A New Litany

at their (common) fixed points. However, in anticipation of confronting similar questions for more general (e.g., cubic) equations, it is of interest to bring technology to bear on this question.

Using a modern spreadsheet program such as Excel, we are able to generate graphs of these functions, together with the line y =x. In creating such a spreadsheet we would use "locked references" to refer to the cells containing values for a, b, c, L and R. Then, after studying the slopes of G₁(x) and G₂(x) for one set of values of a, b, and c, we are able to assign them new values. Excel will automatically re-calculate a table of values and re-draw the graphs accordingly.

This feature enables us to use computer technology as a basis for conjecturing which of these iterators is to be preferred. Of course, the challenge of developing a formal proof will remain.

The spreadsheet below illustrates the format that one might use in such an experiment.[If you would like to see the formulas that underlie such a spreadsheet program, click here.]

As indicated by the special case of x² - x - 1 = 0 in the above spreadsheet, it is G₂ that seems to be flatter at at the fixed point x = 1.61803... .

Exercise. Using a spreadsheet or graphing calculator to generate such graphs, confirm that G₂ appears to remain flat at its fixed points, even as we vary the values of a, b, and c.

The flatness of G₂ at its fixed points makes it a remarkably effective tool for solving quadratic equations withoutrecourse to the quadratic formula. For this reason, it seems appropriate to give it a place alongside the quadratic formula in the algebra curriculum.

That is, in the computer age it seems appropriate to supplement the litany minus bee, plus or minus the square root of bee squared minus four ay cee, all over two ay with:

Ay ex squared minus cee, all over two ay ex plus bee.

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