Engeln-Müllges G., Uhlig F. Numerical Algorithms with C 1996
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Engeln-Müllges G., Uhlig F. Numerical Algorithms with C 1996
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Computer Numbers, Error Analysis, Conditioning,
Stability of Algorithms and Operations Count
Definition of Errors
Decimal Representation of Numbers
Sources of Errors
Input Errors
Procedural Errors
Error Propagation and the Condition of a Problem
The Computational Error and Numerical Stability
of an Algorithm
Operations Count, et cetera
Nonlinear Equations in One Variable
Introduction
Definitions and Theorems on Roots
General Iteration Procedures
How to Construct an Iterative Process
Existence and Uniqueness of Solutions
Convergence and Error Estimates of Iterative Procedures
Practical Implementation
Order of Convergence of an Iterative Procedure
Definitions and Theorems
Determining the Order of Convergence Experimentally
Newton's Method
Finding Simple Roots
A Damped Version of Newton's Method
Newton's Method for Multiple Zeros; a Modified Newton's
Method
Regula Falsi
Regula Falsi for Simple Roots
Modified Regula Falsi for Multiple Zeros
Simplest Version of the Regula Falsi
Steffensen Method
Steffensen Method for Simple Zeros
Modified Steffensen Method for Multiple Zeros
Inclusion Methods
Bisection Method
Pegasus Method
Anderson-Bjorck Method
The King and the Anderson-Bjorck-King Methods,
the Illinois Method
Zeroin Method
Efficiency of the Methods and Aids for Decision Making
Roots of Polynomials
Preliminary Remarks
The Horner Scheme
First Level Horner Scheme for Real Arguments
First Level Horner Scheme for Complex Arguments
Complete Horner Scheme for Real Arguments
Applications
Methods for Finding all Solutions of Algebraic Equations
Preliminaries
Muller's Method
Bauhuber's Method
The Jenkins-'!raub Method
The Laguerre Method
Hints for Choosing a Method
Direct Methods for Solving Systems of Linear Equations
The Problem
Definitions and Theoretical Background
Solvability Conditions for Systems of Linear Equations
The Factorization Principle
GauE Algorithm
GauE Algorithm with Column Pivot Search
Pivot Strategies
Computer Implementation of GauB Algorithm
GauB Algorithm for Systems with Several Right Hand Sides
Matrix Inversion via GauB Algorithm
Linear Equations with Symmetric Strongly Nonsingular
System Matrices
The Cholesky Decomposition
The Conjugate Gradient Method
The GauB - Jordan Method
The Matrix Inverse via Exchange Steps
Linear Systems with Tridiagonal Matrices
Systems with Tridiagonal Matrices
Systems with Tridiagonal Symmetric Strongly
Nonsingular Matrices
Linear Systems with Cyclically Tridiagonal Matrices
Systems with a Cyclically Tridiagonal Matrix
Systems with Symmetric Cyclically Tridiagonal
Strongly Nonsingular Matrices
Linear Systems with Five-Diagonal Matrices
Systems with Five-Diagonal Matrices
Systems with Five-Diagonal Symmetric Matrices
Linear Systems with Band Matrices
Solving Linear Systems via Householder Transformations
Errors, Conditioning and Iterative Refinement
Errors and the Condition Number
Condition Estimates
Improving the Condition Number
Iterative Refinement
Systems of Equations with Block Matrices
Preliminary Remarks
GauB Algorithm for Block Matrices
GauB Algorithm for Block Tridiagonal Systems
ther Block Methods
The Algorithm of Cuthill-McKee for Sparse Symmetric
Matrices
Recommendations for Selecting a Method
Iterative Methods for Linear Systems
Preliminary Remarks
Vector and Matrix Norms
The Jacobi Method
The GauB-Seidel Iteration
A Relaxation Method using the Jacobi Method
