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        AERO20542代做、代寫Python/Java編程

        時間:2024-03-07  來源:合肥網hfw.cc  作者:hfw.cc 我要糾錯



        MECH20042/AERO20542 Numerical Methods and Computing
        Laboratory exercise 1: Direct methods for the solution of
        tridiagonal systems of linear equations
        Solution of systems of linear equations is one of the most frequently encountered problems in
        numerical modelling and simulation. Efficient numerical methods, both in terms of the execution time
        and memory storage are essential to complete this task. Sparse systems of linear equations arise in
        many applications, such as finite element or finite volume solution of differential equations. Sparse
        linear systems have coefficient matrices that are sparse, i.e., a large proportion of the elements are
        equal to zero. Banded matrices are a special class of sparse matrices in which the non-zero coefficients
        are concentrated about the main diagonal.
        Storing sparse matrices in computer memory as two-dimensional arrays is inefficient, as many zero
        elements are kept needlessly in computer memory. Banded matrices can be stored by their diagonals,
        where each diagonal is stored as a one-dimensional array (a vector). With this setup a tridiagonal
        matrix 𝑇 of size 𝑛 × 𝑛

        can be stored using three vectors as follows:
        𝐴 = [𝑎11 𝑎22 ⋯ 𝑎𝑛𝑛]
        𝑇 ∈ 𝑅
        𝑛
        ,
        w**; = [𝑎21 𝑎** ⋯ 𝑎𝑛,𝑛−1]
        𝑇 ∈ 𝑅
        𝑛−1
        ,
        𝐶 = [𝑎12 𝑎23 ⋯ 𝑎𝑛−1,𝑛]
        𝑇 ∈ 𝑅
        𝑛−1
        .
        The Gaussian elimination technique applied to a tridiagonal system 𝑇𝒙 = 𝒇 is particularly simple,
        because only the non-zero elements in the sub-diagonal held in vector w**; need to be eliminated. This
        algorithm, known as the Thomas algorithm, proceeds as follows:
        FORWARD ELIMINATION BACKSUBSTITUTION
        𝑎𝑖𝑖 = 𝑎𝑖𝑖 −
        𝑎𝑖,𝑖−1
        𝑎𝑖−1,𝑖−1
        𝑎𝑖−1,𝑖 w**9;𝑛 =
        𝑓𝑛
        𝑎𝑛𝑛
        𝑓𝑖 = 𝑓𝑖 −
        𝑎𝑖,𝑖−1
        𝑎𝑖−1,𝑖−1
        𝑓𝑖−1 w**9;𝑖 =
        1
        𝑎𝑖𝑖
        (𝑓𝑖 − 𝑎𝑖,𝑖+1 w**9;𝑖+1)
        𝑖 = 2, … , 𝑛 𝑖 = 𝑛 − 1, … ,1
        TASK 1. Calculate the number of arithmetic operations that are required to solve a tridiagonal system
        𝑇𝒙 = 𝒇 of size 𝑛 using the Thomas algorithm. Based on this result, determine the asymptotic
        complexity of the Thomas algorithm, and compare it to the asymptotic complexity of the standard
        Gaussian elimination.
        TASK 2. Rewrite the Thomas algorithm in terms of the arrays 𝐴,w**;, and 𝐶 introduced to store the matrix
        𝑇 efficiently.
        TASK 3. Implement the Thomas algorithm from TASK 2 as a Python function. The input parameters to
        the function should be the coefficient matrix 𝑇 (stored as three arrays 𝐴,w**;, and 𝐶) and the right-hand
        side vector 𝒇. The output should be the solution vector 𝒙. The coefficient matrix and the right-hand
        side should be defined in the main script and passed to the function that solves the system.
        TASK 4. Test your code by solving the linear system of size 𝑛 = 10 with the values 𝐴 = 2, and w**; = 𝐶 =
        −1. Set the right-hand side to 𝒇 = 𝟏. To verify the correctness of your code, compare the solution
        vector obtained from the Thomas algorithm to that obtained by applying the direct solver
        numpy.linalg.solve(). For the latter, the coefficient matrix should be assembled.
        TASK 5. Solve five linear systems 𝑇𝒙 = 𝒇 with 𝐴 = 2, w**; = 𝐶 = −1 and 𝒇 = 𝟏 varying the problem size
        𝑛 between 106
        and 108
        . Record the execution times in seconds for each case. To accomplish this task,
        explore the Python function timer() from the package timeit (refer to the code for matrix
        multiplication covered in lectures). Plot a graph where the obtained execution times are represented
        as the function of the problem size 𝑛. What are your conclusions about the cost of the Thomas
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