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MATH143 SJSU Numerical Analysis and Scientific Computing Report

Please correct the size of the plate from 0≤x≤12; 0≤y≤6; to 0≤x≤11; 0≤y≤5.I have uploaded the file that I have to submit it.

Math-CS-143M
Project-4 (30 points)
Due: Thursday 12/5/2019
A hot plate is shown. The temperature on the left boundary is shown. The steady-state
temperature at an interior point can be obtained by solving Laplace’s Equation,
2
2
2
+ 2 = 0. In the language of Numerical Linear Algebra, this equation can be
converted to a system of linear equations. A typical equation to be solved is:
+ + +
T5 = 8 6 4 2 4 = Average of neighbors = “Average of North, East, South & West”.
After multiplying through by 4 this equation (centered at T5) will become: T8+T6+T2+T4 – 4T5 = 0. Similarly the
+ + +30
equation centered at T4 is : T4 = 7 5 4 1 .
The actual project is to solve the steady-state heat flow problem for a hot plate, 0≤x≤12; 0≤y≤6 and compute
temperatures at the interior points. We will take a grid of squares (length=1 and width=1), so that there are 40
“unknown” temperatures in our plate, T1, T2, T3 , …., T40. These points must be labeled in the same pattern as
shown in the example above. The boundary temperatures are given as follows:
On the x-axis, the temperature is 150oC. On the y-axis, the temperature is 50oC. On the vertical line x = 12, the
temperature is 40oC. On the horizontal line y = 6, the temperature is 100oC.
Our system of linear equations will look like Ax=b, where x= column vector [T1 T2 T3 …., T40]T. You will
calculate the matrix A and the vector b.
Turn in the following on one page only:
1) Write or print the 10×10 submatrix of A=[aij] for 1≤i≤10; 1≤j≤10. Also write the first ten entries of vector b.
2) With a starting vector vo= [1 1 1 1….1]T, apply the usual power method to find the dominant eigenvalue
of matrix A. Estimate  with a tolerance of 0.01. Print the number of iterations required. Print the converged
. Print the transpose of your eigenvector (all 40 components), so the components will appear in successive
lines (not as a column vector). Only use two decimal digits while printing.
If you don’t see a convergence pattern, change your starting vector.
3) Use Gauss-Seidel Method (Algorithm 7.2 of our book) to solve the system Ax=b, with tolerance 0.01.
Take the starting vector as xo=[1 -1 1 -1 1 -1… 1 -1]T. Print the transpose of your solution vector (all 40
components), so the components will appear in successive lines (not as a column vector). Only use two
decimal digits while printing. How many iterations were required?
If you don’t see a convergence pattern, change your starting vector.
Do not attach your computer program.

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