Consequences of unwittingly reinventing an idea Say you have an idea and you can't find any paper or book that explicitly states that idea. You don't know for sure if you're right in assuming that you came up first with that idea so you decide to publish it. What are the consequences if you're wrong and the idea has already been published. Do the reviewers point that out for you before the paper gets published? Is your career over because of that? 

To be more specific, I have an idea for constructing iterative formulas for solving systems of nonlinear equations for any convergence rate. I also have an elegant proof for that. The problem is that I have found just one article that explicitly states that idea just for functions of one variable, but doesn't expand on it.( just one paragraph and then continues with something else)
Ignorance is not plagiarism. If you write that "To the best of my knowledge" the algorithm is new, you imply that you researched the matter duly (I would suggest to consult some experts if you are kind of an outsider to the field) and found no prior publication. If such a publication exists, it will not be plagiarism. It may be embarrassing, but not a career killer by any means (see first comment to your question). 

However, if you know that a suggestion exists for the one-dimensional case, you should state that: "This algorithm has been previously suggested for the one-dimensional case by XXXX." You can also qualify this statement to better highlight what your work adds: "...but no proof of convergence was offered." or "Here I generalize the algorithm to n dimensions and provide a proof of convergence."