Solving Problems Using Linear Systems

So, the convergence assessment answers at least two questions out of four. The question “where to start” doesn’t make much sense without “when to stop”. If it means 0.001 decibels and we want to find if our equipment makes any noise within 0.1-decibel error, then the starting point we have on a plot is a great finishing point already. But we can only do that if we have the proper criteria.You see, we don’t really have a notion of where the solution is.

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If we then project this new point to another line, it will get even closer.

We will just repeat it iteration after iteration, that’s why it’s called an iterative algorithm, until we done.

Systems of linear equations are usually taught in high school or college. Let’s say you have two linear equations with two variables each.

Chances are, you already know how to solve them in several ways of which you have never had to use any in practice. One reason is since the subject is widely known, it makes a perfect specimen to show a bunch of algorithmic stuff on. If these equations constitute a system, then there may be some combination of variables’ values for which both equations hold true. Generally, for the system of two linear equations in two variables each, there is one solution. It’s easy to imagine because every linear equation is the same as a plane equation, which is in our two-dimensional case a simple straight line.

These questions are worth a series of lectures each on their own, but it’s rather easy to illustrate them using a thing as simple as a linear system. It is easy conceptually, but there are corner cases, computational problems, and complexity constraints, — all the things that make programming interesting.

When two lines intersect anywhere in space, they only intersect once.

For our case, 1 equation with 2 variables, every point on the line the remaining equation specifies is a solution.

Because yes, it satisfies all the equations just fine.

If the angle between lines is right, then the convergence is ideal. If the angle gets sharper, it takes more steps to go the same length in space, so we might say that the convergence gets worse.

And if the lines are parallel, then the algorithm does not converge at all.

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