4. LU Factorization Method
In this
method, the coefficient matrix [A] is a multiplication result of its upper
matrix [U] and Lower matrix [L] i.e. [A]=[U]*[L].
Since [A]*{x}={b}, then [U]*[L]*{x}={b}.Recall that [U]*{x}={D}, then
[L]*{D}={b}, so {D} can be computed. Last step is that since [U]*{x}={D}, then
{x} can be computed as well
Using MatLab to determine the roots of a system of linear Equation Using LU
Factorization Method
Thanks to (lu)
function in MatLab, LU factorization method can be executed quickly with shortcodes
as follows:
5. Gauss-Seidel Method
Gauss-Seidel is the most common iterative method used to find roots
of a system if linear equations. If the equations are diagonally non-zero
(preferably diagonally predominant), then the first equation can be solved for
(x1) and the second equation can be solved for (x2), and
the third equation can be solved for the (x3) and so on. To start
the iteration, all x’s are zeros. These zeros are substituted to solve (x1),
then this (x1) is used to solve (x2), then (x1 and x2)
are used to solve the (x3), and so on.
To illustrate this
Note that these equations
are not diagonally predominant. In order to get them diagonally dominant,
equation 2 should come first, and equation 3 come next.
Using MatLab to determine the roots of linear Equations using Gauss-Seidel.
In order to compute
Gauss-Seidel in MatLab, modifications in the equation should be done first
Part (2): System of Nonlinear Equations
In contrast to linear
equations, nonlinear equations are curves, and their solution is the
intersection of these curves.
System of nonlinear
equations can be solved by different methods, but here we will focus on Newton
Raphson and Excel Solver Methods.
1. Computing the Nonlinear roots using Newton Raphson Method
For a single
nonlinear equation, Newton Raphson Law is
In order to solve
this equation by Calculator or MatLab Function, this equation must be simplified
Manually by Calculator.
The equation can be rearranged to be
2. Using Excel Solver to determine the roots of Nonlinear Equations
Excel Solver can be
used efficiently to find the roots of a system on the nonlinear equation. But
in order to solve two equations at once, we have to choose one equation to be the
main equation while adding the other one to as a constraint
equation=0
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