hello internet this is oscar veliz again for this video i'll be going over how to use secant
method to solve systems of nonlinear equations i'll go over the background and history before
visualizing how the method works and then providing some discussion for context you
should have some familiarity with seeking method i recommend watching my award-winning
video on this topic as well as newton's method for systems of nonlinear equations
recall that with secant method we're given two starting points and we draw the secant
line that travels between them where it crosses the x-axis we create our next point then using
our last two points repeat the process and draw another secant to find our next point
and so on and so forth until we reach an approximation close enough to our root for the rest of this
video i'm going to keep referring back to the secant method for simultaneous nonlinear
equations by wolfe this method was also found by bittner published in german covered in
convergence of multi-point integer methods by tornheim
let's look at secant method but from a different angle x3 we can write as the sum of one-half
of x1 plus one-half of x2 x4 can be written as the sum of one fifth of x two plus four
fifths of x three and x five can be written as the sum of negative four twenty-firsts
of x three plus twenty-five twenty-firsts of x four but where are these numbers coming from
to find those numbers i have been solving this system each iteration for example one
half of x1 plus a half of x2 given that a half of the function at x1 plus the half of
the function x2 gives the sum of zero and that a half plus a half gives us one the same
is true of x4 that one-fifth of the function at x2 plus four-fifths of the function x3
gives us zero and one-fifth plus four-fifths is one to solve this i'm using this linear
system each iteration finally the last step that we showed is also from that linear system
both of these methods of using either secant method as we know it or solving a linear system
are doing the same thing we can rewrite our system like this
we can use this second method to solve systems of nonlinear equations
here's a nonlinear example that we'll be using as in my earlier videos on this topic with
these four solutions
our first step is to rewrite our system in terms of zero where our solutions all map
to the origin let's take our first starting point at 1 1 and because it's a secant method
we're going to need another point for example 1 2.
plug this into our linear system and this doesn't work that's because for d dimensions
we're going to need d plus 1 points
let's take a third point at the value 1.5 2 and now set up this linear system which
we can rewrite we can use any linear system solver to solve this once we do we'll get
these values for p which we can then plug in to our summation for our next value for
x given this value for x4 and plugged into F looks like that
then we take our last three points notice how we no longer need the value of 1 1 and
use this in our linear system we solve the linear system like before and used our new
summation to give this value for x5 plotted right there
and then repeat the process with the last three points until our distance between our
steps becomes less than some epsilon at the norm or our function is close enough at that
value or reach the maximum number of iterations let's look at a numeric example here we use
an ending value of 10 to the minus 6 as our epsilon from these three starting points it
only takes a few more iterations to actually converge on a solution notice we haven't used
the jacobian once let's talk about the all important order for seeking method in nonlinear
systems it actually depends in tornheim's paper he's able to prove that the order depends
on this function meaning the higher the dimensionality the slower it converges with 1 it converges
1.618 normal secant method but as you increase dimensionality the order slows down
this might surprise you but the method i've discussed is not the only secant method richard
brent describes some efficient algorithms for solving systems of nonlinear equations
describing other secant methods but he also doesn't compare this to the well-known broyden's
method brent also does not compare this version of secant method citing those three papers
from earlier as well as this additional one by barnes which is actually really good noting
that the math and the assumptions you have to make make comparison difficult and interestingly
enough he actually thanks wolfe for his suggestions in the acknowledgement
in summary you can use secant method to solve systems of nonlinear equations giving you
a superlinear order the method should be credited to wolfe and bittner and there actually isn't
a jacobian we'd also need to solve a linear system each iteration and start with d + 1
starting points there are other classes of "secant method" but those deserve their own
videos you could also make fractals using this method i'll leave that as an exercise
for the viewer the code that i wrote will be hosted on github as always thank you for
watching you all really seem to enjoy my earlier video on newton's method for systems of nonlinear
equations and i hope you found this one interesting as well please continue to give me feedback
on this series and thank you so much for your support