Newton’s Method and Loops:
Solving equations numerically:
For the subsequently few lectures we will focus on the problem of solving an equation:
f(x) = 0.
As you erudite in calculus, the final step in several optimization problems is to solve an equation ofthis form where f is the derivative of a function F that you desire to maximize or minimize. In realengineering problems the function you wish to optimize able to come from a large variety of sources,including formulas, solutions of differential equations, experiments, or simulations.
Newton iterations:
We will signify an actual solution of equation by x∗. There are three methods which you may perhapshave discussed in Calculus: the bisection method the secant method with Newton’s method. Allthree depend on commencement close (in some sense) to an actual solution x∗.
Recall Newton’s method youmust know that the basis for Newton’s method is approximation ofa function by it linearization at a point that is:
f(x) ≈ f(x0) + f′(x0)(x − x0).
Since we wish to find x therefore that f(x) = 0 set the left hand side (f(x)) of this approximation equalto 0 along with solve for x to obtain
x≈ x0−f(x0)/f′(x0)
We begin the method with the initial guess x0, which we hope is fairly close to x∗. Then we definea sequence of points {x0, x1, x2, x3, . . .}from the formula:
xi+1 = xi−f(xi)/f′(xi)
If f(x) is reasonably well-behaved near x∗as well asx0 is close enough to x∗subsequently it is a fact that the sequence will converge to x∗and will do it very quickly.
The loop for ... end:
In order to do Newton’s method we requirerepeating the calculation in a number of times. Thisis accomplished in a program using a loop which signifies a section of a program which is repeated.The simplest manner to accomplish this is to count the number of times through. In Mat labafor ... end statement makes a loop as in the subsequent simple function program:
function S = mysum(n)% gives the sum of the first n integersS = 0; % start at zero% The loop:for i = 1:n % do n timesS = S + i; % add the current integerend % end of the loop
Call this function in the command window as:
>mysum(100)
The result will be the total of the first 100 integers. All for ... end loops have the similar format itbegins with for followed by an index (i) and a range of numbers (1:n). Afterward come the commandsthat are to be repeated. Last come the end command.
Loops are one of the major ways that computers are made to do calculations that humans cannot.
Any computation that involves a repeated process is easily done by a loop.
At the moment let’s do a program that does n steps (iterations) of Newton’s method. We will requires to inputthe initial guess, the function, its derivative, and the number of steps. The output will be the absolutevalue of xthat isxn. If we are merely interested in the final approximation not the intermediate stepswhich is usually the case in the real world then we can use a single variable x in the program andchange it at each step:
function x = mynewton(f,f1,x0,n)% Solves f(x) = 0 by doing n steps of Newton’s method starting at x0.% Inputs: f -- the function, input as an inline% f1 -- it’s derivative, input as an inline% x0 -- starting guess, a number% n -- the number of steps to do% Output: x -- the approximate solutionformat long % prints more digitsformat compact % makes the output more compactx = x0; % set x equal to the initial guess x0for i = 1:n % Do n timesx = x - f(x)/f1(x) % Newton’s formula, prints x tooend
In the command window define an inline function: f(x) = x3− 5 i.e.
> f = inline(’x^3 - 5’)
and define f1 to be its derivative, i.e.
> f1 = inline(’3*x^2’).
Then run my Newton on this function. By trial as well as error what is the lowest value of n for which theprogram converges (stops changing). Through simple algebra the true root of this function is 3√ 5. Howaccurate is the program’s answer to the true value?
Convergence:
Newton’s method converges rapidly when f′(x∗) is nonzero and finite, and x0 is close enough to x∗that the linear approximation is valid. Let us take a look at what is able to go wrong.
For f(x) = x1/3 we have x∗= 0 but f′(x*) = ∞. If you try
> f = inline(’x^(1/3)’)> f1 = inline(’(1/3)*x^(-2/3)’)> x = mynewton(f,f1,0.1,10)
then x explodes.For f(x) = x2 we have x= 0 but f′(x) = 0. If you try
> f = inline(’x^2’)> f1 = inline(’2*x’)> x = mynewton(f,f1,1,10)
Then x does converge to 0 however not that rapidly.
If x0 isn’t close enough to xthat the linear approximation is valid, then the iteration gives some x1 that may or mayn’t be any better than x0. If we keep iterating afterwards either
• xn will ultimately get close to xand the method will then converge (rapidly) or• The iterations will not approach x.
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 paulings method assignment help-homework help by online calculation of atomic radius tutors
Explosives tutorial all along with the key concepts of Classification of Explosives-According to Velocity and According to Sensitivity, Characteristics of explosives, Production of explosives and Applications of Explosives
tutorsglobe.com advantages of monopoly assignment help-homework help by online monopoly tutors
Algebraic Method for Combinational Logic tutorial all along with the key concepts of Getting Truth Table from Boolean Expression, Getting Boolean Expression from Truth Table, Exclusive- OR gate, Exclusive -NOR gate, Half Adder, Full Adder, Designing Circuits Using NAND Gates
www.tutorsglobe.com - free tutorials on structured analysis development method, design dfd data flow diagrams in theory of system analysis methods in programming languages, get assignment help - homework help by tutors.
Cost Audit Programmes - Inventory, Opening and closing stocks, Stores issue procedure, Work-in-progress, Labour, Capacity utilisation, Overheads.
Theory and lecture notes of Lock management all along with the key concepts of Lock management and Transaction management. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Lock management.
tutorsglobe.com producers equilibrium assignment help-homework help by online economies of large scale production tutors
tutorsglobe.com mode of infection assignment help-homework help by online emerging viral infections in humans tutors
Very low Frequency Radiation tutorial all along with the key concepts of VLF Radiation, VLF transmissions, Detecting VLF fields, Magnetic field effects, Electric field effects, Elliptical polarization, Coupling
the digital x-ray machine contains an x-ray tube and driver to source x-ray. the x-ray passes by the body of the patient and the digital camera (situated on another side of the patient) takes the resulting image.
tutorsglobe.com hypothalamus assignment help-homework help by online the brain tutors
tutorsglobe.com prokaryotes assignment help-homework help by online prokaryotic and eukaryotic cell tutors
metals-general characteristics tutorial all along with the key concepts of physical properties of metals, chemical properties of metals, occurrence of metals in nature, electrochemical or activity series, extraction of metals
theory and lecture notes of theory of ttl logic family all along with the key concepts of totem pole in ttl, ttl transistor switching trouble, solution of ttl switching trouble and assignment help. tutorsglobe offers homework help, assignment help and tutor’s assistance on theory of ttl logic family.
1937245
Questions Asked
3689
Tutors
1450775
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!