The LDLT Solution

posted on 12 Dec 2014 22:36 by unchain in Knowlage directory Knowledge

The LDLT Solution

1 Description of The LDLT Solution

LDLT-decomposition is a generalization of for symmetric matrices similar with Cholesky But for LDLT [A] doen’t need to be positive definite. (Cholesky decomposition which can use only for symmetric positive definite matrices)

LDLT-decomposition of the matrix A is a decomposition of the form [L][D][L]T , it mean that [A] = [L][D][L] , when [L] is  lower triangular matrix and [D] is  D is diagonal matrix

                                                     

                                                          [A] = [L][D][L]  

 


2 Step to solve linear equation by LDLT Solution

            We propose to introduce LDLT-decomposition by studying the step to solve the equation [A]{x} = {b}

 

Step 1 : Factorization Matrices

            Transform [A] = [L][D][L] by using

           

 

 

**Remember that diagonal of [L] always = 1**

 

Step 2 : Forward Solution and diagonal scaling 

            From                            [A]{x} = {b}

            And                              [A] = [L][D][L] 

            Then                             [L][D][L] {x} ={b}                                                     (3.3)

            Let’s Define                 [L]T{x} = {y}                                                                  (3.4)

                                                 [D]{y} = {z}                                                               (3.5)

 


 

 

 

            We want to find {y} , we need to know {z} , solve for {z} by using Eq.(3.6)

                From Eq.(3.3) become


 

 

            After we get {z} , we can use Eq.(3.5) to find {y}

 

Step 3 : Backward Solution

            Use Eq.(3.4) to solve for {x}

                       

 


 

 

3 Example for LDLSolution

            Solve the following solve simultaneous equations for the unknown vector x by using LDLSolution

                                   [A]{x} = {b} ; where


 

 

 

Solution

Step 1 : Factorization Matrices

By using Eq. (3.1) and (3.2) we will get

 


 

 

 

Step 2 : Forward Solution and diagonal scaling 

From [L]{z} = {b}


 

 

 

Diagonal scaling [D]{y} = {z}

                                      

 

 

 

Step 3 : Backward Solution

            Use Eq.(3.4) to solve for {x}

                                



 

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