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 موضوع: كتاب Numerical Methods for Engineers Sixth Edition الأحد 18 ديسمبر 2022, 10:53 pm  

أخواني في الله أحضرت لكم كتاب Numerical Methods for Engineers Sixth Edition Steven C. Chapra Berger Chair in Computing and Engineering Tufts University Raymond P. Canale Professor Emeritus of Civil Engineering University of Michigan
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Contents Preface Xiv Guided Tour Xvi About the Authors Xviii Part One Modeling, Pt . Motivation Computers, and Pt . Mathematical Background Error Analysis Pt . Orientation Chapter Mathematical Modeling and Engineering Problem Solving . A Simple Mathematical Model . Conservation Laws and Engineering Problems CHAPTER Programming and Software . Packages and Programming . Structured Programming . Modular Programming . Excel . MATLAB . Mathcad . Other Languages and Libraries Problems CHAPTER Approximations and RoundOff Errors . Significant Figures . Accuracy and Precision . Error Definitions . RoundOff Errors Problems ivCONTENTS v CHAPTER Truncation Errors and the Taylor Series . The Taylor Series . Error Propagation . Total Numerical Error . Blunders, Formulation Errors, and Data Uncertainty Problems EPILOGUE: PART ONE PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References PART TWO ROOTS OF PT . Motivation EQUATIONS PT . Mathematical Background PT . Orientation CHAPTER Bracketing Methods . Graphical Methods . The Bisection Method . The FalsePosition Method . Incremental Searches and Determining Initial Guesses Problems CHAPTER Open Methods . Simple FixedPoint Iteration . The NewtonRaphson Method . The Secant Method . Brent’s Method . Multiple Roots . Systems of Nonlinear Equations Problems CHAPTER Roots of Polynomials . Polynomials in Engineering and Science . Computing with Polynomials . Conventional Methods vi CONTENTS . Müller’s Method . Bairstow’s Method . Other Methods . Root Location with Software Packages Problems CHAPTER Case Studies: Roots of Equations . Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) . Greenhouse Gases and Rainwater (Civil/Environmental Engineering) . Design of an Electric Circuit (Electrical Engineering) . Pipe Friction (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART TWO PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References PART THREE LINEAR ALGEBRAIC PT . Motivation EQUATIONS PT . Mathematical Background PT . Orientation CHAPTER Gauss Elimination . Solving Small Numbers of Equations . Naive Gauss Elimination . Pitfalls of Elimination Methods . Techniques for Improving Solutions . Complex Systems . Nonlinear Systems of Equations . GaussJordan . Summary Problems CHAPTER LU Decomposition and Matrix Inversion . LU Decomposition . The Matrix Inverse . Error Analysis and System Condition Problems CONTENTS vii CHAPTER Special Matrices and GaussSeidel . Special Matrices . GaussSeidel . Linear Algebraic Equations with Software Packages Problems CHAPTER Case Studies: Linear Algebraic Equations . SteadyState Analysis of a System of Reactors (Chemical/Bio Engineering) . Analysis of a Statically Determinate Truss (Civil/Environmental Engineering) . Currents and Voltages in Resistor Circuits (Electrical Engineering) . SpringMass Systems (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART THREE PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References PART FOUR OPTIMIZATION PT . Motivation PT . Mathematical Background PT . Orientation CHAPTER OneDimensional Unconstrained Optimization . GoldenSection Search . Parabolic Interpolation . Newton’s Method . Brent’s Method Problems CHAPTER Multidimensional Unconstrained Optimization . Direct Methods . Gradient Methods Problems viii CONTENTS CHAPTER Constrained Optimization . Linear Programming . Nonlinear Constrained Optimization . Optimization with Software Packages Problems CHAPTER Case Studies: Optimization . LeastCost Design of a Tank (Chemical/Bio Engineering) . LeastCost Treatment of Wastewater (Civil/Environmental Engineering) . Maximum Power Transfer for a Circuit (Electrical Engineering) . Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART FOUR PT . TradeOffs PT . Additional References PART FIVE CURVE FITTING PT . Motivation PT . Mathematical Background PT . Orientation CHAPTER LeastSquares Regression . Linear Regression . Polynomial Regression . Multiple Linear Regression . General Linear Least Squares . Nonlinear Regression Problems CHAPTER Interpolation . Newton’s DividedDifference Interpolating Polynomials . Lagrange Interpolating Polynomials . Coefficients of an Interpolating Polynomial . Inverse Interpolation . Additional Comments . Spline Interpolation . Multidimensional Interpolation Problems CONTENTS ix CHAPTER Fourier Approximation . Curve Fitting with Sinusoidal Functions . Continuous Fourier Series . Frequency and Time Domains . Fourier Integral and Transform . Discrete Fourier Transform (DFT) . Fast Fourier Transform (FFT) . The Power Spectrum . Curve Fitting with Software Packages Problems CHAPTER Case Studies: Curve Fitting . Linear Regression and Population Models (Chemical/Bio Engineering) . Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) . Fourier Analysis (Electrical Engineering) . Analysis of Experimental Data (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART FIVE PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References PART SIX NUMERICAL PT . Motivation DIFFERENTIATION PT . Mathematical Background AND PT . Orientation INTEGRATION CHAPTER NewtonCotes Integration Formulas . The Trapezoidal Rule . Simpson’s Rules . Integration with Unequal Segments . Open Integration Formulas . Multiple Integrals Problems x CONTENTS CHAPTER Integration of Equations . NewtonCotes Algorithms for Equations . Romberg Integration . Adaptive Quadrature . Gauss Quadrature . Improper Integrals Problems CHAPTER Numerical Differentiation . HighAccuracy Differentiation Formulas . Richardson Extrapolation . Derivatives of Unequally Spaced Data . Derivatives and Integrals for Data with Errors . Partial Derivatives . Numerical Integration/Differentiation with Software Packages Problems CHAPTER Case Studies: Numerical Integration and Differentiation . Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) . Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) . RootMeanSquare Current by Numerical Integration (Electrical Engineering) . Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART SIX PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References PART SEVEN ORDINARY PT . Motivation DIFFERENTIAL PT . Mathematical Background EQUATIONS PT . Orientation CONTENTS xi CHAPTER RungeKutta Methods . Euler’s Method . Improvements of Euler’s Method . RungeKutta Methods . Systems of Equations . Adaptive RungeKutta Methods Problems CHAPTER Stiffness and Multistep Methods . Stiffness . Multistep Methods Problems CHAPTER BoundaryValue and Eigenvalue Problems . General Methods for BoundaryValue Problems . Eigenvalue Problems . Odes and Eigenvalues with Software Packages Problems CHAPTER Case Studies: Ordinary Differential Equations . Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) . PredatorPrey Models and Chaos (Civil/Environmental Engineering) . Simulating Transient Current for an Electric Circuit (Electrical Engineering) . The Swinging Pendulum (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART SEVEN PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References PART EIGHT PARTIAL PT . Motivation DIFFERENTIAL PT . Orientation EQUATIONS xii CONTENTS CHAPTER Finite Difference: Elliptic Equations . The Laplace Equation . Solution Technique . Boundary Conditions . The ControlVolume Approach . Software to Solve Elliptic Equations Problems CHAPTER Finite Difference: Parabolic Equations . The HeatConduction Equation . Explicit Methods . A Simple Implicit Method . The CrankNicolson Method . Parabolic Equations in Two Spatial Dimensions Problems CHAPTER FiniteElement Method . The General Approach . FiniteElement Application in One Dimension . TwoDimensional Problems . Solving PDEs with Software Packages Problems CHAPTER Case Studies: Partial Differential Equations . OneDimensional Mass Balance of a Reactor (Chemical/Bio Engineering) . Deflections of a Plate (Civil/Environmental Engineering) . TwoDimensional Electrostatic Field Problems (Electrical Engineering) . FiniteElement Solution of a Series of Springs (Mechanical/Aerospace Engineering) Problems EPILOGUE: PART EIGHT PT . TradeOffs PT . Important Relationships and Formulas PT . Advanced Methods and Additional References APPENDIX A: THE FOURIER SERIES APPENDIX B: GETTING STARTED WITH MATLAB APPENDIX C: GETTING STARTED