L_PPDER

=L_PPDER(pp_array)

Description

L_PPDER returns the derivative of a piecewise polynomial as another piecewise polynomial. Each derivative of a polynomial reduces the degree by one, so the pp_array returned by L_PPDER(pp_array) will have one less column of coefficients.

The derivative of a polynomial can be calculated using the L_POLYDER function, so finding the derivative of an entire piecewise polynomial is simply a matter of finding the derivative of each individual segment and assembling a new piecewise polynomial structure.

pp_array structure: A piecewise polynomial is defined within the bounds [x₁,x_N] where the first column of the data structure array is the column vector of N break points. The other columns are the coefficients for each polynomial segment. The following example is a piecewise polynomial data structure described within the L_PPVAL function documentation.

L_PPDER Example

The following example graphs the first derivative of the piecewise polynomial shown above. Note that in this case the first derivative is not continuous.

=LET(
    pp_array, HSTACK({0;4;10;15},{0,2,-1,1,10; 0,0,-5,0,126; -1,5,1,4,-54}),
    xs, L_LINSPACE(0,15,100),
    dydx, L_PPVAL(L_PPDER(pp_array),xs)
)
Result: (see graph below)

See the example below showing the use of L_PPDER with cubic splines.

Lambda Formula

This code for using L_PPDER in Excel is provided under the License as part of the LAMBDA Library, but to use just this function, you may copy the following code directly into your spreadsheet.

/**
* Returns the derivative of a Piecewise Polynomial as another piecewise polynomial
*/
L_PPDER = LAMBDA(pp_array,
LET(doc,"https://www.vertex42.com/lambda/ppder.html",
    breaks,INDEX(pp_array,0,1),
    pieces,ROWS(breaks)-1,
    coeffs,DROP(pp_array,-1,1),
    newcoeffs,REDUCE(0,SEQUENCE(pieces),LAMBDA(acc,i,LET(
        deriv,L_POLYDER(CHOOSEROWS(coeffs,i)),
        IF(i=1,deriv,VSTACK(acc,deriv))
    ))),
    MAP(HSTACK(breaks,newcoeffs),LAMBDA(cell,IFERROR(cell,"")))
));

Name: L_PPDER
Description: Returns the Derivative of a Piecewise Polynomial
Arguments: pp_array
Function:
[in the works]

L_PPDER Examples

Example: First and Second Derivative of a Cubic Spline

The L_CSPLINE function creates a cubic spline that has a continuous first derivative. Consider the following example from the L_CSPLINE documentation:

Example control points and resulting cubic piecewise polynomial array

Comparison of CSPLINE with the "Smoothed Line" option in Excel

Using L_LINSPACE, L_PPDER and L_PPVAL like above, we can graph the first and second derivatives of this example cubic spline. We can see that it is indeed C¹ continuous (first derivative is continuous), but it is not C² continuous (second derivative is not continuous).

Example: First and Second Derivative of a Natural Cubic Spline

A Natural Cubic Spline is defined to be C² continuous. The following example is a natural cubic spline fit to the same points as the example above. Note that the second derivative is continuous in this case, and is zero at the end points.

=LET(
    x_cpts, {2;3;4;5;6;7;8;9;10;11;12},
    y_cpts, {4;4;2;3;1;1.5;5;2;2;4.5;4},
    pp_spline, L_NSPLINE(x_cpts,y_cpts),
    x_interp, L_LINSPACE(2,12,101),
    y_interp, L_PPVAL(pp_spline,x_interp),
    dy_interp, L_PPVAL( L_PPDER(pp_spline),x_interp),
    d2y_interp, L_PPVAL( L_PPDER(L_PPDER(pp_spline)),x_interp),
    HSTACK(x_interp,y_interp,dy_interp,d2y_interp)
)

Result: (see graph below)

See Also

PPVAL, CSPLINE, PPINT, POLYDER