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Unformatted text preview: Applied linear algebra and numerical analysis Session 24 Prof. Ulrich Hetmaniuk Department of Applied Mathematics March 3, 2010 Normal Equations Consider a matrix A ∈ R m × n of full rank. Theorem A vector x minimizes the residual norm k b Ax k 2 if and only if r ⊥ R ( A ) , A T r = or equivalently A T Ax = A T b The system has a unique solution if and only if A has full rank. Solving the Normal Equations Consider a matrix A ∈ R m × n of full rank. • Algorithm • Form the matrix A T A and the vector A T b . • Compute the Cholesky factorization A T A = R T R . • Solve the lower triangular system R T w = A T b . • Solve the upper triangular system Rx = w . • A T A is symmetric positive definite (invertible). • Cost mn 2 + n 3 3 . • If κ denotes the condition number of matrix A , then the matrix A T A has condition number κ 2 . x x computed k x k = O ( κ 2 ε machine ) . Solving the Normal Equations Consider a matrix A ∈ R m × n of full rank....
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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.
 Winter '07
 Leveque
 Numerical Analysis

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