Wed. Jul 6th, 2022

# Q1. What is numerical equation and how does it help us?

Jun 4, 2022

## Q2. What are some of the traditional methods for solving a given equation? (keywords: numerical equation, numerical solver, computer algebra system)

A numerical equation is a mathematical equation that has been converted to a numeric form. A numerical solver is software or hardware designed to perform computations and solve equations. Computer algebra systems are used, for example, in symbolic mathematics. In the solution of computer algebra problems that involve polynomials, one often uses “symbolic computation” which is easy to do with a CAS but difficult to do with a computer algebra system (CAS) without explicitly converting polynomials to their numeric form.
Various methods are used to numerically solve equations. A common method consists of rearranging the equation and then performing an appropriate mathematical operation. Such operations may include raising a term to a power, integrating, or taking the derivative of a function.
For example:
This is almost always done by using Newton’s method for finding roots of nonlinear equations, which can be iterated to convergence.
Many numerical methods can be formulated as iterative algorithms that solve a sequence of nonlinear equations (often polynomial equations) at each iteration. For example, Newton’s method is an iterative algorithm for approximating the solution to a nonlinear equation.

## Q3. Can you share a brief explanation of the advantages and disadvantages of using MATLAB for numerical solutions to equations?

A numerical solver is software or hardware designed to perform computations and solve equations. There are many different kinds of numerical solver software including: deterministic methods, stochastic methods, and metaheuristic methods. The main advantage of using Matlab/Octave as a numerical solver is that it is a multi-purpose computational tool (with built-in plotting capabilities). The disadvantage of using Matlab for numerical solutions to equations is that it is quite computationally expensive and complex to learn how to use.
Some common problems that can be solved with MATLAB are: approximating the solution to a non-linear equation, finding the root of a quadratic equation, finding the inverses of polynomials , solving systems of linear equations, numerical differentiation, extrapolating data to infinity, performing error minimization.
The purpose of this article is to create a well-rounded understanding of the type of problems that can be solved with MATLAB. This should provide an in-depth understanding of the capabilities and limitations of MATLAB as well as to provide sufficient background knowledge to have a basic understanding on how to solve equations. This article will describe the advantages and disadvantages, methods, time analysis and examples to solve equations with MATLAB.
This article will cover both linear (single variable) equations and non-linear (multiple variables) equations. The reader should be familiar with the idea of an initial value problem (IVP) and single-variable root finding.

## Q4. Describe briefly what you mean when you use MATLAB for Computational Mathematics Research in Engineering/Science/Technology/Theology and Education etc.

Matlab is one of several computer languages used in computational mathematics and numerical analysis. Its capabilities include linear algebra, interpolation and approximation, data processing, polynomial and rational functions, transcendental functions, numerical differentiation and integration, matrix operations such as addition and multiplication, vector operations such as element-wise vector additions or scalar-vector products. These capabilities are often used in engineering and science and technology, but also in Theology and Education.
Matlab is one of several computer languages used in computational mathematics and numerical analysis. Its capabilities include linear algebra, interpolation and approximation, data processing, polynomial and rational functions, transcendental functions, numerical differentiation and integration, matrix operations such as addition and multiplication, vector operations such as element-wise vector additions or scalar-vector products. These capabilities are often used in engineering and science and technology, but also in Theology and Education.
“Matlab | Mathematica” : http://www.mathworks.com/products/matlab-help
{{What is Matlab?:http://www.mathworks.com/products/matlab-help/#whatis}}
There are a number of GUI programs that provide similar functionality, such as:
“Mathematica | Wolfram Research” : http://www.wolfram.com/?software=Mathematica
“Python | Python Tools for Education” : http://www.python-ideas.org/
{{Python|http://www.python-ideas.org/}}

## Q5. What is MATLAB notation used for when solving equations? How do we use it to solve problems in a more efficient way?

Matlab notations are used to solve problems in a more efficient way. Matlab notations include: variables, functions and plots. These notations are used by engineers and scientists to analyze problems before they can solve them. Matlab uses standard scientific notation, as defined by the IEEE standard for binary floating-point arithmetic (IEEE 754-1985). Also, there is a button in the command window toolbar and in the MATLAB editor (called Edit/Run, on the main menu bar) is a button called System Info, with the submenu Items/System Information. This submenu allows one to obtain information about the computer system and its versions and capabilities.
The following example illustrates the use of standard scientific notation (and two other notations). Suppose that we have The chief engineer is in the hallway and he wants to know how many people are in the building. He also wants to know how many of them are engineers. The following is a possible solution:
\begin{align*} N &= \frac{5}{3} \\ E &= \frac{1}{0.7} \\ N &= E \times 100 &\times 400 \\ 200 &= 1.64 \\ \hline N > 100 \text{ and } N > 50 \\ \hline \end{align*}
The chief engineer wants to know the total number of engineers in the building, so he multiplies the number of people by 0.7 times 100 and divides by 400. The result of this multiplication is the total number of engineers in the building.

## How Do We Solve Equations with Numbers?: Gradient Descent Method

Differential equations are solved by first finding a solution to an equation involving first-order differential equations of the form: $$y’=f(x,y)$$ where $$f(x,y)$$ is a function of two variables and $$y$$ is a function of one variable. If $$f(x,y)=0$$, then the solution to this second order differential equation is also zero. This method involves going through all possible values for the first variable $$x$$ and for each value for $$x$$, finding the value of the second variable $$\Delta y$$ that satisfies this equation. This can be done by solving the equations $$y’=0$$ and $$f'(x,y)=0$$, and seeing which values satisfy these equations.