Career, Family And Living For The Lord
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A Twenty-Five Year History

This chapter presents the mathematics of my years at the Newport News Shipbuilding and Dry Dock Company, also known as the "Shipyard." During that time, I worked a great deal with linear regression and some with financial applications, and I wrote my first-ever program to compute and display the amortization schedule for a typical loan payment. From those early days as a Mathematician, I acquired a great passion for the discipline of math. Mathematics has been, for me, highly challenging, but it has also been very rewarding. There is tremendous satisfaction in solving a problem on paper and then in using a computer to actually run "your" program to do all of the number crunching. I am thankful that the Lord has allowed me so many years and so many opportunities to do the works of a Mathematician.

In the early going of this book, as it was in the early going of my career, my focus will mainly be on mathematics. During my years at the Shipyard, I was already a Christian, but I was not yet what I would describe as "totally sold out for the Lord." At that time, I had had a personal relationship with Christ for several years, since December 1967. Yet I still had not learned very much about Him, and I still was not really trying to live for Him. The Lord is faithful to teach us about Him, as we have the desire and willingness to learn. Over the next few years, however, my desire to learn more about Him would not be all that it could or should have been, and it would not be until July 1973 that the focus of my whole life would actually begin to shift.

Therefore, in these opening chapters, I would like to encourage all non-Mathematicians and all non-technical types to just "read" past all of the complicated formulas and equations because they are not that important! My intent in writing these early chapters is to document what I once considered, and what I even still consider, to be the very fascinating and exciting work that I did as a young Mathematician. In a few pages, the Mathematician side of me will begin to diminish, and the Christian side and family side will start to emerge. This book has been written to uplift and glorify Jesus as my Savior and Lord. It has also been written to present some of the many adventures of my family. I truly love the power and majesty of mathematics, and I even like to talk about it. But my sincere hope is that this book will show Jesus, rather than mathematics, in a truly unique and satisfying way.

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Table 1. Maxima and minima theory
------------------------------------------------------------------------------------------------------------------------------ Theorem 1: Let the function f be defined for a < x < b and have a relative maximum or minimum at x = c, where a < c < b. If the derivative f'(x) exists as a finite number at x = c, then f'(c) = 0. [1]

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One of my most exciting problems came as a result of the above theorem (see Table 1). While at the Shipyard, from March 1971 until August 1972, my assignment was to prepare a five-year financial forecast of the company's overhead costs. To do that, I used linear and multiple linear regression. I even developed my own version of a polynomial regression, but the results from that were not very positive. Regression analysis is based on a least squares mentality. The overhead costs, which we were trying to project, were the estimated future costs for the company. We were given twenty-five years worth of historical cost information and were then instructed to identify any possible mathematical relationships which might exist between those "known" costs and the overhead costs for the company during that same period.

One example of how this worked is in the case of the variable, "time." If you desire to project a given cost over time, then the element "time," itself, is usually a good independent variable because it takes into account the effects of inflation. For our work, we used the numeric "year number" as the value for time. Hence, for the year 1971, we used the value "1971," and so forth. Another independent variable, which also showed a strong correlation to total overhead costs, was the number of employees. Obviously, the more employees, the higher the overhead costs, and vice versa. So, we initially set out to examine both of these variables, plus about twenty or thirty more, over the twenty-five years for which we had been given sample data.

Figures 1 (not shown here) illustrates the principle of least squares. The idea is that you have a Cartesian coordinate system with scattered points displayed above and below the straight line. Of course, the straight-line equation is not known until after you have actually done the least squares computation. To understand the nature of the problem which we were given, let each yi in the diagram below, shown as the vertical axis, represent the total overhead costs for the company for a given year of the specific five-year period. Remember, of course, that we were using twenty-five years of such data, so we had to consider twenty-five such points. Next, let each xi, shown below as the horizontal axis, represent an independent variable, such as the aforementioned "time" or "number of employees." Note, for example, that, for x1, the cost, y1, was relatively high when compared to the next two observations. Yet, the application of least squares takes into account these very types of skewed variations and then tries to find the best straight-line solution to average them out.

