Preface xi -- 1 Errors in Numerical Analysis 1 -- 1.1 Enter Data Errors, 1 -- 1.2 Approximation Errors, 2 -- 1.3 Round-Off Errors, 3 -- 1.4 Propagation of Errors, 3 -- 1.4.1 Addition, 3 -- 1.4.2 Multiplication, 5 -- 1.4.3 Inversion of a Number, 7 -- 1.4.4 Division of Two Numbers, 7 -- 1.4.5 Raising to a Negative Entire Power, 7 -- 1.4.6 Taking the Root of pth Order, 7 -- 1.4.7 Subtraction, 8 -- 1.4.8 Computation of Functions, 8 -- 1.5 Applications, 8 -- Further Reading, 14 -- 2 Solution of Equations 17 -- 2.1 The Bipartition (Bisection) Method, 17 -- 2.2 The Chord (Secant) Method, 20 -- 2.3 The Tangent Method (Newton), 26 -- 2.4 The Contraction Method, 37 -- 2.5 The Newton-Kantorovich Method, 42 -- 2.6 Numerical Examples, 46 -- 2.7 Applications, 49 -- Further Reading, 52 -- 3 Solution of Algebraic Equations 55 -- 3.1 Determination of Limits of the Roots of Polynomials, 55 -- 3.2 Separation of Roots, 60 -- 3.3 Lagrange's Method, 69 -- 3.4 The Lobachevski-Graeffe Method, 72 -- 3.4.1 The Case of Distinct Real Roots, 72 -- 3.4.2 The Case of a Pair of Complex Conjugate Roots, 74 -- 3.4.3 The Case of Two Pairs of Complex Conjugate Roots, 75 -- 3.5 The Bernoulli Method, 76 -- 3.6 The Bierge-Vi`ete Method, 79 -- 3.7 Lin Methods, 79 -- 3.8 Numerical Examples, 82 -- 3.9 Applications, 94 -- Further Reading, 109 -- 4 Linear Algebra 111 -- 4.1 Calculation of Determinants, 111 -- 4.1.1 Use of Definition, 111 -- 4.1.2 Use of Equivalent Matrices, 112 -- 4.2 Calculation of the Rank, 113 -- 4.3 Norm of a Matrix, 114 -- 4.4 Inversion of Matrices, 123 -- 4.4.1 Direct Inversion, 123 -- 4.4.2 The Gauss-Jordan Method, 124 -- 4.4.3 The Determination of the Inverse Matrix by its Partition, 125 -- 4.4.4 Schur's Method of Inversion of Matrices, 127 -- 4.4.5 The Iterative Method (Schulz), 128 -- 4.4.6 Inversion by Means of the Characteristic Polynomial, 131 -- 4.4.7 The Frame-Fadeev Method, 131 -- 4.5 Solution of Linear Algebraic Systems of Equations, 132 -- 4.5.1 Cramer's Rule, 132 -- 4.5.2 Gauss's Method, 133.

