Math 449 - Homework 6 - Solutions

1) Exercise 7, section 4.4, page 229.


f(x)=3*(sin(pi*x/6))^2

(a) Divided-difference table:

k | x_k  f[x_k] f[ , ] f[ , , ] f[ , , , ] f[ , , , , ]
__|____________________________________________________
0 | 0.0  0.00
  | 
1 | 1.0  0.75   0.75
  |
2 | 2.0  2.25   1.50    0.375   
  | 
3 | 3.0  3.00   0.75   -0.375   -0.250
  |
4 | 4.0  2.25  -0.75   -0.750   -0.125     0.03125



(b) Write down the Newton polynomials P_1(x), P_2(x), P_3(x), and P_4(x).

P_0(x) = 0.00

P_1(x) = 0.00 + 0.75(x-0)

P_2(x) = 0.00 + 0.75(x-0) + 0.375(x-0)(x-1)

P_3(x) = 0.00 + 0.75(x-0) + 0.375(x-0)(x-1) - 0.250(x-0)(x-1)(x-2)

P_4(x) = 0.00 + 0.75(x-0) + 0.375(x-0)(x-1) - 0.250(x-0)(x-1)(x-2) + 0.03125(x-0)(x-1)(x-2)(x-3)

So we have: 

P_4(x) = 0.75x + 0.375x(x-1) - 0.25x(x-1)(x-2) + 0.03125x(x-1)(x-2)(x-3)


(c) At x=1.5 and 3.5 we have

x      1.5        3.5
____________________________
P_0    0.0        0.0  

P_1    1.1250     2.6250

P_2    1.4062     5.9062

P_3    1.5000     2.6250

P_4    1.5176     2.8301

f(x)   1.5000     2.7990


(d) The values of f(x) at the interpolated points 1.5 and 3.5 are shown

on the last row of the previous table.


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2) Exercise 1 (a, b) of section 4.5, page 241.

We start with 

T_2(x) = 2x^2 - 1

T_3(x) = 4x^3 - 3x 

and use the recurrence relation  T_k = 2xT_(k-1) - T_(k-2)

to obtain T_4 and T_5:

(a) T_4(x) = 2xT_3(x) - T_2(x) 
           
           = 2x(4x^3 - 3x) - (2x^2 - 1)
           
           = 8x^4 - 8x^2 + 1
           

(b) T_5(x) = 2xT_4(x) - T_3(x)
     
           = 2x(8x^4 - 8x^2 + 1) - (4x^3 - 3x)
           
           = 16x^5 - 20x^3 + 5x.           
 

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3) Algorithms and programs 4 (a) of section 4.5, p. 242.

Compute the coefficients  {c_k : k=0, 1, 2, ...} for the Chebyshev polynomial approximation

P_N(x) to f(x)=log(x+2) over [-1,1] for N=4.

See below for instructions on using program 4.3.

(a) C = [0.6238    0.5359   -0.0718    0.0128   -0.0025]

To obtain the Chebyshev coefficients, use the program cheby below.

For example, for f(x)=log(x+2) and N=4, enter at the prompt the command line:

[C, X, Y] = cheby('log(x+2)', 4)

The values of the vector C are the the coefficients c_0, c_1, etc.

One way in which you can plot the function

P_N(x) = c_0T_0(x) + c_1T_1(x) + ... + c_NT_N(x)

is as shown below the program.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [C,X,Y]=cheby(fun,n,a,b)
%Input   - fun is the string function to be approximated
%        - n is the degree of the Chebyshev interpolating polynomial
%        - a is the left endpoint
%        - b is the right endpoint
%Output  - C is the coefficient list for the polynomial
%        - X contains the abscissas
%        - Y contains the ordinates

if nargin==2, a=-1;b=1;end

d=pi/(2*n+2);
C=zeros(1,n+1);

for k=1:n+1
    X(k)=cos((2*k-1)*d);
end

X=(b-a)*X/2 + (a+b)/2;
x=X;
Y=eval(fun);

for k=1:n+1
    z=(2*k-1)*d;
    for j=1:n+1
        C(j)=C(j) + Y(k)*cos((j-1)*z);
    end
end
C=2*C/(n+1);
C(1)=C(1)/2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%To plot the Chebyshev polynomial and the function%%%%%%%%%%%
N=4;
C=cheby('log(x+2)',N);
x=-1:0.001:1;
[s m]=size(x);
T=zeros(N+1,m);
T(1,:)=1;
T(2,:)=x;
for i=3:N+1
    T(i,:)=2*x.*T(i-1,:)-T(i-2,:);
end
P=C*T;
plot(x,P)
hold on
z=-1:.1:1; 
plot(z,log(z+2),'o')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     
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4) Exercise 2 (a, b) of section 4.6, page 248.

(a) Find the Pade' approximation R_(1,1)(x) for f(x)=ln(1+x)/x.

ln(1+x)/x = 1- x/2 + x^2/3 - ... = R_(1,1)(x) = (p_0 + p_1x)/(1 + q_1x)

Multiplying both sides by 1 + q_1x gives:

(1- x/2 + x^2/3 - ...)(1 + q_1x) = p_0 + p_1x + terms of oder 3 or greater.

This gives:

(1 - p_0) + (q_1 - 1/2 - p_1)x + (1/3 - q_1/2)x^2 = 0, 

so q_1=2/3, p_0=1, p_1=2/3-1/2=1/6.

Therefore R_(1,1)(x)=(1 + x/6)(1 + 2x/3)=(6 + x)/(6 + 4x).


(b) The result of part (a) gives

ln(1 + x) = (6x + x^2)/(6 + 4x) + terms of order 4 or greater.

Therefore R_(2,1)(x) = (6x + x^2)/(6 + 4x) is the Pade' approximation of 

ln(1+x).

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5) Algorithms and programs 1 (a, b) of section 4.6, page 250.

See the graphs of e^x - T_4(x) and e^x - R_(2,2)(x) on the separate file.

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