Build accurate Newton interpolation polynomials from tabulated data quickly. Check values, coefficients, and errors clearly. Download reports, tables, and graphs for easy review today.
This sample uses points from the function y = x³ + 1.
| x | y |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 9 |
| 3 | 28 |
Newton interpolation builds a polynomial from tabulated points. It uses divided differences. The general form is:
P(x) = f[x0] + f[x0,x1](x - x0) + f[x0,x1,x2](x - x0)(x - x1) + ...
Each divided difference is computed recursively:
f[xi,...,xj] = (f[xi+1,...,xj] - f[xi,...,xj-1]) / (xj - xi)
The calculator finds the divided difference table, builds the interpolation polynomial, evaluates it at the target x, expands the polynomial, and estimates the derivative from the expanded expression.
Newton interpolation is useful when data comes as measured pairs. It creates a polynomial that passes through each entered point. This helps estimate missing values between known observations. It is common in mathematics, engineering, data analysis, and numerical methods.
The calculator starts with your x and y values. It checks that both lists have the same length. It also checks that x values are distinct. After validation, it builds the divided difference table. The first row of that table gives the Newton coefficients. Those coefficients define the interpolation function.
You receive the interpolated value at the selected x position. You also receive the polynomial in Newton form. That form is ideal for understanding the construction process. The tool also expands the result into a standard polynomial. This makes the function easier to read, differentiate, and graph.
The divided difference table shows how each coefficient was produced. This is helpful for learning. It also helps verify manual calculations. When you study numerical methods, this table is often the most important intermediate step.
The graph displays the entered data points and the interpolating curve. It also marks the evaluated target point. This gives a quick visual check. You can see whether the estimate fits the pattern of the data and how the polynomial behaves across the selected range.
Interpolation is strongest near the data range. Very high degree polynomials can oscillate. Estimates far outside the known points may be unreliable. Use enough points for accuracy, but avoid unnecessary degree when a simpler fit is sufficient.
CSV export is good for spreadsheets and reports. PDF export is useful for printouts, submissions, and documentation. Together, they make the calculator practical for students, teachers, analysts, and technical users who need reusable results.
It builds a polynomial passing through all entered data points. The calculator then uses that polynomial to estimate y for any chosen x value.
No. Newton divided difference interpolation works with unequal spacing. You only need distinct x values and matching y values.
Repeated x values make denominators zero in divided differences. That breaks the interpolation process and makes the coefficients undefined.
It is the step-by-step table used to compute Newton coefficients. It shows how each higher-order difference is formed from lower-order values.
Yes. The calculator accepts integers and decimals. You can separate values with commas, spaces, or new lines.
It is the slope of the interpolating polynomial at the selected x. This helps estimate local rate of change from the generated function.
Use caution. Interpolation is best inside the data range. Predictions outside the known points can become unstable, especially for higher-degree polynomials.
The CSV file saves tables, coefficients, points, and key outputs. The PDF button saves the visible result section as a report.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.