{"id":156064,"date":"2022-03-26T01:20:32","date_gmt":"2022-03-25T19:50:32","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/numerical-analysis-method-types-computation-and-trapezoid-rule\/"},"modified":"2024-08-03T17:24:57","modified_gmt":"2024-08-03T11:54:57","slug":"numerical-analysis-method-types-computation-and-trapezoid-rule","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/","title":{"rendered":"Numerical Analysis &#8211; Method, Types, Computation and Trapezoid Rule"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#An_Introduction_to_Numerical_Analysis\" title=\"An Introduction to Numerical Analysis\">An Introduction to Numerical Analysis<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Numerical_Method\" title=\"Numerical Method\">Numerical Method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Different_Types_of_Numerical_Methods\" title=\"Different Types of Numerical Methods\">Different Types of Numerical Methods<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#1_The_bisection_method\" title=\"1. The bisection method:\">1. The bisection method:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#2_The_Newton-Raphson_Method\" title=\"2. The Newton-Raphson Method:\">2. The Newton-Raphson Method:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#3_The_secant_method\" title=\"3. The secant method:\">3. The secant method:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#4_The_fixed_point_iteration_method\" title=\"4. The fixed point iteration method:\">4. The fixed point iteration method:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#5_The_conjugate_gradient_method\" title=\"5. The conjugate gradient method: \">5. The conjugate gradient method: <\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Numerical_Methods\" title=\"Numerical Methods\">Numerical Methods<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Numerical_Computation\" title=\"Numerical Computation\">Numerical Computation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Numerical_Computing_Characteristics\" title=\"Numerical Computing Characteristics\">Numerical Computing Characteristics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Numerical_Computing_Processors\" title=\"Numerical Computing Processors\">Numerical Computing Processors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Trapezoidal_Law\" title=\"Trapezoidal Law\">Trapezoidal Law<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Introduction_to_Finite_Element_Method\" title=\"Introduction to Finite Element Method\">Introduction to Finite Element Method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Two_Features_of_the_Fem_are_Mentioned_below\" title=\"Two Features of the Fem are Mentioned below\">Two Features of the Fem are Mentioned below<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Typical_Classes_of_Engineering_Problems_That_Can_be_Solved_Using_Fem_are\" title=\"Typical Classes of Engineering Problems That Can be Solved Using Fem are\">Typical Classes of Engineering Problems That Can be Solved Using Fem are<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/#Finite_Element_Method_MATLAB\" title=\"Finite Element Method MATLAB\">Finite Element Method MATLAB<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"An_Introduction_to_Numerical_Analysis\"><\/span>An Introduction to Numerical Analysis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/maths\/numerical-analysis\/\">Numerical analysis<\/a> is the study of algorithms that can be used to solve mathematical problems that arise in scientific and engineering applications. These problems are often too difficult to solve analytically, so numerical methods must be used.<\/p>\n<p>Numerical analysis has two main goals. The first goal is to develop accurate approximations to the solutions of mathematical problems. The second goal is to develop efficient algorithms that can solve these problems quickly.<\/p>\n<p>Numerical analysis is used in a wide variety of applications, including physics, engineering, and finance.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156063 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/numerical-analysis-method-types-computation-and-trapezoid-rule.jpg\" alt=\"Numerical Analysis - Method, Types, Computation and Trapezoid Rule\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/numerical-analysis-method-types-computation-and-trapezoid-rule.jpg?v=1648237827 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/numerical-analysis-method-types-computation-and-trapezoid-rule-300x212.jpg?v=1648237827 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Numerical_Method\"><\/span>Numerical Method<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>Numerical methods are a class of algorithms that use numerical approximation to solve mathematical problems. They are commonly used in physics, engineering, and mathematics.<\/li>\n<li>Numerical methods can be used to solve problems that are too difficult to solve analytically. They can also be used to obtain a more accurate answer than is possible with analytical methods.<\/li>\n<li>Numerical methods typically use a computer to solve mathematical problems. The computer calculates a series of approximate solutions to the problem, and then selects the best solution.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Different_Types_of_Numerical_Methods\"><\/span>Different Types of Numerical Methods<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>There are many different types of numerical methods. Some of the most common include:<\/p>\n<h3><span class=\"ez-toc-section\" id=\"1_The_bisection_method\"><\/span>1. The bisection method:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>This method uses a line segment to divide a given interval into two equal parts. The method then determines the midpoint of the interval and checks to see if it falls within the given interval. If it does, the given interval is the solution. If it does not, the interval is divided into two new parts and the process is repeated.