# Computing with Polynomials

Computing with Polynomials

Simple Fixed-Point Iteration 146 6.2 The Newton-Raphson Method 151 6.3 The Secant Method 157 6.4 Brent’s Method 162 6.5 Multiple Roots 166 6.6 Systems of Nonlinear Equations 169 Problems 173

CHAPTER 7 Roots of Polynomials 176

7.1 Polynomials in Engineering and Science 176 7.2 Computing with Polynomials 179 7.3 Conventional Methods 182

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7.4 Müller’s Method 183 7.5 Bairstow’s Method 187 7.6 Other Methods 192 7.7 Root Location with Software Packages 192 Problems 202

CHAPTER 8 Case Studies: Roots of Equations 204

8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 204 8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 207 8.3 Design of an Electric Circuit (Electrical Engineering) 209 8.4 Pipe Friction (Mechanical/Aerospace Engineering) 212 Problems 215

EPILOGUE: PART TWO 226 PT2.4 Trade-Offs 226 PT2.5 Important Relationships and Formulas 227 PT2.6 Advanced Methods and Additional References 227

PART THREE

LINEAR ALGEBRAIC PT3.1 Motivation 231 EQUATIONS 231 PT3.2 Mathematical Background 233

PT3.3 Orientation 241

CHAPTER 9 Gauss Elimination 245

9.1 Solving Small Numbers of Equations 245 9.2 Naive Gauss Elimination 252 9.3 Pitfalls of Elimination Methods 258 9.4 Techniques for Improving Solutions 264 9.5 Complex Systems 271 9.6 Nonlinear Systems of Equations 271 9.7 Gauss-Jordan 273 9.8 Summary 275 Problems 275

CHAPTER 10 LU Decomposition and Matrix Inversion 278 10.1 LU Decomposition 278 10.2 The Matrix Inverse 287 10.3 Error Analysis and System Condition 291 Problems 297

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CHAPTER 11 Special Matrices and Gauss-Seidel 300

11.1 Special Matrices 300 11.2 Gauss-Seidel 304 11.3 Linear Algebraic Equations with Software Packages 311 Problems 316

CHAPTER 12 Case Studies: Linear Algebraic Equations 319

12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering) 319 12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) 322 12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering) 326 12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) 328 Problems 331

EPILOGUE: PART THREE 341 PT3.4 Trade-Offs 341 PT3.5 Important Relationships and Formulas 342 PT3.6 Advanced Methods and Additional References 342

PART FOUR

OPTIMIZATION 345 PT4.1 Motivation 345 PT4.2 Mathematical Background 350 PT4.3 Orientation 351

CHAPTER 13 One-Dimensional Unconstrained Optimization 355

13.1 Golden-Section Search 356 13.2 Parabolic Interpolation 363 13.3 Newton’s Method 365 13.4 Brent’s Method 366 Problems 368

CHAPTER 14 Multidimensional Unconstrained Optimization 370

14.1 Direct Methods 371 14.2 Gradient Methods 375 Problems 388

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CHAPTER 15 Constrained Optimization 390

15.1 Linear Programming 390 15.2 Nonlinear Constrained Optimization 401 15.3 Optimization with Software Packages 402 Problems 413

CHAPTER 16 Case Studies: Optimization 416

16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering) 416 16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 421 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) 425 16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering) 429 Problems 431

EPILOGUE: PART FOUR 438 PT4.4 Trade-Offs 438 PT4.5 Additional References 439

PART FIVE

CURVE FITTING 441 PT5.1 Motivation 441 PT5.2 Mathematical Background 443 PT5.3 Orientation 452

CHAPTER 17 Least-Squares Regression 456

17.1 Linear Regression 456 17.2 Polynomial Regression 472 17.3 Multiple Linear Regression 476 17.4 General Linear Least Squares 479 17.5 Nonlinear Regression 483 Problems 487

CHAPTER 18 Interpolation 490

18.1 Newton’s Divided-Difference Interpolating Polynomials 491 18.2 Lagrange Interpolating Polynomials 502 18.3 Coeffi cients of an Interpolating Polynomial 507 18.4 Inverse Interpolation 507 18.5 Additional Comments 508 18.6 Spline Interpolation 511 18.7 Multidimensional Interpolation 521 Problems 524

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CHAPTER 19 Fourier Approximation 526

19.1 Curve Fitting with Sinusoidal Functions 527 19.2 Continuous Fourier Series 533 19.3 Frequency and Time Domains 536 19.4 Fourier Integral and Transform 540 19.5 Discrete Fourier Transform (DFT) 542 19.6 Fast Fourier Transform (FFT) 544 19.7 The Power Spectrum 551 19.8 Curve Fitting with Software Packages 552 Problems 561

