How to numerically solve an unsolvable equation

Pretty much. There are plenty of equations whose solutions cannot be derived algebraically.

The essence of an iterative solution is that a trial value is inserted into the equation and the equation is then evaluated. There will be a numerical disagreement between the LHS and the RHS of the equation in all probability. Another trial value, hopefully a refinement of the previous trial value, is inserted and the process repeats, until the numerical disagreement between the LHS and the RHS of the equation is less than some arbitrary tolerance.

Hydraulics and friction due to fluid flow are two examples of fields where iterative solutions to problems are used quite a bit.

Real science and engineering is often messy like that. It's a far cry from algebra class where the solutions to an equation fall out all nice and neat. :)

 

Just to clarify this, Each iteration of Newton's method alternates from a value that is greater than the zero to a value that is lesser than the value. This is because Newton's Method takes the input value (i.e., the guess) and finds out where the tangent line at x crosses the x-axis, which becomes the new guess. If you draw some arbitrary curve you will see (as you should intuit) that this gives the "alternating" answers. This method converges (i think guaranteed - albeit somewhat slowly) because the tangent line at the zero point clearly only intersects at the zero .

EDIT: Newton's method is also the method your typically TI calculator uses to find zeroes.

 

Hello Delcross, welcome to PF :)

A small rectification: Newton is not a root bracketing method (like bisection or regula falsi)

Simple counter-example: ##f(x) = e^x - 5## starting from ##x_0## = 4:

Code (Text):

Iter          x                     f(x)

0           4                   49.60
1           3.092               17.01
2           2.319                5.16
3           1.811                1.114860
4           1.628                0.09573
5           1.609616601          0.00089
6           1.609437928          8.0E-08

ln(5)       1.609437912
 

And perhaps some more info: Newton-Raphson is very widely used.

Regula Falsi and secant methods amount to NR equivalents but with (initially coarse) numerical derivatives (so they end up in digital noise if you're not careful).

Newton converges very rapidly once going (roughly doubles the number of significant digits per iteration). If it converges. Contra-indications are: wrong sign of ##f'## or (even worse) ##f'(x_0)=0##. So you want to have good derivatives and start near a solution. Maybe by starting off with one of the other, more robust methods.

 

Thanks for correcting about bracketing. But, Newton's method is not always quadratically convergent and has issues with multiple roots. It can be modified to include second derivatives to regain its quadratic convergence, but even then you also add many more steps to perform each iteration. Other methods such as Steffenson's method give you quadratic convergence without having to calculate any derivatives.

 

@Del: we agree, but I'm afraid we are venturing into details. Tried to avoid that while remaining correct (with terms like "roughly", "good derivatives" and "start near a solution"). NR really is a workhorse in practice, with everybody who runs into trouble refining it to suit him/her.
Fortunately a lot of physics and chemistry equations don't behave all that pathologically, but I certainly took an extremely tame example.