Homework Answers
1) (d) bisection divides the interval into half by calculating
mid point and evaluating function values on it. hence:
mid-point = -0.75
function value = (-0.75)^3-(-0.75)-11 = -10.6719
2) (d) open methods use gradients to approximate the next guess
3) (c) Newton raphson requires the knowledge of exact derivative
4) (c) As both the methods choose initial bracketing points having opposite sign of function value. Neither bisection or false position can determine the root
5) (c) Newton raphson method can diverge
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What is the function value at the estimated root after one iteration of the...
L. Determine the real root(s) off(x)--5xs + 14x3 + 20x2 + 10x a. Graphically on a graph paper. b....
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Consider the function xtan x -1 defined over all x. Sketch the function to get an idea of the roots 1 find the first couple of roots using bisection to a precision of machine epsilon 2 after straddli...
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Newton's method is always the slowest algorithm (takes the most iterations) for finding a root. True...
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Please MATLAB for all coding with good commenting. (20) Consider the function f(x) = e* -...
Please MATLAB for all coding with good commenting.
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45-3. Modify the code used in Example 4 to find the root only at f(x)<0.01 using...
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clearvars close all clc tol = 0.0001; % this is the tolerance for root identification xold...
clearvars
close all
clc
tol = 0.0001; % this is the tolerance for root identification
xold = 0.5; % this is the initial guess
test = 1; % this simply ensures we have a test value to enter the loop below.
%If we don't preallocate this, MATLAB will error when it trys to start the
%while loop below
k = 1; %this is the iteration counter. Similar to "test" we need to preallocate it
%to allow the while loop to...
1. This question concerns finding the roots of the scalar non-linear function f(x) = r2-1-sinx (1...
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Newton-Raphson scheme ? Calculus Suppose you want to find zeros of the function f(x)102212 and plan to use the Newton-Raphson scheme. (a) Write down the Newton-Raphson algorithm for this. That is, wri...
Newton-Raphson scheme
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Calculus Suppose you want to find zeros of the function f(x)102212 and plan to use the Newton-Raphson scheme. (a) Write down the Newton-Raphson algorithm for this. That is, write down explicitly a formula for computing your (n+1)st guess Tn+1 given your nth guess rn for a root. In other words, deter- mine the recurrence relation resulting from using this particular function f. (b) Modifying Algorithm 2.2 as required, find the values through x7 if you choose an...
5.1.2 Open Methods - Newton-Raphson Method Xi+1= xi – FOTO Matlab Code Example:4 function mynewtraph (f,...
5.1.2 Open Methods - Newton-Raphson Method Xi+1= xi – FOTO Matlab Code Example:4 function mynewtraph (f, f1,x0,n) Xx0; for ilin x = x - f(x)/f1(x); disp (li if f(x) <0.01 f(x))) break end end end Matlab Code from Chapra function [root, ea, iter)=newtraph (func,dfunc, xr, es,maxit,varargin) newtraph: Newton-Raphson root location zeroes 8 [root, ea, iter)-newtraph (func, dfunc, xr, es,maxit,pl,p2, ...): $uses Newton-Raphson method to find the root of fune input: func- name of function 8dfunc = name of derivative of...
4) (16 points) The function f(x)= x? – 2x² - 4x+8 has a double root at...
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l. Determine the real root(s) off(x)--5xs + 14x3 + 20x2 + 10x a. Graphically on a graph paper. b. Using Bisection method c. Using False Position method to determine the root, employing initial guesses of x-2 d. Using the Newton Raphson methods to determine the root, employing initial guess to determine the root, employing initial guesses ofxn-2 and Xu-4 and Es= 18%. and r 5.0 andas answer. 1%. was this method the best for these initial guesses? Explain your xo--l...
Consider the function xtan x -1 defined over all x. Sketch the function to get an idea of the roots 1 find the first couple of roots using bisection to a precision of machine epsilon 2 after straddling a root, find its value using the Newton-Raphson method. 3 after straddling a root, find its value using the secant method 4 after straddling a root, find its value using the false position method. Determine the order of the methods and comment...
Newton's method is always the slowest algorithm (takes the most iterations) for finding a root. True O False The shape of the function influences the performance of the False Position Method. O True O False The Bisection Method can fail to converge if f (201) a and f (xu) have opposite signs. True False
Please MATLAB for all coding with good commenting.
(20) Consider the function f(x) = e* - 3x. Using only and exactly the four points on the graph off with x-coordinates -1,0, 1 and 2, use MATLAB's polyfit function to determine a 3' degree polynomial that approximates f on the interval (-1, 2]. Plot the function f(x) and the 360 degree polynomial you have determined on the same set of axes. f must be blue and have a dashed line style,...
45-3. Modify the code used in Example 4 to find the root only at f(x)<0.01 using Newton-Rephson Method without showing any iteration. Also find the root of equation, f(x) = x 9-3x -10, take initial guess, Xo=2 العقدة College of 9:05 mybb.qu.edu.ca Numerical Methods (Lab.) GENG 300 Summer 2020 5.1.2 Open Methods - Newton-Raphson Method f(x) *1+1 = x; - Matlab Code Example:4 function mynewtraph.t1.x0,-) XXO for ilin x - x - x)/1 x) disp 1 x) <0.01 break end...
clearvars
close all
clc
tol = 0.0001; % this is the tolerance for root identification
xold = 0.5; % this is the initial guess
test = 1; % this simply ensures we have a test value to enter the loop below.
%If we don't preallocate this, MATLAB will error when it trys to start the
%while loop below
k = 1; %this is the iteration counter. Similar to "test" we need to preallocate it
%to allow the while loop to...
1. This question concerns finding the roots of the scalar non-linear function f(x) = r2-1-sinx (1 mark) (b) Apply two iterations of the bisection method to f(x) 0 to find the positive root. (3 marks) (c) Apply two iterations of the Newton-Raphson method to find the positive root. Choose (3 marks) (d) Use the Newton-Raphson method and Matlab to find the positive root to 15 significant (3 marks) (a) Use Matlab to obtain a graph of the function that shows...
Newton-Raphson scheme
?
Calculus Suppose you want to find zeros of the function f(x)102212 and plan to use the Newton-Raphson scheme. (a) Write down the Newton-Raphson algorithm for this. That is, write down explicitly a formula for computing your (n+1)st guess Tn+1 given your nth guess rn for a root. In other words, deter- mine the recurrence relation resulting from using this particular function f. (b) Modifying Algorithm 2.2 as required, find the values through x7 if you choose an...
5.1.2 Open Methods - Newton-Raphson Method Xi+1= xi – FOTO Matlab Code Example:4 function mynewtraph (f, f1,x0,n) Xx0; for ilin x = x - f(x)/f1(x); disp (li if f(x) <0.01 f(x))) break end end end Matlab Code from Chapra function [root, ea, iter)=newtraph (func,dfunc, xr, es,maxit,varargin) newtraph: Newton-Raphson root location zeroes 8 [root, ea, iter)-newtraph (func, dfunc, xr, es,maxit,pl,p2, ...): $uses Newton-Raphson method to find the root of fune input: func- name of function 8dfunc = name of derivative of...
4) (16 points) The function f(x)= x? – 2x² - 4x+8 has a double root at x = 2. Use a) the standard Newton-Raphson, b) the modified Newton-Raphson to solve for the root at x = 2. Compare the rate of convergence using an initial guess of Xo = 1,2. 5) (14 points) Determine the roots of the following simultaneous nonlinear equations using a) fixed-point iteration and b) the Newton-Raphson method: y=-x? +x+0,75 y + 5xy = r? Employ initial...
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