CHAPTER 20 Case Studies: Curve Fitting 563

20.1 Linear Regression and Population Models (Chemical/Bio Engineering) 563 20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) 567 20.3 Fourier Analysis (Electrical Engineering) 569 20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering) 570 Problems 572

EPILOGUE: PART FIVE 582 PT5.4 Trade-Offs 582 PT5.5 Important Relationships and Formulas 583 PT5.6 Advanced Methods and Additional References 584

PART SIX

NUMERICAL PT6.1 Motivation 587 DIFFERENTIATION PT6.2 Mathematical Background 597 AND PT6.3 Orientation 599 INTEGRATION 587

CHAPTER 21 Newton-Cotes Integration Formulas 603

21.1 The Trapezoidal Rule 605 21.2 Simpson’s Rules 615 21.3 Integration with Unequal Segments 624 21.4 Open Integration Formulas 627 21.5 Multiple Integrals 627 Problems 629

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CHAPTER 22 Integration of Equations 633

22.1 Newton-Cotes Algorithms for Equations 633 22.2 Romberg Integration 634 22.3 Adaptive Quadrature 640 22.4 Gauss Quadrature 642 22.5 Improper Integrals 650 Problems 653

CHAPTER 23 Numerical Differentiation 655

23.1 High-Accuracy Differentiation Formulas 655 23.2 Richardson Extrapolation 658 23.3 Derivatives of Unequally Spaced Data 660 23.4 Derivatives and Integrals for Data with Errors 661 23.5 Partial Derivatives 662 23.6 Numerical Integration/Differentiation with Software Packages 663 Problems 670

CHAPTER 24 Case Studies: Numerical Integration and Differentiation 673

24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) 673

24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) 675

24.3 Root-Mean-Square Current by Numerical Integration (Electrical Engineering) 677

24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 680

Problems 684

EPILOGUE: PART SIX 694 PT6.4 Trade-Offs 694 PT6.5 Important Relationships and Formulas 695 PT6.6 Advanced Methods and Additional References 695

PART SEVEN

ORDINARY PT7.1 Motivation 699 DIFFERENTIAL PT7.2 Mathematical Background 703 EQUATIONS 699 PT7.3 Orientation 705

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CHAPTER 25 Runge-Kutta Methods 709

25.1 Euler’s Method 710 25.2 Improvements of Euler’s Method 721 25.3 Runge-Kutta Methods 729 25.4 Systems of Equations 739 25.5 Adaptive Runge-Kutta Methods 744 Problems 752

CHAPTER 26 Stiffness and Multistep Methods 755

26.1 Stiffness 755 26.2 Multistep Methods 759 Problems 779

CHAPTER 27 Boundary-Value and Eigenvalue Problems 781

27.1 General Methods for Boundary-Value Problems 782 27.2 Eigenvalue Problems 789 27.3 Odes and Eigenvalues with Software Packages 801 Problems 808

CHAPTER 28 Case Studies: Ordinary Differential Equations 811

28.1 Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) 811

28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering) 818 28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering) 822 28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering) 827 Problems 831

EPILOGUE: PART SEVEN 841 PT7.4 Trade-Offs 841 PT7.5 Important Relationships and Formulas 842 PT7.6 Advanced Methods and Additional References 842

PART EIGHT

PARTIAL PT8.1 Motivation 845 DIFFERENTIAL PT8.2 Orientation 848 EQUATIONS 845

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CHAPTER 29 Finite Difference: Elliptic Equations 852

29.1 The Laplace Equation 852 29.2 Solution Technique 854 29.3 Boundary Conditions 860 29.4 The Control-Volume Approach 866 29.5 Software to Solve Elliptic Equations 869 Problems 870

CHAPTER 30 Finite Difference: Parabolic Equations 873

30.1 The Heat-Conduction Equation 873 30.2 Explicit Methods 874 30.3 A Simple Implicit Method 878 30.4 The Crank-Nicolson Method 882 30.5 Parabolic Equations in Two Spatial Dimensions 885 Problems 888

CHAPTER 31 Finite-Element Method 890

31.1 The General Approach 891 31.2 Finite-Element Application in One Dimension 895 31.3 Two-Dimensional Problems 904 31.4 Solving PDEs with Software Packages 908 Problems 912

CHAPTER 32 Case Studies: Partial Differential Equations 915

32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering) 915

32.2 Defl ections of a Plate (Civil/Environmental Engineering) 919 32.3 Two-Dimensional Electrostatic Field Problems (Electrical

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