What are the similarities as well as the essential differences between an algorithm and a formula?

Question

I appreciate the answer, but the dictionary shows similarities in definitions that the answerer didn't address. I note that algorithms are referred to in explaining biological processes as well as mathematical ones. As are formulas. So a more comprehensive answer is being sought.

Answer 1

You mentioned algorithms in relation to biological processes as well as mathematical. There's an article about how they are used in biology at: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1635101&rendertype=abstract

An abstrat of the articke begins as follows:
"Complex biological systems are increasingly understood in terms of the algorithms that guide the behavior of system components and the information pathways that link them. Much attention has been given to robust algorithms, or those that allow a system to maintain its functions in the face of internal or external perturbations. At the same time, environmental variation imposes a complementary need for algorithm versatility, or the ability to alter system function adaptively as external circumstances change. An important goal of systems biology is thus the identification of biological algorithms that can meet multiple challenges rather than being narrowly specified to particular problems."
Formulas are not mentioned in this context, and you may conclude from reading this article that there are fewer similarities between the terms than you initially suspected.

Answer 2

You don't give a context, so I'll answer what I think you are asking with an example.

For the 'formula', take Newton's familiar equation, F = ma. If you have the mass and the acceleration, you can calculate the force simply by multiplying the two.

But if you have a known mass and a known force, especially if the force is time-varying, you need to take a second-order integral of the force to get the position as a function of time. It's not a straightforward calculation. Here is where you want an algorithm. Good ones here would be numerical integration with successive approximation and error analysis. Choose a tiny time step. Apply the force present at the beginning of the time step for the entire time step. Assume it was constant, then calculate the new position. Now do the same with smaller time steps. If the difference is larger than your tolerance for error, start over again with a smaller time step. Repeat until your error tolerance is satisfied. Then move to the next time step and repeat the process. It's a way of finding the value of a function when you can't directly calculate it.

Answer 3

A formula is an expression of the relation between a domain and a range. Given a selection of values from a domain, a formula evaluates to a specific value in the range.

An algorithm is a set of instruction on how to solve a problem.

In some cases, you may want to follow an algorithm in order to evaluate a formula. The formula states that the result will be based on the input, and the algorithm would be the set of steps that are conducted during the evaluation of the formula.

Given a specific problem there may be many algorithms that can be used to solve the problem.