Local error

Local error is the error introduced by one step of a numerical method for a differential equation. In Linear Algebra and Differential Equations, it measures how far a single Euler or Improved Euler step is from the exact solution.

What is the local error?

Local error is the error from one step of a numerical approximation to a differential equation. In this course, you usually see it when you take a method like Euler's Method, move from one point to the next, and compare that one-step estimate to the exact solution value at the same new x-value.

Think of it as the mistake made by a single jump. If you start at a known point and use the slope to predict the next y-value, local error measures how good that one prediction is. It does not describe the whole solution curve yet, just the accuracy of that one step.

For Euler's Method, the local error is tied to the fact that the method uses a tangent line instead of the true curved solution. Since real solutions usually bend, the straight-line step misses a little. Smaller step sizes usually shrink that miss, which is why a smaller h gives a better approximation.

Improved Euler's Method lowers local error by using two slope estimates and averaging them. That extra slope check makes the next point closer to the true curve than basic Euler's Method usually gets. The idea is still step-by-step approximation, but with a better estimate of the slope over the interval.

A useful way to separate the ideas is this: local error is what happens in one step, while the overall error after many steps can build up from lots of local errors. In class, Taylor series often show why the error behaves the way it does, because the method is really trying to imitate the exact solution's expansion but stops short of the full curve.

Why the local error matters in Linear Algebra and Differential Equations

Local error shows you how trustworthy a numerical method is before the errors start stacking up across many steps. In Differential Equations, you often cannot solve an initial value problem exactly, so you rely on step-by-step methods and need a way to judge whether the approximation is actually close.

It also explains why step size matters so much. If you halve the step size in Euler's Method, each step usually gets more accurate, and the local error drops fast enough that the whole graph can look much closer to the true solution. That gives you a concrete reason to choose a smaller h instead of just saying it is "more accurate."

This term also sets up the comparison between Euler's Method and Improved Euler's Method. If two methods start at the same initial condition, the one with smaller local error usually tracks the solution better over time. So when you compare outputs from a homework table or calculator run, local error is part of the logic behind why one method wins.

In linear algebra topics that involve systems of differential equations, the same idea still shows up. Even when you are approximating vector-valued solutions, each step can be checked against the exact behavior of the system, and local error tells you how much the method drifted in that one move.

How the local error connects across the course

Step Size

Step size is the distance you move forward on the x-axis with each numerical step. Smaller step sizes usually reduce local error because the method is trying to follow the curve over a shorter interval. In homework, this is the first thing to adjust when an approximation looks too rough.

Global Error

Global error is the total error after many steps, not just one. Local error feeds into global error, but they are not the same thing, and a method with small local error can still drift if you take many steps. This distinction shows up when you compare a table of values to the exact solution.

Convergence

Convergence describes whether a numerical method gets closer to the true solution as the step size gets smaller. Local error is one reason convergence happens, because each step becomes a better approximation. If a method converges well, you should see the output improve as h decreases.

truncation error

Truncation error is the error made when a method stops a calculation early, such as cutting off a Taylor series after a few terms. Local error is often treated as a type of truncation error for one step of an ODE method. That is why Taylor series language comes up when the class explains it.

Is the local error on the Linear Algebra and Differential Equations exam?

A quiz or problem set usually asks you to compare Euler's Method and Improved Euler's Method, change the step size, or explain why one approximation is closer to the exact solution. You may need to identify the error from a single step, not the whole table, so read the prompt carefully. If the problem gives you a differential equation and an initial condition, the move is often to compute a step, then reason about how the local error would change if h got smaller. When a teacher asks for a written explanation, use the language of one-step approximation, slope estimate, and exact solution value at the next x-value. That keeps you focused on the actual source of the error instead of just saying the answer is "more accurate."

The local error vs Global Error

Local error is the mistake from one step of the method. Global error is the total accumulated difference after many steps. A student often mixes them up because both are about accuracy, but the first is step-by-step while the second is the final drift of the full numerical solution.

Key things to remember about the local error

• Local error is the error made in one step of a numerical method for a differential equation.

• In Euler's Method, local error comes from using a straight-line tangent step instead of the true curve.

• Smaller step sizes usually reduce local error, which makes the numerical approximation closer to the exact solution.

• Improved Euler's Method lowers local error by averaging slope information instead of using only one slope.

• Local error is different from global error, which is the total error after many steps.

Frequently asked questions about the local error

What is local error in Linear Algebra and Differential Equations?

Local error is the difference between the exact solution and the numerical estimate after one step of a method like Euler's Method. It tells you how accurate that single move is, before any later errors have a chance to build up. In this course, it usually comes up when you are comparing approximation methods for differential equations.

How is local error different from global error?

Local error is the error from one step, while global error is the total error after many steps. That means global error includes the accumulation of many local errors along the way. A method can have small local error but still show noticeable global error if you take enough steps.

Why does smaller step size reduce local error?

A smaller step size makes each move cover less distance, so the straight-line estimate stays closer to the true curve. In Euler's Method, that means the tangent line has less chance to drift away from the exact solution during one jump. This is why decreasing h usually makes the graph look more accurate.

How does Improved Euler's Method affect local error?

Improved Euler's Method uses two slope estimates and averages them, so it usually gives a better one-step approximation than basic Euler's Method. That lowers local error for the same step size. If a problem asks why the improved method is closer to the exact solution, local error is the reason to mention.