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Simpson's rule is a very powerful tool to perform numerical integration. To maximise accuracy while requiring fewer divisions, Simpson's rules calculate the integrals using weighting factors. The trapezoidal rule only considers two points, π₯π and π₯π+1, to estimate a trapezoid's area, but Simpson's rules use more than two points (or many strips) in each round (refer figure shown below). The values of π(π₯) at the various points are updated by weighing various criteria in order to decrease the error Simpsonβs 1/3 Rule This approach calculates the area of two strips simultaneously, so three different values of "x" are taken ... Read More
Class variables also known as static variables are declared with the static keyword in a class, but outside a method, constructor or a block. There would only be one copy of each class variable per class, regardless of how many objects are created from it. Static variables are rarely used other than being declared as constants. Constants are variables that are declared as public/private, final, and static. Constant variables never change from ... Read More
Definition and Usage In PHP, Object is a compound data type (along with arrays). Values of more than one types can be stored together in a single variable. Object is an instance of either a built-in or user defined class. In addition to properties, class defines functionality associated with data. Primary (scalar) variables, arrays and other objects can be cast to object data type using casting operator. PHP provides stdClass as a generic empty class which is useful for adding properties dynamically and casting. Syntax To declare an object of a class we need to use new statement class myclass ... Read More
In C language, header files contain the set of predefined standard library functions. The "#include" preprocessing directive is used to include the header files with ".h" extension in the program. Here is the table that displays some of the header files in C language, Sr.No. Header Files & Description 1 stdio.hInput/Output ... Read More
An array is a linear sequential data structure to hold homogeneous data in consecutive memory locations. Like other data structures, an array also must-have features to insert, delete, traverse and update elements in some efficient way. In C++, our arrays are static. There are a few dynamic array structures also available in C++. For a static array, there may be a Z number of elements that can be stored inside that array. And till now we have n elements into it. In this article, we will see how to insert an element at the end of an array (which is ... Read More
In this tutorial, we will see how to model 2D heat conduction equation using Python. A 2D, steady, heat conduction equation with heat generation can be written in Cartesian coordinates as follows β $$\mathrm{\triangledown^{2} T \: + \: \frac{q_{g}}{k} \: = \: \frac{\partial^{2}T}{\partial x^{2}} \: + \: \frac{\partial^{2}T}{\partial y^{2}} \: + \: \frac{q_{g}}{k} \: = \: 0 \:\:\dotso\dotso (1)}$$ This has to be discretized to obtain a finite difference equation. Let us consider a rectangular grid as shown below. The index π runs vertically i.e. row wise whereas the index π runs horizontally i.e. column wise. Any inside node ... Read More
The Taylor Table method is a very efficient and elegant method of obtaining a finite difference scheme for a particular derivative considering a specific stencil size. To understand it one should be very much clear about what is a stencil. Suppose one wants to evaluate $\mathrm{\frac{d^{2}f}{dx^{2}}}$ then in finite difference method the starting point is the Taylor series. Consider the figure shown below for a better understanding of the method. The Taylor series expansion at the point $\mathrm{x_{i} \: + \: h}$ will be: $$\mathrm{f(x_{i} \: + \: h) \: = \: f(x_{i}) \: + \: hf'(x_{i}) \: + ... Read More
Introduction In this tutorial, we will understand different types of tries and their uses. Tries are tree-like data structures that are mostly used for operations like string searching. There are various types of trie and they are used as per the task requirement. Generally, trie tries are of three types: Standard trie, compressed trie, and suffix trie. We elaborate on the meaning of each type of trie. What is Trie A trie is a sorted binary tree also called a digital tree or a prefix tree. It has nodes that are used to store data or alphabets. Each node can ... Read More
Introduction In this tutorial, we implement an approach to sort a string on the basis of the number of matchsticks required for its representation. In this approach, we use N numbers of matchsticks and sort an array. The array can contain numbers, words, or both. Matchsticks are used to arrange them in the shape of a particular number or character. Demonstration 1 Input = Arr = ["1", "3", "4"] Output = The sorted array is 1 4 3 Explanation In the above input array, the array elements are 1, 3, and 4 Number of matchsticks required for ... Read More
Introduction This tutorial deals with the problem of sorting an array alphabetically while each number is converted into words. To translate numbers into words means changing numbers to their number names. Like, 65 is sixty-five. Here, we consider a numerical array, convert all array elements to words and arrange them alphabetically. Print the sorted array elements after converting words to their respective numbers. Demonstration 1 Input = Arr = {13, 1, 6, 7} Output = 1 7 6 13 Explanation The input array elements are 13, 1, 6, 7 The output with alphabetically sorted elements is 1 7 ... Read More