Numerical Methods in Engineering with Python
For engineering students and practicing engineers, this text discusses numerical methods thoroughly and illustrates them with problems involving both hand computation and programming.
TABLE OF CONTENTS Numerical Methods in Engineering with Python
2.3: LU Decomposition Methods
2.3 LU Decomposition Methods
Introduction
It is possible to show that any square matrix A can be expressed as a product of a lower triangular matrix L and an upper triangular matrix U:
| (2.11) |  |
The process of computing L and U for a given A is known as LU decomposition or LU factorization. LU decomposition is not unique (the combinations of L and U for a prescribed A are endless), unless certain constraints are placed on L or U. These constraints distinguish one type of decomposition from another. Three commonly used decompositions are listed in Table 2.2.
| Name | Constraints |
|---|---|
| Doolittle s decomposition | L ii=1, i=1,2, , n |
| Crout s decomposition | U ii=1, i=1,2, , n |
| Choleski s decomposition | L= U T |
After decomposing A, it is easy to solve the equations Ax=b, as pointed out in Art. 2.1. We first rewrite the equations as LUx=b. Upon using the notation Ux=y, the equations become
which can be solved for y by forward substitution. Then
will yield x by the back substitution process.
The advantage of LU decomposition over the Gauss elimination method is that once A is decomposed, we can solve Ax=b for as many constant vectors b as we please. The cost of each additional solution is relatively small, since the forward and back substitution operations are much less time...
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