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Preface xi <p><b>1 Errors in Numerical Analysis 1</b></p> <p>1.1 Enter Data Errors, 1</p> <p>1.2 Approximation Errors, 2</p> <p>1.3 Round-Off Errors, 3</p> <p>1.4 Propagation of Errors, 3</p> <p>1.4.1 Addition, 3</p> <p>1.4.2 Multiplication, 5</p> <p>1.4.3 Inversion of a Number, 7</p> <p>1.4.4 Division of Two Numbers, 7</p> <p>1.4.5 Raising to a Negative Entire Power, 7</p> <p>1.4.6 Taking the Root of pth Order, 7</p> <p>1.4.7 Subtraction, 8</p> <p>1.4.8 Computation of Functions, 8</p> <p>1.5 Applications, 8</p> <p>Further Reading, 14</p> <p><b>2 Solution of Equations 17</b></p> <p>2.1 The Bipartition (Bisection) Method, 17</p> <p>2.2 The Chord (Secant) Method, 20</p> <p>2.3 The Tangent Method (Newton), 26</p> <p>2.4 The Contraction Method, 37</p> <p>2.5 The Newton–Kantorovich Method, 42</p> <p>2.6 Numerical Examples, 46</p> <p>2.7 Applications, 49</p> <p>Further Reading, 52</p> <p><b>3 Solution of Algebraic Equations 55</b></p> <p>3.1 Determination of Limits of the Roots of Polynomials, 55</p> <p>3.2 Separation of Roots, 60</p> <p>3.3 Lagrange’s Method, 69</p> <p>3.4 The Lobachevski–Graeffe Method, 72</p> <p>3.4.1 The Case of Distinct Real Roots, 72</p> <p>3.4.2 The Case of a Pair of Complex Conjugate Roots, 74</p> <p>3.4.3 The Case of Two Pairs of Complex Conjugate Roots, 75</p> <p>3.5 The Bernoulli Method, 76</p> <p>3.6 The Bierge–Vi`ete Method, 79</p> <p>3.7 Lin Methods, 79</p> <p>3.8 Numerical Examples, 82</p> <p>3.9 Applications, 94</p> <p>Further Reading, 109</p> <p><b>4 Linear Algebra 111</b></p> <p>4.1 Calculation of Determinants, 111</p> <p>4.1.1 Use of Definition, 111</p> <p>4.1.2 Use of Equivalent Matrices, 112</p> <p>4.2 Calculation of the Rank, 113</p> <p>4.3 Norm of a Matrix, 114</p> <p>4.4 Inversion of Matrices, 123</p> <p>4.4.1 Direct Inversion, 123</p> <p>4.4.2 The Gauss–Jordan Method, 124</p> <p>4.4.3 The Determination of the Inverse Matrix by its Partition, 125</p> <p>4.4.4 Schur’s Method of Inversion of Matrices, 127</p> <p>4.4.5 The Iterative Method (Schulz), 128</p> <p>4.4.6 Inversion by Means of the Characteristic Polynomial, 131</p> <p>4.4.7 The Frame–Fadeev Method, 131</p> <p>4.5 Solution of Linear Algebraic Systems of Equations, 132</p> <p>4.5.1 Cramer’s Rule, 132</p> <p>4.5.2 Gauss’s Method, 133</p> <p>4.5.3 The Gauss–Jordan Method, 134</p> <p>4.5.4 The LU Factorization, 135</p> <p>4.5.5 The Schur Method of Solving Systems of Linear Equations, 137</p> <p>4.5.6 The Iteration Method (Jacobi), 142</p> <p>4.5.7 The Gauss–Seidel Method, 147</p> <p>4.5.8 The Relaxation Method, 149</p> <p>4.5.9 The Monte Carlo Method, 150</p> <p>4.5.10 Infinite Systems of Linear Equations, 152</p> <p>4.6 Determination of Eigenvalues and Eigenvectors, 153</p> <p>4.6.1 Introduction, 153</p> <p>4.6.2 Krylov’s Method, 155</p> <p>4.6.3 Danilevski’s Method, 157</p> <p>4.6.4 The Direct Power Method, 160</p> <p>4.6.5 The Inverse Power Method, 165</p> <p>4.6.6 The Displacement Method, 166</p> <p>4.6.7 Leverrier’s Method, 166</p> <p>4.6.8 The L–R (Left–Right) Method, 166</p> <p>4.6.9 The Rotation Method, 168</p> <p>4.7 QR Decomposition, 169</p> <p>4.8 The Singular Value Decomposition (SVD), 172</p> <p>4.9 Use of the Least Squares Method in Solving the Linear Overdetermined Systems, 174</p> <p>4.10 The Pseudo-Inverse of a Matrix, 177</p> <p>4.11 Solving of the Underdetermined Linear Systems, 178</p> <p>4.12 Numerical Examples, 178</p> <p>4.13 Applications, 211</p> <p>Further Reading, 269</p> <p><b>5 Solution of Systems of Nonlinear Equations 273</b></p> <p>5.1 The Iteration Method (Jacobi), 273</p> <p>5.2 Newton’s Method, 