Newton's Method--a Numerical and Iterative Process of Finding Difficult Roots of a Function:
JOHN, CHRIST, GALILEO, THE SUN'S TRAVELS, AND THE BIBLE BUGS ME:
1. These subjects are at paragraphs 12, 13, 14, 15, 16, 17 and 18.
SOME ROOTS ARE EASY; SOME NOT SO EASY:
2. We know a root, which is a solution to an equation, occurs when Y = 0; therefore, whatever value of x that occurs that causes Y to be equal to zero is a root( solution) to the equation. Much of mathematics is devoted to finding that value, or those values. With the equation of a line, as for example, Y = 3x + 7, we simply equate Y to zero and solve for x, which is x = -7/3 or about -2.33. With a quadratic of, say, x2 +10x +21, we can factor this to (x + 3)(x + 7), and see that the roots are minus three, and minus 7. We can have an equation as Y = 0 = cos( x), and immediately we know the solution is x = π/2 or 90 degrees. These are easy roots to solve.
3. What about x4 - 17x3 + 28x2 +8x - 10? How do we go about finding the roots to that? Or an equation from my textbook: 48x( 1+ x)60 - (1 + x)60 + 1 = 0? Can you imagine trying to factor that? Newton's method, or also called the Newton-Raphson method, gives us a surprisingly very quick way to solve these equations. The first equation will be solved in this hub because it is the characteristic polynomial of our 4 x 4 DNA matrix at hub#12.1( linked), and we will call it D(x), D of x, with D standing for DNA. The reason this hub was written is so we can find the roots to D(x), because they represent the eigenvalues of the DNA matrix, and those eigenvalues will lead us to the eigenvectors. I'm still working on the hub where this equation is determined.
D(X), THE DNA MATRIX, GRAPHED AT FIG 1:
4. D(x) is graphed at Fig 1, which shows all the pertinent points. The trace function of the Ti-89 calculator gives an estimate of where the zeros are, and where the local extrema are. For the zeros we use Newtons method to get more accurate answers to what the trace function( lst time around) gives. Of course one can continue to zoom in to get more accurate trace-zeros, but depending on the accuracy desired, Newton's method can be quicker. To find the local extrema, we differentiate D(x), and then set the derivative to zero, and solve for the roots of that cubic equation. This gives us the x-values, which are plugged back into D(x) to find the corresponding y-values. All these pertinent points are listed at table 1.
WHAT IF YOU DO NOT HAVE A GRAPHING CALCULATOR?
5. What? If you are going to do much math, then it would pay to do what it takes to purchase one, including mow lawns, work at one of those temporary-pay-per-day jobs, or whatever. They are cheaper than a computer, and well worth the investment.
6. Many techniques are available to graph a function without a graphing calculator: The upper and lower bounds theorem, The local extrema of polynomials theorem, Synthetic division, Factor theorem, Rational zeros theorem, Descartes' rule of signs, Fundamental theorem of algebra, Complete factorization theorem, Zeros theorem, Conjugate roots theorem, and many more, and that is in algebra alone. Calculus has many more techniques that can be used to graph a function without a calculator.
7. Sometimes a graphing calculator may miss the exact behavior of a graph as subtle roots or local extrema, or removable discontinuities. Understanding the mathematics behind a graph will help you to determine these things.
D(X) = f(x), and D′ (X) = f′ (x):
5. Right off the bat I messed up at L1 and L2. From force of habit I wrote f(x) and f′ (x), but I meant to write D(x) and D′ (x); just mentally do the replacement. As I wrote and explained at paragraphs 14 and 15 of hub#12.3( linked), I have a thing against wasting paper.
L1 THROUGH L4 EXPLAINED:
6. D(x) is listed at L1, and its derivative at L2. Newton's method is listed at L3, and our initial guess of the root is xn at L3. This guess is plugged into D(x), and D'(x) at L4, and this fraction is subtracted from xn. This was done at L10, which I'll explain further when we get there.
