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The result is the augmented matrix corresponding to the system of equations:
X+Y = 0
X-Y =2
Residual errors in linear system solutions (Function RSD)
Function RSD calculates the ReSiDuals or errors in the solution of the matrix
equation A⋅x=b, representing a system of n linear equations in n unknowns.
We can think of solving this system as solving the matrix equation: f(x) = b -A⋅x
= 0. Suppose that, through a numerical method, we produce as a first
approximation the solution x(0). Evaluating f(x(0)) = b - A⋅x(0) = e ≠ 0. Thus,
e is a vector of residuals of Function for the vector x = x (0).
To use Function RSD you need the terms b, A, and x(0), as arguments. The
vector returned is e = b - A⋅x(0). For example, using A = [[2,-
1][0,2]], x(0) = [1.8,2.7], and b = [1,6], we can find the vector
of residuals as follows:
The result is e = b - A⋅x(0) = [ 0.1 0.6 ].
Note: If we let the vector Δx = x – x (0), represent the correction in the
values of x (0), we can write a new matrix equation for Δx, namely A⋅Δx = e.
Solving for
Δ
x we can find the actual solution of the original system as x =
x(0) + Δx.