This means that, for each xi, the estimated cost, yi est, is on the line and becomes the estimated overhead cost for that year. The differences between the observed and estimated values of "yi," as can be seen in Figure 1, are shown by the vertical lines drawn from the observed point to the straight line. Appendix A shows the mathematics for computing the slope, which is designated as "m," and also the y-intercept, which is designated as "b," for the straight line, "yi = mxi + b." As should be clear from that appendix, a knowledge of partial differentiation and matrix algebra was essential.

Linda and I were married on the fourth Friday in January 1971. I say the fourth Friday because we have always had difficulty remembering whether it was the Twenty-Second or the Twenty-Third. Actually, it was the Twenty-Second, but we had good reason at that time for not being able to remember the exact date. Everything had happened so quickly! Because I was a Junior at Old Dominion University, going back to school for the second time after having flunked out in June 1967 and because I was also in the Navy's Reserve Officer Candidate program, we had not really planned to get married until after my college and commissioning. However, my low morals and sexual indiscretions changed all that! I ended up taking my final examination for the first semester of my Junior year on the Twenty-First of January, and we were married in Elizabeth City, North Carolina on the next day. Because I had loved Linda almost from the start, though, I was not really bothered by that dramatic change in our plans.

She was already living in a house in Cherry Acres, so I moved in with her and her three children - Michael, Debbie, and Pam. Her first husband had signed the mortgage on that property with her, so in the first few months after we were together, I went through the necessary financial steps to have the home legally changed over into our names. From the very start of our life together, however, we were very, very poor! I can remember, in March 1971, adding up our total monthly income, comparing it to our monthly bills, and seeing that the results were horrendous! We were not even close to being able to pay our bills, and added to that already miserable financial picture was the fact that Melody, "our" first child, was due on the Fifteenth of August. We started out in very bad shape, and financially, things only got worse. As a result, it would be a very long time before we would ever be able to see any kind of measurable improvement.

As I have already stated, Linda and I started our married life together as a family of five. Mike was eleven, Debbie was ten, and Pam was eight. From the beginning, however, I had always loved and viewed her kids as my own. So, when we were married, I had actually wanted them to be mine. My biggest problem as a parent, though, was that I had never been a parent before. Thus, while they might have been at the point in their life when they needed and wanted "veteran" parents, as it were, in me, they were only getting a rookie. Because of that, things did not always go smoothly, but I still thought that the Cherry Acres years, in general, went pretty well.

During the Summer of 1971, I had had to go to Newport, Rhode Island to attend my first session in Officer Candidate School. During that Summer, which was also the final three months of Linda's pregnancy with Melody, I was away while Linda stayed at home to tend to the family. Her sister-in-law, Betty, and she spent almost every day together doing different things with the kids. I have always been thankful for Betty, first because she introduced Linda to me at the bowling alley, second because she lent us her birth date, November 15, as the "fake" wedding date which we used to cover our marital indiscretion, third because she drove us to Elizabeth City for our wedding, and fourth because she has always been so good to us. In my opinion, she has been and still is one in a million.

Melody was due on the Fifteenth of August, but she was not born until the First of September. We had thought that she might be born on the day before the first of September, but that turned out to be a false alarm. I can still recall Linda's embarrassment when the hospital staff sent her back home. For me, the idea of Melody's birth was very scary, so my determined eagerness to get to the hospital on time had contributed greatly to our being in the hospital a day early. We were living about thirty minutes away, and I was terrified by the thought that I might be delivering a baby on the side of the road somewhere. Therefore, by the Thirty-First of August, when Melody was already two weeks late, I was ready to head for the hospital at the first pain. As it turned out, my anxieties were not totally unfounded! When Melody was finally born, I had gotten Linda to the hospital at 2:25am, and Melody was "officially" born at 2:30am. Linda never even made it to the Delivery Room, but of course, we were still billed for the whole event, Delivery Room and all! I tried to protest that we had been billed for services not rendered, but it was to no avail. We still had to pay!

