Numerical Analysis
Autor Timothy Saueren Limba Engleză Hardback – 21 oct 2025
Conceput special pentru nivelul de licență, volumul Numerical Analysis de Timothy Sauer reprezintă o resursă modernă ce echilibrează rigoarea matematică și aplicabilitatea computațională. Considerăm că forța acestui manual rezidă în capacitatea de a introduce teme avansate, precum compresia datelor și analiza erorilor, într-un limbaj accesibil studenților de la facultățile de matematică, informatică sau inginerie. Structura textului facilitează o învățare activă prin utilizarea sistematică a mediului MATLAB®, esențial pentru demonstrarea algoritmilor în timp real. Fiecare capitol include secțiunea distinctivă „Reality Check”, o explorare extinsă a unor studii de caz care pot servi drept punct de plecare pentru proiecte individuale sau de grup. Această abordare transformă teoria abstractă în soluții pentru probleme concrete de calcul științific. În această a treia ediție, publicată de Pearson, găsim îmbunătățiri semnificative bazate pe feedback-ul academic, inclusiv capitole noi despre factorizarea Cholesky, metodele GMRES și tratarea numerică a ecuațiilor cu derivate parțiale neliniare. Cititorii familiarizați cu Numerical Mathematics de Alfio Quarteroni vor aprecia la Timothy Sauer o abordare mai orientată către procesul de învățare la nivel de licență, deși ambele lucrări utilizează MATLAB® ca instrument principal. În timp ce volumul lui Quarteroni pune un accent deosebit pe fundamentele matematice vaste, Numerical Analysis se concentrează pe o curbă de învățare progresivă, fiind ideal pentru cursurile introductive care necesită totuși o bază solidă pentru metode iterative și stabilitate numerică.
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Specificații
ISBN-10: 013469645X
Pagini: 674
Dimensiuni: 208 x 260 x 42 mm
Greutate: 1.57 kg
Ediția:3. Auflage
Editura: Pearson
De ce să citești această carte
Recomandăm această carte studenților de la licență care doresc să stăpânească metodele numerice prin practică directă în MATLAB®. Cititorul câștigă nu doar o înțelegere teoretică a algoritmilor, ci și abilitatea de a analiza erorile de calcul și de a implementa soluții pentru probleme complexe de inginerie. Este un manual de referință care face trecerea de la calculul simbolic la cel numeric într-un mod structurat și aplicat.