4.5.3 The Gauss-Jordan Method, 134 -- 4.5.4 The LU Factorization, 135 -- 4.5.5 The Schur Method of Solving Systems of Linear Equations, 137 -- 4.5.6 The Iteration Method (Jacobi), 142 -- 4.5.7 The Gauss-Seidel Method, 147 -- 4.5.8 The Relaxation Method, 149 -- 4.5.9 The Monte Carlo Method, 150 -- 4.5.10 Infinite Systems of Linear Equations, 152 -- 4.6 Determination of Eigenvalues and Eigenvectors, 153 -- 4.6.1 Introduction, 153 -- 4.6.2 Krylov's Method, 155 -- 4.6.3 Danilevski's Method, 157 -- 4.6.4 The Direct Power Method, 160 -- 4.6.5 The Inverse Power Method, 165 -- 4.6.6 The Displacement Method, 166 -- 4.6.7 Leverrier's Method, 166 -- 4.6.8 The L-R (Left-Right) Method, 166 -- 4.6.9 The Rotation Method, 168 -- 4.7 QR Decomposition, 169 -- 4.8 The Singular Value Decomposition (SVD), 172 -- 4.9 Use of the Least Squares Method in Solving the Linear Overdetermined Systems, 174 -- 4.10 The Pseudo-Inverse of a Matrix, 177 -- 4.11 Solving of the Underdetermined Linear Systems, 178 -- 4.12 Numerical Examples, 178 -- 4.13 Applications, 211 -- Further Reading, 269 -- 5 Solution of Systems of Nonlinear Equations 273 -- 5.1 The Iteration Method (Jacobi), 273 -- 5.2 Newton's Method, 275 -- 5.3 The Modified Newton's Method, 276 -- 5.4 The Newton-Raphson Method, 277 -- 5.5 The Gradient Method, 277 -- 5.6 The Method of Entire Series, 280 -- 5.7 Numerical Example, 281 -- 5.8 Applications, 287 -- Further Reading, 304 -- 6 Interpolation and Approximation of Functions 307 -- 6.1 Lagrange's Interpolation Polynomial, 307 -- 6.2 Taylor Polynomials, 311 -- 6.3 Finite Differences: Generalized Power, 312 -- 6.4 Newton's Interpolation Polynomials, 317 -- 6.5 Central Differences: Gauss's Formulae, Stirling's Formula, Bessel's Formula, Everett's Formulae, 322 -- 6.6 Divided Differences, 327 -- 6.7 Newton-Type Formula with Divided Differences, 331 -- 6.8 Inverse Interpolation, 332 -- 6.9 Determination of the Roots of an Equation by Inverse Interpolation, 333 -- 6.10 Interpolation by Spline Functions, 335.

6.11 Hermite's Interpolation, 339 -- 6.12 Chebyshev's Polynomials, 340 -- 6.13 Mini-Max Approximation of Functions, 344 -- 6.14 Almost Mini-Max Approximation of Functions, 345 -- 6.15 Approximation of Functions by Trigonometric Functions (Fourier), 346 -- 6.16 Approximation of Functions by the Least Squares, 352 -- 6.17 Other Methods of Interpolation, 354 -- 6.17.1 Interpolation with Rational Functions, 354 -- 6.17.2 The Method of Least Squares with Rational Functions, 355 -- 6.17.3 Interpolation with Exponentials, 355 -- 6.18 Numerical Examples, 356 -- 6.19 Applications, 363 -- Further Reading, 374 -- 7 Numerical Differentiation and Integration 377 -- 7.1 Introduction, 377 -- 7.2 Numerical Differentiation by Means of an Expansion into a Taylor Series, 377 -- 7.3 Numerical Differentiation by Means of Interpolation Polynomials, 380 -- 7.4 Introduction to Numerical Integration, 382 -- 7.5 The Newton-Cˆotes Quadrature Formulae, 384 -- 7.6 The Trapezoid Formula, 386 -- 7.7 Simpson's Formula, 389 -- 7.8 Euler's and Gregory's Formulae, 393 -- 7.9 Romberg's Formula, 396 -- 7.10 Chebyshev's Quadrature Formulae, 398 -- 7.11 Legendre's Polynomials, 400 -- 7.12 Gauss's Quadrature Formulae, 405 -- 7.13 Orthogonal Polynomials, 406 -- 7.13.1 Legendre Polynomials, 407 -- 7.13.2 Chebyshev Polynomials, 407 -- 7.13.3 Jacobi Polynomials, 408 -- 7.13.4 Hermite Polynomials, 408 -- 7.13.5 Laguerre Polynomials, 409 -- 7.13.6 General Properties of the Orthogonal Polynomials, 410 -- 7.14 Quadrature Formulae of Gauss Type Obtained by Orthogonal Polynomials, 412 -- 7.14.1 Gauss-Jacobi Quadrature Formulae, 413 -- 7.14.2 Gauss-Hermite Quadrature Formulae, 414 -- 7.14.3 Gauss-Laguerre Quadrature Formulae, 415 -- 7.15 Other Quadrature Formulae, 417 -- 7.15.1 Gauss Formulae with Imposed Points, 417 -- 7.15.2 Gauss Formulae in which the Derivatives of the Function Also Appear, 418 -- 7.16 Calculation of Improper Integrals, 420 -- 7.17 Kantorovich's Method, 422 -- 7.18 The Monte Carlo Method for Calculation of Definite Integrals, 423.