THE DERIVATION OF NEWTON'S METHOD:
7. It is based on tangent lines, and the line equations associated with them. When a function is differentiated you get a value that represents a slope at a specific point. For example, in the function of Y = X2, the derivative is dy/dx = 2x. If you want to know what is the slope of a tangent line at x = 7, then it is 2 times 7 or 14; therefore, for every x value going to the right 1 unit, the Y value increased by 14 units( the run is 1, and the rise is 14). A typical line equation is shown at L5, where m is the slope, and b is the Y-intercept. On the right side of L5 is the equation for slope, and by transposing this and isolating x we arrive at Newton's method.
L6 THROUGH L10 EXPLAINED:
8. So the transposing is done at L6, and we are reminded at L6 that Y is a function of xn . At L7 we are reminded that the slope, m, is a function of the derivative of a function; therefore, f′(xn) replaces m. L7 has everything in place; therefore, all we have to do is isolate x, and we arrive at L9, which is Newton's method. Our first guess for the 1st root is -0.62, and that value of x is plugged into L4, which gives the value of xn+1 to be -0.619946, which is iterated back into the formula as our next xn. This is repeated until you get the accuracy you want, and this is done for each of the three other roots. Fig. 2 at L11 gives you a visual picture of how this works. Each tangent crosses the x-axis getting closer and closer to the actual root, as each iteration is completed.
INTEREST ON A CAR---L11 AND L12:
9. In James Stewart's outstanding textbook, Calculus, 5th ed, on page 294 he suggests solving the 2nd equation at paragraph 3 above using Newton's method. That was done at L12, with the denominator of the fraction representing the derivative of the numerator. My first guess was 0.01 for x because if it crosses the x-axis it will do so very quickly, since (1 + x)60 = (1+0.01)60 = 1.0160= 1.8167. And 1.0260 = 3.281. When we multiply these by the 48 factor these skyrocket as x gets larger; therefore, x = 0.01 is a good first guess if it crosses the x-axis. Plugging 0.01 into Newton's formula gives x = 0.00822. This becomes our next iteration, so we plug this in and get x = 0.00768. For the 3rd iteration we get 0.007629. For the 4th we get 0.007628603, and for the 5th we have 0.00762860281.
WHAT DOES 0.00762860281 REPRESENT?
10. The problem in Stewart's text is if a car dealer sells you a car for $18,000 at $375 per month for 5 years, at which time it will be paid off, then x represents the monthly interest rate the dealer is charging you. We solved for x and now know that the yearly interest rate is 9.154%, which is found by multiplying the number by 12. The graph at fig 2 of L11 is the graph of f(x) in the numerator as it crosses the x-axis.
THE EIGENVALUES--THE ROOTS OF OUR 4 X 4 DNA MATRIX:
11. The first equation at paragraph 3 is the characteristic polynomial of our DNA matrix, and the roots of that equation represent the eigenvalues of that matrix. Those roots, found by Newton's method, are: ( -0.6199463, 0.5444593, 1.960012, 15.11547). As I said, I'm still working on the hub that determines the characteristic polynomial. It feels like if I had a dollar for everytime I made a mistake while doing the calculations for that hub, then I could pay off our national debt!
THE APOSTLE JOHN; IS HE A LIAR OR A WRITER INSPIRED BY GOD?
12. It is so appropriate that this paragraph turned out to be paragraph 12, for the 12 Apostles of Jesus Christ. The last hub( #12.24) linked, gave some very disturbing evidence against the Apostle John. Paragraph 13 suggested John was a liar; par. 14, 15 and 16 apparently proved it; par. 17 implied God is discredited, but par. 18 stood firm on faith, and par. 18 and 19 began to vanquish the conundrum. We will continue par. 18 and 19 here.
FACT: GOD IS THE AUTHOR OF OUR BIBLE:
13. If you are a Christian, then this is not an opinion--you believe it to be a fact, and it is indeed a fact. If you own a King James Bible( I've only studied from the KJV; therefore, I'm not qualified to comment on any other) then you own a book authored by the Creator of the universe, and all that it contains. This is a remarkable fact that merits reflection. Ponder this; the One who created the oceans, mountains, stars and galaxies has written a Book intended for you!
BUT SOMETIMES GOD'S WORD BUGS ME!
