is proven by the efforts of a late nineteenth century mathematician named Charles Babbage (1791-1871) who is credited with developing plans for calculating and analytical engines that encompass many features in modern day stored program computers. Like many of today's inventors, Charles Babbage repeatedly re-invented his machines. Unfortunately, this did not allow any of his engines to be constructed. It is this spirit that has propelled technology to new heights in the last quarter of the twentieth century.

The Babylonian sexagesimal system is credited as the first known place-value system of which our decimal system is based upon. This made large calculations much easier to compute due the previous need of having different symbols for ten, one hundred etc. The ability to handle larger numbers has and will always be a focal part of design. It is this ability that led to the birth of machines designed to solve problems. The turn of the twentieth century saw electromechanical calculating devices developed and invention of the computer came during the 1940's as a need of the war, that is, ballistic calculations. Later in the 1940s the invention of the transistor made the microprocessor possible in the 1960s.

Technology today is based primarily on microchip design. The microchip development is rapidly approaching a need for a "new" idea. We have traveled a long distance since the introduction of the super computer of the 1970s. The idea of doing more than 160 million operations in a second seemed astounding, it would seem slow by today's standards. As the "Information Superhighway" grows every second so does our technological development. Time will tell if the question posed by David Hilbert (1862-1943) can be answered by computers; "Is mathematics decidable, that is, is there a mechanical method that can be applied to any mathematical assertion and (at least in principle) will eventually tell whether that assertion is true or not?". Time will tell.

From the earliest times of the approximating "pi" by the Greek mathematician Archimedes (in 250 B.C.) to finite element methods used in developing nanomachines (in 2000 A.D.), computational methods (also called numerical methods) has been the key to technological development.

Numerical methods have provided the foundation for almost everything that has been engineered to this date. Numerical methods by definition are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations. Numerical methods inevitable involve large numbers of tedious mathematical calculations.

Numerical methods have helped humans emerge out of prehistoric times and move forward to new accomplishments. Without the development and use of numerical methods, people like Archimedes, Newton and a host of many other scientists and engineers would not have brought us this far in the
technological revolution.

The use of numerical methods to solve complex problems in Physics, Chemistry, Astronomy and many other numerous sciences contributed to are development and understanding of the world around us. Numerical methods were first used predominately in the field of civil engineering, to build buildings, bridges and dams. With the coming of the industrial revolution numerical methods in all fields of study became very important. Problems that were brought out by the industrial revolution needed more involved solutions than were ever needed before. Numerical methods pulled us through the industrial revolution allowed humans to make the greatest technological jump in history. This jump in technology led to the approach of the computer revolution in the mid to late twentieth century, which would again change the world.

With the onset of computers in the early twentieth century the use and development of numerical methods jumped tremendously. Now, work intensive, complicated problems could now be solved using computers. The problems and solution methods now do not have to be simplified to retrieve a good solution. Before the onset of computers and the digital revolution, solutions had to be graphically linear, and fairly simple to manually compute. Today, complex problems can be modeled and solutions computed using computers, which could never have been done before. Numerical methods are an integral part of everyone's life, whether you work with them or not, you are directly affected by them and their solutions.
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Compiled by David Bearce, Fall 2000

With the rapid evolution of the computer industry, computational methods is playing an even more significant part in everyday life of the average person, not to mention the life of a prospective engineer. From the simple solving of simultaneous equations for the most efficient load of different products to carry on a large truck spanning to the proper roll and trajectory of the space shuttle after lift off, the need for speedy calculations is paramount in the production of a company as well as the entire country.

Computers play a very large part in the development of computational methods. Some parts the computer plays are more obvious than others. For an obvious example, the computer can perform many more calculations in a shorter span of time than that of any person with just a calculator (or a slide rule for the old school). This ability of the computer leaves more time to be spent on the creative side of design of a problem thus leading to better technological advances. This fact leads to a more subtle part that the computer plays in the development of computational methods.

The more time an engineer has to spend on the creative development of a given project, more time is devoted to aspects of a project such as an increased safety factor, a streamlined materials list, or a more efficient gear ratio for a given mechanism thus improvements become more likely to occur. This fact leads to better research passed down from engineer to junior to carry on with current ideas. The possibilities become increasingly available as more development and time is spent on the creative portion rather the "number crunching" portion of a project.

Most people are aware of the computation methods used in the past. In conjunction, it is difficult to find an engineering student that respects those methods. While the former computational methods appear archaic, they were the necessary building blocks for the technological advances that we experience today. Most of us would admit that it was better them (from the old school) than us. However, we should not take for granted the abilities the former computational methods have provided us with today and we should persevere with the tenacity that our predecessors dreamed we would.
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Compiled by Thomas Wasik , Fall 2000

The pre computer times saw a very limited use of numerical methods to solve engineering problems. Quite often, a great deal of energy and time was spent to come up with solutions using analytical or exact methods.

The reason for a limited use of numerical methods was the relatively large number of tedious arithmetic calculations that have to be performed. This fact alone kept its popularity to minimum. With the availability of computers, widespread usage of numerical methods began.

By using numerical methods, engineers no longer have to waste a significant amount of their time on the solution technique itself. Moreover, this switch to numerical methods as a main way to solve mathematical problems allows engineers to spend more time on identifying the problem and making necessary improvements. With computer programming becoming easier, the need to buy special software is declining and more engineers resort to writing their own programs aimed at solving specific problems.

