# Answer on Functional Analysis Question for Ze

Question #19312

__Unconstrained Function__

F = x₁² + 1/x₁ + x₂ + 1/x₂

Question(a) At x₁ =2 & x₂ = 2, calculate gradient of F.

Question(b) And then, calculate the direction of the deepest descent.

Question(c) Using the direction of the steepest descent from (b), update the design by the standard formula : x¹ = x⁰ + αS¹

Evaluate x₁ , x₂ & F for α=0, 0.2, 0.5, & 1.0 and plot the curve of F versus α.

Question(d) Write the equation for F in terms of α alone. Discuss the character of the function.

Question(e) From (d), calculate dF/dα at α=0.

Question(f) Calculate the scalar product ∇F•S using the results of (a) and (b) and compare this with the result of (e).

F = x₁² + 1/x₁ + x₂ + 1/x₂

Question(a) At x₁ =2 & x₂ = 2, calculate gradient of F.

Question(b) And then, calculate the direction of the deepest descent.

Question(c) Using the direction of the steepest descent from (b), update the design by the standard formula : x¹ = x⁰ + αS¹

Evaluate x₁ , x₂ & F for α=0, 0.2, 0.5, & 1.0 and plot the curve of F versus α.

Question(d) Write the equation for F in terms of α alone. Discuss the character of the function.

Question(e) From (d), calculate dF/dα at α=0.

Question(f) Calculate the scalar product ∇F•S using the results of (a) and (b) and compare this with the result of (e).

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