14. Yes I admit it. I was bugged by God's question to Job( see par. 18 of hub#12.24), and I was bugged by the arithmetic concerning John's statement, and I have been bugged many other times besides these two instances. But the fact is that God has inundated me--and everyone else--with so many evidences that the Bible is truly inspired by God, and so many evidences that He is the Creator that these evidences anchor one's faith--through obedience--so securely it becomes immovable and yields to nothing. Plus God gives to each of us innate faith, and we can choose to build upon that faith, or we can choose to destroy it. I knew there was an answer to God's question to Job, and I knew there was an answer to John's statement.
THE TRUE MEANING OF JOHN'S STATEMENT:
15. John did not mean just the 33 years Christ was on earth. The anticipatory problem with John 21:25 was vanquished at John 1:1-3. By Christ everything has been created( John 1:3; Colossians 1:16, 17; Ephesians 3:9). This fact alone could fill the earth with typewritten pages describing the infinite thought and reason that went into the design of this universe, and the complexities that had to be worked out, overcome, and brought together to congruently harmonize into a structured whole. Do I believe God is capable of speaking the universe into existence in an instant( Hebrews 11:2)? Of course I believe this. But I do not believe that John was referring to the amount of books required for God to understand what was accomplished in creating this universe by and through Jesus Christ. God does not need us to tell Him what He did by and through Christ. I believe John was referring to the amount of books it would require to get us to understand just what Jesus Christ achieved in creating this universe. Creation is unfathomably structured and complex--the achievement of Jesus Christ--and far beyond our science, mathematics, and understanding. A spiders web is at least 5 times stronger than steel, pound for pound. Many thousands of chemical reactions occur in a cell, and they are perfectly timed and synchronized, and the inner ear is a marvel of engineering and design. Yes the earth itself could not contain the books that could be written concerning what Jesus Christ accomplished when He made the universe.
JOHN'S STATEMENT WAS INSPIRED BY GOD, AND IT IS CONSERVATIVE:
16. Just the physical creation has been addressed; we haven't even touched the creation of the spiritual realm by Christ. Additionally, Scripture tells us that Christ is eternal past( Psalm 90:2 with John 8:58 and Exodus 3:14; then also Micah 5:2 and Hebrews 6:20 with 7:3); therefore, "there are many other things which Jesus did,"--- going back a very, very long time--- "the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen."
NEITHER SCIENCE NOR MATH WILL EVER TRUMP GOD'S WORD:
17. The amount of things that can be proven with science and math are endless. They are both extremely powerful tools of mankind, but our understanding is limited and our knowledge very finite. I read an article a long time ago about when Isaac Newton( 1642--1727) wrote his laws of motion then the philosophers and "theologians" claimed that Newton's laws proved there was no free will and everyone and everything was, and is, fated. Of course this contradicted God's Word that makes it extremely clear we do have free will to choose life or death, heaven or hell, obey or not obey. Then many years later quantum mechanics comes along. What is the essence of quantum mechanics? It is probability. There is an innate indeterminism to God's creation, and it has nothing to do with the accuracy of our measuring instruments. It is a quality of creation. This brought back free will. I believe the spirit in each of us makes these sovereign decisions; nevertheless, quantum mechanics--centuries later--validated God's Word.
THE SAME THING HAPPENED WITH GALILEO GALILEI:
18. The same thing happened with Galileo Galilei( 1564--1642). People believed the Bible says the sun goes from one end of heaven to the other( Psalm 17:6). Galileo basically said no it does not, and I have proof. He argued that the earth goes around the sun. People argued that Galileo's observational proof was contradicting God's Word. Galileo, and others before and after him, were right; the earth goes around the sun, but they were wrong concerning the sun being relatively stationary. Years later we discover the sun does indeed go from one end of heaven to the other. It travels a distance of 166,504 light years( 9.77 x 1017miles or nearly 1 million, million, million miles) around our galaxy. Plus it is traveling at a rate of 600Km/s( 373 miles per second or 1,342,145 miles per hour) toward the Great Attractor( linked)-- see the section "cosmology-large scale dynamics" --and finally as God stretches space( Isaiah 44:24), the sun is being moved with the space; therefore, it is literally going from one end of heaven to the other in more ways than one. Incidentally, notice the Bible predicted the stretching of space long before we discovered this.
CONCLUSION: CAN WE TRUST THE BIBLE, GOD'S WORD"?
19. Yes, absolutely!!!
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