Lagrange Multipliers in Mathematics

Lagrange Multipliers in MathematicsLagrange multipliers arise as a method for exploit (or minimising) a function that is subject to one or more constraints. It was invented by Lagrange as a method of solving hassles, in particular a problem about the moons app bent motion relative to the earth. He wrote his work in a paper c aloneed Mechanique analitique (1788) (Bussotti, 2003)This appendix will only sketch the technique and is found upon in miscellanyation in an appendix of (Barnett, 2009).Suppose that we have a function which is constrained by . This problem could be solved by rearranging the function for x (or possibly y), and substituting this into . At which point we could then treat as a normal maximisation or minimisation problem to find the maxima and minima.virtuoso of the advantages of this method is that if there argon several constraint functions we can deal with them all in the same manner rather than having to do lots or rearrangements.Considering only f as a funct ion of two variables (and ignoring the constraints) we know that the points where the derivative vanish areNow g can also be minimised and this will allow us to express the equation above in terms of the dxsSince these are linear functions we can add them to find another solution, and traditionally is used to getWhich is 0 only when bothWe can generalise this easily to any number of variables and constraints as followsWe can then solve the miscellaneous equations for the s. The process boils down to finding the extrema of this function As an example imagine that we have a sporting 8 sided die. If the die were fair we would expect an just roll of . Let us imagine that in a large number of trials we keep getting an average of 6, we would start to suspect that the die was not fair. We can now estimate the relative probabilities of each outcome from the entropy since we knowWe can use Lagranges method to solve this equation subject to the constraints that the occur probability s ums to one and the expected symbolize (in this case) is 6. The method tells us to minimise the functionWhere the first part is the entropy and the other two parts are our constraints on the probability and the mean of the rolls. Differentiating this and setting it equal to 0 we getNow if we do an integration we know that this value must be a constant function of since the derivative is 0, also since each of the terms in the summation is 0 we must also have a solution of the formOrWe know that the probabilities sum to 1 givingWhich can be put into (A2.1) to getWhich doesnt look too much better (perhaps even worse). We still have one final constraint to use which is the mean valueWe can use (A2.2) and re-arrange this to findWhich also doesnt seem to be an progress until we realise this is just a polynomial in If a root, exists we can then use it to find . I did not do it that way by hand, I used maple to find the solution to the polynomial. (the script is below) I also deliberate the probabilities for a fair dice as a comparison and test.fair dice mu = 4.5unfair dice mu = 6p10.125p10.32364p20.125p20.04436p30.125p30.06079p40.125p40.08332p50.125p50.11419p60.125p60.15650p70.125p70.21450p80.125p80.29398lambda = 0lambda = -0.31521Table A2. 1 comparison of probabilities for a fair and biased 8sided dice. The bias dice has a mean of 6. Equation also appears in the thermodynamics section.Because can be used to generate the probabilities of the source symbols I think that it would be possible to use this value to condition the alphabet i.e. take a message from an unknown source and classify the language by finding the closest matching from a list (assuming that the alphabets are the same size). I havent done that but think that the same approach as the dice example above would work (the mean would be calculated from the message and we would need more sides).When we have a totally random source, and in this case the probability of each character is the same . This is easily seen from (A2.2) as all the exponentials contribute a 1 and we are left with Where m is the size of the alphabet all the symbols are equally probable in this case.