A Relaxation Method using the GauB-Seidel Method
Iteration Rule
Estimate for the Optimal Relaxation Coefficient,
an Adaptive SOR Method
Systems of Nonlinear Equations
General Iterative Methods
Special Iterative Methods
Newton Methods for Nonlinear Systems
The Basic Newton Method
Damped Newton Method for Systems
Regula Falsi for Nonlinear Systems
Method of Steepest Descent for Nonlinear Systems
Brown's Method for Nonlinear Systems
Choosing a Method
Eigenvalues and Eigenvectors of Matrices
Basic Concepts
Diagonalizable Matrices and the Conditioning of the
Eigenvalue Problem
Vector Iteration
The Dominant Eigenvalue and the Associated
Eigenvector of a Matrix
Determination of the Eigenvalue Closest to Zero
Eigenvalues in Between
The Rayleigh Quotient for Hermitian Matrices
The Krylov Method
Determining the Eigenvalues
Determining the Eigenvectors
Eigenvalues of Positive Definite Tridiagonal Matrices,
the QD Algorithm
Transformation to Hessenberg Form, the LR and
QR Algorithms
Transformation of a Matrix to Upper Hessenberg Form
The LR Algorithm
The Basic QR Algorithm
Eigenvalues and Eigenvectors of a Matrix via the QR Algorithm
Decision Strategy
Linear and Nonlinear Approximation
Linear Approximation
Statement of the Problem and Best Approximation
Linear Continuous Root-Mean-Square Approximation
Discrete Linear Root-Mean-Square Approximation
Normal Equations for Discrete Linear Least Squares
Discrete Least Squares via Algebraic Polynomials
and Orthogonal Polynomials
Linear Regression, the Least Squares Solution Using
Linear Algebraic Polynomials
Solving Linear Least Squares Problems using
Householder Transformations
Approximation of Polynomials by Chebyshev Polynomials
Best Uniform Approximation
Approximation by Chebyshev Polynomials
Approximation of Periodic Functions and the FFT
Root-Mean-Square Approximation of Periodic Functions
Trigonometric Interpolation
Complex Discrete Fourier Transformation (FFT)
Error Estimates for Linear Approximation
Estimates for the Error in Best Approximation
Error Estimates for Simultaneous Approximation
of a Function and its Derivatives
Approximation Error Estimates using Linear Projection
Operators
Nonlinear Approximation
Transformation Method for Nonlinear Least Squares
Nonlinear Root-Mean-Square Fitting
Decision Strategy
Polynomial and Rational Interpolation
The Problem
Lagrange Interpolation Formula
Lagrange Formula for Arbitrary Nodes
Lagrange Formula for Equidistant Nodes
The Aitken Interpolation Scheme for Arbitrary Nodes
Inverse Interpolation According to Aitken
Newton Interpolation Formula
Newton Formula for Arbitrary Nodes
Newton Formula for Equidistant Nodes
Remainder of an Interpolation and Estimates of the
Interpolation Error
Rational Interpolation
Interpolation for Functions in Several Variables
Lagrange Interpolation Formula for Two Variables
Shepard Interpolation
Hints for Selecting an Appropriate Interpolation Method
Interpolating Polynomial Splines for Constructing
Smooth Curves
Cubic Polynomial Splines
Definition of Interpolating Cubic Spline Functions
Computation of Non-Parametric Cubic Splines
Computing Parametric Cubic Splines
Joined Interpolating Polynomial Splines
Convergence and Error Estimates for Interpolating
Cubic Splines
Hermite Splines of Fifth Degree
Definition of Hermite Splines
Computation of Non-Parametric Hermite Splines
Computation of Parametric Hermite Splines
Hints for Selecting Appropriate Interpolating or
Approximating Splines
Cubic Fitting Splines for Constructing Smooth Curves
The Problem
Definition of Fitting Spline Functions
Non-Parametric Cubic Fitting Splines
Contents XIX
Parametric Cubic Fitting Splines