275</p> <p>5.3 The Modified Newton’s Method, 276</p> <p>5.4 The Newton–Raphson Method, 277</p> <p>5.5 The Gradient Method, 277</p> <p>5.6 The Method of Entire Series, 280</p> <p>5.7 Numerical Example, 281</p> <p>5.8 Applications, 287</p> <p>Further Reading, 304</p> <p><b>6 Interpolation and Approximation of Functions 307</b></p> <p>6.1 Lagrange’s Interpolation Polynomial, 307</p> <p>6.2 Taylor Polynomials, 311</p> <p>6.3 Finite Differences: Generalized Power, 312</p> <p>6.4 Newton’s Interpolation Polynomials, 317</p> <p>6.5 Central Differences: Gauss’s Formulae, Stirling’s Formula, Bessel’s Formula, Everett’s Formulae, 322</p> <p>6.6 Divided Differences, 327</p> <p>6.7 Newton-Type Formula with Divided Differences, 331</p> <p>6.8 Inverse Interpolation, 332</p> <p>6.9 Determination of the Roots of an Equation by Inverse Interpolation, 333</p> <p>6.10 Interpolation by Spline Functions, 335</p> <p>6.11 Hermite’s Interpolation, 339</p> <p>6.12 Chebyshev’s Polynomials, 340</p> <p>6.13 Mini–Max Approximation of Functions, 344</p> <p>6.14 Almost Mini–Max Approximation of Functions, 345</p> <p>6.15 Approximation of Functions by Trigonometric Functions (Fourier), 346</p> <p>6.16 Approximation of Functions by the Least Squares, 352</p> <p>6.17 Other Methods of Interpolation, 354</p> <p>6.17.1 Interpolation with Rational Functions, 354</p> <p>6.17.2 The Method of Least Squares with Rational Functions, 355</p> <p>6.17.3 Interpolation with Exponentials, 355</p> <p>6.18 Numerical Examples, 356</p> <p>6.19 Applications, 363</p> <p>Further Reading, 374</p> <p><b>7 Numerical Differentiation and Integration 377</b></p> <p>7.1 Introduction, 377</p> <p>7.2 Numerical Differentiation by Means of an Expansion into a Taylor Series, 377</p> <p>7.3 Numerical Differentiation by Means of Interpolation Polynomials, 380</p> <p>7.4 Introduction to Numerical Integration, 382</p> <p>7.5 The Newton–Cˆotes Quadrature Formulae, 384</p> <p>7.6 The Trapezoid Formula, 386</p> <p>7.7 Simpson’s Formula, 389</p> <p>7.8 Euler’s and Gregory’s Formulae, 393</p> <p>7.9 Romberg’s Formula, 396</p> <p>7.10 Chebyshev’s Quadrature Formulae, 398</p> <p>7.11 Legendre’s Polynomials, 400</p> <p>7.12 Gauss’s Quadrature Formulae, 405</p> <p>7.13 Orthogonal Polynomials, 406</p> <p>7.13.1 Legendre Polynomials, 407</p> <p>7.13.2 Chebyshev Polynomials, 407</p> <p>7.13.3 Jacobi Polynomials, 408</p> <p>7.13.4 Hermite Polynomials, 408</p> <p>7.13.5 Laguerre Polynomials, 409</p> <p>7.13.6 General Properties of the Orthogonal Polynomials, 410</p> <p>7.14 Quadrature Formulae of Gauss Type Obtained by Orthogonal Polynomials, 412</p> <p>7.14.1 Gauss–Jacobi Quadrature Formulae, 413</p> <p>7.14.2 Gauss–Hermite Quadrature Formulae, 414</p> <p>7.14.3 Gauss–Laguerre Quadrature Formulae, 415</p> <p>7.15 Other Quadrature Formulae, 417</p> <p>7.15.1 Gauss Formulae with Imposed Points, 417</p> <p>7.15.2 Gauss Formulae in which the Derivatives of the Function Also Appear, 418</p> <p>7.16 Calculation of Improper Integrals, 420</p> <p>7.17 Kantorovich’s Method, 422</p> <p>7.18 The Monte Carlo Method for Calculation of Definite Integrals, 423</p> <p>7.18.1 The One-Dimensional Case, 423</p> <p>7.18.2 The Multidimensional Case, 425</p> <p>7.19 Numerical Examples, 427</p> <p>7.20 Applications, 435</p> <p>Further Reading, 447</p> <p><b>8 Integration of Ordinary Differential Equations and of Systems of Ordinary Differential Equations 451</b></p> <p>8.1 State of the Problem, 451</p> <p>8.2 Euler’s Method, 454</p> <p>8.3 Taylor Method, 457</p> <p>8.4 The Runge–Kutta Methods, 458</p> <p>8.5 Multistep Methods, 462</p> <p>8.6 Adams’s Method, 463</p> <p>8.7 The