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10 Nov 2019
Project on Linear System of Equations, and LV factorization This project studies a problem on heat transfer, where a steady-state temperature distribution of a thin plate is sought when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross-section of a metal beam with very negligible heat flow in the direction perpendicular to the plate. Let T_1, T_2, ..., T_8 denote the temperatures at the 8 interior nodes of the mesh In the figure. The temperature at a node is approximately equal to the average of 4 nearest nodes - to The left, above right and below. a), Write the system of equations in the matrix from Ax = b. (The matrix A should be a band matrix. Such matrices occur in a variety of applications, and often are large with many rows and columns, but relatively narrow bands. Finding A^-1 and then computing x=A^-1 b is not an efficient way of solving this problem, as you will find in the next two steps.) b). Find an LU factorization of A, and then use it to solve this system. (You will notice that both factor matrices are band matrices with two nonzero diagonals below or above the main diagonal.) c). Obtain A^-1 and note that A^-1 is a dense matrix with no band structure. When A is large, L and U can be stored in much less space than A^-1. This is another reason for preferring LU factorization of A to A^- 1 itself.
Project on Linear System of Equations, and LV factorization This project studies a problem on heat transfer, where a steady-state temperature distribution of a thin plate is sought when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross-section of a metal beam with very negligible heat flow in the direction perpendicular to the plate. Let T_1, T_2, ..., T_8 denote the temperatures at the 8 interior nodes of the mesh In the figure. The temperature at a node is approximately equal to the average of 4 nearest nodes - to The left, above right and below. a), Write the system of equations in the matrix from Ax = b. (The matrix A should be a band matrix. Such matrices occur in a variety of applications, and often are large with many rows and columns, but relatively narrow bands. Finding A^-1 and then computing x=A^-1 b is not an efficient way of solving this problem, as you will find in the next two steps.) b). Find an LU factorization of A, and then use it to solve this system. (You will notice that both factor matrices are band matrices with two nonzero diagonals below or above the main diagonal.) c). Obtain A^-1 and note that A^-1 is a dense matrix with no band structure. When A is large, L and U can be stored in much less space than A^-1. This is another reason for preferring LU factorization of A to A^- 1 itself.