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
Assume R1=2, R2=4, R3=5, V1=6, and V2=2. We get the following system of three linear simultaneous equations.
The solution to these three equations produces the current flows in the network.
aij : known coefficient
Xj : unknown variable
Ci : known contant
The linear system can be written in a matrix-vector form:
Combining A and C, it can be expressed as:
can be expressed as:
It can be solved by substitution.
With substitution, we get
Phase 1: forward pass
Phase 2: back substitution
Written in terms of individual equations:
Represented in the matrix form:
X1 = 1, X2 = 2, X3 = 3, X4 = 4
3.Loop over rows( i + 1) to n below the pivot row and reduce the elements in each row as follows:
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:
Thus, the L and U matrices are
Matrix L can be computed as follows:
We can obtain the following:
Therefore, the L matrix is
The validity can be verified as
Rearrange each equation as follows:
Assume an initial estimate for the solution: X1=X2=X3=1. First iteration:
The solution is shown in the next table.
Table: Example of Jacobi Iteration
Assume an initial solution estimate of X1=X2=X3=1.
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| :
The solution is
in which I is the identity or unit matrix and both P and I are square matrices.
Let the elements of P-1 and P be denoted asqij and pij.