Furthermore, as a result of better efficiency and productivity, the technological development is steamrolling ahead with more momentum than ever before. The benefits of improved technology and scientific breakthroughs are evident at all aspects of life.

The lives of millions of people have been altered forever (in a positive way) due to technological progress. The booming economy is one and probably most noticeable example. If we keep making such a leaping progress, we can expect very revolutionary changes in near future.

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Compiled by D.E.H., Fall 2000
From the earliest times of the approximating "pi" by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development. Numerical methods are applied to simulate physical phenomena. Computational methods are also often divided into elementary methods such as integrating a function, solving linear system of equations, or finding the root of an equation.

In the development of numerical methods, simplifications need to be made to progress towards a solution. For example, computers cannot generally represent exact numbers therefore, general functions may need to be approximated by polynomials. Thus, numerical methods do not usually give the exact answer to a problem but rather tend to get closer to a solution with each iteration.

Numerical analysis is applied to solve problems in the following areas: fluid mechanics, reactive flows, solid mechanics, astro-physics, optimization, and parallelization to name a few. For example, in the last twenty years, the application of numerical methods for least squares problems has influenced technological advances especially in the capacity for automatic data capturing and computing. In particular, tremendous progress has been made in numerical methods for generalized and modified least squares problems and direct and iterative methods for sparse problems. In addition, least squares problems of large size are now routinely solved. This is only an example of the reason why computational methods is considered the essence of technological advances.

Compiled by Scott Scheulein , Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 B.C.) to finite elements methods used in developing nanomachines (in 2000 A.D.), computational methods (also called numerical methods) has been the key to technological development. For example, the use of computational methods has greatly advanced the area of Computational Physics. The use of computational methods had probably doubled our supply of knowledge in the subject as well as the speed at which the knowledge is obtained.

Computational methods can be applied to many areas of physics. Since most of the programs are written for research, the need to purchase expensive equipment for experimental purposes becomes antiquated. This proves to be especially beneficial in the area of Theoretical Physics. For example, with respect to the study of Quantum Magnetism, the exact diagonalization of a Heisenburg spin chain can be calculated. The use of computational methods for this field of study saves millions of dollars on research equipment and the time of some really bored graduate students. Computational methods also has tremendous promise for Chaos theory studies. For a human being to account for all the variables that lead to Chaos in a complex system is almost intractable.

The use of computational methods in physics does not only have to be applied to theoretical physics. The field can also be easily applied to Biophysics. Computational methods can be used for the discovery and implementations of algorithms that facilitate the understanding of biological process. This could provide information about DNA, RNA, and protein sequences. This information could save or improve millions of lives. As illustrated, the use of computational methods in physics can save money, time, and more importantly lives. It can be applied to nearly every branch of physics. It has without a doubt been the key to technological development in physics.

Compiled by Jean-Paul Deeb, Fall 2000

From the earliest times of approximating pi by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in nanomachines (in 2000AD), computational methods (also known as numerical methods) has been the key to technological development.

As we walk down the street we are unaware of the implications of numerical methods in what we see. For centuries scientists have been evolving numerical methods to help form a functional society. Few understand the revolutions that have taken place over time. Artists like Leonardo Da Vinci and his work in numerous areas helped advance computational methods.

These advancements were involved in his paintings, as well as his inventions and his studies of the human body. His predecessor, Brunelleschi, was an architect that revolutionized Renaissance art. His calculations on the idea of linear perspective were based on a vanishing point that gave the vision of three dimensions on a two dimensional surface. These numerical methods are the basis for perspective used in paintings today as well as the foundation for current graphical and three dimensional computer modeling software.

Innovators like Galileo and Copernicus questioned religious beliefs of science and risked their lives and livelihoods to prove the existence of different phenomenon by computational methods. Their studies on the motion of heavenly bodies are still the foundation for space research and physics. This allowed Newton and his followers to expanded their ideas and forge ahead with new methods of thought and computation.

Their numerical studies proved new laws that governed our lives and how things behave in the world we live. All of this leads us to the modern era of Einstein and the computer revolution. Where machines that were never thought to be possible are now in every household or assisting our everyday lives. Computational methods are the foundation that proves anything is possible within the limits of our universe.

Numerical methods created by these early thinkers gave us the foundation and framework to use computers to their full potential today. This may allow us to create new boundaries for tomorrow.

So the next time you use the internet, cell phone, or on board GPS, think about the computational methods involved in each of these everyday products and it will make you wonder about the possibilities of tomorrow.
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Compiled by W. C. , Fall 2000

From the earliest times of approximating 'pi' by the Greek Mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000AD), computational methods (also called numerical methods) has been the key to technological development.

Computational methods has been used in not only all disciplines of engineering, but also in biology in which it is used in the modeling of the ocean, currents ,and tidal flows. The impact of computational methods has been profound because not every equation can be solved using exact methods, if we were limited to the use of only exact methods it would be like trying to work with your hands tied behind your back. The time factor involved with trying to solve such equations is increased, and involves the use of very lengthy and tedious algebraic computations, but since the invention of the computer the time and aggravation involved has been significantly reduced.

The impact of the computer on numerical methods is that the engineer must really only do the procedures once when programming the computer. The computer will follow the program exactly as written and give the answer in the required fashion; the computer will have the same units and precision every time. The draw back to this is that the computer is only as good as the person who wrote the program. The computer performs exactly what it is told to do every time, and in doing so, it significantly reduces the amount of error in the calculation. If the computer is programmed to do so it can even calculate the error in the calculation.

