Question

Given a function y = f(x), X = (x0M, . . . ,xn)T with yi f(x) and xo < x1 < . . . < Xn, and one value a, write a Matlab function [y]=[][P1(X,n,a) to do the following: 1. Calculate the coefficients of the interpolation polynomial P(x) for the points (20,yo) (x %) using Newton divided-differences as in Algorithm 3.2. 2. For the given value a, calculate y = P(o) y is the output of your function. 3. Draw the graphs of y = f(x) and y Po(x) on the interval [20, aal together on one coordinate system.

The first two parts should be solved by Matlab. This is from an intro to Numerical Analysis Class and I have provided the Alog 3.2 in below. Please write the whole codes for me.

Alog3.2

% NEWTONS INTERPOLATORY DIVIDED-DIFFERENCE FORMULA ALGORITHM 3.2
% To obtain the divided-difference coefficients of the
% interpolatory polynomial P on the (n+1) distinct numbers x(0),
% x(1), ..., x(n) for the function f:
% INPUT: numbers x(0), x(1), ..., x(n); values f(x(0)), f(x(1)),
% ..., f(x(n)) as the first column Q(0,0), Q(1,0), ...,
% Q(N,0) of Q, or may be computed if function f is supplied.
% OUTPUT: the numbers Q(0,0), Q(1,1), ..., Q(N,N) where
% P(x) = Q(0,0) + Q(1,1)*(x-x(0)) + Q(2,2)*(x-x(0))*   
% (x-x(1)) +... + Q(N,N)*(x-x(0))*(x-x(1))*...*(x-x(N-1)).
syms('OK', 'FLAG', 'N', 'I', 'X', 'Q', 'A', 'NAME', 'INP','OUP', 'J');
syms('s','x');
TRUE = 1;
FALSE = 0;
fprintf(1,'Newtons form of the interpolation polynomial \n');
OK = FALSE;
while OK == FALSE
fprintf(1,'Choice of input method:\n');
fprintf(1,'1. Input entry by entry from keyboard \n');
fprintf(1,'2. Input data from a text file \n');
fprintf(1,'3. Generate data using a function F \n');
fprintf(1,'Choose 1, 2, or 3 please \n');
FLAG = input(' ');
if FLAG == 1 | FLAG == 2 | FLAG == 3
OK = TRUE;
end
end
if FLAG == 1
OK = FALSE;
while OK == FALSE
fprintf(1,'Input n \n');
N = input(' ');
if N > 0
OK = TRUE;
X = zeros(N+1);
Q = zeros(N+1,N+1);
for I = 0:N
fprintf(1,'Input X(% d) and F (X(% d)) ', I, I);
fprintf(1,'on separate lines \n');
X(I+1) = input(' ');
Q(I+1,1) = input(' ');
end
else
fprintf(1,'Number must be a positive integer \n');
end
end
end
if FLAG == 2
fprintf(1,'Has a text file been created with the data in two columns?\n');
fprintf(1 'Enter Y or N \n');
A = input(' ','s');
if A == 'Y' | A == 'y'
fprintf(1,'Input the file name in the form - ');
fprintf(1,'drive:\\ name.ext\n');
fprintf(1,'For example: A:\\ DATA.DTA\n');
NAME = input(' ','s');
INP = fopen(NAME,'rt');
OK = FALSE;
while OK == FALSE
fprintf(1,'Input n \n');
N = input(' ');
if N > 0   
X = zeros(N+1);
Q = zeros(N+1,N+1);
for I = 0:N
X(I+1) = fscanf(INP, '% f',1);
Q(I+1,1) = fscanf(INP, '% f',1);
end
fclose(INP);
OK = TRUE;
else
fprintf(1,'Number must be a positive integer \n')
end
end
else
fprintf(1,'Please create the input file in two column ');
fprintf(1,'form with the X values and \n');
fprintf(1,'F(X) values in the corresponding columns.\n');
fprintf(1,'The program will end so the input file can ');
fprintf(1,'be created.\n');
OK = FALSE;
end
end
if FLAG == 3
fprintf(1,'Input the function F(x) in terms of x \n');
fprintf(1,'For example: cos(x)\n');
s = input(' ');
F = inline(s,'x');
OK = FALSE;
while OK == FALSE
fprintf(1,'Input n \n');
N = input(' ');
if N > 0
X = zeros(N+1);
Q = zeros(N+1,N+1);
for I = 0:N
fprintf(1,'Input X(% d)\n', I);
X(I+1) = input(' ');
Q(I+1,1) = F(X(I+1));
end
OK = TRUE;
else
fprintf(1,'Number must be a positive integer \n');
end
end
end
if OK == TRUE
fprintf(1,'Select output destination \n');
fprintf(1,'1. Screen \n');
fprintf(1,'2. Text file \n');
fprintf(1,'Enter 1 or 2\n');
FLAG = input(' ');
if FLAG == 2
fprintf(1,'Input the file name in the form - drive:\\ name.ext\n');
fprintf(1,'For example: A:\\ OUTPUT.DTA\n');
NAME = input(' ','s');
OUP = fopen(NAME,'wt');
else
OUP = 1;
end
fprintf(OUP, 'NEWTONS INTERPOLATION POLYNOMIAL \n \n');
% STEP 1
for I = 1:N
for J = 1:I
Q(I+1,J+1) = (Q(I+1,J) - Q(I,J)) / (X(I+1) - X(I-J+1));
end
end
% STEP 2
fprintf(OUP, 'Input data follows:\n');
for I = 0:N
fprintf(OUP,'X(% d) = %12 .8f F(X(% d)) = %12 .8f \n',I,X(I+1),I,Q(I+1,1));
end
fprintf(OUP, '\n The coefficients Q(0,0), ..., Q(N,N) are:\n');
for I = 0:N
fprintf(OUP, '%12 .8f \n', Q(I+1,I+1));
end
if OUP ~= 1
fclose(OUP);
fprintf(1,'Output file % s created successfully \n',NAME);
end
end