WITH MATHCAD BIBLIOGRAPHY INDEX INDEX A Accuracy, Accuracy and precision, – AdamsBashforth formulas, – Adams formulas AdamsBashforth formulas, – AdamsMoulton formulas, – closed formulas, – NewtonColes, contrasted, , open formulas, – AdamsMoulton formulas, – Adaptive fourthorder RK method, , – Adaptive quadrature, – Adaptive RungeKutta methods, – adaptive fourthorder RK method, , – adaptive stepsize control, pseudocode, , RungeKutta Fehlberg, – stepsize control, – Adaptive stepsize control, Adding large and small number, – Addition, ADI scheme, – Advanced methods/additional references background material, – curve fitting, linear algebraic equations, – numerical differentiation and integration, ODEs, – optimization, PDEs, roots of equations, – Advectiondispersion equation, Algorithms, Alternatingdirection implicit (ADI) scheme, – Alzheimer’s disease, Amplitude, Analysis of experimental data, – Analytical solution, , Andrade’s equation, Angular frequency, Antidifferentiation, Approximate percent relative error, Approximation function, Approximations and roundoff errors, – accuracy and precision, – error definitions, – iterative calculation, – roundoff error. See Roundoff error significant figures, – Areal integral, Arithmetic manipulations of computer numbers, – Arithmetic mean, Artificial neural network, Ascent methods, Assemblage property matrix, Augmentation, Auxiliary conditions, Axiallydispersed plugflow reactor, – Axially loaded column, – , B B splines, Back substitution, – Background material, – accuracy and figures, – additional references, – blunders, – computer objectives, computer programming, – . See also Computer programming condition, conservation laws, – data uncertainty, error, – error definitions, – error propagation, – formulation errors, important relationships/formulas, iterative calculation, – mathematical model, – overview, , roundoff error. See Roundoff error scope/preview, , significant figures, – stability, study objectives, Taylor series. See Taylor series total numerical error, – truncation errors, Backward deflation, Backward difference approximation, Backward Euler’s method, Backward finitedivideddifference formula, Bairstow’s method, – Banded matrix, , Base number system, Base number system, Base system, Basic feasible solution, Basic variables, Best fit, – BFGS, , Bias, Bibliography, – Bilinear functions, Bilinear interpolation, – Binary chopping, Binary system, Binding constraints, Bingham plastics, Biofilm, Bisection method, – algorithm, alternative names, error estimates, – falseposition method, compared, – minimizing function evaluations, pseudocode, termination criteria, Bit reversal, Bits, Blasius formula, Blood, Blunders, – Bolzano’s method, Book, overview, – Boole’s rule, , Boundary conditions, – Boundaryvalue problems, – eigenvalues, – finitedifference methods, – nonlinear twopoint problems, – other techniques, – shooting method, – Boussinesq’s equation, Bracketing methods, – bisection method, – falseposition method, – graphical methods, – incremental searches, initial guesses, modified false position, – open methods, compared, Breadth of application, Break loop, – Brent, Richard, Brent’s method algorithm, – inverse quadratic interpolation, – optimization, , pseudocode, , roots of equations, – Brent’s root finding method, Brent’s rootlocation method, BroydenFletcherGoldfarbShanno (BFGS), , Bubble sort, – Buckling load, BulirschStoer method, Butcher’s fifthorder RK method, Butterfly network, C Calculator, Calculus, . See also Numerical differentiation and integration Canned programs, CASE structure, , Case studies analysis of experimental data, – chaos, – currents and voltages in resistor circuits, – curve fitting, – deflections of a plate, – electrical circuit design, – equilibrium and minimum potential energy, – INDEX Fourier analysis, – greenhouse gases, – heat calculations, – heat transfer, – ideal/nonideal gas laws, – leastcost design of tank, – leastcost treatment of wastewater, – linear algebraic equations, – linear regression and population models, – mass balance of reactor, – maximum power transfer for circuit, – numerical integration/differentiation, – ODEs, – optimization, – PDEs, – pipe friction, – population models, – predatorprey models, – rainwater, – reactor, – , – reactors, – rootmeansquare current, – roots