EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 9780100254145
Author: Chapra
Publisher: YUZU
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Chapter 6, Problem 26P
Fir Prob. 6.25, the root can be located with fixed-point iteration as
or as
Only one will converge for initial guesses of
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For the DE: dy/dx=2x-y y(0)=2 with h=0.2, solve for y using each method below in the range of 0 <= x <= 3:
Q1) Using Matlab to employ the Euler Method (Sect 2.4)
Q2) Using Matlab to employ the Improved Euler Method (Sect 2.5
close all
clear all
% Let's program exact soln
for i=1:5
x_exact(i)=0.5*i-0.5;
y_exact(i)=-x_exact(i)-1+exp(x_exact(i));
end
plot(x_exact,y_exact,'b')
% now for Euler's
h=0.5
x_EM(1)=0;
y_EM(1)=0;
for i=2:5
x_EM(i)=x_EM(i-1)+h;
y_EM(i)=y_EM(i-1)+(h*(x_EM(i-1)+y_EM(i-1)));
end
hold on
plot (x_EM,y_EM,'r')
% Improved Euler's Method
h=0.5
x_IE(1)=0;
y_IE(1)=0;
for i=2:1:5
kA=x_IE(i-1)+y_IE(i-1);
u=y_IE(i-1)+h*kA;
x_IE(i)=x_IE(i-1)+h;
kB=x_IE(i)+u;
k=(kA+kB)/2;
y_IE(i)=y_IE(i-1)+h*k;
end
hold on
plot(x_IE,y_IE,'k')
Consider the function p(x) = x² - 4x³+3x²+x-1. Use Newton-Raphson's method with initial guess of 3. What's the updated value of the root at the end of the second
iteration?
Type your answer...
A root of the function
f(x) = x3 – 10x² +5
lies close to x = 0.7. Doing three iterations, compute this root using the Newton-
Raphson method with an initial guess of x=1).
Newton-Raphson iterative equation is given as:
f(x;)
Xi+1 = Xị -
f'(xi)
Chapter 6 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 6 - 6.1 Use simple fixed-point iteration to locate the...Ch. 6 - 6.2 Determine the highest real root of...Ch. 6 - Use (a) fixed-point iteration and (b) the...Ch. 6 - Determine the real roots of f(x)=1+5.5x4x2+0.5x3:...Ch. 6 - 6.5 Employ the Newton-Raphson method to determine...Ch. 6 - Determine the lowest real root of...Ch. 6 - 6.7 Locate the first positive root of
Where x...Ch. 6 - 6.8 Determine the real root of, with the modified...Ch. 6 - 6.9 Determine the highest real root of:...Ch. 6 - 6.10 Determine the lowest positive root...
Ch. 6 - 6.11 Use the Newton-Raphson method to find the...Ch. 6 - 6.12 Given
Use a root location technique to...Ch. 6 - You must determine the root of the following...Ch. 6 - Use (a) the Newton-Raphson method and (b) the...Ch. 6 - 6.15 The “divide and average” method, an old-time...Ch. 6 - (a) Apply the Newton-Raphson method to the...Ch. 6 - 6.17 The polynomial has a real root between 15...Ch. 6 - Use the secant method on the circle function...Ch. 6 - You are designing a spherical tank (Fig. P6.19) to...Ch. 6 - 6.20 The Manning equation can be written for a...Ch. 6 - 6.21 The function has a double root at. Use (a)...Ch. 6 - 6.22 Determine the roots of the following...Ch. 6 - 6.23 Determine the roots of the simultaneous...Ch. 6 - Repeat Prob. 6.23 except determine the positive...Ch. 6 - A mass balance for a pollutant in a well-mixed...Ch. 6 - Fir Prob. 6.25, the root can be located with...Ch. 6 - 6.27 Develop a user-friendly program for the...Ch. 6 - Develop a user-friendly program for the secant...Ch. 6 - 6.29 Develop a user-friendly program for the...Ch. 6 - 6.30 Develop a user-friendly program for Brent’s...Ch. 6 - 6.31 Develop a user-friendly program for the...Ch. 6 - 6.32 Use the program you developed in Prob. 6.31...
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