Let \(f(x)\) be a function that is continuous on a closed interval \([a,b]\), and where \(f(a)\) and \(f(b)\) take opposite signs (one value is positive and the other negative). By the Intermediate Value Theorem, this function must have a root on the interval \([a,b]\). The bisection method is an algorithm that allows us to approximate this root.
After choosing a starting interval \([a,b]\) on which the required conditions exist, we calculate the midpoint \(c = \frac{a+b}{2}\) of the interval. This midpoint becomes the current approximation of the root. If \(f(a)\) and \(f(c)\) take opposite signs, the root must be within the interval \([a,c]\). Otherwise, \(f(b)\) and \(f(c)\) will take opposite signs, and we must have a root in the interval \([c,b]\). We then repeat this process using whichever of these intervals contains the root, dividing the interval in half at each stage. After enough iterations, the intervals will grow narrow enough that repeating the process does not result in any significant change to the approximation.

Enter a matrix in the text field. Surround each row with square brackets, and the entire matrix with another pair of square brackets. For instance, to enter the matrix \(\begin{bmatrix}2&1&3\\4&9&7\end{bamtrix}\) enter [[2,1,3],[4,9,7]]. Spaces and extra lines will be ignored, but your matrix may be rejected if it is not a rectangular array, or if it contains values other than numbers in decimal form. Non-numerical characters and missing elements will be treated as 0.

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