CTB_HORNER

This function evaluates a polynomial at a specified scalar input using Horner’s method, which is a numerically efficient nested form for polynomial evaluation.

For coefficients a_0, a_1, \dots, a_n, Horner’s form evaluates a_0x^n + a_1x^{n-1} + \dots + a_n with reduced arithmetic operations and good computational efficiency.

Excel Usage

=CTB_HORNER(coeffs, x)

coeffs (list[list], required): Polynomial coefficients ordered from highest power to constant (-).
x (float, required): Point at which to evaluate the polynomial (-).

Returns (float): Polynomial value at the specified point, or an error message if invalid.

Example 1: Linear polynomial

Inputs:

coeffs		x
1	3	2

Excel formula:

=CTB_HORNER({1,3}, 2)

Expected output:

5

Example 2: Quadratic polynomial

Inputs:

coeffs			x
1	0	-1	3

Excel formula:

=CTB_HORNER({1,0,-1}, 3)

Expected output:

8

Example 3: Constant polynomial

Inputs:

coeffs	x
2.5	10

Excel formula:

=CTB_HORNER({2.5}, 10)

Expected output:

2.5

Example 4: Coefficients provided as a column

Inputs:

coeffs	x
1	1.5
2
3

Excel formula:

=CTB_HORNER({1;2;3}, 1.5)

Expected output:

8.25

Python Code

Show Code

from ht.conv_tube_bank import horner as ht_horner

def ctb_horner(coeffs, x):
    """
    Evaluate a polynomial using Horner's method.

    See: https://ht.readthedocs.io/en/latest/ht.conv_tube_bank.html

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

    Args:
        coeffs (list[list]): Polynomial coefficients ordered from highest power to constant (-).
        x (float): Point at which to evaluate the polynomial (-).

    Returns:
        float: Polynomial value at the specified point, or an error message if invalid.
    """
    try:
        def to2d(x_value):
            return [[x_value]] if not isinstance(x_value, list) else x_value

        coeffs = to2d(coeffs)

        if not all(isinstance(row, list) for row in coeffs):
            return "Error: coeffs must be a 2D list"

        flat = []
        for row in coeffs:
            for val in row:
                try:
                    flat.append(float(val))
                except (TypeError, ValueError):
                    return "Error: coeffs must be numeric"

        if not flat:
            return "Error: coeffs must contain at least one value"

        return ht_horner(flat, x)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

coeffs *

Polynomial coefficients ordered from highest power to constant (-).

x *

Point at which to evaluate the polynomial (-).