Solving Ax=b with Pivoting Solving Ax=b with Gaussian Elimination - - PowerPoint PPT Presentation

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Jan 20, 2023 198 likes •243 views

Numerical and Scientific Computing with Applications David F . Gleich CS 314, Purdue September 29, 2016 In this class: Solving Ax=b with Pivoting Solving Ax=b with Gaussian Elimination and LU and partial pivoting Next class

SLIDE 1

Operations in solving Ax=b

G&C – Chapter 7 Numerical and Scientific Computing with Applications David F . Gleich CS 314, Purdue September 29, 2016

Solving Ax=b with Pivoting

Next class

QR Factorization & Least Squares

G&C – Chapter 7.6 Next next class In this class:

Solving Ax=b with

Gaussian Elimination and LU and partial pivoting

SLIDE 2

Solving Ax=b

We use Gaussian Elimination to solve Ax=b We record the steps in a matrix factorization A = LU So that we can “replay” them more efficiently.

SLIDE 3

Solving Ax=b

But – there are some issues with this!

Pivoting is necessary to make this work on

the computer!

Swapping rows to avoid dividing by

(i) zero or (ii) small numbers THEOREM A matrix is non-singular if and

nly if the pivoted LU decomposition

Operations in solving Ax=b

Solving Ax=b with Pivoting

QR Factorization & Least Squares

Solving Ax=b

We use Gaussian Elimination to solve Ax=b We record the steps in a matrix factorization A = LU So that we can “replay” them more efficiently.

Solving Ax=b

But – there are some issues with this!

the computer!

(i) zero or (ii) small numbers THEOREM A matrix is non-singular if and

succeeds without dividing by zero