Do 3 steepest descent steps when:f(x) = X1 2 + cx/. Xo = [c I ]T. Show that 2 stepsgive

What happens if you apply the method of steepest descent to \(f(x)=x_{1}^{2}+x_{2}{ }^{2}\)? Text Transcription: f(x) = x_{1}^{2} + x_{2}{ }^{2}

Do 3 steepest descent steps when: \(f(\mathbf{x})=0.5 x_{1}^{2}+0.7 x_{2}^{2}-x_{1}+4.2 x_{2}+1\), \(x_{0}= [-1 1]^{\top}\) Text Transcription: f(x) = 0.5 x_1^2 + 0.7x_2^2 - x_1 + 4.2x_2 + 1 x_0 = [-1 1]^{top}

Do 3 steepest descent steps when: \(f(\mathbf{x})=x_{1}^{2}+0.1 x_{2}^{2}+8 x_{1}+x_{2}+22.5\), \(x_{0}=[2 -1]^{\top}\) Text Transcription: f(x) = x_1^2 + 0.1 x_2^2 + 8x_1 + x_2 + 22.5 x_0 = [2 -1]^{top}

Do 3 steepest descent steps when: \(f(\mathbf{x})=0.2{x_{1}}^{2}+x_{2}^{2}-0.08 x_{1}, x_{0}=\left[\begin{array}{ll}4 & 4\end{array}\right]^{\top}\) Text Transcription: f(x) = 0.2{x_1}^2 + x_2^2 - 0.08x_1, x_0 = [4 4]^{top}

Do 3 steepest descent steps when: \(f(\mathbf{x})=x_{1}{ }^{2}-x_{2}{ }^{2}, x_{0}= [2 1]^{\top}\) .5 steps. First guess. Then compute. Sketch your path. Text Transcription: f(x) = x_{1}{ }^{2} - x_{2}{ }^{2}, x_{0} = [2 1]^{top}

Do 3 steepest descent steps when: \(f(x)=x_{1}{ }^{2}+c x_{2}^{2}, \mathbf{x}_{0}= [c \quad 1]^{\top}\). Show that 2 steps give \([c 1]^{\top}\) times a factor, \(-4 c^{2} /\left(c^{2}-1\right)^{2}\). What can you conclude from this about the speed of convergence? Text Transcription: f(x) = x_{1}{ }^{2} + cx_{2}^{2}, x_0 = [c 1]^{top} [c 1]^{top} -4 c^{2} /(c^2 - 1)^2

Do 3 steepest descent steps when: \(f(x)=x_{1}^{2}-x_{2}, x_{0}=\left[\begin{array}{ll}1 & 1\end{array}\right]^{\top}\). Sketch your path. Predict the outcome of further steps. Text Transcription: f(x) = x_{1}^{2} - x_{2}, x_{0} = [1 1]^{top}

Do 3 steepest descent steps when: \(f(\mathbf{x})=a x_{1}+b x_{2}\), any \(\mathbf{x}_{0}\). First guess, then compute. Text Transcription: f(x) = ax_1 + bx_2 x_0

Steepest Descent. (a) Write a program for the method. (b) Apply your program to \(f(\mathbf{x})=x_{1}^{2}+4 x_{2}^{2}\), experimenting with respect to speed of convergence depending on the choice of \(\mathbf{x}_{0}\). (c) Apply your program to \(f(\mathbf{x})=x_{1}{ }^{2}+x_{2}^{4}\) and to \(f(x)=x_{1}^{4}+x_{2}^{4}, x_{0}=\left[\begin{array}{ll}2 & 1\end{array}\right]^{\top}\). Graph level curves and your path of descent. (Try to include graphing directly in your program.) Text Transcription: f(x) = x_1^2 + 4x_2^2 f(x) = x_{1}{ }^{2} + x_2^4 f(x) = x_1^4 + x_2^4, x_0 = [2 1]^top

Verify that in Example 1. successive gradients are orthogonal. What is the reason?

Do 3 steepest descent steps when: \(f(x)=3 x_{1}^{2}+2 x_{2}^{2}-12 x_{1}+16 x_{2}, x_{0}=[1 1]^{\top}\) Text Transcription: f(x) = 3x_1^2 + 2x_2^2 - 12x_1 + 16x_2, x_{0} = [1 1]^{top}

Do 3 steepest descent steps when: \(f(x)=x_{1}{ }^{2}+2 x_{2}{ }^{2}-x_{1}-6 x_{2}, \mathbf{x}_{0}= [0 0]^{\top}\) Text Transcription: f(x) = x_{1}{ }^{2} + 2 x_{2}{ }^{2} - x_1 - 6x_2, x_0 = [0 0]^{\top}\)