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For what reasons do we implement and Df(xk)wk = − f(xk) (*) Xk+1 = xk + Wk in a step of the multivariable Newton's method rather than the mathematically correct formula Xk+1 = Xk − Df(xk)-f(xk)? (**)

For what reasons do we implement and Df(xk)wk = − f(xk) (*) Xk+1 = xk + Wk in a step of the multivariable Newton's method rather than the mathematically correct formula Xk+1 = Xk − Df(xk)-f(xk)? (**)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Which of the following statements are true?

 

For what reasons do we implement and Df(xk)uk = −f(xk) (*) Xk+1 = Xk tuk in a step of the multivariable Newton's method rather than the mathematically correct formula Xk+1 = Xk − Df(x)'f(x)? (**)
Transcribed Image Text:For what reasons do we implement and Df(xk)uk = −f(xk) (*) Xk+1 = Xk tuk in a step of the multivariable Newton's method rather than the mathematically correct formula Xk+1 = Xk − Df(x)'f(x)? (**)
a. When we solve the system (*) for wê using LU or PALU factorisation, the resulting truncation error is lower than the truncation error resulting from the computation (**). b. Solving the system (*) for wk is cheaper (has a lower computational cost) than computing Df(xk)¯¹. c. The matrix Df(x)-¹ may not be invertible. Solving the system (*) for wk avoids this problem. d. The matrix Df(x₁)-¹ can be ill-conditioned. Solving the system (*) for wk avoids this problem.
Transcribed Image Text:a. When we solve the system (*) for wê using LU or PALU factorisation, the resulting truncation error is lower than the truncation error resulting from the computation (**). b. Solving the system (*) for wk is cheaper (has a lower computational cost) than computing Df(xk)¯¹. c. The matrix Df(x)-¹ may not be invertible. Solving the system (*) for wk avoids this problem. d. The matrix Df(x₁)-¹ can be ill-conditioned. Solving the system (*) for wk avoids this problem.
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