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Table of Contents
Intro
Preface
Preface to Second Edition
Contents
1 Introduction to Scientific Computing
1.1 Unexpected Results
1.2 Floating-Point Number System
1.2.1 Normal Floats
1.2.2 Rounding
1.2.3 Non-Normal Floats
1.2.4 Significance
1.2.5 Flops
1.2.6 Functions
1.3 Arbitrary-Precision Arithmetic
1.4 Explaining, and Possibly Fixing, the Unexpected Results
1.5 Error and Accuracy
1.5.1 Over-computing?
1.6 Multicore Computing
2 Solving A Nonlinear Equation
2.1 The Problem to Solve
2.2 Bisection Method
2.2.1 Convergence Theory
2.3 Newton's Method
2.3.1 Picking x0
2.3.2 Order of Convergence
2.3.3 Failure
2.3.4 Examples
2.3.5 Convergence Theorem
2.4 Secant Method
2.4.1 Convergence Theorem
2.5 Other Ideas
2.5.1 Is Newton's Method Really Newton's Method?
3 Matrix Equations
3.1 An Example
3.2 Finding L and U
3.2.1 What Matrices Have a LU Factorization?
3.3 Implementing a LU Solver
3.3.1 Pivoting Strategies
3.3.2 LU and Gaussian Elimination
3.4 LU Method: Summary
3.4.1 Flops and Multicore Processors
3.5 Vector and Matrix Norms
3.5.1 Matrix Norm
3.6 Error and Residual
3.6.1 Correct Significant Digits
3.6.2 The Condition Number
3.6.3 A Heuristic for the Error
3.7 Positive Definite Matrices
3.7.1 Cholesky Factorization
3.8 Tri-Diagonal Matrices
3.9 Sparse Matrices
3.10 Nonlinear Systems
3.11 Some Additional Ideas
3.11.1 Yogi Berra and Perturbation Theory
3.11.2 Fixing an Ill-Conditioned Matrix
3.11.3 Insightful Observations about the Condition Number
3.11.4 Faster than LU?
3.11.5 Historical Comparisons
4 Eigenvalue Problems
4.1 Review of Eigenvalue Problems
4.2 Power Method
4.2.1 General Formulation
4.3 Extensions of the Power Method
4.3.1 Inverse Iteration
4.3.2 Rayleigh Quotient Iteration
4.4 Calculating Multiple Eigenvalues
4.4.1 Orthogonal Iteration
4.4.2 Regular and Modified Gram-Schmidt
4.4.3 QR Factorization
4.4.4 The QR Method
4.4.5 QR Method versus Orthogonal Iteration
4.4.6 Are the Computed Values Correct?
4.5 Applications
4.5.1 Natural Frequencies
4.5.2 Graphs and Adjacency Matrices
4.5.3 Markov Chain
4.6 Singular Value Decomposition
4.6.1 Derivation of the SVD
4.6.2 Interpretations of the SVD
4.6.3 Summary of the SVD
4.6.4 Consequences of a SVD
4.6.5 Computing a SVD
4.6.6 Low-Rank Approximations
4.6.7 Application: Image Compression
5 Interpolation
5.1 Information from Data
5.2 Global Polynomial Interpolation
5.2.1 Direct Approach
5.2.2 Lagrange Approach
5.2.3 Runge's Function
5.3 Piecewise Linear Interpolation
5.4 Piecewise Cubic Interpolation
5.4.1 Cubic B-splines
5.5 Function Approximation
5.5.1 Global Polynomial Interpolation
5.5.2 Piecewise Linear Interpolation
5.5.3 Cubic Splines
5.5.4 Chebyshev Interpolation
Preface
Preface to Second Edition
Contents
1 Introduction to Scientific Computing
1.1 Unexpected Results
1.2 Floating-Point Number System
1.2.1 Normal Floats
1.2.2 Rounding
1.2.3 Non-Normal Floats
1.2.4 Significance
1.2.5 Flops
1.2.6 Functions
1.3 Arbitrary-Precision Arithmetic
1.4 Explaining, and Possibly Fixing, the Unexpected Results
1.5 Error and Accuracy
1.5.1 Over-computing?
1.6 Multicore Computing
2 Solving A Nonlinear Equation
2.1 The Problem to Solve
2.2 Bisection Method
2.2.1 Convergence Theory
2.3 Newton's Method
2.3.1 Picking x0
2.3.2 Order of Convergence
2.3.3 Failure
2.3.4 Examples
2.3.5 Convergence Theorem
2.4 Secant Method
2.4.1 Convergence Theorem
2.5 Other Ideas
2.5.1 Is Newton's Method Really Newton's Method?
3 Matrix Equations
3.1 An Example
3.2 Finding L and U
3.2.1 What Matrices Have a LU Factorization?
3.3 Implementing a LU Solver
3.3.1 Pivoting Strategies
3.3.2 LU and Gaussian Elimination
3.4 LU Method: Summary
3.4.1 Flops and Multicore Processors
3.5 Vector and Matrix Norms
3.5.1 Matrix Norm
3.6 Error and Residual
3.6.1 Correct Significant Digits
3.6.2 The Condition Number
3.6.3 A Heuristic for the Error
3.7 Positive Definite Matrices
3.7.1 Cholesky Factorization
3.8 Tri-Diagonal Matrices
3.9 Sparse Matrices
3.10 Nonlinear Systems
3.11 Some Additional Ideas
3.11.1 Yogi Berra and Perturbation Theory
3.11.2 Fixing an Ill-Conditioned Matrix
3.11.3 Insightful Observations about the Condition Number
3.11.4 Faster than LU?
3.11.5 Historical Comparisons
4 Eigenvalue Problems
4.1 Review of Eigenvalue Problems
4.2 Power Method
4.2.1 General Formulation
4.3 Extensions of the Power Method
4.3.1 Inverse Iteration
4.3.2 Rayleigh Quotient Iteration
4.4 Calculating Multiple Eigenvalues
4.4.1 Orthogonal Iteration
4.4.2 Regular and Modified Gram-Schmidt
4.4.3 QR Factorization
4.4.4 The QR Method
4.4.5 QR Method versus Orthogonal Iteration
4.4.6 Are the Computed Values Correct?
4.5 Applications
4.5.1 Natural Frequencies
4.5.2 Graphs and Adjacency Matrices
4.5.3 Markov Chain
4.6 Singular Value Decomposition
4.6.1 Derivation of the SVD
4.6.2 Interpretations of the SVD
4.6.3 Summary of the SVD
4.6.4 Consequences of a SVD
4.6.5 Computing a SVD
4.6.6 Low-Rank Approximations
4.6.7 Application: Image Compression
5 Interpolation
5.1 Information from Data
5.2 Global Polynomial Interpolation
5.2.1 Direct Approach
5.2.2 Lagrange Approach
5.2.3 Runge's Function
5.3 Piecewise Linear Interpolation
5.4 Piecewise Cubic Interpolation
5.4.1 Cubic B-splines
5.5 Function Approximation
5.5.1 Global Polynomial Interpolation
5.5.2 Piecewise Linear Interpolation
5.5.3 Cubic Splines
5.5.4 Chebyshev Interpolation