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AMS 147 Computational Methods and Applications

Exercise #8

1. What is the cost (number of operations) of calculating, in the straightforward way, the

Fourier transform of a vector of N components?

Answer: 2N 2

2. What is the cost (number of operations) of using the fast recursive algorithm to calculate the

Fourier transform of a vector of N components ( N = 2 p )?

Answer: 3

2N log2 N( )

3. Suppose F(N) satisfies

F 1( ) = 0

F N( ) = 2FN

2+

N

2, for N > 1 and

N

2= integer

Find F(N) when N is a power of 2 ( N = 2 p ).

Answer: F N( ) =N

2log2 N( )

4. Suppose L is an N N lower triangular matrix.

What is the cost (number of operations) of solving L x = b?

Answer: N 2

5. Suppose U is an N N upper triangular matrix.

What is the number of operations of solving U x = b?

Answer: N 2

AMS 147 Computational Methods and Applications

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6. Suppose L is an N N lower triangular matrix and U is an N N upper triangular matrix.

What is the cost (number of operations) of solving LU( ) x = b ?

Answer: 2N 2

7. Suppose A is an N N dense matrix.

What is the cost (number of operations) of doing LU decomposition on A?

Answer: 2

3N 3