Numerical Computation #

This section discusses numerical computation, with a focus on scientific and technical calculations.

Numerical integration

Numerical integration # This section introduces methods for performing definite or improper integrals using computers. Gaussian quadrature (derivation) The Gaussian quadrature method is an optimal numerical integration technique for integrands of the form (polynomial) × (specific weight function). It allows for the exact integration of integrands of the form (2N-1 degree polynomial) × (weight function) by referencing only N points, despite the restriction on the positions of the points used for integration.

Initial value problem

Initial value problem # This section explains numerical methods for solving initial value problems. Implicit Runge-Kutta Method Implicit methods have the advantage of higher stability compared to explicit methods. They are particularly effective for ‘stiff’ problems where the solution changes rapidly or when long-term integration is required. However, these methods tend to have higher computational costs. The Implicit Runge-Kutta method is one such implicit method. It is known as an efficient solution for stiff problems, along with multistep methods and predictor-corrector methods.

Arithmetic Operations

Arithmetic Operations # This section covers efficient methods for common calculations frequently encountered in scientific and technical computing, as well as computations specific to floating-point arithmetic. Square Root of Complex Numbers Square Root of Complex Numbers # This section demonstrates specific methods for calculating the square root of a complex number. The result can be expressed as follows: $$ \begin{eqnarray} \sqrt{a+ib} = \left\{ \begin{aligned} &\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}}+i~\text{sgn}(b)\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}} &&(-\pi\lt\text{arg}(a+ib)\lt\pi)\\ &-\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}}+i~\text{sgn}(b)\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}} &&\left(\substack{-2\pi\lt\text{arg}(a+ib)\lt -\pi \\ \pi\lt\text{arg}(a+ib)\lt 2\pi}\right) \end{aligned} \right.