Use the Bisection Method to find the first root of f(x) = -12 – 21x + 18x² – 2.75a. In doing so, use the initial guesses (bracket) of a = – -1 and b = 0 with a stopping criterion of 1% relative error (also known as relative or approximation error), as given by xnew pold Crel = x 100% < a = 1% xnew where xew is the midpoint for the current interval (really, the current estimate of the root) and xold is that for the latest previous iteration. After you have implemented the Bisection Method to find the first root, plot the function f(x) from x = -2 to x = 7 (i.e. these are the x-limits). Using the hold on command, plot your estimate of the root on top of the original plot using a single marker that is of a different style and color than the main plot (for example, plot the range of values in blue with x-markers, and overlay the first root with a red circle marker). This plotting serves as a "sanity check" for the reasonableness of your results. Hints: • You can treat f as an anonymous function so that you can easily pass values of x to f as an argument without having to re-define the expression for f. Example syntax: my_fun = @(x) 2 + 5*x • If you are going to pass a vector of input values to your function f (for example, when plotting) then make sure to use the "dot" operator for element-wise operations (e.g., "*" instead of "*" etc. to avoid non-conformable errors). Explore the documentation for the built-in function plot or the plotting examples from class to beautify your figure.
Use the Bisection Method to find the first root of f(x) = -12 – 21x + 18x² – 2.75a. In doing so, use the initial guesses (bracket) of a = – -1 and b = 0 with a stopping criterion of 1% relative error (also known as relative or approximation error), as given by xnew pold Crel = x 100% < a = 1% xnew where xew is the midpoint for the current interval (really, the current estimate of the root) and xold is that for the latest previous iteration. After you have implemented the Bisection Method to find the first root, plot the function f(x) from x = -2 to x = 7 (i.e. these are the x-limits). Using the hold on command, plot your estimate of the root on top of the original plot using a single marker that is of a different style and color than the main plot (for example, plot the range of values in blue with x-markers, and overlay the first root with a red circle marker). This plotting serves as a "sanity check" for the reasonableness of your results. Hints: • You can treat f as an anonymous function so that you can easily pass values of x to f as an argument without having to re-define the expression for f. Example syntax: my_fun = @(x) 2 + 5*x • If you are going to pass a vector of input values to your function f (for example, when plotting) then make sure to use the "dot" operator for element-wise operations (e.g., "*" instead of "*" etc. to avoid non-conformable errors). Explore the documentation for the built-in function plot or the plotting examples from class to beautify your figure.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
Transcribed Image Text:Use the Bisection Method to find the first root of f(x) = -12 – 21x + 18x² – 2.75a³. In
doing so, use the initial guesses (bracket) of a = –
-1 and b = 0 with a stopping criterion
of 1% relative error (also known as relative or approximation error), as given by
xnew
pold
Crel =
x 100% < E =
1%
xnew
where xew is the midpoint for the current interval (really, the current estimate of the root) and
xold is that for the latest previous iteration.
After you have implemented the Bisection Method to find the first root, plot the function f(x)
from x = -2 to x = 7 (i.e. these are the x-limits). Using the hold on command, plot your
estimate of the root on top of the original plot using a single marker that is of a different
style and color than the main plot (for example, plot the range of values in blue with x-markers,
and overlay the first root with a red circle marker). This plotting serves as a "sanity check"
for the reasonableness of your results.
Hints:
You can treat f as an anonymous function so that you can easily pass values of x
to f as an argument without having to re-define the expression for f. Example syntax:
my_fun = @(x) 2 + 5*x
• If you are going to pass a vector of input values to your function f (for example, when
plotting) then make sure to use the "dot" operator for element-wise operations (e.g., ".*"
instead of "*" etc. to avoid non-conformable errors).
Explore the documentation for the built-in function plot or the plotting examples from
class to beautify your figure.
Expert Solution
Step 1
Following is the Matlab code for the given problem:
%clear the command window
clc;
clear all;
%given function
f=@(x)-12-21*x+18*x^2-275*x^3;
low=-1;
high=0;
tol=0.00000001;
% Evaluate both ends of the interval
y1 = feval(f, low);
y2 = feval(f, high);
i = 0;
% Display error and finish if signs are not different
if y1 * y2 > 0
disp('Have not found a change in sign. Will not continue...');
m = 'Error'
return
end
% Work with the limits modifying them until you find
% a function close enough to zero.
disp('Iter low high x0');
while (abs(high - low) >= tol)
i = i + 1;
% Find a new value to be tested as a root
m = (high + low)/2;
y3 = feval(f, m);
if y3 == 0
fprintf('Root at x = %f \n\n', m);
return
end
fprintf('%2i \t %f \t %f \t %f \n', i-1, low, high, m);
% Update the limits
if y1 * y3 > 0
low = m;
y1 = y3;
else
high = m;
end
end
% Show the last approximation considering the tolerance
w = feval(f, m);
fprintf('\n x = %f produces f(x) = %f \n %i iterations\n', m, y3, i-1);
%plot
hold on;
fplot(f,[-2 7]);
xlabel('x');
ylabel('f(x)');
hS1 =
scatter(m,f(m),300,'*','MarkerEdgeColor',[0 .5 .5],'MarkerFaceColor',[0 .7 .7]);
Code demo for reference:
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