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Comparing Methods for Solving Linear Systems

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Quantum Computing
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You’ve learned a few different methods for solving a system of linear equations, probably with just two or three variables and equations. If you want, you can apply the same procedures to solve a larger number of equations. But how large of a system can you handle? Five equations? Ten? Twenty? How about a thousand, or a million? Well, time to bring your laptop out!

Methods That Computers Use

Solving linear systems of equations is a common problem that often arises with a very large number of equations. While systems of three or four equations can be readily solved by hand, computers are usually used to help solve larger systems. Computational algorithms for finding such solutions are an important part of numerical linear algebra.

What you’ve done by hand so far (eliminating, substituting, etc.) is what the computer does too. The difference is that they store the numbers in array forms, known as matrices, and work based on these. The elimination and substitution method, when applied on matrix representations, is known as the Gaussian elimination method. However, this is not the only method computers use; there are also methods based on matrix inversions or Cramer’s rule. Different methods have different speeds and different degrees of precision.

Read more about the various methods that computers use here:
http://en.wikipedia.org/wiki/System_of_linear_equations#Solving_a_linear_system

Explore More

The best-known classical computer algorithms require times that are proportional to the number of variables in a linear set of equations. But with rapidly growing data sets, keeping up with the enormous numerical burdens is becoming difficult. A recently proposed quantum algorithm shows that quantum supercomputers could solve linear systems at an exponential speedup over traditional computers.

Learn more about these computers and their capabilities with the links below.

http://physicsworld.com/cws/article/news/2013/jun/12/quantum-computer-solves-simple-linear-equation

http://web.mit.edu/newsoffice/2009/quantum-algorithm.html

http://phys.org/news/2013-06-quantum-algorithm-linear-equations.html

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