Numerical Mathematics & Scientific Computation

Numerical Mathematics & Scientific Computation......Page 0
Contents......Page 1
Introduction......Page 7
Common Ideas and Concepts......Page 8
Numerical Algorithms......Page 19
Numerical Solution of Differential Equations......Page 38
Monte Carlo Methods......Page 47
Linear Algebra and Matrix Computations......Page 57
Basic Concepts in Error Estimation......Page 81
Computer Number Systems......Page 87
Accuracy and Rounding Errors......Page 98
Error Propagation and Condition Numbers......Page 126
Some Basic Facts about Series......Page 149
Difference Operators and Operator Expansion......Page 190
Acceleration of Convergence......Page 242
Continued Fractions and Pade Approximants......Page 305
The Interpolation Problem......Page 323
Interpolation Formulas and Algorithms......Page 330
Interpolation where values of derivatives are used......Page 345
Spline Functions......Page 351
Polynomial Interpolation of Analytic Functions......Page 374
Trigonometric Interpolation and FFT......Page 384
Examples of Interpolation in Nonlinear Function Spaces......Page 387
Introduction......Page 389
Classical Quadrature Rules......Page 390
Special Transformations......Page 398
Multiple Integrals......Page 401
Introduction......Page 413
Fixed-Point Iteration......Page 419
Newton's Method......Page 424
Methods Based on Interpolation......Page 435
Attainable Accuracy and Multiple Roots......Page 444
Zero of Polynomials......Page 449
Minimizing a Scalar Function......Page 459
Appendix -  Guide to Literature......Page 473
Index......Page 478
Contents......Page 486
Introduction......Page 494
Gaussion Elimination and LU Factorization......Page 495
Banded Systems......Page 516
Symmetric Matrices......Page 521
Block Matrix Algorithms......Page 535
Direct Methods for General Sparse Systems......Page 547
Perturbation Theory and Error Analysis......Page 557
Structured Systems......Page 576
Introduction......Page 584
The Method of Normal Equations......Page 592
Least Squares and the SVD......Page 601
Gram-Schmidt Orthogonalization......Page 614
Orthogonal Factorizations......Page 622
Rank Deficient and Ill-Posed Problems......Page 638
Modified Least Squares Problems......Page 653
Some Generalized Problems......Page 663
Basic Properties......Page 682
Canonical Forms and Matrix Functions......Page 692
Perturbation Theory and Eigenvalue Bounds......Page 708
Jacobi Methods......Page 722
The Power Method......Page 729
Transformation to Condensed Form......Page 739
The LR and QR Algorithms......Page 750
Subspace Methods for Large Eigenvalue Problems......Page 771
Generalized Eigenvalue Problems......Page 783
Classical Iterative Methods......Page 796
Stationary Iterative Methods......Page 801
Successive Overrelaxation Methods......Page 811
Convergence Acceleration......Page 820
Projection Methods......Page 825
Krylov Subspace Methods......Page 831
Nonsymmetric Problems......Page 840
Preconditioned Iterative Methods......Page 855
Preconitioners......Page 860
Systems of Nonlinear Equations......Page 874
Unconstrained Optimization......Page 893
Nonlinear Least Squares Problems......Page 899
Constrained Linear Optimization......Page 916
Appendix: Calculus in Vector Space......Page 932
Appendix. Guide to Literature in Linear Algebra......Page 944
Index......Page 949
Initial Value Problems for ODEs Theoretical Background......Page 959
Control of Step Size and Numerical Stability......Page 985
One step Methods......Page 1012
Linear Multistep Methods......Page 1025
Extrapolation Methods......Page 1038
Second Order Equations and Boundary Problems......Page 1044
Qualitative Theory of ODEs......Page 1059
General Concepts and Theorems for ODEs......Page 1083
Introduction......Page 1109
Partial Differential Equations......Page 1112
Variational Methods......Page 1124
Integral Equations......Page 1130
15. Guide to Literature and Software......Page 1135