Hello_World

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A set of simultaneous equations, such as:
2x + 3y = 12 [1] 5x - 6y = 3 [2] can be solved by methods of elimination, substitution, matrixes. Investigate a fourth way - iterative guessing. Equations [1] and [2] can be written: x = 6 - 3/2y y = 5/6x - 1/2 Suppose we start with a guess of x = 0 and y = 0. This gives the new values of: x = 6 y = -0.5 Are these new values better guesses? Try... this is known as iteration. x = 6.76 y = 4.5 Try with new: x = -0.75 y = 5.125 These equations appear to go nowhere. A different re-arrangement allows a different iteration: y = 4 - 2/3x x = 3/5 + 6/5y If the iterations start with x = 0 and y = 0: After 60 iterations, you end up with x = 3.004 and y = 2.003 which is close to the correct answer (x = 3, y = 2) Why did the first attempt fail and the second attempt succeed (albeit slowly)? Consider this pair of equations: x - 2y = -2 2x + y = 2 Rearranged as: y = 1/2(x + 2) x = 1/2(2 - y) After 8 iterations you get x = .398 and y = 1.195 (close enough to x = .4 and y = 1.2). Why did it work so much better for this than the first set? Investigate this method with the aim of coming up with a rule that will enable you, given a pair of sim equations: ax + by = c dx + ey = f to determine which way will converge to a true solution. Can you derive any rules that will enable you to predict how rapidly the iterative scheme will converge? |

I'm at a loss. Anyone know or have ideas to try out?