The world is not a perfect place and so mathematics is not a perfect science; which is why there is a need for computational methods to estimate modeled situations that are not perfect. Computational Methods can be very useful, but these methods are used for the approximation of exact solutions which introduces certain trade-offs. The trade-offs introduced include errors, limitations of application, and others. These trade-offs stem from certain types of input that produce outputs that are not of the desired precision.
Computational methods is indeed a very useful tool for the engineer; a tool which as made many technological innovations possible. Computational methods is the application of mathematics; in which the theory finds it's practical purpose.
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Compiled by Phillip Hoffmann, Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 B.C.) to finite elements used in developing nanomachines (in 2000 A.D.), computational methods (also called numerical methods) has been the key to technological development.

In the past, solving problems using numerical methods was slow since there were no computers to help expedite the process. In the pre-computer era, there were three methods for problem solving, each having their advantages and flaws.

The first pre-computer era method was analytical or exact methods. This format was good for linear models, simple geometry, and problems with low dimensionality. The weak points were that most problems were non-linear, and involved complex shapes and processes.

The second pre-computer era method used graphical solutions made up of plots or nomographs. This method can solve complex problems, but are not very precise, are tedious without a computer, and are limited to three dimensions or less.

The third pre-computer method used calculators and slide rules. This method was also good for complex problems, but was tedious and long, and it was easy to make a mistake with a long computation.

The overall problem with these methods in the past, was that a large amount of time was spent focusing on the calculation, instead of the principles of the problem at hand. Especially with longer problems, it would be easy to make a simple mistake, resulting in wrong results, and more time spent looking for the mistake, along with time consuming recalculations, if the mistakes were caught.

Today with widespread availability, and relatively low costs, computers easily facilitate the computing of complex calculations that accompany numerical methods.

With the power of computers, it is advantageous to use numerical methods and computer programming to write programs that may not exist, or are expensive to buy. It may also be the case that such software does not exist, in which case a program must be written.

Finding someone with that understands the material, and has the ability to write a program may be time consuming, expensive, along with the issue of when he is available to do the work with respect to your deadlines, or goals.

Computers paired with numerical/computational methods have made many calculations possible in the world of engineering

Compiled by Cyntra Sonnylal, Fall 2000
From the earliest times of the approximating "pi" by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000AD), computational methods (also called numerical methods) has been the key to technological development.

What makes numerical methods so essential today is its close ties to engineering needs. Before the introduction of the computer, calculator or even slide rule, problem solving took the form of analytical methods. This method was limiting in the type of problems encountered. It could be used on problems that were only linear and simplistic in setup. Most engineering applications are nonlinear and progressively more complicated in geometries.

Numerical methods has the advantage of using arithmetic to break down intricate, involved problems. These problems are extremely tedious to work by hand, but are perfectly well suited for computer methods.

Numerical methods goes hand in hand with computers because of the ease to which computers process complicated problems. Just as a computer program needs to be carefully written, with step by step functions, calculations, and decision structures, so too are numerical methods used to break down difficult, higher mathematics to basic arithmetic calculations using a step by step layout. This type of layout is well suited to programming and so the connection between computers and numerical methods is made. With this union comes the ease of interpretation and evaluation of data and calculation results.

Where, in the past, the majority of time was spent acquiring results to only a few calculations, now almost countless trials can be accomplished within seconds of inputting data.

Faster computers have meant faster results, improved technology has meant a reduction in calculation errors and a broadening of problem solving capabilities by the new computers. No matter how advanced technology has become, it is thanks to the underlying simplistic of numerical methods that this technology has reached so far.
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Compiled by Greg Rosa, Fall 2000

Archimedes devised the first method to calculate pi to any degree of accuracy. The Archimedes' method was a numerical method that involved polygons. One polygon was inscribed within the circle, while the other circumscribed around the circle. He observed that as the polygons approached the limiting circle, the area between the two polygons went to zero. He calculated the value for pi using a 96-sided polygon as 3.14084 < pi < 3.142858. His numerical method was adopted by the Romans and used in their structural arches.

In Asia, the Chinese mathematician named Lui Hui took Archimedes' method a step farther. He used a 3,072 sided polygon to find pi was about 3.1416, which was extremely accurate. During the second millennium A.D., Sir Isaac Newton, with the help of others, invented calculus. Using this new branch of math, many formulas have been worked out to find pi.

With the advent of the computer, the accuracy of pi can be taken to unimaginable accuracy. In 1949 an ENIAC computer computed pi to 2,037 decimals in 70 hours. As computers continued to increase with speed and accuracy, so did pi. Currently the value of pi is known to 6.4 billion places.

Throughout time, the value of pi has been sought out. Pi has been so important because it is used in almost every analysis of rotational movement and circular properties. As technology expands, so does societies' demand for more accurate calculations. The importance of pi can be seen throughout history in great structures such as the dome of the Parthenon, and in the vaulted ceilings of gothic cathedrals. In the future pi will continue to be essential in mathematical calculations.

Compiled by Giancarlo Battaglini, Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes ( in 250 BC ) to finite element methods used in developing nanomachines ( in 2000 AD ), computational methods ( also called numerical methods ) has been the key to technical development.

