**Algorithm 3.1**

**Matrix Generation** 
    *FormMatrixB.m*
    ! Generate 4x4 matrix B from 3x3 matrix A.
    
    *FormMatrixQ.m*
    ! Generate 3x3 matrix Q from eigen vector v.

**Lambda Calculation**
    *EstimateLambda1.m*
    ! Estimate the dominant eigenvalue lambda1.
        --- *PolyMethod.m*
            ! Use the characteristic polynomial formula.
        --- *NewtonMethod.m*
            ! Use Newton's method.
            --- *HornerPoly.m*
                ! Use Horner's method to calculate polynomial formula.

**Eigvec Calculation**
    *CalcBpEigvec.m*
    ! Calculate 2x2 Matrix Bp's eigen vector.

**Matrix Decomposition**
    *LU.m*
    ! complete or partial pivoting
    ! A=LU, PA=LU, P1AP2=LU

    *LDL.m*
    ! Bunch-Parlett pivoting
    ! A=LDL', P'AP=LDL'

    *QR.m*
    ! Gram-Schmidt or Householder Transformation
    ! A=QR

    *SVD.m*
    ! singular value decomposition
    ! A=USV

```
MATLAB or Octave command line
> A = [0.1,0.2,0.3;0.1,-0.1,0;0.3,0.2,0.1;]; % 3x3 matrix 
> [Q,H] = Complete(A); % polar decomposition of A with Higham's Method 
```