Solving Sets of Equations

• Solving sets of linear equations is the most frequently used numerical procedure when real-world situations are modeled.
• The methods for solving ordinary and partial-differential equations depend on them.
  • Matrices and Vectors. Reviews concepts of matrices and vectors in preparation for their use in this chapter.
  • Elimination Methods. Describes two classical methods that change a system of equations to forms that allow getting the solution by back-substitution and shows how the errors of the solution can be minimized.
  • The Inverse of a Matrix and Matrix Pathology. Shows how an important derivative of a matrix, its inverse, can be computed. It shows when a matrix cannot be inverted and tells of situations where no unique solution exists to a system of equations.
  • Ill-Conditioned Systems. Explores systems for which getting the solution with accuracy is very difficult. A number, the condition number, is a measure of such difficulty; a property of a matrix, called its norm, is used to compute its condition number. A way to improve an inaccurate solution is described.
  • Iterative Methods.This section describes how a linear system can be solved in an entirely different way, by beginning with an initial estimate of the solution and performing computations that eventually arrive at the correct solution. An iterative method is particularly important in solving systems that have few nonzero coefficients.
  • Parallel Processing. Tells how parallel computing can be applied to the solution of linear systems. An algorithm is developed that allows a significant reduction in processing time.