MIT6_079F09_lec15

# MIT6_079F09_lec15 - Convex Optimization Boyd Vandenberghe 9 Numerical linear algebra background matrix structure and algorithm complexity solving

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Convex Optimization — Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization block elimination and the matrix inversion lemma solving underdetermined equations 9–1

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Matrix structure and algorithm complexity cost (execution time) of solving Ax = b with A R n × n for general methods, grows as n 3 less if A is structured (banded, sparse, Toeplitz, . . . ) ﬂop counts ﬂop (ﬂoating-point operation): one addition, subtraction, multiplication, or division of two ﬂoating-point numbers to estimate complexity of an algorithm: express number of ﬂops as a (polynomial) function of the problem dimensions, and simplify by keeping only the leading terms not an accurate predictor of computation time on modern computers useful as a rough estimate of complexity Numerical linear algebra background 9–2
vector-vector operations ( x , y R n ) inner product x T y : 2 n 1 ﬂops (or 2 n if n is large) sum x + y , scalar multiplication αx : n ﬂops matrix-vector product y = Ax with A R m × n m (2 n 1) ﬂops (or 2 mn if n large) 2 N if A is sparse with N nonzero elements 2 p ( n + m ) if A is given as A = UV T , U R m × p , V R n × p matrix-matrix product C = AB with A R m × n , B R n × p mp (2 n 1) ﬂops (or 2 mnp if n large) less if A and/or B are sparse (1 / 2) m ( m + 1)(2 n 1) m 2 n if m = p and C symmetric Numerical linear algebra background 9–3

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Linear equations that are easy to solve diagonal matrices ( a ij = 0 if i n ﬂops = j ): x = A 1 b = ( b 1 /a 11 , . . . , b n /a nn ) lower triangular ( a ij = 0 if j > i ): n 2 ﬂops x 1 := b 1 /a 11 x 2 := ( b 2 a 21 x 1 ) /a 22 x 3 := ( b 3 a 31 x 1 a 32 x 2 ) /a 33 . . . x
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## This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.

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MIT6_079F09_lec15 - Convex Optimization Boyd Vandenberghe 9 Numerical linear algebra background matrix structure and algorithm complexity solving

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