of equations, – sailboat, – series of springs, – simulating transient current for electric current, – springmass system, – statistically determinate truss, – steadystate analysis of system of reactors, – swinging pendulum, – transient responses of reactor, – truss, – twodimensional electrostatic field problems, – work, calculation of, – CashKarp RK method, , Casson region, Casson relationship, Catenary cable, Centered difference approximation, Centered finitedivideddifference formula, Central limit theorem, Chaos, – Chaotic solutions, Characteristic, Characteristic equation, Characteristic value, Chebyshev economization, Cholesky decomposition, – Chopping, Classical fourthorder RK method, – Closed integration formulas, , Coefficient of determination, Coefficient of thermal conductivity, Coefficient of thermal diffusivity, Coefficient of variation (c.v.), Colebrook equation, , Collocation approach, Column, Columnsum norm, Column vectors, Commercial programming libraries, Comparison of onedimensional methods, – Complete pivoting, Composite integration formulas, Computer, Computer program, Computer programming, – algorithms, CASE structure, , commercial programming libraries, computer program, countcontrolled loop, – DOEXIT construct, , DOFOR loop, – Excel. See Excel flowchart, highlevel languages, IF/THEN/ELSE structure, , IF/THEN/ELSEIF structure, , IF/THEN structure, , logical representation, – loops, – Mathcad. See Mathcad MATLAB. See MATLAB modular programming, – numerical library, other languages, packages, – pseudocode, , repetition, – selection, – sequence, structured programming, – Computer representation of numbers, – Computer software. See Excel; Mathcad; MATLAB Condition, Condition number, , – Confidence intervals, – Confidence intervals for linear regression, – Conjugate directions, Conjugate gradient method, Conservation laws, – Conservation of charge, Conservation of energy, Conservation of mass, , Conservation of momentum, Consistency index, Constant of integration, Constitutive equation, Constrained optimization, , – linear programming, – nonlinear, simplex method, – Continuous Fourier series, – Controlvolume approach, – Convergence, , , Cooley, J. W., CooleyTukey algorithm, – Corrector, Corrector equation, Corrector modifier, Correlation coefficient, Countcontrolled loop, – Covariance, n Cramer’s rule, – , CrankNicolson algorithm, CrankNicolson method, – Creep rate, Critically damped case, Crout decomposition, – Cubic interpolating polynomial, Cubic splines, , , – Cumulative normal distribution, Current balance, Currents and voltages in resistor circuits, – Curvature, Curve fitting, – advanced methods/references, analysis of experimental data, – approaches, case studies, – computer objectives, – engineering practice, Excel, – Fourier analysis, – Fourier approximation. See Fourier approximation heat transfer, – important relationships/formulas, INDEX interpolation. See Interpolation leastsquares regression. See Leastsquares regression linear regression and population models, – Mathcad, – mathematical background, – MATLAB, – methods, compared, noncomputer methods, overview, – population models, – scope/preview, – sinusoidal functions, – statistics, – study objectives, , tradeoffs, – Curvilinear interpolation, c.v., D Danckwerts boundary condition, Data distribution, Data uncertainty, DavidonFletcherPowell (DFP), Decimal number system, Decimation in time, , Decimation in frequency, Decision loop, Definite integration, n Deflation, Deflections of a plate, – Degrees of freedom, Dekker, Theodorus, del f, Dependent variable, , Derivative, Derivative boundary conditions, – , Derivative form, Derivative meanvalue theorem, Descent methods, Descriptive models, Design, Design of electric circuit, – Design vector, Determinant, – Determinant evaluation, DFP, DFT, – Diagonal matrix, Diagonally dominant, Differential calculus, Differential equation defined, ordinary. See Ordinary differential equations partial. See Partial differential equations what is it, Differentiation, – . See also Numerical differentiation and integration Dirac delta function, Direct approach, , – Direct methods, – Directional derivative, Dirichlet boundary condition, , , Dirichlet condition, n Discrete Fourier transform (DFT), – Discretization errors, , Discriminant, Distributedparameter