Chapter 5 Simultaneous Linear Equations. Many engineering and scientific problems can be formulated in terms of systems of simultaneous linear equations.
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I: current R: resistance
The solution to these three equations produces the current flows in the network.
aij : known coefficient
Xj : unknown variable
Ci : known contant
Combining A and C, it can be expressed as:
It can be solved by substitution.
With substitution, we get
Phase 1: forward pass
Phase 2: back substitution
Represented in the matrix form:
X1 = 1, X2 = 2, X3 = 3, X4 = 4
1. For the last row n:
2. For rows (n-1) through 1,
LUX=C, we have LE=C and UX=E
Applying LU decomposition:
We can obtain the following:
The validity can be verified as
Rearrange each equation as follows:
The solution is shown in the next table.
Jocobi iteration method requires 13 iterations to reach the accuracy of 3 decimal places. Gauss-Seidel iteration method needs 7 iterations.
If we solve both equations individually for X1 we get
Cramer’s rule for obtaining Xi:
An example of |Ai| :
in which I is the identity or unit matrix and both P and I are square matrices.