10. Consider applying the method of steepest descent to a positive definite quadratic function with Hessian...

 

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10. Consider applying the method of steepest descent to a positive definite quadratic function with Hessian G. Suppose that x(1) (***) can be expressed as x(1) = x* + s where s is an eigenvector of G with eigenvalue A. Show that g(1) = s and that, if an exact line search in the direction of steepest descent is assumed, then the method terminates in one iteration. Hence show that the method terminates in one iteration for any x(1) if G is a multiple of a unit matrix. If x(1) cannot be expressed as above, then it can be expressed as x(1) = x*+ m 1 Hist where m > 1, and for all i, u, 0 and the s; are eigenvectors of G corre- sponding to distinct eigenvalues A. In this case show that the method does not terminate in one iteration. 10. Consider applying the method of steepest descent to a positive definite quadratic function with Hessian G. Suppose that x(1) (***) can be expressed as x(1) = x* + s where s is an eigenvector of G with eigenvalue A. Show that g(1) = s and that, if an exact line search in the direction of steepest descent is assumed, then the method terminates in one iteration. Hence show that the method terminates in one iteration for any x(1) if G is a multiple of a unit matrix. If x(1) cannot be expressed as above, then it can be expressed as x(1) = x*+ m 1 Hist where m > 1, and for all i, u, 0 and the s; are eigenvectors of G corre- sponding to distinct eigenvalues A. In this case show that the method does not terminate in one iteration.