On the basic side of the spectrum Archimedes constant 'pi' has been used to find the area and the circumference of a circle. This mathematical definition of the area and circumference of a circle equaling 'pi' and 2'pi' respectively for a circle with a radius of 1, led to the mathematical representation of the volume and surface area of a sphere. Beyond this, Archimedes constant has many infinite series, infinite products definite integrals and continued fraction representations as well as an infinite radical expression. According to Jonathan and Peter Borwein; it requires a mere 39 digits of 'pi' in order to compute the circumference of a circle of radius 2*10^25 meters ( an upper bound on the distance traveled by a particle moving at the speed of light for 20 billion years, and as such an upper bound for the radius of the universe ) with an error of less than 10^-12 meters ( a lower bound for the radius of a hydrogen atom). As you can see 'pi' is of great significance, and to extreme accuracy, for many mathematical model approximations.

Today numerical methods are used to simulate physical models like motion, forces, collision, deformation, compression inertia, etc. And with computers growing faster and faster due to the effects of computational methods, previously unobtainable tests and analysis can be done and new technological developments extracted. An example of this is the use of nanotechnology to
solve even the greatest problems like the population crisis, pollution, and global warming. Some say numerical methods are the only realistic solution to the enormous problems facing humanity today. For example, think in the terms of atoms in a structure that is only one millionth of a meter across. Thousands of atoms can fit along this length and billions in a cube of this dimension. But inside this one micron object there is a possibility of a thousand fold higher level of detail. If you could get down to that nanometer level, and craft the object with atomic precision, the power of your ability to control the behavior of this object would be immense.

It seems that by just searching numerical or computational methods on your computer, it is related to just about all of our understandings of what surrounds us, even the key to technological development.
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Compiled by Jack Al-Kahwati, Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes to finite elements methods used in nanomachines, computational methods has been the key to technological developments. In the last decade NASA has utilized the power of computational methods to obtain an edge on current space technology. A computational methods for space branch has recently been added to the government agency. The objective of this branch is to develop new Computational Fluid Dynamics software for the analysis and design of fluid systems and to provide computational support to others.

One of the many research conducted at NASA utilizing computational methods included the development of specialized software to model the situations encountered in space and specific aero propulsion problems. In 1992, the NASA's Langley 8-Foot High-Temperature Tunnel was used to test components of hypersonic vehicles for thermodynamic, and flow properties of the equilibrium chemically reacting oxygen- enriched methane-air combustion. This computer code calculated the fuel, air, and oxygen mass flow rates and test section flow properties for Mach 7, Mach 5, and Mach 4 nozzle configurations for a given combustor and mixer conditions.

In 1996 NASA presented a computational procedure for the solution of frictional contact problems for aircraft tires. Experimental measurements were taken to define the response of the Space Shuttle nose-gear tire to inflation- pressure loads and to inflation-pressure loads combined with normal static loads against a rigid flat plate. Numerical results were obtained for the Space Shuttle nose-gear tire subjected to inflation pressure loads and combined inflation pressure and contact loads against a rigid flat plate. The experimental measurements and the numerical results were then compared.

In the past engineering problems in thermodynamics and fluid dynamics had to be solved using simple models with assumptions such as ideal gases, and adiabatic processes. Today computational methods are able to solve gigantic problems with several dependent and independent variables that nature has included. In the future, computational methods will play an even greater role in the field of engineering. This will place even more emphasis on formulation and interpretation of mathematical models rather than the antiquated emphasis on complicated solutions. With the advent of quantum computers, the power of computational methods will increase so much that perhaps creating a true model of the weather would be a common task.
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Compiled by Scott Avoy , Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes to finite elements methods used in nanomachines, computational methods has been the key to technological developments. One of the first documented uses of a numerical method was Archimedes approximation of pi circa 250 BC. He used a method that is still considered to be accurate however he lacked to computing power to solve this method quickly. It is hard to fathom approximating pi using his method without the use of a computer. He must have taken a great deal of time to perform this approximation enough times to get a suitable answer. This is the exact reason why numerical methods did not gain tremendous popularity until the advent of the computer.

Although the process of numerical methods does simplify some problems in terms of the complexity of mathematical operations to be performed it also adds a degree of difficulty to the problem. This added degree comes from the necessary repetition of the procedure to arrive at an approximation. By using computers to solve our numerical methods this added complexity can be overcome and more accurate answers can be obtained in a very small amount of time. One important item to remember about numerical methods is that they do not give an exact answer. Due to the nature of the procedure only an approximation can be obtained. These approximations are however still quite accurate. So accurate in fact that they are often used to predict behaviors of complex systems.

The numerical method is ideally suited to computers because in many cases the approximation can be achieved using only addition, subtraction, multiplication, and division. This however is completely dependent on the ability of the programmer to adequately translate the more complex mathematics. With this in mind look for the use of numerical methods to continue to grow in technical fields as computer power and programming skills increase.

Compiled by C.R, Fall 2000

From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements used in developing nanomachines (in 2000AD), computational methods (also called numerical methods) has been the key to technological development. As man's understanding of the world grew, he needed a better way to represent the world. When man drew the wheel, he had to figure out how to draw the perfect wheel. Once that was accomplished, he had to develop the idea of how he could impart this knowledge to his fellow man. Mathematics was the key to this knowledge. Archimedes figured out what pi represented and what its approximate value is. As time passed, other mathematicians developed knew ideas and methods to better represent the world. Better reproducible results were attained.