Answer 1

Answer A

interpolation_polynomial(i,points)
    coefficients = [1/denominator(i,points)]
    for k = 0 to points.length-1
        if k == i
            next k
        new_coefficients = [0,0,...,0] // length k+2 if k < i, k+1 if k > i
        if k < i
            m = k
        else
            m = k-1
        for j = m downto 0
            new_coefficients[j+1] += coefficients[j]
            new_coefficients[j] -= points[k]*coefficients[j]
        coefficients = new_coefficients
    return coefficients

Answer B

% Clear all windows and variables
clc
% Input the values for mass, height, friction coefficient, and gravity
m=18; % Mass (kilograms)
h=10; % Height (meters)
mu=0.55; % Friction coefficient (no units)
g=9.81; % Gravitational acceleration (m/s^2)
% Input a reasonable range for x
x=[-100:100];
% Calculate F given the range of x
F=@(x) (mu.*m.*g.*(h.^2 + x.^2).^(1/2))/(x+mu.*h);
% Plot F as a function of x
fplot(F,[0,100])
title('Force versus Distance')
xlabel('Distance (meters)')
ylabel('Force (newtons)')
grid on
% Find the x value that gives F=90
??????????????

Answer C

% This figure will be used to plot the structure of the graph represented
% by the current A matrix.
figure

dt = 2*pi/N_robots;
t = dt:dt:2*pi;
x = cos(t); y = sin(t);

agents=[       2    2.5;
               0.5 2.0;
               0.5 1.0;
               2.0 0.5;
               3.5 1.0;
               3.5 2.0;];

agents = p0;      

agents = [x' y'];

% plot(agents(:,1),agents(:,2),'s','MarkerSize', 20, 'MarkerFaceColor', [1 0 1])
grid on
%xlim([0 4])
%ylim([0 3])
hold on
index=1;
% The following prints the non-directed graph corresponding to the A matrix
for i=1:N_robots
    for j=index:N_robots
        if A(i,j) == 1
            arrowline(agents([i j],1),agents([i j],2),'arrowsize',600);         
%             plot(agents([i j],1),agents([i j],2),'--','LineWidth',2.5);
        end
    end
end
set(gca,'FontSize',fontsize2)

title('Structure of the Graph','interpreter', 'latex','FontSize', 18)