LBG - FS2024 – Solution 8

Problem 1: Henderson’s Rule

Use the following pedigree to come up with the final version of Hendersons rules.

Task

• Get matrices \(L^{-1}\) and \(D^{-1}\)
• Do the matrix multiplication to get \(A^{-1}\)
• Describe rules to come up with \(A^{-1}\)

Solution

• Matrix \(L^{-1}\)

• Matrix \((L^{-1})^T\)

• Matrix \(D^{-1}\)

• Matrix multiplication

• Multiplication diagram

• Extract Rules

  • Add \((D^{-1})_{ii}\) to \((A^{-1})_{ii}\)
  • Subtract \((D^{-1})_{ii}/2\) from \((A^{-1})_{is}\), \((A^{-1})_{si}\), \((A^{-1})_{id}\) and \((A^{-1})_{di}\)
  • Add \((D^{-1})_{ii}/4\) to \((A^{-1})_{ss}\), \((A^{-1})_{dd}\), \((A^{-1})_{sd}\) and \((A^{-1})_{ds}\)