Leapfrog to these times. Numerical methods has been used to develop sound practical solutions to an almost infinite amount of things. Man has used these methods to expand his knowledge and understanding of the Universe. In the past few decades, the moon and space flight have been conquered. The bottom of the oceans have been explored. We can hop on a plane and be in Paris in a few hours. We can escape to almost anywhere on this planet in a matter of a few hours. These are technological advancements that can all be traced back to numerical methods.

The laws governing all these technological advancements can be complicated at times. Numerical Methods solves these problems for us in a practical manner. Without numerical methods we would have less leisure time. Computers and machines are doing a lot of our work.
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Compiled by Panyada Sachs , Fall 2000

From the earliest times of approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000AD) , computational methods (also called numerical methods) has been the key to technological development.
The history of the number pi provides some interesting examples of the use of computation methods to solve a problem that has fascinated mankind for thousands of years.

Some of the first techniques involve circumscribing an sided polygon in a circle so that the circumference of a circle of known diameter could be calculated geometrically. This allows pi to be hand calculated to 12-20 digits as far back as the time of Archimedes. Much later on in history various power series expansions were shown to converge to pi. Each of the algorithms converges to pi but some are more efficient than others. The first method produces a single digit of pi for every iteration where the last method produces 14 digits for every term or iteration. So it definitely pays to have the better algorithm if you need to program a computer to produce a billion digits of pi. The current record for calculating pi belongs to the Japanese at 51,539,600,000 decimal digits and it still does not seem to be repeating.

Nanotechnology is the science of using computer controlled tools to precisely measure and position individual atoms to construct nanometer scale machinery. A common tool in nanotechnology is the Atomic Force Microscope. The AFM uses computers to precisely control a very fine stylus to nudge individual atoms to specific positional coordinates. Under computer control it should be possible to construct mechanistic components such as a gear. The computer would utilize various computation algorithms to 'draw' the circular shape of a gear and then draw the vectors that define a single gear tooth. Each subsequent tooth would simply be a rotated and translated version of the original.

Computational Methods are one of the corner stones of modern technology tools. It extends the ability of human beings to be able to calculate 'pi' to over billions correct digits and advances the ability for nanomachines to be able to move atoms in to a precise position. In general, the modern technology could not be developed without the use of Computational Methods.
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Compiled by J. M., Fall 2000

Everywhere you look today you can see the products of computational methods. If you look at transportation computational methods allowed cars and planes to be built. When you see the buildings and bridges we know that computational methods allowed them to be built. If we also look at the appliances in our homes that we use every day like a refrigerator or a microwave, computational methods has been used there as well. Who today could not recognize that the way we communicate across vast distances has been a product of the computational methods.

When we look into the past of computational methods, we see contributions by great men like Newton, Archimedes, and Euclid. These men and others asked questions of the world around them and worked to discover the answers. They asked questions like-why do the planets act in a predictable manner, and then they worked out the answers using computational methods. They asked why does the apple fall and they developed the tools (computational methods) to answer that question. They first dared to ask the questions that many had asked before them but they had the vision and they developed the tools to finally answer those elusive questions. They could see where other could not see and developed ways to show the world what they could see. Those "ways" were computational methods.

Today we still have questions that need answers. We are still trying to make sense of the universe around us and we still need computational methods to help us answer our questions. In fact, our need for computational methods may be more necessary today than ever before. In our technology-rich environment we now have the use of fast computers that can crunch numbers at alarming speeds. We can compute more information using computers but we must use the correct computational methods to get the answers we desire. Who will be the next person to ask the hard questions that could not be answered before? Who will be the new breed of innovators who have the vision to develop the tools we need to answer the questions that have eluded generation after generation. I do not know who they will be, but I do know that computational methods will be the avenue through which those questions will be answered just as they have through out our past.
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Compiled by Gabe Moreno , Fall 2000

From the earliest times of the approximation of 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in the development of nanomachines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development.

According to a college professor, all problems in everyday life are mathematically related in one way or another. While all problems are can not be solved mathematically, some can be solved by applying mathematical concepts from knowledge you have acquired to solve problems you may encounter in the field. In old times, mathematicians like Archimedes used numerical methods to approximate the value of 'pi' using the area of an n-sided polygon, where n is an increasing number of sides. Archimedes found an approximation for 'pi' without the aid of such intricate calculators as we have today. Imagine where the world would be today if the minds of that time had these calculation devices available to them.

As students of engineering, we are taught the basic forms of mathematics that will later enable us to solve many problems. A system has determined the relative order of classes that would best prepare the student for subsequent courses. It is the responsibility of both the teacher to teach the material as well as for the student to learn. The student will then be asked to apply that knowledge for future technological advancement.

It is difficult to determine who of the number of students in the world will be mediocre and who will be like Archimedes or Einstein. Whoever and wherever they are, they will most definitely be using some sort of numerical method to further the technological advancement of this world. We are far from the technological wall and with the great minds in our midst, that wall will probably never even exist.

Compiled by S.B. , Fall 2000

From the earliest times of the approximating 'Pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000 AD) computational methods has been the key to technological development. In broad terms computational methods is about using computers to analyze and solve scientific problems.

The idea of using machines to solve mathematical problems can be traced as far as the early 17th century with the first multi-purpose computing device by Charles Babbage. Since then scientists have work together in a unanimous effort to develop a better and more reliable machine.

