Question

1. a Discretize a continuous variable $x$ into 5 points, $x_0, x_1, x_2, x_3$, and $x_4$. Explicitly write down the matrices $A$, $B$ and $R = A^{-1}B$. You should write down the matrices by hand without substituting any numerical value for $\omega$ (Please notice the boundary of A and B, namely $A[0,:] =[1,0,0,0,0]$ and for B is: $B[4,:] = [0,0,0,0,-\omega]).$ 1. b Using $\omega = 0.9$, write a program to calculate the powers of $R$. Use $R^n_{max}$ to denote the element in $R^n$ with the largest magnitude, in Python, it is `np.max(np.abs(R))`. Plot $R^n_{max}$ versus $n$ for $n$ from 0 to 10. You can use `np.matmul()` repeatedly to calculate powers of a matrix, do not use R**n. From the plot and the definition of the matrix $R$, comment on whether the Gauss-Seidal method is stable.

          1. a Discretize a continuous variable $x$ into 5 points, $x_0, x_1, x_2, x_3$, and $x_4$. Explicitly write down the
matrices $A$, $B$ and $R = A^{-1}B$. You should write down the matrices by hand without substituting any
numerical value for $\omega$ (Please notice the boundary of A and B, namely $A[0,:] =[1,0,0,0,0]$ and for B is:
$B[4,:] = [0,0,0,0,-\omega]).$
1. b Using $\omega = 0.9$, write a program to calculate the powers of $R$. Use $R^n_{max}$ to denote the element in
$R^n$ with the largest magnitude, in Python, it is `np.max(np.abs(R))`. Plot $R^n_{max}$ versus $n$ for $n$ from 0 to
10. You can use `np.matmul()` repeatedly to calculate powers of a matrix, do not use R**n. From the plot
and the definition of the matrix $R$, comment on whether the Gauss-Seidal method is stable.

1. a Discretize a continuous variable x into 5 points, x0, x1, x2, x3, and x4. Explicitly write down the
matrices A, B and R = A^-1B. You should write down the matrices by hand without substituting any
numerical value for ω (Please notice the boundary of A and B, namely A[0,:] =[1,0,0,0,0] and for B is:
B[4,:] = [0,0,0,0,-ω]).
1. b Using ω = 0.9, write a program to calculate the powers of R. Use R^nmax to denote the element in
R^n with the largest magnitude, in Python, it is `np.max(np.abs(R))`. Plot R^nmax versus n for n from 0 to
10. You can use `np.matmul()` repeatedly to calculate powers of a matrix, do not use R**n. From the plot
and the definition of the matrix R, comment on whether the Gauss-Seidal method is stable.

Added by Esther S.

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