48 | | Now, we can either assume the independent variables u, rho, etc. of these functions for alpha, lambda, and K are 1) reference states which we set, or 2) some general averaged versions of the variables, which we have to solve for. Toro goes through the alebraic analysis for the case of the Euler equations. The results are as follows: |

| 48 | Now, we can either assume the independent variables u, rho, etc. of these functions for alpha, lambda, and K are 1) reference states which we set, or 2) some general averaged versions of the variables, which we have to solve for. Toro goes through the algebraic analysis for the general case for the Euler equations. The results are as follows. |

| 49 | |

| 50 | For the x-split, 3-dimensional Euler equations, we have the following eigen values and vectors: |

| 51 | |

| 52 | {{{#!Latex |

| 53 | \lambda_1 = u-a, \lambda |

| 54 | }}} |

| 55 | |

| 56 | |