While at the Shipyard, in addition to performing mathematical analysis, I also had the opportunity to do some of my own computer programming. But because I was totally new to computers, I felt like I had to sneak in the back door, as it were, to write my very first program.

One of my functions as a Math Clerk was to enter data into a statistical analyzer program and then to review the results. Occasionally, I would even be required to type in a BASIC program for my boss and run it to get the results. Even though I do not now recall even one of the programs that I had entered for Hope, I still learned a lot about computer data entry from doing that work and also about how to use the BASIC Interpreter to run a program. After doing a few job-related tasks, I finally got up the courage to try to write my own computer program, only I was not sure that the Shipyard would really want me to do that on their time. So, I went to the computer lab one day during lunch and entered a program which was very similar to the one shown in Figure 2.

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FIGURE 2. MY FIRST PROGRAM - AMORT.BAS
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My First Program - AMORT.BAS

REM get the input data => principle, annual interest, number of years
      INPUT "Enter mortgage principle (xxxxxx.xx):"; pn
      INPUT "Enter annual interest rate (xx.x%):  "; ai
      ai = ai / 1200
      INPUT "Enter number years to pay debt (xx): "; iy
      IF pn = 0 THEN 99

REM compute annuity rate
      ar = 1 / (1 + ai) ^ (12 * iy)
      af = (1 - ar) / ai

REM compute monthly payment
      pm = pn / af
      PRINT pm, ar, af

REM display the amortization schedule
      OPEN "mort_dat" FOR OUTPUT AS #1
      PRINT #1, "Month   Balance   Interest  Towards Principle  Ending Balance"
      FOR i = 1 TO 12 * iy
      PRINT #1, "  ";
      PRINT #1, USING "###"; i;
      PRINT #1, "  ";
      PRINT #1, USING "######.##"; pn;
      PRINT #1, "  ";
      PRINT #1, USING "####.##"; ai * pn;
      PRINT #1, "       ";
      PRINT #1, USING "####.##"; pm - ai * pn;
      PRINT #1, "          ";
      PRINT #1, USING "######.##"; pn - pm + ai * pn

      pn = pn - pm + ai * pn
      NEXT i
      CLOSE #1

99    STOP
      END

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To say the least, I was a very novice programmer, yet I still had a genuine desire to learn everything that I could about the rules and techniques of programming. I did not know enough about problem solving at that point to even know what kind of program to write, therefore I had to improvise. Since I was in the process of buying a house, I thought that a simple amortization type of program would be good practice. At the time, though, I did not even know how to set up a payment schedule or anything like that, much less about how to handle simple interest and principle payments. Thus, that first program gave me ample opportunity to learn many new things, both with computers and also with financial mathematics!

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Table 2. Results From AMORT.BAS
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Principle:                  $79500.00        (  not included in code of Figure 2
            Interest Rate:                        6.5%    (  not included in code of Figure 2
            Number of Years of Loan:       1       (  not included in code of Figure 2

Month   Balance   Interest  Towards Principle  Ending Balance
    1      79500.00   430.63         6429.94                73070.06
    2      73070.06   395.80         6464.77                66605.30
    3      66605.30   360.78         6499.79                60105.51
    4      60105.51   325.57         6534.99                53570.52
    5      53570.52   290.17         6570.39                47000.13
    6      47000.13   254.58         6605.98                40394.15
    7      40394.15   218.80         6641.76                33752.39
    8      33752.39   182.83         6677.74                27074.65
    9      27074.65   146.65         6713.91                20360.74
   10      20360.74   110.29         6750.28                13610.46
   11      13610.46    73.72         6786.84                 6823.62
   12       6823.62    36.96         6823.60                    0.02

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After I typed in my program, I had a few syntax errors, but I had already gotten practice working with those from some of Hope's programs. I carefully corrected each mistake and continued through in that fashion until all of my syntax errors were eliminated. Finally, after correcting some faulty logic, I held in my hand the first-ever computer program that I had ever written. The results of a sample run with that program are shown in Table 2. I was so pleased by my success that I used my first chance that afternoon to tell Hope what I had done. Being happy that I had shown so much initiative, she told me that I could write more programs if I liked. She also told me that I could write some programs that would attempt to perform other aspects of our mathematical analysis. For me, that was like waving a red flag in front of the bull, and I charged!