One of the first commercial uses of mechanical computers was by the US census bureau to tabulate date for the 1890 census. In the mid 1900's the use of computers became a very advantageous tool to calculate ballistic in World War II. Later decommissioned, computers were used for civil purposes such as the design of wind tunnels and weather predictions.

By the early 1960's the use of integrated circuits (IC) microprogramming would allow computers to do calculations at a rate of one million floating-point operations per second. Shortly after Ken Thompson of Bell company would put everything together and came up with the operating system UNIX. At this point scientist started mathematics into computers and well into the 1980's great developments in software included very high-level languages. Having a language complex enough to work with computers awoke NASA who shortly became very active in the research and development of computers. The scale of integration in semiconductors continued at an incredible pace.

By the 1990's it was possible to build chips with one million components and semiconductors memories became standard on all computers. Manufacturers along with scientists had set themselves the goal of achieving teraflops (10^12 arithmetic operations per second). The momentum is building and the rate of development of computers is so fast that they get obsolete in a very short period of time. The combination of computers and mathematics have made possible to come up with different method to compute and calculate arithmetic operations very fast and accurate.

Now more than ever technology seems to be in fashion, everyone wants to have the smallest but yet fastest computer in the market. This trend has created a dependence on computers forcing us to use languages that can be understood by these machines.
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Compiled by Nathaniel Collier , Fall 2000

From the earliest times of the approximating of "pi" by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in nanomachines (in 2000AD), computational methods (also called numerical methods) has been the key to technological development.

When one considers all of the great ancient wonders such as the Pyramids or the Parthenon, one does not view them as great mathematical problems. The truth is that great calculation and precision were necessary to build such feats--without a fraction of the calculation technology that we have today.

The builders of the past relied heavily on numerical methods developed over a period of centuries. The Greeks used these methods to calculate their arches and analyze the planets. Even in the more recent past, numerical methods have been an important aspect of development. Consider the technological monster called the "Titanic". Although computational technology has improved drastically since the Egyptians, it still was not what it is today. Imagine the calculations
involved in determining the configuration of the gears in the engine room or in discovering how much fuel was necessary to power the vessel across the ocean. All of this was done mostly by hand and age-long tradition in computational methods.

Today it is the same story, but the difference is that we have developed technology, which uses those same methods at astounding speed. This speed increases the complexity of the problems we can solve. This in turn has increased development greatly. These machines have made things like landing on the moon with pinpoint accuracy a possibility.

As illustrated, behind the great wonders of the world in all of time, was a lot of figuring and calculations that depended on numerical methods (whether done by computer or by hand) to determine answers.
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Compiled by Hamza Begdouri , Fall 2000

Since the early ages, computational methods have been the tool of scientists and researchers to test theories in the world of classic physics. Computational methods is how we can get as close as possible to the true real or complex values of things. Especially, how far in the precision do we allow our selves to go?

Archimedes, a great Greek scientist and mathematician was one the earliest to use computational methods to develop machinery and war machines that found a huge success among his people. However, his biggest accomplishment known to men is his approximation of ' pi '. Some theories say that that the old Egyptians must have computed pi to be able to build the Pyramids. But Archimedes is the only person proven to have done it first and with documentation showing the use of computational methods.

In the present days computational methods are used in many domains, including fluid flow and electromagnetism, etc. Computational methods have allow us to persevere in the study of finite element that it use in the building of nanomachinery. More and more software are used to facilitate these applications for example, Visual Basic. All depending on the computer used capacities the approximations is different, that shows that computational methods have a large interval of precision that we can shorten depending on our needs.

Finally, in our days firms specialized in computational methods are created making new versions of software available for more precision in the computation. This is a great tool for science that existed for hundreds of years and still will for many years to come.

Compiled by GWF, Fall 2000

From the earliest times of approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite element methods used in developing nanomachines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development. An important figure in this development was Abu Ja'far Muhammad ibn Musa Al-Khwarizmi, born in 780 AD in Baghdad. From the mispronunciation of his name we get the term algorithm, and he is supposed to have written one of the first books on algebra.

Al-Khwarizmi studied at an academy called the House of Wisdom in Baghdad. This academy was founded shortly after 813 AD, by the emperor of Islam; here Greek philosophical and scientific works were translated, and observatories were built for the study of astronomy. Al-Khwarizmi translated Greek scientific manuscripts, and studied and wrote on algebra, geometry and astronomy. Al-Khwarizmi's most famous work was the Hisab al-jabr w'al-muqabala. It is from this title that we have the term "algebra" and as mentioned above, it is one of the first texts on this subject.

In this text, he discusses solutions to linear and quadratic equations, all without using the symbology known to us today! In this book, all discussions and solutions were expressed in words! His solutions to these equations were done by both algebraic methods and geometrical methods. Also present in this text were rules for finding the area of shapes such as the circle and also finding the volume of objects such as the sphere, cone, and pyramid.

Another famous text written by Al-Khwarizmi is the Algoritmi de numero Indorum (Latin title) or Al-Khwarizmi on the Hindu Art of Reckoning (English title). (The original Arabic text is lost). Obviously, this is the book that gave rise to the term algorithm, and describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. The first use of zero as a place holder in positional base notation is usually attributed to Al-Khwarizmi and this work. Also known to be present in this text were methods for arithmetic calculation, and methods to find square roots.

