Mathematica has a short tutorial entitled "Arbitrary-Precision Numbers." You can find it by searching for "tutorial/ArbitraryPrecisionNumbers" in Help/Wolfram Documentation or by typing SetPrecision, highlighting it, pressing F1, then find the link for the tutorial at the bottom of the page.

I made prog run, but it still does not converge. I believe that the coefficients to the ExplicitRungeKutta method may be incorrect. The four changes I made are:

1. There was no initial value for point, so the inner While never executed.

2. Since the RHS of ClassicalRungeKuttaCoefficients[4, prec] was never executed, the coefficients had no value. ClassicalRungeKuttaCoefficients[4, prec] is a non-starter; you cannot define a constant in the definition of a function like that. In any case, it was not being called as a function in NDSolve, so the coefficients were not defined. "point = y[1] /. f[[1]];" was doing exactly what it was supposed to; however, f had no meaningful value;

3. The lack of a ';' after the definition of NDSolve, meant that f was interpreted as a multiplication of the Print stmt after it; so, f had the value garbage * null, null being what Print returns.

4. As you sent it, prog would not define because of an imbalance in [ ]. I made several changes to balance the brackets. You might want to check that the logic still makes sense.

Recommendations:

You may be better off using the AccuracyGoal and/or PrecisionGoal options to NDSolve before using SetPrecision there. In any case, SetPrecison conflicts with the "MinPrecison = 50;" stmt in the beginning of the definition of prog. In theory, $MinPrecison controls the number of digits of all computations in the Module. Mathematica has a short tutorial (entitled, "Arbitrary-Precision Numbers") on the use of SetPrecision and SetAccuracy built-in. Just highlight one or the other and press F1. A link to the tutorial is at the bottom.

I put in code to save and restore the value of $MinPrecision.

When you add Print stmts to the routine, put in a label, such as Print["Point: ", point, "] instead of Print[point].

When you define a While or other kind of loop, try to remember to put the closing "];" on its own line and perhaps even comment it, thusly:

]; (* End inner while *)

That way, Mathematica will indent the lines properly for you so you can clearly see where the loops start and end. It is almost impossible to put in too many comments. A month from now, you will not remember what you did if you don't put in comments. Consider you are building a library of working code.

I suggest you make NDSolve and the RungeKutta method work outside of the prog Module. The edit prog, and try again.

Instead of defining a function, such as ClassicalRungeKuttaCoefficients[prec_Integer] := ..., within a module consider using anonymous functions instead:

Function[prec, body]

Then put the Function[prec, body] within the NDSolve. Anonymous functions are much, much faster in execution. But that is frosting on the cake. Make it work first, then make it faster.

Attachments:

All I was trying to show you was the basic paradigm for writing a program that finds an answer by iteration. There are several ways of doing this. As a former PhD student in Computer Science, I cannot imagine not having taken courses in numerical analysis (where iteration paradigms are in the first chapter), differential equations, and infinite series, plus many courses in business administration.

General Douglas MacArthur was ultimately Supreme Commander of Allied Forces in the Pacific Theater in World War II and military governor of Japan. He is widely credited, even in Japan, with being the father of modern Japan. His instructions from the American War Department were simple: "Make Japan look like the United States." In his autobiography, Reminisces, MacArthur makes the point over and over that he was a product of the education the Army afforded him. He showed again and again how this course or that course led directly to solutions to important problems.

I watched the videos on YouTube about the making of Fury, a movie about tank warfare in Europe in World War II. Fury's director said that the US sent thousands of young men to their deaths by burning them alive in those large metal boxes. I could not believe it. How is it possible that a culture that espouses progress, freedom for oneself and others, and concern for the poor kill so many of its young men in what was obviously a second rate weapon? A book on tank warfare shows it was simple. An isolationist population produced an isolationist congress that refused to fund much tank research; the US did not know how to make a good tank until the end of WW II. There are several other factors, not the least of which were lack of time and money, but one of the most important was that tank warfare doctrine was heavily influenced by men with a cavalry background, which said that the basic unit had to be light and fast. So, initially American tanks were light, fast, and prey to a single shot from a German anti-tank weapon or heavy panzer tank. My point is that the top American Ordnance and Armor planners simply could not escape their backgrounds to enable them to interpret the evidence supplied by the German blitzkrieg and the fall of France.

