Stepwise Minimization Calculator

Calculator Input

Formula Used

The calculator evaluates this polynomial objective:

f(x) = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀

For each pass, it samples points inside the current interval:

xᵢ = lower bound + i × step size

For minimization, the best point is:

x* = arg min f(xᵢ)

For maximization, the best point is:

x* = arg max f(xᵢ)

After each pass, the search window is narrowed:

new interval = x* ± step size × window multiplier

The next step size is:

new step = old step ÷ refine factor

The derivative check uses:

f'(x) = 4a₄x³ + 3a₃x² + 2a₂x + a₁

How to Use This Calculator

1. Enter the polynomial coefficients. Use zero for unused terms.
2. Choose whether you want to minimize or maximize the function.
3. Enter left and right bounds for the search interval.
4. Set the initial step size for the first rough search.
5. Choose refinement passes for deeper stepwise improvement.
6. Use a larger refine factor for faster narrowing.
7. Press the calculate button to view results above the form.
8. Download the result table as CSV or PDF when needed.

Example Data Table

Stepwise Minimization Guide

What It Does

Stepwise minimization is a practical search method for one variable functions. It does not need symbolic solving. It scans an interval, finds the best sampled point, and then narrows the next search around that point. This makes it useful when a curve is easy to evaluate but hard to solve by hand.

Input Choices

The calculator above uses a polynomial objective. You can enter linear, quadratic, cubic, or quartic coefficients. Set unused coefficients to zero. Then choose the left and right bounds. These bounds define the allowed search area. The first step size controls how many points are tested in the first pass. A smaller step gives more detail. A larger step runs faster.

Refinement Logic

Each refinement pass builds a smaller window around the current best point. The refine factor reduces the step size after every pass. The window multiplier controls how much space is kept around the best point. Together, these options balance speed and accuracy. The tolerance stops the search when changes become very small.

Reading the Output

The result table shows every pass. It lists the active interval, step size, sampled points, best x value, best function value, and improvement. This makes the calculation transparent. You can see how the answer moves from a rough estimate toward a polished minimum.

Using the Chart

The chart helps confirm the decision visually. The curve shows function values across the original interval. Markers show the best point from each pass. When markers move toward a low area, the search is behaving well. If markers jump unexpectedly, try a smaller first step or wider refinement window.

Accuracy Notes

Stepwise minimization is not always a perfect global optimizer. A rough first step may miss a narrow valley. A bad interval may exclude the true answer. For best results, use realistic bounds and run several searches with different step sizes. Compare the exported results when precision matters.

Common Uses

This tool is helpful for cost planning, tuning, estimation, scoring models, and simple optimization lessons. It also supports maximum search by reversing the comparison. Export the table when you need records for reports, audits, examples, or classroom notes. Because each step is visible, beginners can learn the logic while advanced users can test assumptions quickly and safely.

FAQs