Other works by Al-Khwarizmi include the Sindhind zij which consisted of calendars; calculations of the true positions of the sun, moon and planets, tables of sines and tangents; spherical astronomy; astrological tables; and parallax and eclipse calculations. He also wrote on the astrolabe, the sundial, the Jewish calendar, geography and a political history containing horoscopes of famous people.

The most famous work by Al-Khwarizmi is most certainly the text on algebra, but as he was well versed in the field of mathematics, he has made more than this one major contribution to the field of numerical methods and modern mathematics. The knowledge and genius present so many centuries ago is truly amazing.

Compiled by A. R. , Fall 2000

From the earliest times of approximating 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in developing nanomachines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development. The earlier efforts of using numerical methods to solve large numbers of tedious arithmetic calculations by hand eventually led to the idea of using machines for this purpose.

Mathematicians such as Wilhelm Schickhard, Blaise Pascal, and Gottfried Leibnitz designed and implemented calculators that were capable of addition, subtraction, multiplication, and division as far back as the early 1600's. Additionally, in the mid 1800's, the efforts of Charles Babbage produced the beginning of two programmable computing devices, the Difference Engine and the Analytical Engine. Unfortunately, the technology at that time was not reliable enough and the devices were only partially completed.

These machines were the inspiration for George Scheutz and his son, Edward to begin work on a smaller version in 1853. By 1853, they had constructed a machine that would win a gold medal at the Exhibition of Paris in 1855. Their machine could process fifteen digit numbers and calculate fourth-order differences. It was sold to the Dudley Observatory in Albany, New York, which used it to calculate the orbit of Mars. This in turn led to one of the first commercial uses of mechanical computers by the US Census Bureau in 1890. This punch-card equipment which tabulated data was designed by Herman Hollerith, who's company later merged with a competitor in 1924 and became International Business Machines.

From these humble and heroic beginnings, technology surged forward to the early development of electronic machines that became electronic computers which had the capacity to store programs. The ability to store instructions in the same medium as data was an incredible breakthrough that allowed future designers to concentrate on the countless improvements to the internal structure of the machine that have brought us to where we are in technology today.
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Compiled by B. F, Fall 2000

From the earliest of times of the approximation of "pi" by the Greek mathematician Archimedes (in 250BC) to the finite elements methods used in developing nanomachines (in 2000 AD), computational methods (also called numerical methods) have been key to technological development. All of the accomplishments over the last one hundred years can be related to the development of mathematics.

Early in the twentieth century the invention of the airplane revolutionized the transportation industry. People would no longer have to ride on trains to travel from place to place. The evolution of the airplane coincided with the evolution of mathematics. As computers and circuitry became more advanced ,aircrafts began to fly higher and faster than ever imagined. As aircrafts evolved, the next goal was to put a man in space. The Russians were the first to launch a rocket in to space; it was called Sputnik. With the advancement of computers and physics, the United States was able to put a man on the moon in the 1960s.

There have been many trips into space since then. There have even been probes launched into space to travel to other planets and survey their atmospheres and take pictures. For example, the Voyager mission took pictures of several planets in our solar system. As the Voyager probe took pictures of the last planet it would encounter, it began to travel into interstellar space, in hopes it would someday be found by intelligent life. Attached to the Voyager spacecraft was a numerical code to communicate with any extraterrestrial life it may encounter.

The movie "Contact" was about this type of communication with extraterrestrial life. The radio signal from the extraterrestrials was found to be numerical in nature. Therefore mathematicians had to use computational methods to decipher the message. The mathematicians were having difficulty breaking the mathematical code until they looked at it in three dimensions. In doing so they were able to decipher the code and see a set of blue prints for a star gate type of machine. One day in the future this type of thing might happen. With the advancement of physics and mathematics we might be the ones sending a signal to a far-off planet.
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Compiled by E.L , Fall 2000
From the earliest times of the approximation of 'pi' by the Greek mathematician Archimedes (in 250 BC) to finite elements methods used in the development of nanomachines (in 2000 AD), computational methods (also called numerical methods) has been the key to technological development.

    Even today in the 21st century, the idea of numerical methods is the sole foundation to solving many if not most of our mathematical problems. The aspects and applications that are involved in our everyday life all incorporate numerical methods of some sort. This is due to the fact that all problems in the world are mathematical.

    As engineering students, we are taught to depend on numerical methods for mathematical problem solving. In physics, the area of projectile motion involves numerical methods in deriving equations used in solving problems. Just like Archimedes who was considered one of the three greatest mathematicians of all time along with Newton and Gauss, developed numerical methods to further the technology of their time, we as engineering students take classes in numerical methods to further develop the technology of our own time. Just as great inventors such as Einstein who developed E=mc^2, and Newton who discovered the law if gravity, used numerical methods to further their theories, we as young inventors use numerical methods of our own times to further our own theories.

Though Archimedes had many great inventions, he considered his purely theoretical work to be his true calling with numerous accomplishments. His approximation of 'pi' between 3-1/2 and 3-10/71 was the most accurate of his time and he devised a new way to approximate square roots. Archimedes further developed the infinite number of rectangles and added the areas together and created integration.

We still use the concepts these great men developed as a basis for our technological advancement

Compiled by JCR , Fall 2000

Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number pi. In today's world of high technology and precision instruments, it is hard to believe this civilization cannot exactly solve a problem as simple as dividing a circle's circumference by its diameter.