system, Distributed variable problems, , Distribution coefficient, Divide and average method, Divideddifference table, Division, DOEXIT construct, , DOFOR loop, – Doolittle decomposition, Double integral, – Double precision, Double root, Drag coefficient, Dynamic instability, E Ease of application, Eigenvalue, , Eigenvalue problems, – axially loaded column, – , boundaryvalue problem, – Given’s method, Hotelling’s method, Householder’s method, intermediate eigenvalue, Jacobi’s method, – largest eigenvalue, – LR method, mathematical background, physical background, – polynomial method, – power method, – QR method, smallest eigenvalue, Eigenvector, Electrical circuit design, – Element property, Element stiffness matrix, Elimination methods, – , Elimination of unknowns, – Elliptic equations, – boundary conditions, – controlvolume approach, – derivative boundary conditions, – flux distribution for heated plate, – heated plate with insulated edge, – heated plate with irregular boundary, – irregular boundaries, – Laplace equation, – Laplacian difference equation, – Liebmann’s method, – secondary variables, software, – solution technique, – temperature of heated plate, – Embedded RK method, Energy balance, Engineering practice. See Case studies Engineering problem solving accuracy and precision, – blunders, – computer programming, – . See also Computer programming condition, conservation laws, – data uncertainty, error, – error definitions, – error propagation, – formulation errors, iterative calculation, – mathematical model, – phases, process, roundoff error. See Roundoff error significant figures, – stability, Taylor series. See Taylor series total numerical error, – truncation errors, twopronged approach, Engineering problemsolving process, Entering variable, Enzymatic reactions, Epilimnion, Equalarea graphical differentiation, , Equality constraints, Equilibrium and minimum potential energy, – Ergun equation, INDEX Error. See Approximations and roundoff errors; Truncation errors and Taylor series Error analysis and system condition, – Error definitions, – Error estimates for interactive methods, – Error propagation, – Estimated mean, Estimation, Euclidean norm, – EulerCauchy method, Euler’s formula, Euler’s identity, , Euler’s method, – algorithm, – backward, basic equation, C, computer program, – equation and basic approach, – error analysis, – Excel, Fortran function, fundamental source of error, Heun’s method, higherorder Taylor series methods, implicit, – MATLAB, pseudocode, , solving differential equation, step size, – systems of equations, Taylor series, , Even function, Exact method, Exact solution, Excel, – builtin numerical capabilities, condition number, conjugate gradient approach, curve fitting, – Data Analysis Toolpack, – Dirichlet boundary conditions, double precision, Euler’s method, evaluation of ex using infinite series, – factorial, , Goal Seek, iterative calculation, leastcost treatment of wastewater, – Liebmann method, linear equations, – linear programming, – maximum power transfer, – nonlinear optimization, – nonlinear regression, , – nonlinear system of equations, – optimization, – parachute optimization problem, – parachutist problem, , PDEs, – random search, reaction kinetics, – regression fits of data, roots of equations, – roots of quadratic, Solver, – Trendline, – VBA macro, visualization tools, Explicit methods, – , Exponent, Exponential model, Extended midpoint rule, Extended precision, – Extrapolation, Extreme point, F Factored form of polynomial, Factorial, , Factorization, Falling parachutist problem. See Parachutist problem Falseposition formula, Falseposition method, – bisection, compared, – derivation, modified false position, – name, pitfalls, – secant method, compared, – Fanning friction factor, – Faraday’s law, , , Fast Fourier transform (FFT), – Feasible extreme points, Feasible solution space, FFT, – Fibonacci numbers, Fick’s first law, Fick’s law of diffusion, Finitedifference approximations, – Finite difference approximations of higher derivatives, – Finite difference methods. See Elliptic equations; Parabolic equations Finitedifference methods, – Finite divided difference, , , , Finitedivideddifference approximations of derivatives, – Finiteelement method, – approximation functions, – assembly, , – , – boundary conditions, , , – direct