With Hope's permission, I began studying the mathematics of multiple linear regression. She was also a Mathematician, plus she, too, was curious about what we might be able to accomplish with a little extra effort. Therefore, with her encouragement and even under her tutelage, I started reading about partial differentiation, least squares theory, curve fitting techniques, and matrix algebra. I learned not only how to write multiple linear regression computer application programs, but more importantly, I learned why the least squares method even works!

One of the things which I like about mathematics is that most problems can be defined by and reduced to a fairly simple system of equations. Then, once those equations are correct and in place, they will always give you the correct answer! A psychologist might analyze this kind of statement and conclude that mathematicians are basically insecure and that we do not deal well with uncertainty. I would not argue with either of those assessments. Part of my passion for this field through the years has been in the fact that numbers always provide a certain amount of security and also that they usually have legitimate value and usefulness.

In Appendix B, I have shown the solution for a multiple linear regression problem. The difference between multiple linear regression and linear regression is simply in the number of independent variables. The linear regression solution, which was developed in the earlier sections and shown in Appendix A, only dealt with one independent variable, xi. With multiple linear regression, the solution is still based on a straight-line equation, but it is multi-dimensional and deals with more than one variable. From a mathematical perspective, this problem is not much different either way, except that with the addition of one or more independent variables, the solution becomes much more complex!

Hope was almost always very good to me and also very supportive. Often, she gave me a lot of freedom to charge into uncharted waters. One example of this was when she let me try to develop my own version of polynomial regression. After having studied and worked with multiple linear regression and least squares theory for some time and in some depth, I thought that it might be worthwhile to try to implement a polynomial regression program of my own. She agreed! The mathematical derivation for this type of least squares application is given in Appendix C.

I learned a couple of useful things from developing my own polynomial least squares application. First, by getting a little more hands-on work with the computer, I learned even more about automated data processing and writing programs. I may have only been learning those things through on-the-job training, but I was still beginning to acquire a measure of expertise. Second, my confidence and expertise as a Mathematician were also increasing. From the start and even though I still had not yet earned my first college degree, I had done everything for my polynomial regression problem on my own. I had derived my own equations, I had written my own polynomial least squares program, I had debugged that program and desk-checked my answers, and I had done my own numerical study of our twenty-five years of data to identify the best candidates for a polynomial least squares.

That single effort contributed greatly to my professional development. I was still young but was already well on my way to becoming what I had wanted to be since the Fifth Grade - a real Mathematician! As a Fifth Grader, I had gotten a hold of a college Trigonometry book from one of my older sister's boy friends, a young man named Skip. He had shown me how to work basic ladder problems, and even at that early age, I had known that I would probably have a very special love for the power of mathematics. While still at the Shipyard, I was able to live out that youthful fantasy, yet the personal benefit of doing that simple polynomial least squares program did not stop with just the fulfillment of a childhood fantasy! Hope had wanted me to learn FORTRAN, so she made me teach it to the others in the office. None of my co-workers knew very much about computers, therefore I had to learn most of it on my own. I learned FORTRAN and then taught the class; hence, starting to develop some other skills which would come in handy in the future; that of learning, putting together lesson plans, and teaching others.

Everything went very well for me with my polynomial regression experiment, except that is, with the experiment, itself. From my efforts, I learned that a polynomial regression past the third order was not feasible for our computer because the large numbers would sometimes overload the word size of the General Electric 600 computer. Occasionally, when that happened, the computer program would abort. At other times, the computer would round off and/or truncate significant digits from key values. In almost all of the cases, the results were either inconsistent or completely wrong! Therefore, we ended up scrapping the idea of a polynomial regression for our particular application. Yet, that lack of success with that single problem did not tarnish or even diminish my sense of accomplishment over having tried something which was new and relatively unknown. In my mind, the whole effort had still been a huge success!