And yet this value has puzzled mathematicians for nearly four thousand years. Not until the invention of powerful computers and numerical methods, the mystery of pi was hidden from those brave soul dedicated to it study. The current record of 51 billions digits is a testament to the power of brain and computers.

The same computer power which permits the computation of pi to the 51 billion decimal place allows engineers to use explicit finite element programs to analyze the nonlinear dynamic response of three-dimensional inelastic structures. By taking advantage of this technology, engineers untrained in a mathematical application such as finite elements analysis can run an finite element analysis software program that performs calculations automatically and will save valuable time in the design process. The finite element analysis used to take weeks to accomplish just an approximation since many problems were just too complex to attempt a solution without using computers.

Nanomachines are emerging from this developments in computer power and computational techniques. Because building a prototype of a very small machine is so difficult, it is better to simulate one with computational methods. The machine can then be tested and evaluated before it is actually built. Nanotechnology is moving ahead so fast it is almost impossible to be up to date in this field. In the future, machines will be cruising through the human circulatory system administering drugs and cleaning obstructions. This is a quantum leap in technology thanks to finite elements analysis.
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Compiled by Jose Busquets, Fall 2000
From the earliest times of the approximating 'pi' by the Greek mathematician Archimedes (in 250 B.C.) to finite elements methods used in developing nanomachines (in 2000 A.D.), computational methods (also called numerical methods), has been the key to technological development. In Archimedes times, he was considered one of the most brilliant mathematicians. The theorem for the buoyancy of a material suspended under water was named after Archimedes. Unfortunately, for Archimedes and all the other earlier mathematicians, they did not have computers or calculators to do a lot of the tedious differentiating and derivations.

Everything that was calculated had to be developed first by formulas and numerical methods. Thanks to computers we can input a complex integration with its appropriate limits and these marvelous machines will give us the precise area under any curve. Until recently, these calculations had to be done by hand because there were not any programs to complete these more difficult problems.

Although the terms sequence and series are sometimes used interchangeably, they represent different mathematical concepts. A sequence is a function whose domain is the set of natural numbers and a series is the summation of the terms in the sequence. Series are a major part of the development of modern mathematics. Series are mainly used to simplify functions that are too hard to just represent them with a symbol. Series are also the basis for approximating 'pi' and 'e'. Series are used in making programs. They are also used in probability, investments, etc. Ruffini, in 1805, realized that it was impossible to solve quintic or higher degree equations. His result made it necessary for us to rely on numerical methods and also graphical methods.

One of the most important inventions of the twentieth century was the electronic digital computer. It was invented by John Atanasoff in 1940. He was trying to solve a system of twenty nine equations. It was too difficult for John to solve by hand so he developed the first electronic digital computer. That was the first problem it solved for us and now there is no limit to what these machines are doing for us. These computers can solve 600,000 equations simultaneously and also compute millions and even billions of arithmetic operations in a single second.

These are only a few examples of the improvements that numerical methods have done to move us towards the future. From Archimedes approximating 'pi' in 250 B.C., to finite elements used in developing nanomachines in 2000 A.D., there is no stopping the power of mathematics. "It is not enough to have a good mind; the main thing is to use it well."; this quote by Rene Descartes says it all. As long as we keep developing new ideas and techniques, there is no limit to how far we will go.
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Compiled by Sal Nsheiwat, Fall 2000

From the earliest times of the approximating "pi" by the Greek mathematician Archimedes ( in 250 BC) to finite elements methods used in developing nanomachines ( in 2000AD), computational methods (also called numerical methods ) has been the key technological development. In the 3rd century BC, Archimedes considered inscribed and circumscribed regular polygons of 96 sides and concluded an estimated value of "pi" to be 3+10/71 < 3+1/7. The Borcariot Pfaff algorithm ( a = 2*3^.5 ,b=3 ) gave Archimedes' estimate on the fourth iteration that was the most accurate value given for "pi" at that time. He also, broke the sections bounded by geometric figures such as parabolas and eclipses into infinite numbers of rectangles and added the areas together. In other words, he started integration and anticipated differential calculus.

Today, we can find the value of "pi" to the thousands of decimal places because of Archimedes invention in mathematics. We are able to calculate "pi" and several other values to almost an exact number due to differential equations and integration. Moreover, today the computational methods that Archimedes started are the basis in developing some of our greatest and most complicated technological advancements we have, nanomachines. Studies have been developed to create nanomachines that are able to make fusion between two hydrogen atoms. Nanomachines can be constructed on the micron scale ( 10^ (-6) meter), made of parts as small as (10^ (-9) meter) with smooth, super hard surfaces made of atomically flawless diamonds.

Numerical methods are used in calculating the compressible fluid flow, heat changes and several other factors that affect the nanomachines. Numerical methods are extremely important to calculate things to the most accurate degree. I mean nanomachines are being built as small as 10 ^(-6) meters, and their parts even smaller, leaving very little room for error.

Let us keep in mind that before the 1940s, computers were not available. As a result, the use of differential equations was not as easy as plugging it into a personal computer and getting an answer. People then spent more time figuring out how to approach and solve the problem, than they did to analyze it.

Nonetheless, we as a society are using Archimedes' discoveries in our greatest technological advancements. In other words, it all started with Archimedes in 250 BC, and today, 2,250 years later, we operate on the grounds that he has established for us.