Root-finding algorithms | Computer arithmetic algorithms
Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted , , or ) of a real number. Arithmetically, it means given , a procedure for finding a number which when multiplied by itself, yields ; algebraically, it means a procedure for finding the non-negative root of the equation ; geometrically, it means given two line segments, a procedure for constructing their geometric mean. Every real number has two square roots. The principal square root of most numbers is an irrational number with an infinite decimal expansion. As a result, the decimal expansion of any such square root can only be computed to some finite-precision approximation. However, even if we are taking the square root of a perfect square integer, so that the result does have an exact finite representation, the procedure used to compute it may only return a series of increasingly accurate approximations. The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system. The most common analytical methods are iterative and consist of two steps: finding a suitable starting value, followed by iterative refinement until some termination criterion is met. The starting value can be any number, but fewer iterations will be required the closer it is to the final result. The most familiar such method, most suited for programmatic calculation, is Newton's method, which is based on a property of the derivative in the calculus. A few methods like paper-and-pencil synthetic division and series expansion, do not require a starting value. In some applications, an integer square root is required, which is the square root rounded or truncated to the nearest integer (a modified procedure may be employed in this case). The method employed depends on what the result is to be used for (i.e. how accurate it has to be), how much effort one is willing to put into the procedure, and what tools are at hand. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result). Procedures for finding square roots (particularly the square root of 2) have been known since at least the period of ancient Babylon in the 17th century BCE. Heron's method from first century Egypt was the first ascertainable algorithm for computing square root. Modern analytic methods began to be developed after introduction of the Arabic numeral system to western Europe in the early Renaissance. Today, nearly all computing devices have a fast and accurate square root function, either as a programming language construct, a compiler intrinsic or library function, or as a hardware operator, based on one of the described procedures. (Wikipedia).
How to take the square root of three different types of numbers, root(4), root(18)
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number
Explain how to take the root of a number even or odd using prime factorization, root
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number
Learn how to simplify square root of a number when it is multiplied by another number
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist Simplify the Square Root Expressions
Simplify by prime factorization the square root of a prime number
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist Simplify the Square Root Expressions
Use Prime Factorization to Simplify the Square Root of a Number, sqrt(32)
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number
How to take the square root of a number using prime factorization, sqrt(64)
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number
Taking the Square Root of a Decimal
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number
How To Take the Square Root of a Number Using the Identify Element
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number
Lec-15 Solution of a System of Linear Algebraic Equations-Part-5
Lecture series on Numerical Methods and Computation by Prof.S.R.K.Iyengar, Department of Mathematics, IIT Delhi. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist Core - Numerical Methods and Computation
Lecture 2A: Higher-order Procedures
MIT 6.001 Structure and Interpretation of Computer Programs, Spring 2005 Instructor: Harold Abelson, Gerald Jay Sussman, Julie Sussman View the complete course: https://ocw.mit.edu/6-001S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE18841CABEA24090 Higher-order Procedures
From playlist MIT 6.001 Structure and Interpretation, 1986
Lecture 12: Square Roots, Newton's Method
MIT 6.006 Introduction to Algorithms, Fall 2011 View the complete course: http://ocw.mit.edu/6-006F11 Instructor: Srini Devadas License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.006 Introduction to Algorithms, Fall 2011
8. Quasi-Newton-Raphson Methods
MIT 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2015 View the complete course: http://ocw.mit.edu/10-34F15 Instructor: James Swan This lecture continued on the topic of solving nonlinear equations, introducing quasi Newton-Raphson method and Boyden's method. License: Cr
From playlist MIT 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2015
Mod-03 Lec-13 Second Order Linear Equations Continued I
Ordinary Differential Equations and Applications by A. K. Nandakumaran,P. S. Datti & Raju K. George,Department of Mathematics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in.
From playlist IISc Bangalore: Ordinary Differential Equations and Applications | CosmoLearning.org Mathematics
CMPSC/Math 451. March 6, 2015. Newton's method, secant method. Wen Shen
Wen Shen, Penn State University. Lectures are based on my book: "An Introduction to Numerical Computation", published by World Scientific, 2016. See promo video: https://youtu.be/MgS33HcgA_I
From playlist Numerical Computation spring 2015. Wen Shen. Penn State University.
MegaFavNumbers | The magic number and the legendary fast inverse square root hack.
Hi! I'm Rodrigo Aldana. This is my contribution to the #MegaFavNumbers project. This video is based on a presentation I gave some time ago about the fast inverse square root algorithm but now focused on the related magic number 1597463007. I want to make something clear: 1597463007 is not
From playlist MegaFavNumbers
How to solve a quadratic equation by using the square root method with irrational solutions
πLearn how to solve quadratic equations using the square root method. It is important to understand that not all quadratics have to be solved using factoring or quadratic formula. When we only have one variable but it is squared we can just use inverse operations to isolate the variable.
From playlist Solve Quadratic Equations by Factoring
Lecture 1A: Overview and Introduction to Lisp
MIT 6.001 Structure and Interpretation of Computer Programs, Spring 2005 Instructor: Harold Abelson, Gerald Jay Sussman, Julie Sussman View the complete course: https://ocw.mit.edu/6-001S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE18841CABEA24090 Overview and Introductio
From playlist MIT 6.001 Structure and Interpretation, 1986
The Newton Fractal Explained | Deep Dive Maths
A Newton fractal is obtained by iterating Newton's method to find the roots of a complex function. The iconic picture of this fractal is what I call The Newton Fractal, and is generated from the function f(z)=z^3-1, whose roots are the three cube roots of unity. What is the history of th
From playlist Deep Dive Maths
Simplifying the Square Root of a Number Using a Factor Tree, Sqrt(80)
π Learn how to find the square root of a number. To find the square root of a number, we identify whether that number which we want to find its square root is a perfect square. This is done by identifying a number which when raised to the 2nd power gives the number which we want to find it
From playlist How to Simplify the Square Root of a Number