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2_The_Newton-Raphson_Method\"><\/span>2. The Newton-Raphson Method:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>This method uses the slope of a line tangent to a curve at a given point to find the root of the equation. It is a more efficient version of the bisection method.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3_The_secant_method\"><\/span><span style=\"font-size: 14pt;\">3. The secant method:<\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>This method uses a line segment to divide a given interval into two unequal parts. The method then determines the midpoint of the interval and checks to see if it falls within the given interval. If it does, the given interval is the solution. If it does not, the interval is divided into two new parts and the process is repeated.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4_The_fixed_point_iteration_method\"><\/span>4. The fixed point iteration method:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>This method uses a function to find a fixed point of another function. It is used to solve systems of equations.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5_The_conjugate_gradient_method\"><\/span><span style=\"font-size: 14pt;\">5. The conjugate gradient method: <\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>This method uses the gradient of a function to find the minimum or maximum of the function.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Numerical_Methods\"><\/span>Numerical Methods<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>Numerical methods are a class of mathematical techniques used to solve problems that cannot be solved using analytical methods. Numerical methods rely on the use of approximations and iterative procedures to calculate solutions to mathematical problems.<\/li>\n<li>One of the most common numerical methods is the method of successive approximation. This method involves calculating a solution to a problem using a first approximation, then calculating a new solution using a second approximation that is closer to the true solution. This process is repeated until the solution is sufficiently close to the true solution.<\/li>\n<li>Another common numerical method is the method of iteration. This method involves calculating a solution to a problem using a first approximation, then calculating a new solution using a second approximation that is closer to the true solution. This process is repeated until the solution is sufficiently close to the true solution. However, the solution is not necessarily fixed, and it may be possible to continue iterating to find an even better solution.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Numerical_Computation\"><\/span>Numerical Computation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>Numerical computation is the process of solving mathematical problems using a computer. The computer takes a set of instructions, called a algorithm, and a set of data, and calculates a result.<\/li>\n<li>The most basic type of numerical computation is arithmetic. The computer calculates the result of adding, subtracting, multiplying, and dividing two numbers. This type of computation is used in many applications, from simple tasks like balancing a checkbook to more sophisticated operations like solving a differential equation.<\/li>\n<li>Another common type of numerical computation is solving equations. The computer finds the solutions to equations by systematically trying different values for the variables until it finds a solution that satisfies all the constraints of the equation. This type of computation is used in many scientific and engineering applications.<\/li>\n<li>Numerical computation can also be used to generate graphs of data. The computer takes a set of data points and calculates the points on a graph that correspond to them. This type of computation is used in many scientific and engineering applications.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Numerical_Computing_Characteristics\"><\/span>Numerical Computing Characteristics<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>Numerical computing is a form of computing where mathematical operations are performed on numerical data. These operations are typically performed in a serial manner, meaning that each number is processed one at a time. This type of computing is often used for scientific and engineering applications, where precise calculations are required.<\/li>\n<li>There are several characteristics that are important for numerical computing. One is the accuracy of the results. It is important that the calculations produce results that are as accurate as possible. This can be affected by the type of number representation that is used, as well as the precision of the calculations.<\/li>\n<li>Another important characteristic is the speed of the calculations. The faster that the calculations can be performed, the more efficient the system will be. This is particularly important for systems that are used in real-time applications, such as those found in scientific and engineering fields.<\/li>\n<li>A third important characteristic is the amount of memory that is required. The more memory that is required, the more expensive the system will be. It is important to select a system that has the right amount of memory for the application that is to be used.