CUBIC_HERMITE_SPLINE

Overview

The CUBIC_HERMITE_SPLINE function performs piecewise cubic interpolation of a 1-D function, matching both function values and first derivatives at each data point. This method is useful for smooth curve fitting in engineering, graphics, and scientific computing, where both the value and slope of the function are known at each sample point. The interpolation is $C^1$ continuous (smooth in both value and first derivative) and is based on the Hermite form of the cubic polynomial:

H(x) = h_{00}(t) y_0 + h_{10}(t) (x_1 - x_0) dydx_0 + h_{01}(t) y_1 + h_{11}(t) (x_1 - x_0) dydx_1

where $h_{00}$ , $h_{10}$ , $h_{01}$ , and $h_{11}$ are Hermite basis functions, $y_0$ , $y_1$ are function values, and $dydx_0$ , $dydx_1$ are derivatives at the endpoints of each interval. For more details, see the SciPy CubicHermiteSpline documentation and the Wikipedia article .

This function is a simplified wrapper for the SciPy CubicHermiteSpline class, supporting only 1D arrays and not exposing advanced options like axis or extrapolate.

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=CUBIC_HERMITE_SPLINE(x, y, dydx, x_new)

x (2D list, required): Table of one column, strictly increasing x-coordinates of data points.
y (2D list, required): Table of one column, function values at each x.
dydx (2D list, required): Table of one column, first derivatives at each x.
x_new (2D list, required): Table of one column, x-coordinates where interpolation is desired.

The function returns a 2D list (column vector) of interpolated values at each x_new, or an error message (list of one string) if the input is invalid.

Examples

Example 1: Interpolate a Sine Curve

Inputs:

x	y	dydx	x_new
0	0.0	1.0	0.5
1	0.84147	0.5403	0.75
2	0.909297	-0.4161	1.5
3	0.14112	-0.98999	2.5

Excel formula:


=CUBIC_HERMITE_SPLINE({0;1;2;3}, {0;0.84147;0.909297;0.14112}, {1;0.5403;-0.4161;-0.98999}, {0.5;0.75;1.5;2.5})

Expected output:

Result
0.4782
0.6809
0.9949
0.5969

Example 2: Linear Data

Inputs:

x	y	dydx	x_new
0	0	1	0.5
1	1	1	1.5
2	2	1	2.0

Excel formula:


=CUBIC_HERMITE_SPLINE({0;1;2}, {0;1;2}, {1;1;1}, {0.5;1.5;2.0})

Expected output:

Result
0.5
1.5
2.0

Example 3: Quadratic Data

Inputs:

x	y	dydx	x_new
0	0	0	0.5
1	1	2	1.5
2	4	4	2.0

Excel formula:


=CUBIC_HERMITE_SPLINE({0;1;2}, {0;1;4}, {0;2;4}, {0.5;1.5;2.0})

Expected output:

Result
0.25
2.25
4.0

Example 4: Flat Data

Inputs:

x	y	x_new
0	2	0.5
1	2	1.0
2	2	1.5

Excel formula:


=CUBIC_HERMITE_SPLINE({0;1;2}, {2;2;2}, {0;0;0}, {0.5;1.0;1.5})

Expected output:

Result
2.0
2.0
2.0

Python Code


from scipy.interpolate import CubicHermiteSpline as scipy_cubic_hermite_spline
 
def cubic_hermite_spline(x, y, dydx, x_new):
    """
    Piecewise cubic interpolation matching values and first derivatives at each point.
 
    Args:
        x: 2D list, strictly increasing x-coordinates of data points.
        y: 2D list, function values at each x.
        dydx: 2D list, first derivatives at each x.
        x_new: 2D list, x-coordinates where interpolation is desired.
 
    Returns:
        2D list (column vector) of interpolated values at each x_new, or a list of one string error message if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input shapes
    try:
        x_flat = [float(row[0]) for row in x]
        y_flat = [float(row[0]) for row in y]
        dydx_flat = [float(row[0]) for row in dydx]
        x_new_flat = [float(row[0]) for row in x_new]
    except Exception:
        return [["Invalid input: all arguments must be 2D lists of numbers."]]
    if not (len(x_flat) == len(y_flat) == len(dydx_flat)):
        return [["Invalid input: x, y, dydx must have the same length."]]
    if len(x_flat) < 2:
        return [["Invalid input: at least two data points required."]]
    if any(x2 <= x1 for x1, x2 in zip(x_flat, x_flat[1:])):
        return [["Invalid input: x must be strictly increasing."]]
    try:
        spline = scipy_cubic_hermite_spline(x_flat, y_flat, dydx_flat)
        y_new = spline(x_new_flat)
        return [[round(float(val), 4)] for val in y_new]
    except Exception as e:
        return [[f"scipy.interpolate.CubicHermiteSpline error: {e}"]]

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CUBIC_HERMITE_SPLINE

Overview

Usage

Examples

Python Code

Live Notebook

Live Demo