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Slides_2010_03_03

# Slides_2010_03_03 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis Session 24 Prof. Ulrich Hetmaniuk Department of Applied Mathematics March 3, 2010

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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 = 0 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 ) .

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Solving the Normal Equations Consider a matrix A R m × n of full rank.
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