Homework5 - Department of Mathematics University of Houston...

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University of Houston Numerical Analysis I Dr. Ronald H.W. Hoppe Numerical Analysis I (5th Homework Assignment) Exercise 17 ( Reformulation of linear least squares problems ) Let A R m × n , m > n, rank A = n, b R m . The linear least squares problem ( * ) k Ax - b k 2 = min can be formulated as the linear algebraic system ( ** ) ˆ I m A A T 0 ! ˆ r x ! = ˆ b 0 ! , where I m stands for the m × m unit matrix and r := b - Ax R m is the residual. (i) Using the normal equations, show that the component x of the solution of ( ** ) solves the linear least squares prolem ( * ). (ii) Given a decomposition of A according to ( ) A = Q ˆ R 0 ! with an orthogonal matrix Q R m × m and a regular upper triangular matrix R R n × n show that by orthogonal row and column transformations the linear system ˆ I m A A T 0 ! ˆ p z ! = ˆ f g ! can be transformed to the form I n 0 R 0 I m - n 0 R T 0 0 h d z = f
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Homework5 - Department of Mathematics University of Houston...

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