Chapter 3 Solution of Linear Systems - Math/CS

64 CHAPTER 3. SOLUTION OF LINEAR SYSTEMS
Problem 3.3.3. Suppose
⎡
A = ⎣
1 3 −4
0 −1 5
2 0 4
⎤
⎡
⎦ and b = ⎣
(a) Use Gaussian elimination without row interchanges to find the A = LU factorization.
(b) Use the A = LU factorization of A to solve Ax = b.
Problem 3.3.4. Suppose
A =
⎡
⎢
⎣
4 8 12 −8
−3 −1 1 −4
1 2 −3 4
2 3 2 1
⎤
1
3
2
⎥
⎦ and b =
(a) Use Gaussian elimination without row interchanges to find the A = LU factorization.
(b) Use the A = LU factorization of A to solve Ax = b.
3.3.2 P A = LU Factorization
As we know, the practical implementation of Gaussian elimination uses partial pivoting to determine
if row interchanges are needed. In this section we show that this results in a modification of the
the LU factorization. Row interchanges can by represented mathematically as multiplication by a
permutation matrix. A permutation matrix is obtained by interchanging rows of the identity matrix.
Example 3.3.3. Consider the matrix P obtained by switching the first and third rows of a 4 × 4
identity matrix:
⎡
⎤
⎡
⎤
1 0 0 0
0 0 1 0
⎢
I = ⎢ 0 1 0 0 ⎥ switch
⎢
⎥
⎣ 0 0 1 0 ⎦ −→ P = ⎢ 0 1 0 0 ⎥
⎣ 1 0 0 0 ⎦
rows 1 and 3
0 0 0 1
0 0 0 1
Then notice that multiplying any 4 × 4 matrix on the left by P has the effect of switching its first
and third rows. For example,
⎡
0
⎢
P A = ⎢ 0
⎣ 1
0
1
0
1
0
0
⎤ ⎡
0 2
0 ⎥ ⎢
⎥ ⎢ 5
0 ⎦ ⎣ 1
−3
2
1
0
−4
−1
⎤
1
7 ⎥
−1 ⎦
0 0 0 1 0 3 8 −6
=
⎡
1
⎢ 5
⎣ 2
1
2
−3
−1
−4
0
⎤
−1
7 ⎥
1 ⎦
0 3 8 −6
Suppose we apply Gaussian elimination with partial pivoting by rows to reduce a matrix A to
upper triangular form. Then the matrix factorization that is computed is not an LU decomposition
of A, but rather an LU decomposition of a permuted version of A. That is,
P A = LU
where P is a permutation matrix representing all row interchanges. It is nontrivial to derive this
factorization, which is typically done in more advanced treatments of numerical linear algebra.
However, finding P , L and U is not difficult. In general, we proceed as in the previous subsection
for the LU factorization, but we keep track of the row swaps as follows:
⎡
⎤
⎢
⎣
⎦ .
3
60
1
5
⎤
⎥
⎦ .