Solving Linear Systems Using LDL^T Factorization: Step-by-Step Guide

In summary, the LDL^T factorization is used to solve a linear system by first decomposing the matrix into a product of a lower triangular matrix (L), an upper triangular matrix (LT), and a diagonal matrix (D). From there, back substitution is used to solve for the unknowns in the system. First, DLTX=B is solved by dividing by the diagonal elements of D, then LTX=C is solved by back substitution, and finally X=D is solved by back substitution again.
  • #1
stunner5000pt
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Use the [itex] LDL^T [/itex] factorization to solve the following linear system
[tex] \left(\begin{array}{cccc|1}4&1&-1&0&7\\1&3&-1&0&8\\-1&-1&5&2&-4\\0&0&2&4&6\end{array}\right)[/tex]
now i know ihow to get a matrix in the form LDL^T. But i was wondering how one would go about solving from there?



Please help!
 
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  • #2
stunner5000pt said:
Use the [itex] LDL^T [/itex] factorization to solve the following linear system
[tex] \left(\begin{array}{cccc|1}4&1&-1&0&7\\1&3&-1&0&8\\-1&-1&5&2&-4\\0&0&2&4&6\end{array}\right)[/tex]
now i know ihow to get a matrix in the form LDL^T. But i was wondering how one would go about solving from there?
Please help!
That should be straight forward- the Cholesky decomposition is supposed to be the hard part! L here is a lower triangular matrix, LT is upper triangular, and D is diagonal, so going from LDLTX= A to DLTX= B is just a matter of "back substitution", starting from the value you get immediately in the last row and working up.
Since D is diagonal, going from DLTX= B to LTX= C is just dividing by the diagonal elements. Finally, since LT is upper triangular, going from LTX= C to X= D is again back substitution, this time working from the top row down.
 

1. What is LDL^T factorization and why is it useful for solving linear systems?

LDL^T factorization is a method used to decompose a matrix into a lower triangular matrix (L), a diagonal matrix (D), and the transpose of a lower triangular matrix (L^T). It is useful for solving linear systems because it allows us to simplify the matrix into smaller, easier to solve parts.

2. What are the steps involved in solving a linear system using LDL^T factorization?

The steps involved in solving a linear system using LDL^T factorization are:

  1. Perform the LDL^T factorization on the coefficient matrix
  2. Solve for the lower triangular matrix (L)
  3. Solve for the diagonal matrix (D)
  4. Transpose the lower triangular matrix (L^T)
  5. Multiply all three matrices together to get the solution to the linear system

3. Can any matrix be solved using LDL^T factorization?

No, a matrix must meet certain conditions in order to be solved using LDL^T factorization. The matrix must be symmetric and positive definite, meaning all of its eigenvalues are positive. If these conditions are met, then the matrix can be solved using LDL^T factorization.

4. Is there a specific order in which the steps of LDL^T factorization should be performed?

Yes, there is a specific order in which the steps of LDL^T factorization should be performed. The factorization should be performed first, followed by solving for the lower triangular matrix (L), then the diagonal matrix (D), and finally the transpose of the lower triangular matrix (L^T).

5. Are there any advantages of using LDL^T factorization over other methods of solving linear systems?

Yes, there are several advantages of using LDL^T factorization over other methods of solving linear systems. These include:

  • It is numerically stable and efficient for solving large, sparse matrices
  • It is more efficient than other factorization methods, such as LU decomposition
  • It can be used to solve systems with multiple right-hand sides, making it useful for solving systems of equations with many variables

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