Introduction

Numerical values are fundamental for quantifying and measuring quantities in fields like math, physics, engineering, and finance. They represent numbers in mathematics, physical quantities in science and engineering, and financial data. Various methods, such as Newton's method, are employed to compute or approximate these values, each with its pros and cons. Analyzing these numerical values with mathematical and statistical tools helps gain insights and make informed decisions across different disciplines.

Newton-Raphson Method:

A numerical technique used to find approximate solutions to real-valued functions. It is commonly used to find the roots of an equation.

Lagrange Interpolation:

A method for constructing a polynomial that passes through a given set of data points. It is used for function approximation.

Secant Method:

An iterative method for finding the roots of a function. It approximates the derivative using finite differences.

Newton Interpolation and Divided Differences:

A method to interpolate a set of data points using a polynomial. It involves calculating divided differences to construct the interpolating polynomial.

Cubic Spline:

A piecewise interpolation method that uses cubic polynomials to smoothly connect data points. It's commonly used in computer graphics and numerical analysis.

Numerical Differentiation:

A process of estimating the derivative of a function at a given point using finite difference formulas.

Richardson Extrapolation:

A method for improving the accuracy of numerical approximations by taking weighted averages of successive approximations with different step sizes.

Numerical Integration:

The process of approximating definite integrals of functions. It's used when analytical integration is not possible.

Composite Numerical Integration:

A technique that breaks down an interval into smaller subintervals and applies a numerical integration method within each subinterval.

Romberg Integration:

A method for improving the accuracy of numerical integration by repeatedly applying the Trapezoidal Rule and extrapolation.

Gaussian Quadrature:

A highly accurate method for numerical integration that uses specific weighted nodes and weights to approximate integrals.

Euler's Method:

A simple numerical technique for solving ordinary differential equations (ODEs) by approximating the solution step by step.

Modified Euler's Method:

An improvement of Euler's Method that uses a weighted average of the function values at the endpoints of the step to increase accuracy.

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