2-lu - Chapter 2 Linear Equations One of the problems...

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Chapter 2 Linear Equations One of the problems encountered most frequently in scientific computation is the solution of systems of simultaneous linear equations. This chapter covers the solu- tion of linear systems by Gaussian elimination and the sensitivity of the solution to errors in the data and roundoff errors in the computation. 2.1 Solving Linear Systems With matrix notation, a system of simultaneous linear equations is written Ax = b. In the most frequent case, when there are as many equations as unknowns, A is a given square matrix of order n , b is a given column vector of n components, and x is an unknown column vector of n components. Students of linear algebra learn that the solution to Ax = b can be written x = A - 1 b , where A - 1 is the inverse of A . However, in the vast majority of practical computational problems, it is unnecessary and inadvisable to actually compute A - 1 . As an extreme but illustrative example, consider a system consisting of just one equation, such as 7 x = 21 . The best way to solve such a system is by division: x = 21 7 = 3 . Use of the matrix inverse would lead to x = 7 - 1 × 21 = 0 . 142857 × 21 = 2 . 99997 . The inverse requires more arithmetic—a division and a multiplication instead of just a division—and produces a less accurate answer. Similar considerations apply February 15, 2008 1
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2 Chapter 2. Linear Equations to systems of more than one equation. This is even true in the common situation where there are several systems of equations with the same matrix A but different right-hand sides b . Consequently, we shall concentrate on the direct solution of systems of equations rather than the computation of the inverse. 2.2 The MATLAB Backslash Operator To emphasize the distinction between solving linear equations and computing in- verses, Matlab has introduced nonstandard notation using backward slash and forward slash operators, “ \ ” and “/”. If A is a matrix of any size and shape and B is a matrix with as many rows as A , then the solution to the system of simultaneous equations AX = B is denoted by X = A \ B. Think of this as dividing both sides of the equation by the coefficient matrix A . Because matrix multiplication is not commutative and A occurs on the left in the original equation, this is left division . Similarly, the solution to a system with A on the right and B with as many columns as A , XA = B, is obtained by right division , X = B/A. This notation applies even if A is not square, so that the number of equations is not the same as the number of unknowns. However, in this chapter, we limit ourselves to systems with square coefficient matrices. 2.3 A 3-by-3 Example To illustrate the general linear equation solution algorithm, consider an example of order three: 10 - 7 0 - 3 2 6 5 - 1 5 x 1 x 2 x 3 = 7 4 6 .
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This note was uploaded on 02/20/2012 for the course ECON 101 taught by Professor Burkhauser during the Spring '08 term at Cornell University (Engineering School).

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2-lu - Chapter 2 Linear Equations One of the problems...

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