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Direct Solution Methods

Characteristics

• 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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