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AMS 147 Computational Methods and Applications

Exercise #9

1. Write out the 1-norm of vector x = x1, x2 ,, xN( )

T.

Answer:

1-norm: x1

= x jj =1

N

2. Write out the 2-norm of vector x = x1, x2 ,, xN( )

T.

Answer:

2-norm: x2

= x j2

j =1

N1

2

3. Write out the infinity-norm of vector x = x1, x2 ,, xN( )

T.

Answer:

-norm: x = maxj

x j

4. Write out the norm of matrix A in terms of vector norms.

Answer:

A = maxx 0

Ax

x

5. Write out the condition number of matrix A in terms of matrix norms.

Answer:

cond A( ) = A 1 A

AMS 147 Computational Methods and Applications

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6. Consider a linear system A x = b with condition number cond(A) < 10.

We use Gauss elimination without pivoting to solve A x = b.

Is the relative error in the numerical solution guaranteed to be small? Why?

Answer:

No. The relative error in the numerical solution is not guaranteed to be small because Gauss

elimination without pivoting is not a good numerical method.

7. Consider a linear system A x = b with condition number cond(A) = 1020.

We use Gauss elimination with pivoting to solve A x = b.

Is the relative error in the numerical solution guaranteed to be small? Why?

Answer:

No. The relative error in the numerical solution is not guaranteed to be small because the

condition number is very large.

8. Let A be an N N matrix. For N = 106, what is the amount of RAM memory required for

storing all elements of matrix A as double precision real numbers?

Answer:

To store all 1012 elements of matrix A as double precision real numbers, it requires

1012 64 bits = 1012 8 Bytes = 8000 GB