Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.1

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.1

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10.1 Gaussian Elimination with Partial Pivoting 10.2 Iterative Methods for Solving Linear Systems 10.3 Power Method for Approximating Eigenvalues 10.4 Applications of Numerical Methods NUMERICAL METHODS 10 570 arl Gustav Jacob Jacobi was the second son of a successful banker in Potsdam, Germany. After completing his secondary schooling in Potsdam in 1821, he entered the University of Berlin. In 1825, having been granted a doctorate in mathematics, Jacobi served as a lecturer at the University of Berlin. Then he accepted a position in mathematics at the University of Königsberg. Jacobi’s mathematical writings encom- passed a wide variety of topics, including elliptic functions, functions of a complex variable, functional determinants (called Jacobians), differential equations, and Abelian functions. Jacobi was the first to apply elliptic functions to the theory of numbers, and he was able to prove a longstanding conjecture by Fermat that every positive integer can be written as the sum of four perfect squares. (For instance, ) He also contributed to several branches of mathe- matical physics, including dynamics, celestial mechanics, and fluid dynamics. In spite of his contributions to applied mathematics, Jacobi did not believe that math- ematical research needed to be justified by its applicability. He stated that the sole end of science and mathematics is “the honor of the human mind” and that “a question about numbers is worth as much as a question about the system of the world.” Jacobi was such an incessant worker that in 1842 his health failed and he retired to Berlin. By the time of his death in 1851, he had become one of the most famous mathemati- cians in Europe. 10 5 1 2 1 1 2 1 2 2 1 2 2 . 10.1 GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING In Chapter 1 two methods for solving a system of n linear equations in n variables were discussed. When either of these methods (Gaussian elimination and Gauss-Jordan elimina- tion) is used with a digital computer, the computer introduces a problem that has not yet been discussed— rounding error. Digital computers store real numbers in floating point form, where k is an integer and the mantissa M satisfies the inequality For instance, the floating point forms of some real numbers are as follows. Real Number Floating Point Form 527 0.00045 0.45 3 10 2 3 2 0.381623 3 10 1 2 3.81623 0.527 3 10 3 0.1 M < 1. ± M 3 10 k , Carl Gustav Jacob Jacobi 1804–1851 C
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The number of decimal places that can be stored in the mantissa depends on the computer. If n places are stored, then it is said that the computer stores n significant digits. Additional digits are either truncated or rounded off. When a number is truncated to n significant digits, all digits after the first n significant digits are simply omitted. For instance, truncated to two significant digits, the number 0.1251 becomes 0.12.
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This note was uploaded on 02/24/2012 for the course MATH 310 taught by Professor Staff during the Spring '08 term at VCU.

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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.1

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