Why Newton’s Method Converges Quadratically: The Hidden Connection Between Linear, Quadratic, and Higher-Order Approximations

Introduction

One of the most interesting features of Newton’s method is that it uses a linear approximation to achieve quadratic convergence.

At first, this might seem surprising.

Newton’s method estimates the root of a function by drawing a tangent line and finding where that tangent intersects the x-axis. A tangent line is a linear approximation, so why does the method achieve such a high rate of accuracy?

The answer lies in understanding what the linear approximation fails to capture.

The difference between the actual function and its tangent-line approximation is primarily governed by the quadratic term near the point of approximation.

This quadratic contribution explains why Newton’s method can reduce the error so dramatically with each successive iteration.

Let’s develop this connection step by step.

1. Begin with the linear approximation

Suppose a function f(x) is being approximated near x = a.

The tangent-line approximation is:

Here:

• f(a) represents the starting value.
• f'(a) represents the slope at the starting point.
• h represents the horizontal change.

Geometrically, this approximation assumes that the function continues along its tangent line.

However, most functions are curved rather than perfectly straight.

Consequently, the tangent line cannot capture the complete behaviour of the function.

It captures the starting value and the instantaneous rate of change, but ignores curvature.

2. What does the quadratic approximation add?

To account for curvature, we introduce another term:

The additional term is:

This term represents the leading contribution of curvature that the linear approximation ignores.

Therefore, the difference between the quadratic and linear approximations is approximately:

This reveals something important.

The difference depends on the square of the horizontal change.

For example, if the horizontal change is reduced by half, the quadratic contribution becomes approximately one-quarter of its previous value.

This is the beginning of the connection with Newton’s method.

3. Does the quadratic approximation capture the complete error?

Not entirely.

The quadratic approximation is more accurate than the linear approximation, but it still ignores higher-order contributions.

A more complete expansion is:

The successive terms represent additional features of the function.

Approximation	Information captured
Linear	Starting value and slope
Quadratic	Starting value, slope and curvature
Cubic	Also captures the next-order variation
Higher-order	Captures progressively finer behaviour

Thus, the quadratic term does not represent the entire difference between the actual function and its tangent line.

It represents the leading contribution to that difference, provided the second derivative is nonzero.

The higher-order terms contribute as well, but become progressively smaller near the approximation point.

4. Now consider Newton’s method

Newton’s method uses the tangent line to estimate the location of a root.

Suppose the current approximation is x_n.

The tangent-line equation is:

To estimate the root, we set y equal to zero:

Rearranging:

This is the familiar Newton iteration formula.

Notice that Newton’s method relies entirely on the linear approximation to determine the next position.

But the actual function does not necessarily reach zero at the same point as its tangent line.

The difference between these two predictions determines the error remaining after the Newton step.

5. Why does the remaining error become quadratic?

Suppose the actual root is r.

The current error is:

Therefore:

Since r is the actual root:

We can expand the function around r:

This is the quadratic approximation. Higher-order terms have temporarily been omitted.

Similarly, the derivative can be approximated as:

Newton’s method gives:

Substituting the approximations and simplifying gives the leading-order result:

This is the fundamental explanation of quadratic convergence.

The new error is approximately proportional to the square of the previous error.

In other words, Newton’s method does not simply reduce the error by a fixed percentage. As the approximation approaches the root, the error can decrease much more rapidly.

6. An illustration of quadratic convergence

Suppose the initial error is 0.1 and, for simplicity, the error approximately follows:

The successive errors become:

Iteration	Error
Initial	0.1
First	0.01
Second	0.0001
Third	0.00000001

Notice how quickly the error disappears.

The number of correct decimal places approximately doubles with each iteration once the method enters its quadratic convergence region.

This is why Newton’s method is particularly powerful when the initial approximation is sufficiently close to a simple root.

7. The deeper geometric interpretation

We can now connect the entire idea geometrically.

Imagine three different representations of the same function near a root:

The tangent line

It captures the starting height and slope but ignores curvature.

The quadratic approximation

It follows the tangent line while also accounting for the leading curvature contribution.

The actual function

It follows the quadratic approximation more closely near the root but may still differ because of higher-order contributions.

When Newton’s method uses the tangent line to predict the root, it does not account for the curvature of the actual function.

The resulting discrepancy is primarily governed by the quadratic term.

As the current approximation gets closer to the actual root, the horizontal distance becomes smaller. Since the leading curvature contribution depends on the square of that distance, the remaining error decreases quadratically.

8. The most important insight

There is a beautiful relationship between the approximation method and the convergence rate:

• Linear approximation provides the direction and distance of the Newton step.
• The quadratic term explains the leading error left behind by that step.
• Higher-order terms explain the smaller corrections that the quadratic approximation itself does not capture.

This leads to an interesting mathematical observation:

Newton’s method uses a linear approximation, but its convergence rate is governed by the leading error that the linear approximation fails to capture.

That leading error is quadratic.

Consequently, although Newton’s method does not use a quadratic approximation to calculate each new point, its convergence behaviour is fundamentally connected to the difference between linear and quadratic approximations.

Conclusion

The power of Newton’s method comes from the relationship between a tangent line and the actual curvature of a function.

A tangent line is not a perfect representation of a curved function. However, sufficiently close to a smooth root, the difference between the two is dominated by a quadratic contribution.

Newton’s method repeatedly corrects its estimate using the tangent line. The remaining error is therefore approximately proportional to the square of the previous error.

This is the geometric and mathematical reason behind quadratic convergence.

The deeper lesson is that sometimes the most important feature of an approximation is not what it captures, but what it leaves behind.