Type splinetool at the command line.

Select Import your own data from the initial screen, and accept the default function. You should see the following display.

The default approximation is the cubic spline interpolant with the not-a-knot end condition.

The vector x of data sites is linspace(0,2*pi,31) and the values are cos(x). This differs from simply providing the vector y of values in that the cosine function is explicitly recorded as the underlying function. Therefore, the error shown in the graph is the error in the spline as an approximation to the cosine rather than as an approximation to the given values. Notice the resulting relatively large error, about 5e-5, near the endpoints.

For comparison, follow these steps:

This procedure results in the display shown below. The right end slope is zero only up to round-off. Bottomline tells you that the tool uses the function csape to create the spline.

The improvement in the error is only about 5e-6.

For further comparison, follow these steps:

Note the deterioration of the approximation near the ends, an error of about 2e-3, which is much worse than with the not-a-knot end conditions.

For a final comparison, follow these steps:

Note the dramatic improvement in the approximation, back to an error of about 5e-6, particularly compared to the natural end conditions.

Type splinetool at the command line or if the tool is already open, click File > Restart.

Select Richard Tapia's drag race data. The data shows the distance traveled by a drag car as a function of time. The message window asks you to estimate the initial acceleration by setting the initial speed to zero. Click OK, or press the space bar or Enter, to remove the message window.

In Approximation method, select complete from the list of End conditions.

Adjust the initial speed by changing the first derivative at the left endpoint to zero.

Look for the value of the initial acceleration, which is given by the value of the second derivative at the left endpoint. You can toggle between the first derivative and the second derivative at this endpoint by clicking on the left end button. The value of the second derivative should be around 187 in the units chosen. Select View > Show 2nd Derivative to see the result graphically.

For comparison, click New, then select Least-Squares Approximation as the Approximation method. With this method, you can no longer specify end conditions. Instead, you may vary the order of the method. Verify that the initial acceleration is close to the cubic interpolation value.

The results of this procedure are shown below.

Type splinetool at the command line or if the tool is already open, click File > Restart.

Select Titanium heat data.

Select Least-Squares Approximation as the Approximation method.

Notice how poorly this selection approximates the data since there are no interior knots. To view the current knots and add new knots, select Knots from Data, breaks/knots, weights. The knots are now listed in Knots, and also displayed in the data graph as vertical lines. There are just the two end knots, each with multiplicity 4.

Right-click in the data graph and select Add Knot. This option displays a crosshairs for you to move with the mouse. Its precise horizontal location is shown in the edit field below the list of knots. A mouse click places a new knot at the current location of the crosshairs. One possible strategy is to add the knot at the place of maximum absolute error, as shown in the auxiliary graph below the data graph.

If you right-click and select Replicate Knot, you increase the multiplicity of the current knot, which is reflected by its repeated occurrence in Knots. To delete a specific knot, first select it in either the list of knots or the data graph, and then right-click in the graph and select Delete Knot.

You can also opt for an approximation using six polynomial pieces, which corresponds to five interior knots. To specify this option, enter 6 as # pieces in Data, breaks/knots, weights.

After you have the five interior knots, try to make the error even smaller by moving the knots. To select the knot you want to move, click its vertical line in the graph, and then use the control below Knots in Data, breaks/knots, weights and observe how the error changes with the movement of the knot. You can also use the edit field to overwrite the current knot location. Also try adjust, which redistributes the current knot sequence.

Use Replicate in List of approximations to save any good knot distribution for later use. Rename the replicated approximation to lstsqr by using Rename. To return to the original approximation, click on its name in List of approximations.

Type splinetool at the command line or, if the tool is already open, click File > Restart.

Select Titanium heat data.

In Approximation method, select Smoothing Spline.

Vary Parameter between 0 and 1, which changes the approximation from the least-squares straight-line approximation to the “natural” cubic spline interpolant.

Vary Tolerance between 0 and some large value, even inf. The approximation changes from the best possible one, the “natural” cubic spline interpolant, to the least-squares straight-line approximation.

As you increase the Parameter value or decrease the Tolerance value, the error decreases. However, a smaller error corresponds to more roughness, as measured by the size of the second derivative. To see this result, select View > Show 2nd Derivative and vary the Parameter and Tolerance values once again.

Modify the weights in the roughness measure to permit a more accurate but less smooth approximation in the peak area while having a smoother, less accurate, approximation away from the peak area.