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Related Concept Videos

Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Piecewise-Defined Functions01:28

Piecewise-Defined Functions

Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function:  uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value function, given...
Types of Functions III01:28

Types of Functions III

Logarithmic and piecewise functions play central roles in mathematical modeling, particularly when capturing nonlinear or segmented behaviors in real-world phenomena. Although these functions differ fundamentally in structure and application, both serve to represent complex relationships in simplified mathematical terms.A logarithmic function is defined as the inverse of an exponential function, expressed as These functions grow quickly for small values of x but slow down as x increases,...
Approximate Integration01:24

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

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Related Experiment Videos

Efficient algorithms for function approximation with piecewise linear sigmoidal networks.

D R Hush1, B Horne

  • 1Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87131, USA.

IEEE Transactions on Neural Networks
|February 8, 2008
PubMed
Summary
This summary is machine-generated.

This study introduces an efficient algorithm for function approximation using piecewise linear sigmoidal nodes. The novel method offers finite step convergence and robust numerical implementation for neural networks.

Related Experiment Videos

Area of Science:

  • Computational Science
  • Machine Learning
  • Artificial Intelligence

Background:

  • Function approximation is a core problem in machine learning.
  • Existing methods like backpropagation have limitations.

Purpose of the Study:

  • To develop a computationally efficient algorithm for function approximation.
  • To introduce a novel method for constructing neural networks with piecewise linear sigmoidal nodes.

Main Methods:

  • A one hidden layer network is built node by node using residual fitting.
  • Individual node fitting is achieved by solving a sequence of quadratic programming problems.

Main Results:

  • The algorithm demonstrates finite step convergence and independence from initial conditions.
  • It exhibits good scaling properties and robust numerical implementation.
  • Empirical results validate the algorithm's characteristics.

Conclusions:

  • The proposed algorithm offers significant advantages over derivative-based methods.
  • It provides an efficient and robust approach to function approximation with neural networks.