Cuprins
CHAPTER 0 Fundamentals
0.1 Evaluating a Polynomial
0.2 Binary Numbers
0.2.1 Decimal to binary0.2.2 Binary to decimal
0.3 Floating Point Representation of Real Numbers
0.3.1 Floating point formats0.3.2 Machine representation
0.3.3 Addition of floating point numbers
0.4 Loss of Significance
0.5 Review of Calculus
Software and Further Reading
CHAPTER 1 Solving Equations
1.1 The Bisection Method
1.1.1 Bracketing a root1.1.2 How accurate and how fast?
1.2 Fixed-Point Iteration
1.2.1 Fixed points of a function1.2.2 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 error1.3.2 The Wilkinson polynomial
1.3.3 Sensitivity of root-finding
1.4 Newton's Method
1.4.1 Quadratic convergence of Newton's Method1.4.2 Linear convergence of Newton's Method
1.5 Root-Finding without Derivatives
1.5.1 Secant Method and variants1.5.2 Brent's Method
Reality Check 1: Kinematics of the Stewart platform
Software and Further Reading
CHAPTER 2 Systems of Equations
2.1 Gaussian Elimination
2.1.1 Naive Gaussian elimination2.1.2 Operation counts
2.2 The LU Factorization
2.2.1 Matrix form of Gaussian elimination2.2.2 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 number2.3.2 Swamping
2.4 The PA = LU Factorization
2.4.1 Partial pivoting2.4.2 Permutation matrices
2.4.3 PA = LU factorization
Reality Check 2: The Euler-Bernoulli Beam
2.5 Iterative Methods
2.5.1 Jacobi Method2.5.2 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 matrices2.6.2 Cholesky factorization
2.6.3 Conjugate Gradient Method
2.6.4 Preconditioning
2.7 Nonlinear Systems of Equations
2.7.1 Multivariate Newton's Method2.7.2 Broyden's Method
Software and Further Reading
CHAPTER 3 Interpolation
3.1 Data and Interpolating Functions
3.1.1 Lagrange interpolation3.1.2 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 formula3.2.2 Proof of Newton form and error formula
3.2.3 Runge phenomenon
3.3 Chebyshev Interpolation
3.3.1 Chebyshev's theorem3.3.2 Chebyshev polynomials
3.3.3 Change of interval
3.4 Cubic Splines
3.4.1 Properties of splines3.4.2 Endpoint conditions
3.5 Bézier Curves
Reality Check 3: Fonts from Bézier curves
Software and Further Reading
CHAPTER 4 Least Squares
4.1 Least Squares and the Normal Equations
4.1.1 Inconsistent systems of equations4.1.2 Fitting models to data
4.1.3 Conditioning of least squares
4.2 A Survey of Models
4.2.1 Periodic data4.2.2 Data linearization
4.3 QR Factorization
4.3.1 Gram-Schmidt orthogonalization and least squares4.3.2 Modified Gram-Schmidt orthogonalization
4.3.3 Householder reflectors
4.4 Generalized Minimum Residual (GMRES) Method
4.4.1 Krylov methods4.4.2 Preconditioned GMRES
4.5 Nonlinear Least Squares
4.5.1 Gauss-Newton Method4.5.2 Models with nonlinear parameters
4.5.3 The Levenberg-Marquardt Method
Reality Check 4: GPS, Conditioning, and Nonlinear Least Squares
Software and Further Reading
CHAPTER 5 Numerical Differentiation and Integration
5.1 Numerical Differentiation
5.1.1 Finite difference formulas5.1.2 Rounding error
5.1.3 Extrapolation
5.1.4 Symbolic differentiation and integration
5.2 Newton-Cotes Formulas for Numerical Integration
5.2.1 Trapezoid Rule5.2.2 Simpson's Rule
5.2.3 Composite Newton-Cotes formulas
5.2.4 Open Newton-Cotes Methods
5.3 Romberg Integration
5.4 Adaptive Quadrature
5.5 Gaussian Quadrature
Reality Check 5: Motion Control in Computer-Aided Modeling
Software and Further Reading
CHAPTER 6 Ordinary Differential Equations
6.1 Initial Value Problems
6.1.1 Euler's Method6.1.2 Existence, uniqueness, and continuity for solutions
6.1.3 First-order linear equations
6.2 Analysis of IVP Solvers
6.2.1 Local and global truncation error6.2.2 The explicit Trapezoid Method
6.2.3 Taylor Methods
6.3 Systems of Ordinary Differential Equations
6.3.1 Higher order equations6.3.2 Computer simulation: the pendulum
6.3.3 Computer simulation: orbital mechanics
6.4 Runge-Kutta Methods and Applications
6.4.1 The Runge-Kutta family6.4.2 Computer simulation: the Hodgkin-Huxley neuron
6.4.3 Computer simulation: the Lorenz equations
Reality Check 6: The Tacoma Narrows Bridge
6.5 Variable Step-Size Methods
6.5.1 Embedded Runge-Kutta pairs6.5.2 Order 4/5 methods
6.6 Implicit Methods and Stiff Equations