Decision Strategy
Two-Dimensional Splines, Surface Splines,
Bezier Splines, B-Splines
Interpolating Two-Dimensional Cubic Splines for
Constructing Smooth Surfaces
Two-Dimensional Interpolating Surface Splines
Bezier Splines
Bezier Spline Curves
Bezier Spline Surfaces
Modified Interpolating Cubic Bezier Splines
B-Splines
B-Spline-Curves
B-Spline-Surfaces
Hints
Akima and Renner Subsplines
Akima Subsplines
Renner Sub splines
Rounding of Corners with Akima and Renner Splines
Approximate Computation of Arc Length
Selection Hints
Numerical Differentiation
The Task
Differentiation Using Interpolating Polynomials
Differentiation via Interpolating Cubic Splines
Differentiation by the Romberg Method
Decision Hints
Numerical Integration
Preliminary Remarks
Interpolating Quadrature Formulas
Newton-Cotes Formulas
The Trapezoidal Rule
Simpson's Rule
The / Formula
Other Newton-Cotes Formulas
The Error Order of Newton-Cotes Formulas
Maclaurin Quadrature Formulas
The Tangent Trapezoidal Formula
Other Maclaurin Formulas
Euler-Maclaurin Formulas
Chebyshev Quadrature Formulas
GauB Quadrature Formulas
Calculation of Weights and Nodes of Generalized Gaussian
Quadrature Formulas
Clenshaw-Curtis Quadrature Formulas
Romberg Integration
Error Estimates and Computational Errors
Adaptive Quadrature Methods
Convergence of Quadrature Formulas
Hints for Choosing an Appropriate Method
Numerical Cubature
The Problem
Interpolating Cubature Formulas
Newton-Cotes Cubature Formulas for Rectangular Regions
Newton-Cotes Cubature Formulas for Triangles
Romberg Cubature for Rectangular Regions
GauB Cubature Formulas for Rectangles
GauB Cubature Formulas for Triangles
Right Triangles with Legs Parallel to the Axis
General Triangles
Riemann Double Integrals using Bicubic Splines
Decision Strategy
Initial Value Problems for Ordinary Differential
Equations
The Problem
Principles of the Numerical Methods
One-Step Methods
The Euler-Cauchy Polygonal Method
The Improved Euler-Cauchy Method
The Predictor-Corrector Method of Heun
Explicit Runge-Kutta Methods
Construction of Runge-Kutta Methods
The Classical Runge-Kutta Method
A List of Explicit Runge-Kutta Formulas
Embedding Formulas
Implicit Runge-Kutta Methods of Gaussian Type
Consistence and Convergence of One-Step Methods
Error Estimation and Step Size Control
Error Estimation
Automatic Step Size Control, Adaptive Methods
for Initial Value Problems
Multi-Step Methods
The Principle of Multi-Step Methods
The Adams-Bashforth Method
The Predictor-Corrector Method of Adams-Moulton
The Adams-Stormer Method
Error Estimates for Multi-Step Methods
Computational Error of One-Step and Multi-Step Methods
Bulirsch-Stoer-Gragg Extrapolation
Stability
Preliminary Remarks
Stability of Differential Equations
Stability of the Numerical Method
Stiff Systems of Differential Equations
The Problem
Criteria for the Stiffness of a System
Gear's Method for Integrating Stiff Systems
Suggestions for Choosing among the Methods
Boundary Value Problems for Ordinary Differential
Equations
Statement of the Problem
Reduction of Boundary Value Problems to Initial
Value Problems
Boundary Value Problems for Nonlinear Differential
Equations of Second Order
Boundary Value Problems for Systems of Differential
Equations of First Order
The Multiple Shooting Method
Difference Methods
The Ordinary Difference Method
Higher Order Difference Methods
Iterative Solution of Linear Systems for Special
Boundary Value Problems
Linear Eigenvalue Problems
Appendix: ANSI C Functions
Bibliography
Literature for Other Topics
- Numerical Treatment of Partial Differential Equations
- Finite Element Method
Index