I have a point in writing this, and it is that you are obviously crippled by not knowing the rudiments of numerical analysis. But there are many other things you may not know: For instance, forecasting is nothing like naïve thinking might make one think. For another example, one of the most significant causes of American homelessness is a young person having a bad first boss who does not lead the young person to enjoy work. Do you know how to forecast or lead young people to be successful in life and work? If you are offered a chance to enter a PhD program, I urge you to seriously consider it. That or something similar is the only way to escape the tyranny of your impoverished background (numerical-analysis-wise). Also, everybody whoever made it had a mentor. Find someone in business, industry, or academia where you want to work who can lead you to study the issues they deal with now and will deal with in the future. The higher your mentor's rank, the higher yours will likely be.

OMG THANK YOU VERY MUCH. It totally works. I will not be able to get it correct by myself. The codes posted here is an alternate code for my research. Sorry I cant disclose it here. Here's the explanation. I'm an engineering master student working on heat transfer. In this case, it's about fluid. For the code, I would need to find the value of "z" so that "y=100". For a given condition, the "Twater" changes and the thermal conductivity of water. Then, "z" will be replace into Tnew equation. If Twater=Tnew, then that is the temperature that we want. At first, I was thinking to try to iterate with If,Do,While loop, with step size 0.001...., but that will take forever. After some trial, Tnew will actually converge, so I wanted to replace Twater with Tnew. This equation is some kind of temperature loop.

I will study the code given and learn more about the commands.

BTW,

Cwater = QuantityMagnitude[
   ThermodynamicData["Water", 
    "ThermalConductivity", {"Temperature" -> 
      Quantity[Twater, "Kelvins"]}]];

Is there any reason to have Cwater outside of the While loop? Well I tried putting it into the While loop and the loop just stop after 2 iterations. It will be helpful for me to understand this.

Hold on..... I suppose when we replace Twater with Tnew, then

While[(Abs[Twater - Tnew] > tol) && (iters++ < iterLim),

will not hold True anymore, that is why the iteration only run once.... So, how do I change it?, or I can just use the conventional method, which is much more "messy"?

n = 5; While[0 < n - 1 < 9, Print[n]; n = n - 0.5]

The code below is the general method, but I am not sure it achieves the correct answer. Start it by "Shift+Enter" or "Keypad Enter" on the cell containing prog. You debug it by putting in print statements until you find the error.

prog := Module[{Twater = 300, Tnew = 0, Cwater, Dwater, Ewater, 
    ClassicalRungeKuttaCoefficients, f, x, y, point, ans, tol = 0.001,
     iters = 0, iterLim = 100},
   Cwater = 
    QuantityMagnitude[
     ThermodynamicData["Water", 
      "ThermalConductivity", {"Temperature" -> 
        Quantity[Twater, "Kelvins"]}]];
   ans = "Twinkle, twinkle, little star,
     How I wonder what you are.
     Up above the world so high,
     Like a diamond in the sky.
     Twinkle, twinkle, little star,
     How I wonder what you are!";
   While[(Abs[Twater - Tnew] > tol) && (iters++ < iterLim),
    Dwater = (5 Cwater)/Pi;
    Ewater = Dwater^3 + 2*Cwater;
    ClassicalRungeKuttaCoefficients[4, prec_] := 
     With[{amat = {{1/2}, {0, 1/2}, {0, 0, 1}}, 
       bvec = {1/6, 1/3, 1/3, 1/6}, cvec = {1/2, 1/2, 1}}, 
      N[{amat, bvec, cvec}, prec]];
    f = ParametricNDSolveValue[{Derivative[1][y][x] == 
        Piecewise[{{(y[x] + x^3 + 3 z - 120*Ewater), 
           0 <= x <= 1}, {(y[x] + x^2 + 2 z), 
           1 <= x <= 2}, {(y[x] + x + z), 2 <= x <= 3}}], y[0] == 0}, 
      y[3.], {x, 0., 3.}, z, 
      Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, 
        "Coefficients" -> ClassicalRungeKuttaCoefficients}, 
      StartingStepSize -> 1/10];
    point = {z /. FindRoot[f[z] == 100., {z, 1}, Evaluated -> False], 
      100.};
    Print[point];
    FindRoot[f[z] == 100., {z, 1}, Evaluated -> False];
    Tnew = 
     170 + 1.89*z /. 
      FindRoot[f[z] == 100., {z, 1}, Evaluated -> False];
    Print["Twater: ", Twater, " Tnew: ", Tnew];
    Twater = Tnew;
    ];  (* End while *)
   Print["The answer is:\n", ans];
   ];

prog