Calculator inputs
Enter function values and first derivatives for each node. The calculator builds the Hermite polynomial automatically.
Example data table
This example uses f(x) = ex, where the derivative equals the function itself. It is useful for checking the calculator output.
| Node x | f(x) | f'(x) | Comment |
|---|---|---|---|
| 0 | 1.000000 | 1.000000 | Starting point for the Hermite polynomial. |
| 1 | 2.718282 | 2.718282 | Second node for repeated-node construction. |
| 2 | 7.389056 | 7.389056 | Third node to increase the polynomial degree. |
| 3 | 20.085537 | 20.085537 | Fourth node for a richer fitted curve. |
Formula used
Hermite interpolation matches both function values and first derivatives at each node. Each xi is repeated in the divided-difference table so the polynomial honors slope information.
Repeated nodes: z2i = z2i+1 = xi
Base entries: Q[2i,0] = Q[2i+1,0] = f(xi)
Derivative entry: Q[2i+1,1] = f'(xi)
Newton form: H(x) = a0 + a1(x-z0) + a2(x-z0)(x-z1) + ...
The diagonal of the divided-difference table supplies the Newton coefficients ak. The calculator also expands the polynomial into standard powers of x for easier reading and plotting.
How to use this calculator
- Select how many distinct nodes you want to use.
- Enter each node x, the function value f(x), and the derivative f'(x).
- Choose the x-value where you want the Hermite estimate.
- Adjust decimal precision, graph padding, and plot density if needed.
- Press the calculate button to build the polynomial and show the result above the form.
- Review the summary table, divided differences, standard polynomial, and graph.
- Use the export buttons to save the report as CSV or PDF.
FAQs
1. What does Hermite interpolation do?
It creates a polynomial that matches known function values and known first derivatives at selected nodes. This makes the fitted curve respect both position and slope information.
2. Why are nodes repeated in the table?
Repeated nodes let the divided-difference table encode derivative conditions. Without repetition, the method would reduce to ordinary Newton interpolation using only function values.
3. Must my x-values be unique?
Yes. Enter each source node only once in the input table. The calculator internally repeats nodes while building the Hermite structure.
4. What derivative order does this version support?
This page supports first derivatives at each node. It is ideal for standard Hermite interpolation where both f(x) and f'(x) are known.
5. Why does the polynomial degree become large?
With n distinct nodes and first derivatives, the Hermite polynomial can reach degree 2n−1. Extra slope conditions increase flexibility and polynomial order.
6. Can I use decimal or negative values?
Yes. The calculator accepts integer, decimal, and negative inputs for nodes, function values, derivatives, and the evaluation point.
7. What does the graph show?
The graph plots the computed Hermite polynomial across the chosen range. It also highlights your source nodes and the requested evaluation point.
8. When is Hermite interpolation better than ordinary interpolation?
Hermite interpolation is stronger when slope information is available. It often produces a curve that follows local behavior more faithfully than value-only interpolation.