CE 5143 Homework 4 Due 10/4/07

David Wagner 2007/10/03 11:40

Calculation by Hand

1. Problem #3.47

Express the following equations in the form [L][U]x = b using the Choleski method, and find the solution of the resulting equations.

delim{[}{matrix{3}{3}{ {16} {-4} {4} {-4} {17} {11} {4} {11} {14} } }{]} delim{lbrace}{matrix{3}{1}{x_1 x_2 x_3} }{rbrace} = delim{lbrace}{matrix{3}{1}{ {-4} {-3} {-16} } }{rbrace}

[See text §3.11.2, Example 3.16, page 179]

delim{[}{A}{]} = delim{[}{matrix{3}{3}{ {16} {-4} {4} {-4} {17} {11} {4} {11} {14} } }{]}

  1. u_11 = sqrt{a_11}=sqrt{16}=4;
  2. u_12 = {a_12}/{u11} = {-4}/{4} =-1;
  3. u_13 = {a_13}/{u11} = {4}/{4} =1;
  4. u_22 = sqrt{a_22 - {u_12}^2 } = sqrt{17 - 1 }=4;
  5. u_23 = {1/u_22}(a_23 - u_12 u_13) = {1/4}(11 - (-1)(1))=3;
  6. u_33 = sqrt{a_33 - sum{k=1}{3-1}{{u_k3}^2} } =sqrt{14 -1^2 -3^2}=sqrt{4}2

delim{[}{U}{]} = delim{[}{matrix{3}{3}{ {4} {-1} {1} {0} {4} {3} {0} {0} {2} } }{]}

[A]x = [L][U]x=b = [U]T[U]x=b = delim{[}{matrix{3}{3}{ {4} {0} {0} {-4} {4} {0} {1} {3} {2} } }{]} delim{[}{matrix{3}{3}{ {4} {-1} {1} {0} {4} {3} {0} {0} {2} } }{]} delim{lbrace}{matrix{3}{1}{x_1 x_2 x_3} }{rbrace} = delim{lbrace}{matrix{3}{1}{ {-4} {-3} {-16} } }{rbrace}

[A]-1[A]x = [A]-1b = [U]-1([U]-1)Tb = x

  1. lambda_11=1/u_11=1/4
  2. lambda_22=1/u_22=1/4
  3. lambda_33=1/u_33=1/2
  4. lambda_12=-{{u_12 lambda_22}/{u_11}}-{{-1/4}/{4}}=1/16
  5. lambda_23=-{{u_23 lambda_33}/{u_11}}=-{{3/2}/{4}}=-{3/8}
  6. lambda_13=-{{u_12 lambda_23 + u_13 lambda_33}/{u_11}}=-{{(-1)(-3/8) + (1)(4/8) }/{4}}={28/8}=-7/32

tabular{111}{111}{ {dot} { (delim{[}{U}{]}^{-1})^T= delim{[}{matrix{3}{3}{ {1/4} {0} {0} {1/16} {1/4} {0} {-{7/32}} {-{3/8}} {1/2} } }{]} } { delim{[}{U}{]}^{-1}= delim{[}{matrix{3}{3}{ {1/4} {1/16} {-{7/32}} {0} {1/4} {-{3/8}} {0} {0} {1/2} } }{]} } {delim{[}{A}{]}^{-1}= delim{[}{matrix{3}{3}{ {0.114257} {0.097656} {-0.109375} {0.097656} {0.203125} {-0.187500} {-0.109375} {-0.187500} {0.250000} } }{]} } }1)

x=[A]-1b delim{lbrace}{matrix{3}{1}{x_1 x_2 x_3} }{rbrace} = delim{[}{matrix{3}{3}{ {0.114257} {0.097656} {-0.109375} {0.097656} {0.203125} {-0.187500} {-0.109375} {-0.187500} {0.250000} } }{]} delim{lbrace}{matrix{3}{1}{ {-4} {-3} {-16} } }{rbrace} =

x=delim{lbrace}{matrix{3}{1}{ {1} {2} {-3} } }{rbrace}
1) Checked with Matrix Multiplication Applet

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