<\/li>\n<li>Finally, the system must be reliable. The calculations must produce accurate results every time, without fail. This is essential for any system that is used in a critical application.<\/li>\n<li>These are some of the most important characteristics for numerical computing systems. It is important to consider each of these when selecting a system for a particular application.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Numerical_Computing_Processors\"><\/span>Numerical Computing Processors<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>A numerical computing processor is a type of computer processor that is specifically designed to carry out mathematical operations. These processors are used in a variety of settings, including scientific and engineering applications, financial analysis, and gaming.<\/li>\n<li>Numerical computing processors are different from traditional processors in that they are able to carry out a wider range of mathematical operations. In addition, they often include features that make them better suited for scientific and engineering applications, such as high-performance vector processors and floating-point units.<\/li>\n<li>There are a number of different numerical computing processors on the market, including the Intel Xeon Phi, the NVIDIA Tesla, and the AMD FirePro.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Trapezoidal_Law\"><\/span>Trapezoidal Law<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Trapezoidal Rule is a numerical approximation technique used to calculate the area under a curve. The rule is based on the assumption that the curve can be approximated by a series of straight lines (i.e. a trapezoid).<\/p>\n<p>The Trapezoidal Rule can be expressed as follows:<\/p>\n<p>Where:<\/p>\n<ul>\n<li>A is the area under the curve<\/li>\n<li>h is the width of the trapezoid<\/li>\n<li>x i is the value of the function at point i<\/li>\n<li>n is the number of points used to calculate the area<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Finite_Element_Method\"><\/span>Introduction to Finite Element Method<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>A finite element method (FEM), in the context of mathematical modeling and numerical simulation, is a numerical technique for solving problems by approximating them with a finite number of discrete objects.<\/li>\n<li>A finite element model is composed of a discrete set of independent elements that are connected by joints. The behavior of the model is determined by the behavior of the elements and the joints. The finite element method is a numerical technique for solving problems by approximating them with a finite number of discrete objects.<\/li>\n<li>The finite element method is used to solve problems in engineering, physics, and mathematics. In engineering, the finite element method is used to solve problems in structures, fluids, and heat transfer. In physics, the finite element method is used to solve problems in elasticity, fluid dynamics, and thermodynamics. In mathematics, the finite element method is used to solve problems in partial differential equations.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Two_Features_of_the_Fem_are_Mentioned_below\"><\/span>Two Features of the Fem are Mentioned below<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>It is a versatile WordPress theme that can be used for any type of website.<\/li>\n<li>It has a clean and modern design that makes it look great on any device.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Typical_Classes_of_Engineering_Problems_That_Can_be_Solved_Using_Fem_are\"><\/span>Typical Classes of Engineering Problems That Can be Solved Using Fem are<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>There are many types of engineering problems that can be solved using fem. Some of these include problems involving stress and strain analysis, fluid flow analysis, thermal analysis, and static and dynamic analysis. Fem can also be used to model complex geometries and to simulate the behavior of physical systems.<\/p>\n<ul>\n<li>Structural analysis<\/li>\n<li>Fluid dynamics<\/li>\n<li>Electromagnetism<\/li>\n<li>Thermodynamics<\/li>\n<li>Quantum mechanics<\/li>\n<li>Statistical mechanics<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Finite_Element_Method_MATLAB\"><\/span>Finite Element Method MATLAB<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>The finite element method is a powerful technique for solving problems that cannot be solved by other methods. It can be used to solve problems with complex geometries and to solve problems with varying degrees of accuracy.<\/li>\n<li>The finite element method works by breaking a problem into a series of smaller problems. These smaller problems can be solved more easily than the original problem. The solutions to the smaller problems are then combined to solve the original problem.<\/li>\n<li>The finite element method can be used to solve problems in physics, engineering, and mathematics.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>An Introduction to Numerical Analysis Numerical analysis is the study of algorithms that can be used to solve mathematical problems [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Numerical Analysis","_yoast_wpseo_title":"%%title%% %%page%%","_yoast_wpseo_metadesc":"Numerical analysis is the study of algorithms that can be used to solve mathematical problems that arise in scientific applications.","custom_permalink":"maths\/numerical-analysis\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Numerical Analysis - 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