approach, – discretization, , , element equations, – , – , – fit of the function to solution, – heated rod, – , method of weighted residuals (MWR), – one dimension, – solution, , , twodimensional problems, – First backward difference, First derivative, First finite divided difference, , First forward difference, Firstorder equation, Firstorder error propagation, – Firstorder splines, – Fixedpoint iteration, – algorithm, , convergence, – divergence, nonlinear equations, – pseudocode, twocurve graphical method, – FletcherReeves conjugate gradient algorithm, Floatingpoint operations, – Floatingpoint representation, – Flops, – Flow balance, Flowchart, Flux distribution for heated plate, – Force balance, , Forcing function, Formulation errors, Forward deflation, – Forward difference approximation, Forward divided difference, Forward elimination of unknowns, – Forward finitedivideddifference formulas, Fourier analysis, – Fourier approximation, – continuous Fourier series, – CooleyTukey algorithm, – discrete Fourier transform (DFT), – INDEX fast Fourier transform (FFT), – Fourier integral, – Fourier transform, – frequency domain, – power spectrum, references, – SandeTukey algorithm, – sinusoidal functions, – time domain, – Fourier integral, – Fourier series, – Fourier transform, – Fourier transform pair, Fourier’s heat law, Fourier’s law, , , Fourier’s law of heat conduction, – , Fourth AdamsBashforth open formula, Fourth AdamsMoulton closed formula, Fourthorder Adams method, , – Fourthorder RK methods, – Frequency domain, – Frequency plane, – Friction factor, Frobenius norm, Fully augmented version, Functional approximation, Functions, Fundamental frequency, Fundamental theorem, G Galerkin’s method, , Gauss elimination, – algorithm, back substitution, – complex systems, determinant evaluation, division by zero, falling parachutist problem, – floatingpoint operations, – forward elimination of unknowns, – GaussJordan, – illconditioned systems, – LU decomposition, – naive, – names, NewtonRaphson method, , nonlinear system of equations, – operation counting, – pitfalls, – pivoting, – pseudocode, , roundoff errors, scaling, , – significant figures, singular systems, GaussJordan, – GaussLegendre formulas, , – GaussNewton method, , – Gauss quadrature, – error analysis, – falling parachutist problem, higherpoint formulas, – method of undetermined coefficients, – threepoint GaussLegendre formula, twopoint GaussLegendre formula, – Gauss Seidel, – algorithm, – convergence, – divergence, formula/equations, – , Jacobi iteration, compared, , PDEs, problem contexts, – pseudocode, relaxation, tradeoffs, General linear least squares, – General solution, Generalized reduced gradient (GRG), Genetic algorithm, Given’s method, Glaucoma, Global truncation error, Global warming, Goal Seek, Golden ratio, Goldensection search, – algorithm, example, golden ratio, initial step, – minimizing number of function evaluations, , pseudocode, , second step, Gossett, W. S., Gradient, – , Gradient methods, – Graphical methods linear equations, – roots of equations, – Graphical solution, Greenhouse gases, – GRG, Grid search, Gross error, H Halfsaturation constant, Halvingdoubling strategy, , Hamming’s method, Harmonics, HazenWilliams equation, Heat balance, Heat calculations, – Heat conduction equation, – Heat transfer, – Heated plate with insulated edge, – Heated plate with irregular boundary, – Heated rod, – , Henry’s constant, Hessenberg form, Hessian, – Heun’s method, – , , , – Highaccuracy differentiation formulas, – Highlevel languages, Higherorder multistep methods, – Higherorder NewtonCotes closed formulas, – Higherorder NewtonCotes formulas, , Higherorder RK methods, Higherorder Taylor series methods, Higherorder temporal approximations, – Higherpoint formulas, – Hilbert matrix, , Histogram, , Hooke’s law, , , , , Hotelling’s method, Householder’s method, Human blood, Humps function, Hyperbolic equations, , Hypolimnion, Hypothesis testing, I Ideal gas law, Ideal/nonideal gas laws, – Ideal vs. nonideal, Identity matrix, IF/THEN/ELSE structure, , IF/THEN/ELSEIF structure, , IF/THEN structure, , Illconditioned, , INDEX Illconditioned systems, – , – Implicit Euler’s method, – Implicit method, , – , Imprecision, Improper integrals, – Improved polygon method, Inaccuracy, Increment