7.18.1 The One-Dimensional Case, 423 -- 7.18.2 The Multidimensional Case, 425 -- 7.19 Numerical Examples, 427 -- 7.20 Applications, 435 -- Further Reading, 447 -- 8 Integration of Ordinary Differential Equations and of Systems of Ordinary Differential Equations 451 -- 8.1 State of the Problem, 451 -- 8.2 Euler's Method, 454 -- 8.3 Taylor Method, 457 -- 8.4 The Runge-Kutta Methods, 458 -- 8.5 Multistep Methods, 462 -- 8.6 Adams's Method, 463 -- 8.7 The Adams-Bashforth Methods, 465 -- 8.8 The Adams-Moulton Methods, 467 -- 8.9 Predictor-Corrector Methods, 469 -- 8.9.1 Euler's Predictor-Corrector Method, 469 -- 8.9.2 Adams's Predictor-Corrector Methods, 469 -- 8.9.3 Milne's Fourth-Order Predictor-Corrector Method, 470 -- 8.9.4 Hamming's Predictor-Corrector Method, 470 -- 8.10 The Linear Equivalence Method (LEM), 471 -- 8.11 Considerations about the Errors, 473 -- 8.12 Numerical Example, 474 -- 8.13 Applications, 480 -- Further Reading, 525 -- 9 Integration of Partial Differential Equations and of Systems of Partial Differential Equations 529 -- 9.1 Introduction, 529 -- 9.2 Partial Differential Equations of First Order, 529 -- 9.2.1 Numerical Integration by Means of Explicit Schemata, 531 -- 9.2.2 Numerical Integration by Means of Implicit Schemata, 533 -- 9.3 Partial Differential Equations of Second Order, 534 -- 9.4 Partial Differential Equations of Second Order of Elliptic Type, 534 -- 9.5 Partial Differential Equations of Second Order of Parabolic Type, 538 -- 9.6 Partial Differential Equations of Second Order of Hyperbolic Type, 543 -- 9.7 Point Matching Method, 546 -- 9.8 Variational Methods, 547 -- 9.8.1 Ritz's Method, 549 -- 9.8.2 Galerkin's Method, 551 -- 9.8.3 Method of the Least Squares, 553 -- 9.9 Numerical Examples, 554 -- 9.10 Applications, 562 -- Further Reading, 575 -- 10 Optimizations 577 -- 10.1 Introduction, 577 -- 10.2 Minimization Along a Direction, 578 -- 10.2.1 Localization of the Minimum, 579 -- 10.2.2 Determination of the Minimum, 580 -- 10.3 Conjugate Directions, 583.

10.4 Powell's Algorithm, 585 -- 10.5 Methods of Gradient Type, 585 -- 10.5.1 The Gradient Method, 585 -- 10.5.2 The Conjugate Gradient Method, 587 -- 10.5.3 Solution of Systems of Linear Equations by Means of Methods of Gradient Type, 589 -- 10.6 Methods of Newton Type, 590 -- 10.6.1 Newton's Method, 590 -- 10.6.2 Quasi-Newton Method, 592 -- 10.7 Linear Programming: The Simplex Algorithm, 593 -- 10.7.1 Introduction, 593 -- 10.7.2 Formulation of the Problem of Linear Programming, 595 -- 10.7.3 Geometrical Interpretation, 597 -- 10.7.4 The Primal Simplex Algorithm, 597 -- 10.7.5 The Dual Simplex Algorithm, 599 -- 10.8 Convex Programming, 600 -- 10.9 Numerical Methods for Problems of Convex Programming, 602 -- 10.9.1 Method of Conditional Gradient, 602 -- 10.9.2 Method of Gradient's Projection, 602 -- 10.9.3 Method of Possible Directions, 603 -- 10.9.4 Method of Penalizing Functions, 603 -- 10.10 Quadratic Programming, 603 -- 10.11 Dynamic Programming, 605 -- 10.12 Pontryagin's Principle of Maximum, 607 -- 10.13 Problems of Extremum, 609 -- 10.14 Numerical Examples, 611 -- 10.15 Applications, 623 -- Further Reading, 626 -- Index 629.