**Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems**

**Direct Solution Methods**

- Generally faster for a matrix with a few thousand equations. The speed advantage continues to higher higher numbers of equations when the spectral radius of the matrix is near one.
- Extra solutions to set of equations for different right hand sides are very inexpensive.
- Accuracy can normally (but not always) be quite high.
- Appropriate when the matrix very complicated and rapid convergence techniques for iterative methods cannot be used.

**Matrix inversion**

For the implicit set of equations

The inverted or "explicit" set of equations is

EXAMPLE

IMPLICIT

EXPLICIT

**Progressive Elimination**

See a discussion extracted from CmpSci 201F notes for a discussion of the details of Gaussian elimination and LU factorization. We can use the efficiencies associated with factorization to cut the level of roundoff error in a given direct solution. However, you should understand that matrices exist for which roundoff can not be controlled within a standard LU factorization (even with pivoting). When the off-diagonal coefficients

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