Consider the general problem: [i]Find the real zeros of an arbitrary polynomial.[/i][br][br]Here is my plan of attack. I will introduce some points on the curve, see how it behaves at those points, and use that information to find points closer to the zeros. [br][br]How should I introduce points? What behavior interests me? How do I isolate an interval that must contain a zero?[br]To understand how this might be done, I begin with a particular case which lends itself to the discovery of answers:

To give the polynomial from arbitrary points P1, P2, P3, P4:[br][br]Let the points be given by[br][math]\;\;\;\;\;{\small P_i =(x_i, y_i)\;\;\;, i = 1, \ldots, 4 }[/math][br]And vector Y be the function values of the four points:[br] [math]\;\;\;\;\;{\small Y = [ y_1 \;\, y_2\;\, y_3\;\, y_4 ] } [/math][br]The [i]i[/i]th row of matrix M be [i]x_i[/i] evaluated at each of the four functions:[br][math]\;\;\;\;\;m_{i*} ={\small [\cos x_i \;\, \sin x_i \;\, \cos 2x_i \;\, \sin 2x_i]} [/math][br]The unknowns are the coefficients:[br] [math]\;\;\;{\small {\rm cf} = [ A \;\, B\;\, C\;\, D ]^T} [/math][br][br]For each point, I have[br][math]{\small y_i = A \cos x_i + B \sin x_i + C \cos 2x_i + D \sin 2x_i} [/math][br][math] \;\;\;\;\; = m_{i*} \cdot {\rm cf} [/math][br][br]The system of four points can be written as the matrix multiplication[br] [math]\;\;\;\;\;[M] {\rm cf} = Y [/math][br][br]Hence, the coefficients are [math]{\small \;\;\;\;\; {\rm cf} = M^{-1}Y} [/math][br][br]I can now pass back and forth between arbitrary points and coefficients. But I would like to do better than this. I want to choose points which give me information about the function and its zeros...[br][br]______________[br]To find the inverse of M, Set M = I, and perform Gaussian elimination on the two at once. When the left hand side is I, the right hand side is the inverse matrix.