Partial Pivoting and

the PA=LU

Factorization

Grayson Ishihara

Math 480

April 15, 2013

Topic

s at

Hand What is Partial

Pivoting? What is the PA=LU Factorization? What kinds of things

can we use these tools

for?

Partia

l

Piv

otin

g Used to solve matrix

equations

Compare absolute

values of entries in and

below pivot position

If |ap1|≥|a i1

|, switch rows

1 and p

Repeat for each pivot

position

LU

Fact

ori

zati

on Writing a matrix A as

the product of a lower

triangular matrix L and

an upper triangular matrix U

LU

Fact

ori

zati

on

LU

Fact

ori

zati

on

Row reduce A into upper triangular U

Keep track of steps in

lower triangular L

LU

Fact

ori

zati

on

Not all matrices allow

such a factorization…

aa Ab adacb = 0, ab = 1 ?

PA

=LU

Facto

rizatio

n Apply permutation

matrix P to A

Find LU Factorization of

the resulting matrix

Ensures all matrices

have LU Factorization

Why I

s It

U

sefu

l?

Solving systems of equations: Matrix form Ax=b

Multiply both sides by

P: PAx=Pb LUx=Pb Two-step solution:

Ly=Pb, Ux=y Easy: back substitution