How to perform Gauss elimination on a 4x4 and 5x5 matrix?

Question

Using spreadsheet software.

Show an example on a given matrix. you must show all your working(on an easy matrix)plz

Answer 1

In an Excel worksheet, follow these steps to set up your worksheet and elminate the coefficient matrix into identity matrix as typically known as the Gaussian Elimination:

Divide equation (3) (the entire one) by the coefficient of x1.
Eliminate x1 by substracting the result in Step 1 from equation (4).
Calculate the value of x2 as a result from Step 2.
Substitute the x2 value from Step 3 into equation (3) to obtain the value of x1. Record the value of x1 in a separate cell.
In the above steps, use cell references throughout instead of the values of the coefficients. Change the coefficient of x1 in equation (3) to 3×10-12, 3×10-13, 3×10-14, 3×10-15, 3×10-16, 3×10-17, 3×10-18, 3×10-19, and 3×10-20. Record the values of x1 everytime you change the coefficient.
Apply the full pivoting technique and redo the Gaussian Elimination above. Use 3×10-18 as the coefficient of x1 in equation (3).

After finishing the exercise above, again change the coefficient of x1 in equation (3) to 3×10-12 ~ 3×10-20 as you did in Step 5 above, but instead we will use the inverse method as outlined in Problem 1 to obtain the solutions. Record the values of x1 and compare them with those you obtained from the elimination procedure above.




A commonly used method for transforming a given matrix into an upper-triangular matrix is Gaussian Elimination.

Gaussian Elimimation
1.procedure GAUSSIAN ELIMINATION (A, b, y)
2.begin
3.for k := 0 to n1 do/* Outer loop */
4.begin−
5.for j := k + 1 to n1 do
6.A[k; j] := A[k; −
j]=A[k; k];/* Division step */
7.y[k] := b[k]=A[k; k];
8.A[k; k] := 1;
9.for i := k + 1 to n1 do
10.begin−
11.for j := k + 1 to n1 do
12.A[i; j] := A[i; j −
]A[i; k]A[k; j]; /* Elimination step */
13.b[i] := b[i]A[i; k]− y[k];×
14.A[i; k] :=−×
0;
15.endfor;/* Line 9 */
16.endfor;/* Line 3 */
17.end GAUSSIAN ELIMINATION


Serial Gaussian Elimination
Gaussian Elimination
The computation has three nested loops – in the kth iteration of the outer loop, the algorithm performs (nk)2 computations. Summing from k = 1::n, we have roughly ( −n3=3) multiplications- subtractions.

Answer 2

Whatchu talkin about, Willis?