Slides_2010_02_17 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 18 Prof. Ulrich Hetmaniuk Department of Applied Mathematics February 17, 2010 Operation counts • The work required to solve a problem on a computer is often measured in terms of the number of fl oating point op erations or flops needed to do the calculation. A floating point operation is multiplication, division, addition or subtraction on floating point numbers (real numbers represented on the computer). Operation counts Example Computing the inner product of two vectors in R n , x T y = x 1 y 1 + x 2 y 2 + ··· + x n y n , requires n multiplications and n- 1 additions, so 2 n- 1 flops. • Computing an inner product will take about 2 times more operations for a vector with 1000 rows than for a vector with 500 rows. • It should take about 2 times more seconds. Operation counts Example Computing y = Ax where A ∈ R m × n and x ∈ R n . The i-th element of y is the inner product of the i-th row of A with x and requires 2 n- 1 flops. There are m rows. So we need to compute y i for i = 1 , 2 ,..., m . The total work is 2 mn- m flops. When the matrix A is square ( m = n ) , the count becomes 2 n 2- n . • Computing a matrix-vector product will take about 4 times more operations (or longer) for a 1000 × 1000 matrix than for a 500 × 500 matrix. Operation counts Example Consider A ∈ R m × r and B ∈ R r × n . Computing the matrix-matrix product C = AB will require the computations of mn entries c ij . Each entry c ij is the inner product of two vectors with r components, which require 2 r- 1 flops. So the total work becomes 2 mnr- mn flops. • Computing a matrix-matrix product will take about 8 times more longer for a 1000 × 1000 matrix than for a 500 × 500 matrix. “Big oh” notation Definition The function W ( n ) is “Big oh” of n k when the ratio W ( n ) / n k remains bounded as n-→ + ∞ : W ( n ) = O ( n k ) ⇔ | W ( n ) | ≤ Cn k for large values of n • If the work required by some algorithm for a system of size...
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This note was uploaded on 03/31/2010 for the course AMATH 352 taught by Professor Leveque during the Winter '07 term at University of Washington.

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Slides_2010_02_17 - Applied linear algebra and numerical...

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