Figure 1: Visual comparison . Steppers (Left) are dense and simple. Servos (Right) are longer and include a visible feedback encoder housing on the rear. The Million Dollar Question: "Which Motor Do I Need?" If you are designing a CNC machine, a packaging robot, or a conveyor system, you face the same dilemma every time: Stepper or Servo? Make the wrong choice, and you face two disasters: The Stepper Trap: Your machine "loses steps" (positional error) without knowing it, scrapping parts. The Servo Trap: You spend $5,000 on a system that could have been done for $500, blowing your budget. This guide bridges the gap between mechanical requirements and electrical reality. 1. The Stepper Motor: The "Digital Ratchet" Think of a Stepper Motor like a very strong, magnetic ratchet. It divides a full rotation into equal steps (typically 200 steps per revolution, or 1.8°). Pros: Incredible Holding Torque: Ste...
Figure 1: Partial Pivoting involves swapping rows to avoid zero pivots.
The "Fatal Flaw" in Basic Code
In our previous post, Gauss Elimination (Part 2), we derived the mathematical foundation of the algorithm. However, if you implement that raw math directly into code (VBA, MATLAB, or C++), your program will eventually crash.
Why? Because the basic algorithm assumes the diagonal element (the pivot) is never zero. In the real world of engineering simulations, zeros happen frequently.
The Problem: Division by Zero
Let's look at a standard solvable system of equations:
The first step of the algorithm is to normalize the first row by dividing by the coefficient of x1 (which is 5). Since 5 is not zero, this works perfectly.
But what if the equations were rearranged?
Mathematically, the order of equations doesn't matter. But computationally, it changes everything. Look at this arrangement:
Here, the coefficient of x1 is 0. The computer attempts to divide by zero, resulting in a runtime error.
The Hidden Danger: Small Numbers
Even if the number isn't zero, but is very small (e.g., 0.000001), it causes Round-off Errors. Dividing by a tiny number creates a massive result, which can cause the computer to lose precision in the remaining digits. This is known as an "Ill-conditioned system."
The Solution: Partial Pivoting
To solve this, we use a technique called Partial Pivoting. Before performing the elimination step on a column, the algorithm must:
- Search: Look down the current column (below the diagonal).
- Identify: Find the row with the largest absolute value.
- Swap: Switch the current row with that row.
By ensuring the largest possible number is always in the pivot position, we avoid division by zero and minimize round-off errors.
Next Part: The Final Code
Now that we understand the logic and the safety features required, we are ready to build the robust Excel VBA tool.
Continue to Part 4:
Solving System of Equations using Gauss Elimination Method (Part 4: The Final Program)
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