Nonlinear Systems - Newton’s Method
An Example:
The LORAN (Long Range Navigation) system computes the position of a boat at sea using signals from fixed transmitters. From the time divergences of the incoming signals the boat obtains differences of distances to the transmitters. This leads to two equations every representing hyperbolas defined by the differences of distance of two points (foci). An illustration of such equations is:
(x2/1862)− {y2/(3002− 18620} = 1 and{(y − 500)2/2792} – {(x − 300)2/(5002− 2792) }= 1 .
Resolving two quadratic equations with two unknowns would require solving a 4 degree polynomial equation. We could do this through hand but for a navigational system to work well it should do the calculations automatically and numerically. We note down that the Global Positioning System (GPS) works on similar principles and should do similar computations.
Vector Notation:
Generally we are able to usually find solutions to a system of equations when the number of unknowns matches the number of equations. Therefore we wish to find solutions to systems that have the form:
f1(x1, x2, x3, . . . , xn) = 0f2(x1, x2, x3, . . . , xn) = 0f3(x1, x2, x3, . . . , xn) = 0.......fn(x1, x2, x3, . . . , xn) = 0.
For ease we can think of (x1, x2, x3, . . . , xn) as a vector x and (f1, f2, . . . , fn) as a vector-valued function f . With this notation we are able to write the system of equations simply as:
f(x) = 0,
That is we wish to find a vector that makes the vector function equal to the zero vector.
While in Newton’s method for one variable we need to start with an initial guess x0. Theoretically the more variables one has the harder it is to find a good initial guess. In fact this should be overcome by using physically reasonable assumptions about the possible values of a solution that is take advantage of engineering knowledge of the problem. Formerly x0 is chosen let
Δx = x1− x0.
Linear Approximation for Vector Functions:
In the single changeable case Newton’s method was derived by considering the linear approximation of the function f at the initial guess x0. From Calculus the subsequent is the linear approximation of f at x0, for vectors and vector-valued functions:
f (x) ≈ f (x0) + Df (x0)(x − x0).
Here Df (x0) is an n × n matrix whose access are the various partial derivative of the components of f. Purposely:
Newton’s Method:
We wish to find x that makes f equivalent to the zero vectors so let’s choose x1 so that:
f (x0) + Df (x0)(x1− x0) = 0.
Since Df (x0) is a square matrix we can solve this equation by:
x1 = x0− (Df (x0))−1f (x0),
Offered that the inverse exists. The formula is the vector equal of the Newton’s method formula we learned before. Nevertheless in practice we never use the inverse of a matrix for computations so we cannot use this formula directly. Rather we are able to do the following First solve the equation:
Df (x0) Δx = −f (x0).
Since Df (x0) is a known matrix and −f (x0) is a known vector, this equation is just a system of linear equations which can be solved efficiently and accurately. Once we have the solution vector ?x we can acquire our improved estimate x1 by
x1 = x0 + Δx.
For following steps we have the following process:
• Solve Df (xi)Δx = −f (xi) for Δx.• Let xi+1 = xi + Δx
Graphs of the equations x3 + y = 1 also y3− x = −1. There is one and only one intersection at (x, y) = (1, 0).
An Experiment:
We will solve the subsequent set of equations:
x3 + y = 1y3− x = −1.
You can simply check that (x, y) = (1, 0) is a solution of this system. Through graphing both of the equations you can as well see that (1, 0) is the only solution.
We are able to put these equations into vector form by letting x1 = x, x2 = y and
f1(x1, x2) = x31+ x2− 1f2(x1, x2) = x32− x1 + 1.
or
Now that we comprise the equation in the vector form write the following script program format long
f = inline(’[ x(1)^3+x(2)-1 ; x(2)^3-x(1)+1 ]’);x = [.5;.5]x = fsolve(f,x)
Save this program as myfsolve.m as well as run it. You will observe that the internal Mat lab solving command fsolve approximates the solution however only to about 7 decimal places. While that would be close sufficient for most applications one would expect that we could do better on such a easy problem.
Next we will execute Newton’s method for this problem. Change your myfsolve program to
% mymultnewton
format long
n=8 % set some number of iterations may need adjusting
f = inline(’[x(1)^3+x(2)-1 ; x(2)^3-x(1)+1]’); % the vector function
Df = inline(’[3*x(1)^2, 1 ; -1, 3*x(2)^2]’); % the matrix of partial derivatives
x = [.5;.5] % starting guess
for i = 1:n
Dx = -Df(x)\f(x); % solve for increment
x = x + Dx % add on to get new guess
f(x) % see if f(x) is really zero
end
Save as well as run this program (as mymultnewton) and you will see that it finds the root exactly (to machine precision) in only 6 iterations. Why is this easy program able to do better than Mat lab’s built-in program?
Latest technology based Matlab Programming Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Matlab Programming help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Matlab Programming, project ideas and tutorials. We provide email based Matlab Programming help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Matlab Programming. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Matlab Programming Homework help and assignment help services. They use their experience, as they have solved thousands of the Matlab Programming assignments, which may help you to solve your complex issues of Matlab Programming. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
tutorsglobe.com mechanism of fracture assignment help-homework help by online bones and joints tutors
tutorsglobe.com derivation of m-m equation assignment help-homework help by online enzyme kinetics tutors
Derivatives of Carboxylic Acids tutorial all along with the key concepts of Properties of Carboxylic Acids, Acyl Chlorides, Acid Anhydrides, Esters, Preparation of esters, Amides, Preparation of Amides, Physical Properties of Amides, Uses of carboxylic acids and their derivatives
www.tutorsglobe.com offers steps for two-phase method, linear programming problems, lpp solution, assignment help and homework help by live online operation research tutors
tutorsglobe.com economic importance of bryophyte assignment help-homework help by online bryophytes tutors
Herbs-Shrubs-Trees tutorial all along with the key concepts of Herbaceous Monocotyledonous Stem, Herbaceous Dicotyledonous Stem, Ephemerals, Annual Plants, Biennial Plants and Perennial Plants
tutorsglobe.com demonstration of osmosis assignment help-homework help by online osmosis tutors
theory and lecture notes of bioelectric signals and electrocardiogram all along with the key concepts of cell membrane potential, action potential, cardiovascular system, electro-stimulation of heart and electrodes. tutorsglobe offers homework help, assignment help and tutor’s assistance on bioelectric signals and electrocardiogram.
www.tutorsglobe.com offers redox reactions homework help, redox reactions assignment help, online tutoring assistance, physical chemistry solutions by online qualified chemistry tutor's help.
tutorsglobe.com hearing aid assignment help-homework help by online ear tutors
motion under gravity tutorial all along with the key concepts of concept of projectile, motion of projectile, vertical and horizontal projections, resultant velocity of a projectile, projection at an angle to the horizontal and applications of projectiles
www.tutorsglobe.com offers concurrent development model homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
there are three major components of the cpu that are as follow - arithmetic-logic unit (alu), control unit (cu), on-board cache memory.
theory and lecture notes of inverse functions all along with the key concepts of graph of inverse function, finding inverses informally and finding inverses formally. tutorsglobe offers homework help, assignment help and tutor’s assistance on inverse functions.
tutorsglobe.com length of stamens assignment help-homework help by online sterile stamen tutors
1963457
Questions Asked
3689
Tutors
1476086
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!