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I have a python assignment and need some help to fix these codes ..... I put the codes and some instructions in a WORD file

Assignment Description

FIRST Currently, the cheb function defined in cheb_tools.py only produces the correct answer for the interval [-1,1]. If you run pytest you'll see that the function test_cheb_big_2 fails. That's because it's trying to test the cheb function on the interval [-3.2, 2.38].

To fix the code, you need to do two things.

1.   The x values need to be transformed (by a linear change of variables) that maps [-1,1] to [x_left,x_right].

  1. The D matrix needs to be rescaled so that it correctly approximates a first derivative on the requested interval.

Hint: The second part is a calculus problem. How does a linear change of variables affect derivatives?





Tools for Chebyshev differentiation.



import numpy as np


def cheb(n, x_left=-1, x_right=1):


    Computes the Chebyshev grid and differentiation matrix on n+1 points.



    n : int

        n+1 is the number of grid points.

    x_left : float

        Location of the left grid point.

    x_right : float

        Location of the right grid point.



    D : array_like

        The (n+1)-by-(n+1) Chebyshev differentiation matrix.

    x : array_like

        The Chebyshev nodes on the interval [-x_left,x_right].



    This function is based on cheb.m from Trefethen's Spectral

    Methods in MATLAB.  Unlike Trefethen's, however, this function

    orders the nodes from left to right.  That is, x[0] = x_left

    and x[-1] = x_right



    # Trefethen's algorithm

    n_range = np.arange(n+1)

    x = np.cos(np.pi*n_range/n)

    c = np.array([2] + (n-1)*[1] + [2]) * (-1)**n_range


    X = np.tile(x, (n+1, 1)).T

    dX = X - X.T


    D = np.outer(c, 1/c)/(dX + np.eye(n+1))

    D -= np.diag(np.sum(D, axis=1))


    # swap orders of nodes so x=-1 is first

    x = x[::-1]

    D = D[::-1, ::-1]


    # change variables for given interval




    return D, x



Next, take a look at the function solve_poisson in elliptic_tools.py. Notice that there is a missing piece to this function (# fill in the top and bottom rows). Just like with the finite difference solver, you need to fill in the top and bottom rows of D2 depending on the type of boundary condition.


Hint: For Neumann conditions, keep in mind that the top and bottom rows of D contain spectral approximations of the first derivative at the left and right endpoints respectively.




Functions for solving 1d elliptic problems.



import numpy as np


def solve_poisson(D, x, f, left, right):


    Solves the 1D Poisson equation u'' = f with the specified boundary conditions.



    D : array_like

        The Chebyshev differentiation matrix.

    x : array_like

        The Chebyshev nodes.

    f : array_like

        The array of right-hand side values (interior only).

    left : tuple

        A tuple of the form (type,val) where type is either 'D' for Dirichlet

        data or 'N' for Neumann data and val is a float containing the

        left boundary value data.

    right : tuple

        Similar to left, but for the right endpoint.



    u : array

        An array of solution values on the grid.



    The array f should not include values at the endpoints.  The returned array

    will have two more elements than f.



    # x has length n+1

    n = len(x) - 1


    # compute the second derivative matrix

    D2 = D.dot(D)


    # zero out the top and bottom rows

    D2[0, :] = 0

    D2[-1, :] = 0


    # initialize the right-hand side vector

    b = np.zeros(n+1)

    b[1:-1] = f

    b[0] = left[1]

    b[-1] = right[1]


    # fill in the the top and bottom rows




    # solve the system

    u = np.linalg.solve(D2, b)

    return u


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