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Table of Contents

Acknowledgements
  • 1. Introduction to Computational Science
    • 1.1 What is Computational Science?
    • 1.2 What Do Computational Scientists Do
    • 1.3 Computational Science and Other Disciplines
    • 1.4 Computational Science Education
    • 1.5 The History of Computational Science
      • 1.5.1 Supercomputing Centers
      • 1.5.2 Computational Grand Challenge Problems
        • 1.5.2.a Grand Challenge Problem: Simulation of X-Ray Clusters
        • 1.5.2.b Grand Challenge Problem: Groundwater Remediation
        • 1.5.2.c Grand Challenge Problem: The Human Genome Project
    • 1.6 Computional Complexity
    • 1.7 Introduction to the Remainder of the Book
  • 2. The Solution of Linear Equations
    • 2.1 A Standard Classroom Example
    • 2.2 A "Real" Real-World Problem
      • 2.2.1 A Simple Model and a Textbook Solution
      • 2.2.2 A Closer Look at the Model and the Solution
      • 2.2.3 Using Information in Mathematical Models
    • 2.3 Using Computers to Solve Systems of Linear Equations
      • 2.3.1 Expressing Linear Systems in Matrix Form
      • 2.3.2 Notation for General Linear System
    • 2.4 Solution of Linear Systems
      • 2.4.1 "Easy" Systems of Linear Equations
      • 2.4.2 Converting Linear Systems from One Form to Another
      • 2.4.3 Gaussian Elimination
      • 2.4.4 Gauss-Jordan Elimination
      • 2.4.5 Multiple Right-Hand Sides
      • 2.4.6 Solving Linear Systems by Using the Inverse
      • 2.4.7 Using Cramer's Rule
    • 2.5 Problems
  • 3. Data Fitting
    • 3.1 Another "Real" Real-World Example
    • 3.2 When Do We Use Interpolation?
    • 3.3 Smoothing
      • 3.3.1 Error
      • 3.3.2 Fitting Data with a Linear Function
      • 3.3.3 An Example
      • 3.3.4 Data Fitting Using Other Measures of Aggregate Error
    • 3.4 Choosing the Function to Fit a Data Set
    • 3.5 Correlation Coefficient and Coefficien of Determination
    • 3.6 Problems
  • 4. Solutions of Nonlinear Equations
    • 4.1 The Method of Bisection
    • 4.2 Newton's Method
    • 4.3 The Secant Method
    • 4.4 The Method of False Position (Regular Falsi)
    • 4.5 A Comparison of Methods
      • 4.5.1 Difficulties
      • 4.5.2 Rates of Convergence
  • 5. The Cost of Doing Business
    • 5.1 Cost of Solving a System of Linear Equations Using Gaussian Elimination
    • 5.2 Cost of Solving a System of Linear Equations Using Gauss-Jordan Elimination
    • 5.3 Cost of Solving Problems with Multiple Right-Hand Sides
      • 5.3.1 Cost of Solving Problems with Multiple Right-Hand Sides Using Gaussina Elimination
      • 5.3.2 Cost of Solving Problems with Multiple Right-Hand Sides Using Gauss-Jordan Elimination
    • 5.4 Cost of Solving a System of Linear Equations Using the Inverse
    • 5.5 Cost of Solving a System of Linear Equations Using Cramer's Rule
    • 5.6 Comparison of the Costs for Various Methods
  • 6. An Introduction to Iterative Methods
    • 6.1 Simple Iterative Methods for Computing Square Roots
      • 6.1.1 An Easy Iteration Equation for square root 2
      • 6.1.2 A Modification of the Previous Iteration Equation
      • 6.1.3 Using Newton's Method to Compute square root of 2
    • 6.2 Iterative Solution of Systems of Linear Equations
      • 6.2.1 The Jacobi Method
      • 6.2.2 The Gauss-Seidel Method
      • 6.2.3 The Conjugate Gradient Method
      • 6.2.4 Two Examples
  • 7. Review Topics
    • 7.1 Matrix Notation
    • 7.2 The Transpose of a Matrix
    • 7.3 Matrix Multiplication
      • 7.3.1 The Inner Product
      • 7.3.2 General Matrix Multiplication
    • 7.4 The Identity Matrix
    • 7.5 The Inverse of a Matrix
    • 7.6 The Determinant of a Matrix
  • 8. Solutions to Problem
    • 8.1 Chapter 2
    • 8.2 Chapter 3
Bibliography

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 Copyright © 2001 Richard Tapia and Cynthia Lanius