Question

\[ f(x, y)=A x^{2}-B x y+C y^{2}+x-y \] (where \( A=4, B=4, C=2, x_{0}=5, y_{0}=6 \) ) to begin your iteration. Expected steps in solution: 1. Identify the minimum for the function using the vectoral application of Newton's method while applying dynamic \( \alpha \). Thus, you need to optimize \( \alpha \) for each iteration step to improve performance and avoid overshoot. Use iterations to find the solution with the accuracy of five significant numbers. 2. Identify the minimum again using the Newton's method with dynamic \( \alpha \). However, use this time numerical derivatives instead of \( \nabla f(x, y) \). When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, \( f^{\prime}\left(x_{i}\right)=\frac{f\left(x_{i}+h\right)-f\left(x_{i}\right)}{h} \). Here \( h \) equals some very small number. For each step \( n \), solve first the \( \mathbf{c}^{(n-1)} \) and optimal \( \alpha \) using the condition \( \varphi^{\prime}(\alpha)=\frac{d f(\alpha)}{d \alpha}=0 \). When taking the derivative of \( f(\alpha) \), please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form \( \varphi^{\prime}(x, y)=\nabla f(x-\alpha \nabla x, y-\alpha \nabla y) \cdot \nabla f(x, y) \). 3. Compare the results obtained using the numerical method with the use of accurate \( \nabla f(x, y) \). Discuss at least the following: - What values of \( h \) allow you to find an equally accurate solution as with the accurate method. Report here the values you have tested for \( h \) (for example \( h= \) \( 0.01, \ldots, 0.0000000001) \) - How does the value of \( h \) influence the needed number of iterations in the Newton's Method to reach the same minimum with the same number of significant numbers as when using the accurate \( \nabla f(x, y) \) ? Discuss and demonstrate this with examples.

          \[
f(x, y)=A x^{2}-B x y+C y^{2}+x-y
\]
(where \( A=4, B=4, C=2, x_{0}=5, y_{0}=6 \) ) to begin your iteration.
Expected steps in solution:
1. Identify the minimum for the function using the vectoral application of Newton's method while applying dynamic \( \alpha \). Thus, you need to optimize \( \alpha \) for each iteration step to improve performance and avoid overshoot. Use iterations to find the solution with the accuracy of five significant numbers.
2. Identify the minimum again using the Newton's method with dynamic \( \alpha \). However, use this time numerical derivatives instead of \( \nabla f(x, y) \). When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, \( f^{\prime}\left(x_{i}\right)=\frac{f\left(x_{i}+h\right)-f\left(x_{i}\right)}{h} \). Here \( h \) equals some very small number. For each step \( n \), solve first the \( \mathbf{c}^{(n-1)} \) and optimal \( \alpha \) using the condition \( \varphi^{\prime}(\alpha)=\frac{d f(\alpha)}{d \alpha}=0 \). When taking the derivative of \( f(\alpha) \), please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form \( \varphi^{\prime}(x, y)=\nabla f(x-\alpha \nabla x, y-\alpha \nabla y) \cdot \nabla f(x, y) \).
3. Compare the results obtained using the numerical method with the use of accurate \( \nabla f(x, y) \). Discuss at least the following:
- What values of \( h \) allow you to find an equally accurate solution as with the accurate method. Report here the values you have tested for \( h \) (for example \( h= \) \( 0.01, \ldots, 0.0000000001) \)
- How does the value of \( h \) influence the needed number of iterations in the Newton's Method to reach the same minimum with the same number of significant numbers as when using the accurate \( \nabla f(x, y) \) ? Discuss and demonstrate this with examples.

f(x, y)=A x^2-B x y+C y^2+x-y

(where A=4, B=4, C=2, x0=5, y0=6 ) to begin your iteration.
Expected steps in solution:
1. Identify the minimum for the function using the vectoral application of Newton's method while applying dynamic α. Thus, you need to optimize α for each iteration step to improve performance and avoid overshoot. Use iterations to find the solution with the accuracy of five significant numbers.
2. Identify the minimum again using the Newton's method with dynamic α. However, use this time numerical derivatives instead of ∇ f(x, y). When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, f^'(xi)=(f(xi+h)-f(xi))/(h). Here h equals some very small number. For each step n, solve first the 𝐜^(n-1) and optimal α using the condition φ^'(α)=(d f(α))/(d α)=0. When taking the derivative of f(α), please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form φ^'(x, y)=∇ f(x-α∇ x, y-α∇ y) ·∇ f(x, y).
3. Compare the results obtained using the numerical method with the use of accurate ∇ f(x, y). Discuss at least the following:
- What values of h allow you to find an equally accurate solution as with the accurate method. Report here the values you have tested for h (for example h= 0.01, …, 0.0000000001)
- How does the value of h influence the needed number of iterations in the Newton's Method to reach the same minimum with the same number of significant numbers as when using the accurate ∇ f(x, y) ? Discuss and demonstrate this with examples.

Added by Samuel O.

Question

Please give Ace some feedback