7.1 Naïve Gaussian Elimination 8.1 The LU Factorization

7.1 Naïve Gaussian Elimination8.1 The LU Factorization

• Motivating Ax=b: Newton's method for systems of nonlinear equations (pp. 96-99) • C&K 7.1: Naive Gaussian Elimination

– See answered problems (1, 2, 3ab, 4, 5, 6, 7abc)

• C&K 8.1: Matrix Factorizations (see pp. 293-302 on LU factorization)– See answered problems (1-8)

A x = b

Motivation: Newton’s Method (for systems of nonlinear equations, pp.96-99)

Motivation: Newton’s Method (for systems of nonlinear equations, pp.96-99)

Motivation: Newton’s Method (for systems of nonlinear equations, pp.96-99)

Example 2 (p. 99)

Cha

pter

7.1

(p.

245

)

A x = b

A x = b

A x = b

Naïve Gaussian Elimination

Naïve Gaussian Elimination

Naïve Gaussian Elimination

Naïve Gaussian Elimination

Back Substitution

Naïve Gaussian Elimination

Naïve Gaussian Elimination

Naïve Gaussian Elimination

Continuing to second equation…

kth step of forward elimination

Naïve GE: Pseudocode

Storing Multipliers (instead of zeros)

Back Substitution

Back Substitution

Back Substitution

LU Factorization Example(supplements chalkboard)

Matrix Interpretation

Column 1

Column 2

Column 3

Recognizing “L”

Verifying LU=A