## Solution in Excel

Equation is … X=10+Log(20*X)

Loop 1 is =10+LOG(20*D4) the D4 is the guessed number above it. (Cell D4 where I guess 100)

Eash loop uses X for the number above it and before 20 loops it stops changing.

Guess a X 100 Changed =

Loop 1 13.30102999566400000000 -86.698970004336000

Loop 2 12.42491526851930000000 -0.876114727144710

Loop 3 12.39532343141620000000 -0.029591837103039

Loop 4 12.39428785897010000000 -0.001035572446172

Loop 5 12.39425157414140000000 -0.000036284828671

Loop 6 12.39425030272310000000 -0.000001271418261

Loop 7 12.39425025817260000000 -0.000000044550490

Loop 8 12.39425025661160000000 -0.000000001561048

Loop 9 12.39425025655690000000 -0.000000000054699

Loop 10 12.39425025655500000000 -0.000000000001917

Loop 11 12.39425025655490000000 -0.000000000000068

Loop 12 12.39425025655490000000 0.000000000000000

Loop 13 12.39425025655490000000 0.000000000000000

Loop 14 12.39425025655490000000 0.000000000000000

Loop 15 12.39425025655490000000 0.000000000000000

Loop 16 12.39425025655490000000 0.000000000000000

Loop 17 12.39425025655490000000 0.000000000000000

Loop 18 12.39425025655490000000 0.000000000000000

Loop 19 12.39425025655490000000 0.000000000000000

Loop 20 12.39425025655490000000 0.000000000000000

July 3rd, 2015 at 12:15 pm

The iterative method that you propose is called ‘The Fixed point iteration’

https://en.wikipedia.org/wiki/Fixed-point_iteration

But beware, there are situations where it fails miserably. For example, the equation that you consider is X=10+Log(20*X). If you solve this equation for the X inside the Log you will get X = (1/20) * 10^(X-10). Now, take your Excel spreadsheet and pick an initial guess of 12.39425. You will see that the method fails…

July 4th, 2015 at 7:19 am

Hi Zorglub. For that inverted equation you get:

X = 5.0000000000575646273258452479962*10^-12

The initialiser X = 0 does the trick.

I showed Ragknot the algorithm (in a private email) several years ago. I wish that I hadn’t. At the time he really did want to know how to find X (for the related Colebrook-White equation). He is/was some sort of computational support man to civil engineers.

Since then I have given him a fantastic fourth order algorithm. But he quibbles because it doesn’t get the very last available digit that Excel supports. The fact that it needs a maximum of three call to Log doesn’t seem to register on him at all. He doesn’t seem to understand rounding errors either.