Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics. Such problems originate generally from real-world applications of algebra, geometry and calculus, and they involve variables which vary continuously; these problems occur throughout the natural sciences, social sciences, engineering, medicine, and business. During the past half-century, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science and engineering, and numerical analysis of increasing sophistication has been needed to solve these more detailed mathematical models of the world. The formal academic area of numerical analysis varies from quite theoretical mathematical studies to computer science issues .With the growth in importance of using computers to carry out numerical procedures in solving mathematical models of the world, an area known as scientific computing or computational science has taken shape during the 1980s and 1990s. This area looks at the use of numerical analysis from a computer science perspective. It is concerned with using the most powerful tools of numerical analysis, computer graphics, symbolic mathematical computations, and graphical user interfaces to make it easier for a user to set up, solve, and interpret complicated mathematical models of the real world.
Historical background.: Numerical algorithms are almost as old as human civilization. The Rhind Papyrus (˜1650 BC) of ancient Egypt describes a rootfinding method for solving a simple equation; . Archimedes of Syracuse (287212 BC) created much new mathematics, including the “method of exhaustion” for calculating lengths, areas, and volumes of geometric figures. When used as a method to find approximations, it is in much the spirit of modern numerical integration; and it was an important precursor to the development of the calculus by Isaac Newton and Gottfried Leibnitz. A major impetus to developing numerical procedures was the invention of the calculus by Newton and Leibnitz, as this led to accurate mathematical models for physical reality, first in the physical sciences and eventually in the other sciences, engineering, medicine, and business. These mathematical models cannot usually be solved explicitly, and numerical methods to obtain approximate solutions are needed. Another important aspect of the development of numerical methods was the creation of logarithms by Napier (1614) and others, giving a much simpler manner of carrying out the arithmetic operations of multiplication, division, and exponentiation. Newton created a number of numerical methods for solving a variety of problems, and his name is attached today to generalizations of his original ideas. Of special note is his work on rootfinding and polynomial interpolation. Following Newton, many of the giants of mathematics of the 18th and 19th centuries made major contributions to the numerical solution of mathematical problems. Foremost among these are Leonhard Euler(1707-1783), Joseph-Louis Lagrange(1736-1813), and Karl Friedrich Gauss (1777-1855). Up to the late 1800’s, it appears that most mathematicians were quite broad in their interests, and many of them were interested in and contributed to numerical analysis. For a general history of numerical analysis up to 1900.
Historical background.: Numerical algorithms are almost as old as human civilization. The Rhind Papyrus (˜1650 BC) of ancient Egypt describes a rootfinding method for solving a simple equation; . Archimedes of Syracuse (287212 BC) created much new mathematics, including the “method of exhaustion” for calculating lengths, areas, and volumes of geometric figures. When used as a method to find approximations, it is in much the spirit of modern numerical integration; and it was an important precursor to the development of the calculus by Isaac Newton and Gottfried Leibnitz. A major impetus to developing numerical procedures was the invention of the calculus by Newton and Leibnitz, as this led to accurate mathematical models for physical reality, first in the physical sciences and eventually in the other sciences, engineering, medicine, and business. These mathematical models cannot usually be solved explicitly, and numerical methods to obtain approximate solutions are needed. Another important aspect of the development of numerical methods was the creation of logarithms by Napier (1614) and others, giving a much simpler manner of carrying out the arithmetic operations of multiplication, division, and exponentiation. Newton created a number of numerical methods for solving a variety of problems, and his name is attached today to generalizations of his original ideas. Of special note is his work on rootfinding and polynomial interpolation. Following Newton, many of the giants of mathematics of the 18th and 19th centuries made major contributions to the numerical solution of mathematical problems. Foremost among these are Leonhard Euler(1707-1783), Joseph-Louis Lagrange(1736-1813), and Karl Friedrich Gauss (1777-1855). Up to the late 1800’s, it appears that most mathematicians were quite broad in their interests, and many of them were interested in and contributed to numerical analysis. For a general history of numerical analysis up to 1900.
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