# Which kind of error occurs in series approximation?

## Which kind of error occurs in series approximation?

The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value).

What symbol do we use for approximation?

The approximation sign “≈” I use for decimal approximations with tilde “~” being a rougher approximation.

### What is erf of infinity?

The error function at infinity is exactly 1 (see Gaussian integral).

What is the error function of infinity?

## What is erf infinity?

The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf z approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

What is approximation error in machine learning?

The approximation error is the error implied by the choice of function class and is defined as the difference in risk obtained by the best model within the function class and the optimal model.

### How is the finite series approximation algorithm used in mathematics?

Notice that the finite series needs to have around 500th order polynomial before it is accurate for double precision floating point numbers. This approximation relies on a magic number initial approximation but is then refined using Newton’s method. This algorithm is found in many high end mathematic programs such as Matlab.

What is the difference between error function and inverse error function?

The error function (see Figure 1) is the result of integrating a normalized Gaussian function (Normal Distribution). The error function, is defined as: Figure 1: The error function. The inverse error function (see Figure 2), , needs to satisfy: Figure 2: The inverse error function. It ranges from .

## Is there a closed form expression of the inverse error function?

Unfortunately, there is no closed form expression (evaluates in a finite number of operations) of the inverse error function. Instead the inverse error function is typically defined by an infinite series expansion: Why is it important? At first, second and even third glance it is very hard to see the importance of the inverse error function.

What is Newton’s method of approximation?

This approximation relies on a magic number initial approximation but is then refined using Newton’s method. This algorithm is found in many high end mathematic programs such as Matlab. The one drawback of this method is that it requires multiple calls to the error function slowing down the calculation.