Result Preview Area
Enter data values and press calculate. Your polynomial, estimate, coefficients, tables, and graph will appear here above the form.
Calculator Input
Enter matching x-values and y-values in the same order. Separate entries with commas, spaces, or new lines.
Example Data Table
This sample follows the rule y = x² + 1. It is useful for checking the calculator quickly.
| x | y |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 5 |
| 3 | 10 |
Formula Used
Newton divided difference interpolation builds a polynomial from ordered data pairs. It works well when points are added progressively, because only new divided differences need updating.
Polynomial form
P(x) = f[x0] + f[x0,x1](x - x0) + f[x0,x1,x2](x - x0)(x - x1) + ...
Divided difference recurrence
f[xi,...,xi+k] = ( f[xi+1,...,xi+k] - f[xi,...,xi+k-1] ) / ( xi+k - xi )
How to Use This Calculator
- Enter all x-values in the first field.
- Enter the matching y-values in the second field.
- Provide the target x-value for interpolation.
- Choose decimal precision for displayed results.
- Optionally define chart minimum, maximum, and sample points.
- Press Calculate Polynomial to generate the result.
- Review the Newton form, expanded form, coefficients, and graph.
- Use the export buttons for CSV or PDF output.
FAQs
1. What does this calculator compute?
It computes the Newton divided difference interpolating polynomial from your entered data points. It also evaluates the polynomial at a target x-value, shows the divided difference table, and plots the fitted curve.
2. Do the x-values need equal spacing?
No. Newton divided differences work with unevenly spaced x-values. The only strict rule is that every x-value must be unique, because repeated x-values make the divided difference denominator zero.
3. Why must x-values be unique?
Each divided difference uses a denominator based on x-value gaps. If two x-values are identical, that denominator becomes zero. Unique x-values keep the interpolation process valid and numerically defined.
4. What is the difference between interpolation and extrapolation?
Interpolation estimates values inside the data range. Extrapolation estimates outside the data range. Interpolation is usually more reliable, while extrapolation can become unstable when the target lies far beyond known points.
5. When is Newton form especially useful?
Newton form is useful when data points are added one at a time. You can extend the polynomial by computing only the new divided differences instead of rebuilding the entire expression from scratch.
6. What does the divided difference table show?
The table organizes first, second, and higher-order divided differences. The top entry of each order becomes a coefficient in the Newton polynomial. It provides a clear view of how the interpolation expression is built.
7. Can I use scientific notation in inputs?
Yes. Numeric entries such as 1e-3 or 2.5e2 are accepted. Separate values with commas, spaces, semicolons, or line breaks, and keep x and y lists aligned by position.
8. What do the CSV and PDF downloads include?
The downloads include summary values, Newton coefficients, and the divided difference table. This makes it easier to keep a calculation record for reports, classroom work, lab notes, or later review.