Adams–Bashforth Methods, 465</p> <p>8.8 The Adams–Moulton Methods, 467</p> <p>8.9 Predictor–Corrector Methods, 469</p> <p>8.9.1 Euler’s Predictor–Corrector Method, 469</p> <p>8.9.2 Adams’s Predictor–Corrector Methods, 469</p> <p>8.9.3 Milne’s Fourth-Order Predictor–Corrector Method, 470</p> <p>8.9.4 Hamming’s Predictor–Corrector Method, 470</p> <p>8.10 The Linear Equivalence Method (LEM), 471</p> <p>8.11 Considerations about the Errors, 473</p> <p>8.12 Numerical Example, 474</p> <p>8.13 Applications, 480</p> <p>Further Reading, 525</p> <p><b>9 Integration of Partial Differential Equations and of Systems of Partial Differential Equations 529</b></p> <p>9.1 Introduction, 529</p> <p>9.2 Partial Differential Equations of First Order, 529</p> <p>9.2.1 Numerical Integration by Means of Explicit Schemata, 531</p> <p>9.2.2 Numerical Integration by Means of Implicit Schemata, 533</p> <p>9.3 Partial Differential Equations of Second Order, 534</p> <p>9.4 Partial Differential Equations of Second Order of Elliptic Type, 534</p> <p>9.5 Partial Differential Equations of Second Order of Parabolic Type, 538</p> <p>9.6 Partial Differential Equations of Second Order of Hyperbolic Type, 543</p> <p>9.7 Point Matching Method, 546</p> <p>9.8 Variational Methods, 547</p> <p>9.8.1 Ritz’s Method, 549</p> <p>9.8.2 Galerkin’s Method, 551</p> <p>9.8.3 Method of the Least Squares, 553</p> <p>9.9 Numerical Examples, 554</p> <p>9.10 Applications, 562</p> <p>Further Reading, 575</p> <p><b>10 Optimizations 577</b></p> <p>10.1 Introduction, 577</p> <p>10.2 Minimization Along a Direction, 578</p> <p>10.2.1 Localization of the Minimum, 579</p> <p>10.2.2 Determination of the Minimum, 580</p> <p>10.3 Conjugate Directions, 583</p> <p>10.4 Powell’s Algorithm, 585</p> <p>10.5 Methods of Gradient Type, 585</p> <p>10.5.1 The Gradient Method, 585</p> <p>10.5.2 The Conjugate Gradient Method, 587</p> <p>10.5.3 Solution of Systems of Linear Equations by Means of Methods of Gradient Type, 589</p> <p>10.6 Methods of Newton Type, 590</p> <p>10.6.1 Newton’s Method, 590</p> <p>10.6.2 Quasi-Newton Method, 592</p> <p>10.7 Linear Programming: The Simplex Algorithm, 593</p> <p>10.7.1 Introduction, 593</p> <p>10.7.2 Formulation of the Problem of Linear Programming, 595</p> <p>10.7.3 Geometrical Interpretation, 597</p> <p>10.7.4 The Primal Simplex Algorithm, 597</p> <p>10.7.5 The Dual Simplex Algorithm, 599</p> <p>10.8 Convex Programming, 600</p> <p>10.9 Numerical Methods for Problems of Convex Programming, 602</p> <p>10.9.1 Method of Conditional Gradient, 602</p> <p>10.9.2 Method of Gradient’s Projection, 602</p> <p>10.9.3 Method of Possible Directions, 603</p> <p>10.9.4 Method of Penalizing Functions, 603</p> <p>10.10 Quadratic Programming, 603</p> <p>10.11 Dynamic Programming, 605</p> <p>10.12 Pontryagin’s Principle of Maximum, 607</p> <p>10.13 Problems of Extremum, 609</p> <p>10.14 Numerical Examples, 611</p> <p>10.15 Applications, 623</p> <p>Further Reading, 626</p> <p>Index 629</p>

<p><b>PETRE TEODORESCU, P<small>H</small>D,</b> is a Professor in the Faculty of Mathematics and Computer Science at the University of Bucharest in Romania and the author of 250 papers and twenty-eight books. </p> <p><b> NICOLAE-DORU STĂNESCU, P<small>H</small>D,</b> is a Professor in the Faculty of Mechanics and Technology at the University of Piteşti in Romania and the author of 200 papers and ten books. <p><b>NICOLAE PANDREA, P<small>H</small>D,</b> is a Professor in the Faculty of Mechanics and Technology at the University of Piteşti in Romania and the author of 250 papers and six books.

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