LotusScript: Calculating Square Roots Without Using the SQR Function

Of course, the LotusScript language includes the SQR function for calculating the square root of a given number. The following algorithm is easy to understand and achieves good accuracy after just a few iterations.

The Heron Method (Babylonian Method)

This code calculates the square root of any positive number without using the built-in Sqrt function. Instead, it uses the Heron method (also known as the Babylonian method) – an iterative algorithm that is over 2,000 years old.

The Code


Function MySquareRoot(number As Double) As Double
    ' Square root using the Heron method (Babylonian method)
    ' without using the Sqrt function
    
    Dim estimate As Double
    Dim previous As Double
    Dim precision As Double
    Dim iteration As Integer
    
    If number < 0 Then
        Error 5  ' Invalid input
    End If
    
    If number = 0 Then
        MySquareRoot = 0
        Exit Function
    End If
    
    precision = 0.0000000001   ' 10 decimal places
    estimate = number / 2      ' Starting value: half the input number
    iteration = 0
    
    Do
        previous = estimate
        ' Heron formula: new value = (old value + number / old value) / 2
        estimate = (estimate + number / estimate) / 2
        iteration = iteration + 1
        Print "Iteration " & iteration & ": " & Format$(estimate, "0.0000000000")
    Loop Until Abs(estimate - previous) < precision
    
    MySquareRoot = estimate
End Function

Call Example


Sub Click(Source As Button)
    Dim number As Double
    Dim result As Double
    
    number = 50
    Print "Calculating square root of " & number & " ..."
    Print "---"
    result = MySquareRoot(number)
    Print "---"
    Print "Result:  " & Format$(result, "0.0000000000")
    Print "Check:   " & Format$(result * result, "0.0000000000")
End Sub

How It Works in Detail

The Heron method is based on a simple idea:

1. Choose a Starting Value

We take number / 2 as the first estimate. The starting value does not need to be particularly good; the method converges extremely quickly regardless.

2. Apply the Iteration Formula

In each round, we calculate:

x_{new} = \frac{x_{old} + \frac{number}{x_{old}}}{2}

Why does this work?

If x_old is too large, then number / x_old is too small – and vice versa.

The average of both therefore always lies closer to the true root.

Mathematically, this is Newton's method applied to the equation $f(x) = x^2 - number = 0$ .

3. Termination Condition

As soon as two consecutive estimates differ by less than the chosen precision (0.0000000001), the loop terminates.

Example: Square Root of 50

The true square root of 50 is 7.0710678118…

Iteration	Estimate	Error
1	26.0000000000	~18.93
2	13.9615384615	~6.89
3	8.7731197691	~1.70
4	7.2376726427	~0.17
5	7.0730031544	~0.002
6	7.0710680509	~0.0000002
7	7.0710678119	~0.0000000001

After only 7 iterations, the method has calculated the root to 10 decimal places.

Key Properties

Quadratic Convergence – The accuracy doubles with each step. Even with a poor starting value, rarely more than 10 iterations are needed.

Robustness – The method works for any positive number, no matter how large or small.

Historical – The method is over 2,000 years old and was already used by the Babylonians – hence the name.