04_linear_algebra

Homework 4

#Problem 2

Part A Outputs

• norm2_H4 = 1.5002
• normf_H4 = 1.5097
• norm1_H4 = 2.0833
• normi_H4 = 2.0833
• norm2_H5 = 1.5671
• normf_H5 = 1.5809
• norm1_H5 = 2.2833
• normi_H5 = 2.2833

Part B Outputs

• norm2_H4i = 1.0341e+04
• normf_H4i = 1.0342e+04
• norm1_H4i = 13620
• normi_H4i = 13620
• norm2_H5i = 3.0414e+05
• normf_H5i = 3.0416e+05
• norm1_H5i = 413280
• normi_H5i = 413280

Part C Outputs

• H4_cond2 = 1.5514e+04
• H4_condf = 1.5614e+04
• H4_cond1 = 2.8375e+04
• H4_condi = 2.8375e+04
• H5_cond2 = 4.7661e+05
• H5_condf = 4.8085e+05
• H5_cond1 = 9.4366e+05
• H5_condi = 9.4366e+05

Problem 3 Function chol_tridiag

• function [d,u] = chol_tridiag(e,f)
• %Function takes two vectors as inputs and
• %calculates the Cholseky factorization of
• %the tridiagonal matrix, where e is off-
• %diagonal vector and f is diagonal vector

• n = length(f);
  

• d = zeros(1,n);
  

• u = zeros(1,n-1);
  

• d(1) = sqrt(f(1));
  

• u(1) = e(1)/d(1);
  

• for i = 2:n-1
  

•     d(i) = sqrt(f(i)-u(i-1)^2);
  

•     u(i) = e(i)/d(i);
  

• end
  

• d(n) = sqrt(f(n)-u(n-1)^2);
• end
• %Outputs two vectors, diagonal of upper
• %matrix (d) and off-diagonal of upper
• %matrix (u)

Problem 4 Function solve_tridiag

• function x = solve_tridiag(d,u,b)
• %This function solves Ax = b with the
• %diagonal and off-diagonal of the Cholesky
• %and Diagonal matrices d and u, respectively

• n = length(d);
  

• x = zeros(1,n);
  

• y = zeros(1,n);
  

• for i = 1:n
  

•     if i == 1
  

•     y(1) = b(1)/d(1);
  

•     else
  

•     y(i) = (b(i)-u(i-1)* y(i-1))/d(i);
  

•     end
  

• end
  

• for i = fliplr(1:n)
  

•     if i == n
  

•         x(i) = y(i)/d(i);
  

•     else
  

•         x(i) = (y(i)-u(i)* x(i+1))/d(i);
  

•     end
  

• end
  

• end

Problem 5 Outputs

• For all k's = 1000, x = 29.2841 and Error = 29.2841
• For k2 = 1000e12, x = Error = 1.1291e+13
• For k2 = 1000e-12, x = Error = 9.0010e+12

Problem 6 Outputs

• x = [0.0392,0.0687,0.0833,0.0981]

Problem 7 Outputs

• ans = [40.522,14.409,2.569]

Problem 8 Table of Outputs

# of Segments	Largest Load (N)	Smallest Load (N)	# of Eigen Values
5	1.1756	0.6180	2
6	1.2000	0.6212	3
10	1.2361	0.6257	7

• As the segment length approaches zero, there would be no eigen values