Computational Science and Engineering

Encompasses the full range of computational science and engineering from modelling to solution, both analytical and numerical. It develops a framework for the equations and numerical methods of applied mathematics. Gilbert Strang has taught this material to thousands of engineers and scientists (and many more on MITs OpenCourseWare 18.085-6). His experience is seen in his clear explanations, wide range of examples, and teaching method. The book is solution-based and not formula-based: it integrates analysis and algorithms and MATLAB codes to explain each topic as effectively as possible. The topics include applied linear algebra and fast solvers, differential equations with finite differences and finite elements, Fourier analysis and optimization. This book also serves as a reference for the whole community of computational scientists and engineers. Supporting resources, including MATLAB codes, problem solutions and video lectures from Gilbert Strangs 18.085 courses at MIT, are provided at math.mit.edu/cse.

Spis treści:
1. Applied Linear Algebra
1.1 Four special matrices
1.2 Differences, derivatives, and boundary conditions
1.3 Elimination leads to K = LDL^T
1.4 Inverses and delta functions
1.5 Eigenvalues and eigenvectors
1.6 Positive definite matrices
1.7 Numerical linear algebra, LU, QR, SVD
1.8 Best basis from the SVD
2. A Framework for Applied Mathematics
2.1 Equilibrium and the stiffness matrix
2.2 Oscillation by Newtons law
2.3 Least squares for rectangular matrices
2.4 Graph models and Kirchhoffs laws
2.5 Networks and transfer functions
2.6 Nonlinear problems
2.7 Structures in equilibrium
2.8 Covariances and recursive least squares
2.9 Graph cuts and gene clustering
3. Boundary Value Problems
3.1 Differential equations of equilibrium
3.2 Cubic splines and fourth order equations
3.3 Gradient and divergence
3.4 Laplaces equation
3.5 Finite differences and fast Poisson solvers
3.6 The finite element method
3.7 Elasticity and solid mechanics
4. Fourier Series and Integrals
4.1 Fourier series for periodic functions
4.2 Chebyshev, Legendre, and Bessel
4.3 The discrete Fourier transform and the FFT
4.4 Convolution and signal processing
4.5 Fourier integrals
4.6 Deconvolution and integral equations
4.7 Wavelets and signal processing
5. Analytic Functions
5.1 Taylor series and complex integration
5.2 Famous functions and great theorems
5.3 The Laplace transform and z-transform
5.4 Spectral methods of exponential accuracy
6. Initial Value Problems
6.1 Introduction
6.2 Finite difference methods for ODEs
6.3 Accuracy and stability for u_t = c u_x
6.4 The wave equation and staggered leapfrog
6.5 Diffusion, convection, and finance
6.6 Nonlinear flow and conservation laws
6.7 Fluid mechanics and Navier-Stokes
6.8 Level sets and fast marching
7. Solving Large Systems
7.1 Elimination with reordering
7.2 Iterative methods
7.3 Multigrid methods
7.4 Conjugate gradients and Krylov subspaces
8. Optimization and Minimum Principles
8.1 Two fundamental examples
8.2 Regularized least squares
8.3 Calculus of variations
8.4 Errors in projections and eigenvalues
8.5 The Saddle Point Stokes problem
8.6 Linear programming and duality
8.7 Adjoint methods in design.