Tuesday, March 20, 2018
Lagrange introduced the Method of Multipliers in his Analytical Mechanics of 1811 but when one first encounters this method it's not very clear why it works. One can deduce the procedure starting with a set of condition equations, Φ, and use least squares.
One ends up with a linear combination of the condition equations with arbitrary coefficients, the dΦ, set equal to zero. Division by dΦ1 removes some of the arbitrariness since the linear combination equals a constant, -Φ1.
Edit (Mar 20): Add dΦ and last sentence.
Monday, March 19, 2018
The gradient correction equation procedure is not free from error and can be forced to fail as seen in the following example.
The procedure works correctly for an initial rough estimate of x=2, y=6 for the zeroes of the Condition Equations Φ.
But the procedure fails along the x=y diagonal.
Saturday, March 17, 2018
This iterative correction for the fit of the parameters for the catenary using the least squares gradient formula works quite well too. The equations of condition for xmin and L are fairly smooth over a wide range as these grid plots show.
And the iteration of the corrections converges rapidly to the optimal values of xmin and L using an initial rough estimate of their values.
Using the formulas for the catenary we can now compute exact values for ymin and y(b).
Friday, March 16, 2018
Using smin and σ=A/(gλ) to rewrite the parametric equations for x and y one can eliminate the mass density λ and show that the curve is in fact a catenary.
Supplemental (Mar 16): One can also show that the tension in the string is,
Thursday, March 15, 2018
Although the parametric equations for the equilibrium curve involve the linear mass density, the fits for A and B don't appear to be affected by changes in the value for the linear mass density, λ. All the values for the minimum agree to 15 decimal places.
At the minimum gλs+B=0 so B=-g λsmin making the argument of the sinh-1u equal to u=gλ(s-smin)/A so there are two parameters, A/gλ and smin appear to be independent.
What we've been calling the equilibrium equations may in fact be the parametric equations for a catenary.
The relative linearity of the equations of condition for the fit of the equilibrium curve allows us to use numerical methods for its solution. We can start with a rough estimate and improve on it by using the estimate to evaluate the condition equations and their derivatives and estimating the necessary correction.
This method works quite well.
Wednesday, March 14, 2018
Cobwebs can be used to solve a pair of simultaneous equations but they are notorious for the problems associated with them. In the preceding blog I estimated the intersection point by using a formula for computing where two straight lines will cross.
In the figure below the two lines were fitted using least squares to obtain the coefficients for the linear equations. The values a and a' above are the points where the lines intersect the y-axis with b and b' being the slopes of the lines. The distance of the intersection point from the origin is x.
A simple explanation of the formula is the distance of the intersection point from the origin is the separation of the lines there, Δa, divided by their rate of approach, Δb. As one gets closer to the intersection point the data becomes relatively more linear but eventually one sees more and more relative error in the estimate as one zooms in.
Supplemental (Mar 15): The equilibrium price is a topic discussed in Economics. See the Wikipedia articles on Economic equilibrium and Bayesian game. One can explain the divergent cobwebs as due to the incomplete information that results from basing estimates on past history in a changing market. It's an example of a flaw in the use of Bayes' theorem where current statistics differ from those of the past. Basically what works here doesn't always work there and the situation spirals out of control.