= = p CSE 541 x x x x = n p 1-norm: x x = - - PowerPoint PPT Presentation

SLIDE 1

Vector Norms

CSE 541 Roger Crawfis

Vector Norms

Measure the magnitude of a vector Measure the magnitude of a vector

Is the error in x small or large?

General class of p norms: General class of p-norms:

1/ p n p

x x ⎛ ⎞ = ⎜ ⎟

∑

1-norm: 2-norm:

1 p i

x x

=

= ⎜ ⎟ ⎝ ⎠

∑

1 1 n i i

x x

=

=∑

( )

1/2 2 2 1 n i i

x x

=

= ∑

• ∞-norm:

( )

maxi

i

x x

∞ =

Properties of Vector Norms p

For any vector norm: For any vector norm:

0 if x x > ≠ for any scalar (triangle inequality) x x x y x y γ γ γ = ⋅ + ≤ +

These properties define a vector norm

(triangle inequality) x y x y + ≤ +

These properties define a vector norm

Matrix Norms

We will only use matrix norms “induced” We will only use matrix norms induced

by vector norms:

Ax max

x

Ax A x

≠

=

1-norm:

1 1

max (max absolute column sum)

n ij j i

A A =

∑

∞-norm:

1 j i=

max (max absolute row sum)

n ij i

A A

∞ =

∑

1 i j=

SLIDE 2

Properties of Matrix Norms p

These induced matrix norms satisfy: These induced matrix norms satisfy:

0 if A A > ≠ for any scalar (triangle inequality) A A A B A B γ γ γ = ⋅ + ≤ + (triangle inequality) A B A B AB A B + ≤ + ≤ ⋅ for any vector Ax A x x ≤ ⋅

Condition Number

If A is square and nonsingular, then If A is square and nonsingular, then If A is singular then cond(A) = ∞

1

cond( ) A A A− = ⋅

If A is singular, then cond(A) = ∞ If A is nearly singular, then cond(A) is large. The condition number measures the ratio of The condition number measures the ratio of

maximum stretch to maximum shrinkage:

1

A A

−

⎛ ⎞ ⎛ ⎞

1

max min

x x

Ax Ax A A x x

− ≠ ≠

⎛ ⎞ ⎛ ⎞ ⋅ = ⋅ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

Properties of Condition Number

For any matrix A cond(A) ≥ 1 For any matrix A, cond(A) ≥ 1 For the identity matrix, cond(I) = 1

F t ti t i d(P) 1

For any permutation matrix, cond(P) = 1 For any scalar α, cond(α A) = cond(A) For any diagonal matrix D,

( ) ( )

cond( ) max / min D D D

( ) ( )

cond( ) max / min

ii ii

D D D =

Errors and Residuals

Residual for an approximate solution y to Residual for an approximate solution y to

Ax = b is defined as r = b – Ay

If A is nonsingular, then ||x – y|| = 0 if and only

g , || y|| y if ||r || = 0.

Does not imply that if ||r||<ε, then ||x-y|| is small.

SLIDE 3

Estimating Accuracy g y

Let x be the solution to Ax = b Let x be the solution to Ax = b Let y be the solution to Ay = c

Th i l l i h th t

Then a simple analysis shows that

cond( ) x y b c A − − ≤

Errors in the data (b) are magnified by

cond( ) A x c ≤

Errors in the data (b) are magnified by

cond(A)

Likewise for errors in A Likewise for errors in A