6.7 Multistep Methods
6.7.1 Generating multistep methods6.7.2 Explicit multistep methods
6.7.3 Implicit multistep methods
Software and Further Reading
CHAPTER 7 Boundary Value Problems
7.1 Shooting Method
7.1.1 Solutions of boundary value problems7.1.2 Shooting Method implementation
Reality Check 7: Buckling of a Circular Ring
7.2 Finite Difference Methods
7.2.1 Linear boundary value problems7.2.2 Nonlinear boundary value problems
7.3 Collocation and the Finite Element Method
7.3.1 Collocation7.3.2 Finite elements and the Galerkin Method
Software and Further Reading
CHAPTER 8 Partial Differential Equations
8.1 Parabolic Equations
8.1.1 Forward Difference Method8.1.2 Stability analysis of Forward Difference Method
8.1.3 Backward Difference Method
8.1.4 Crank-Nicolson Method
8.2 Hyperbolic Equations
8.2.1 The wave equation8.2.2 The CFL condition
8.3 Elliptic Equations
8.3.1 Finite Difference Method for elliptic equations
Reality Check 8: Heat distribution on a cooling fin
8.3.2 Finite Element Method for elliptic equations8.4 Nonlinear partial differential equations
8.4.1 Implicit Newton solver8.4.2 Nonlinear equations in two space dimensions
Software and Further Reading
CHAPTER 9 Random Numbers and Applications
9.1 Random Numbers
9.1.1 Pseudo-random numbers9.1.2 Exponential and normal random numbers
9.2 Monte Carlo Simulation
9.2.1 Power laws for Monte Carlo estimation9.2.2 Quasi-random numbers
9.3 Discrete and Continuous Brownian Motion
9.3.1 Random walks9.3.2 Continuous Brownian motion
9.4 Stochastic Differential Equations
9.4.1 Adding noise to differential equations9.4.2 Numerical methods for SDEs
Reality Check 9: The Black-Scholes Formula
Software and Further Reading
CHAPTER 10 Trigonometric Interpolation and the FFT
10.1 The Fourier Transform
10.1.1 Complex arithmetic10.1.2 Discrete Fourier Transform
10.1.3 The Fast Fourier Transform
10.2 Trigonometric Interpolation
10.2.1 The DFT Interpolation Theorem10.2.2 Efficient evaluation of trigonometric functions
10.3 The FFT and Signal Processing
10.3.1 Orthogonality and interpolation10.3.2 Least squares fitting with trigonometric functions
10.3.3 Sound, noise, and filtering
Reality Check 10: The Wiener Filter
Software and Further Reading
CHAPTER 11 Compression
11.1 The Discrete Cosine Transform
11.1.1 One-dimensional DCT11.1.2 The DCT and least squares approximation
11.2 Two-Dimensional DCT and Image Compression
11.2.1 Two-dimensional DCT11.2.2 Image compression
11.2.3 Quantization
11.3 Huffman Coding
11.3.1 Information theory and coding11.3.2 Huffman coding for the JPEG format
11.4 Modified DCT and Audio Compression
11.4.1 Modified Discrete Cosine Transform11.4.2 Bit quantization
Reality Check 11: A Simple Audio Codec
Software and Further Reading
CHAPTER 12 Eigenvalues and Singular Values
12.1 Power Iteration Methods
12.1.1 Power Iteration12.1.2 Convergence of Power Iteration
12.1.3 Inverse Power Iteration
12.1.4 Rayleigh Quotient Iteration
12.2 QR Algorithm
12.2.1 Simultaneous iteration12.2.2 Real Schur form and the QR algorithm
12.2.3 Upper Hessenberg form
Reality Check 12: How Search Engines Rate Page Quality
12.3 Singular Value Decomposition
12.3.1 Finding the SVD in general12.3.2 Special case: symmetric matrices
12.4 Applications of the SVD
12.4.1 Properties of the SVD12.4.2 Dimension reduction
12.4.3 Compression
12.4.4 Calculating the SVD
Software and Further Reading
CHAPTER 13 Optimization
13.1 Unconstrained Optimization without Derivatives
13.1.1 Golden Section Search13.1.2 Successive parabolic interpolation
13.1.3 Nelder-Mead search
13.2 Unconstrained Optimization with Derivatives
13.2.1 Newton's Method13.2.2 Steepest Descent
13.2.3 Conjugate Gradient Search
Reality Check 13: Molecular Conformation and Numerical Optimization
Software and Further Reading
Appendix A
A.1 Matrix Fundamentals
A.2 Systems of linear equations
A.3 Block Multiplication
A.4 Eigenvalues and Eigenvectors
A.5 Symmetric Matrices
A.6 Vector Calculus
Appendix B
B.1 Starting MATLAB
B.2 Graphics
B.3 Programming in MATLAB
B.4 Flow Control
B.5 Functions
B.6 Matrix Operations
B.7 Animation and Movies
ANSWERS TO SELECTED EXERCISES
BIBLIOGRAPHY
INDEX