Numerical Analysis - Sauer, Timothy

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Résumé :

CHAPTER 0 ????Fundamentals

0.1 Evaluating a Polynomial

0.2 Binary Numbers

? ? 0.2.1 Decimal to binary

? ? 0.2.2 Binary to decimal

0.3 Floating Point Representation of Real Numbers

? ? 0.3.1 Floating point formats
? ? 0.3.2 Machine representation

? ? 0.3.3 Addition of floating point numbers

0.4 Loss of Significance

0.5 Review of Calculus

Software and Further Reading

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CHAPTER 1 ????Solving Equations

1.1 The Bisection Method

? ? 1.1.1 Bracketing a root

? ? 1.1.2 How accurate and how fast?

1.2 Fixed-Point Iteration

? ? 1.2.1 Fixed points of a function ? ? 1.2.2 Geometry of Fixed-Point Iteration ? ? 1.2.3 Linear convergence of Fixed-Point Iteration

? ? 1.2.4 Stopping criteria

1.3 Limits of Accuracy

? ? 1.3.1 Forward and backward error ? ? 1.3.2 The Wilkinson polynomial

? ? 1.3.3 Sensitivity of root-finding

1.4 Newton's Method

? ? 1.4.1 Quadratic convergence of Newton's Method

? ? 1.4.2 Linear convergence of Newton's Method

1.5 Root-Finding without Derivatives

? ? 1.5.1 Secant Method and variants

? ? 1.5.2 Brent's Method

Reality Check 1: Kinematics of the Stewart platform

Software and Further Reading

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CHAPTER 2 ????Systems of Equations

2.1 Gaussian Elimination

? ? 2.1.1 Naive Gaussian elimination

? ? 2.1.2 Operation counts

2.2 The LU Factorization

? ? 2.2.1 Matrix form of Gaussian elimination ? ? 2.2.2 Back substitution with the LU factorization

? ? 2.2.3 Complexity of the LU factorization

2.3 Sources of Error

? ? 2.3.1 Error magnification and condition number

? ? 2.3.2 Swamping

2.4 The PA = LU Factorization

? ? 2.4.1 Partial pivoting ? ? 2.4.2 Permutation matrices

? ? 2.4.3 PA = LU factorization

Reality Check 2: The Euler-Bernoulli Beam

2.5 Iterative Methods

? ? 2.5.1 Jacobi Method ? ? 2.5.2 Gauss-Seidel Method and SOR ? ? 2.5.3 Convergence of iterative methods

? ? 2.5.4 Sparse matrix computations

2.6 Methods for symmetric positive-definite matrices

? ? 2.6.1 Symmetric positive-definite matrices ? ? 2.6.2 Cholesky factorization ? ? 2.6.3 Conjugate Gradient Method

? ? 2.6.4 Preconditioning

2.7 Nonlinear Systems of Equations

? ? 2.7.1 Multivariate Newton's Method

? ? 2.7.2 Broyden's Method

Software and Further Reading

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CHAPTER 3 ????Interpolation

3.1 Data and Interpolating Functions

? ? 3.1.1 Lagrange interpolation ? ? 3.1.2 Newton's divided differences ? ? 3.1.3 How many degree d polynomials pass through n points? ? ? 3.1.4 Code for interpolation

? ? 3.1.5 Representing functions by approximating polynomials

3.2 Interpolation Error

? ? 3.2.1 Interpolation error formula ? ? 3.2.2 Proof of Newton form and error formula

? ? 3.2.3 Runge phenomenon

3.3 Chebyshev Interpolation

? ? 3.3.1 Chebyshev's theorem ? ? 3.3.2 Chebyshev polynomials

? ? 3.3.3 Change of interval

3.4 Cubic Splines

? ? 3.4.1 Properties of splines

? ? 3.4.2 Endpoint conditions

3.5 B?zier Curves

Reality Check 3: Fonts from B?zier curves

Software and Further Reading

CHAPTER 4 ????Least Squares

4.1 Least Squares and the Normal Equations

? ? 4.1.1 Inconsistent systems of e...