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lecture1 - CSE6643/MATH6643 Numerical Linear Algebra Haesun...

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CSE6643/MATH6643: Numerical Linear Algebra Haesun Park [email protected] School of Computational Science and Engineering College of Computing Georgia Institute of Technology Atlanta, GA 30332, USA Lecture 1
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What is Numerical Analysis? Three great branches of science: Theory, Experiment, and Computation The purpose of computing is insight, not numbers (1961 Hamming). Numerical Analysis is Study of Algorithms for Problems of Continuous Mathematics . Ex. Newton’s method, Lagrange interpolation polynomial, Gaussian Elimination, Euler’s method... Computational mathematics is mainly based on two ideas (extreme simplification) Taylor series and linear algebra. Role of Computers in Numerical Computing Computers certainly play a part in numerical computing but even if rounding error vanishes, 95 % of numerical analysis would remain. Most mathematical problems cannot be solved by a finite sequence of elementary operations Need: fast algorithms that converge to ’approximate’ answers accurate to many digits of precision, in science and engineering applications. CSE6643/MATH6643: Numerical Linear Algebra – p.1/9
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Different Types of Problems in Numerical Computing Problem F: can be solved in a finite sequence of elementary operations: Root for polynomial of degree 4: closed form formula exists (Ferrari 1540) Solving linear equations Linear programming Problem I: cannot be solved in a finite sequence of elementary operations: Root for polynomial of degree 5 and higher: no closed form formula exisits (Ruffini and Abel, around 1800) Finding eigenvalues of an n × n matrix with n 5 Minimize a function of several variables Evaluate an integral
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