function, Incremental search methods, , Indefinite integral, Indefinite integration, n Independent variable, , Index, – Indoor air pollution, Inequality constraints, Initial guess, Initial value, Initialvalue problem, , Inner products, Instability, , Integer representation, – Integral calculus, Integral form, Integrand, Integration, . See also Numerical differentiation and integration Integration of equations, – adaptive quadrature, – Gauss quadrature. See Gauss quadrature higherorder error correction of integral estimates, – improper integrals, – NewtonColes algorithm, – Richardson’s extrapolation, – Romberg integration, , – Interactive methods, – Interpolation, – bilinear, – coefficients of interpolating polynomial, equally spared data, , extrapolation, illconditioning, inverse, – Lagrange interpolating polynomials, linear, – multidimensional, – Newton’s interpolating polynomials. See Newton’s interpolating polynomials quadratic, – spline. See Spline interpolation Interpolation function, Interval estimator, Interval halving, Intraocular pressure, Inverse, , – Inverse Fourier transform, Inverse interpolation, – Irregular boundaries, – Iterative approach, Iterative calculation, – Iterative Heun method, Iterative refinement, – J Jacobi iteration, , Jacobian, Jacobi’s method, – JenkinsTraub method, K Kirchhoff’s current law, Kirchhoff’s laws, , , Kirchhoff’s second law, Kirchhoff’s voltage law, , , , Knot, L Lagging phase angle, Lagrange form, Lagrange interpolating polynomials, Lagrange multiplier, , Lagrange polynomial, Laguerre’s method, Laplace equation, , – , , Laplacian difference equation, – Large computations, – Large vs. small systems, Leading phase angle, Leastcost design of tank, – Leastcost treatment of wastewater, – Leastsquares, Leastsquares fit of sinusoid, – Leastsquares fit of straight line, – Leastsquares regression, – GaussNewton method, , – general linear least squares, – linear regression. See Linear regression multiple linear regression, – nonlinear regression, – polynomial regression, – statistical aspects, – Leaving variable, Liebmann’s method, – Line spectra, , Linear algebraic equations, – advanced methods, – approximate technique, , case studies, – Cholesky decomposition, – computer objectives, – Cramer’s rule, – Crout decomposition, – distributed variable problems, , elimination methods, – , elimination of unknowns, – engineering practice, – exact methods, Excel, – Gauss elimination. See Gauss elimination GaussJordan, – GaussSeidel. See GaussSeidel general form, graphical method, – important relationships/formulas, iterative refinement, – LU decomposition. See LU decomposition lumped variable problems, , Mathcad, – mathematical background, – MATLAB, – matrix. See Matrix matrix condition number, – matrix inverse. See Matrix inverse matrix norms, – methods, compared, noncomputer methods, – numerical methods, – overview, – reactors, – references, – resistor circuits, – scope/preview, – springmass system, – study objectives, Thomas algorithm, tradeoffs, – tridiagonal system, – truss, – vector norms, – Linear convergence, Linear equation, Linear interpolation, , , – Linearinterpolation formula, Linear interpolation method, Linear least squares, – Linear leastsquares fit, – , Linear ordinary differential equation, Linear programming, , – graphical solution, – possible outcomes, – INDEX simplex method, – standard form, – Linear regression, – algorithm, best fit, – computer program, – general comments, leastsquares fit of straight line, – linearization of nonlinear relationships, – pseudocode, quantification of error, – statistical assumptions, Linear regression and population models, – Linear splines, – Linear vs. nonlinear, Linearization, Linearization of nonlinear relationships, – Linearization of power equation, – Little, John N., Local truncation error, Logical representation, – Loops, – Lorenz, Edward, Lorenz equation, Lotka, Alfred J., LotkaVolterra equation, , Lower Colorado River, Lower triangular matrix, LR method, LU decomposition, – algorithm, , Crout decomposition, – decomposition phase, forwardsubstitution steps, Gauss elimination, – overview, , pseudocode, , , steps in process, , substitution steps, – Lumpedparameter systems, Lumped variable problems, , M Mfiles, , MacCormack’s method, Machine epsilon, Maclaurin series expansion, , Macro, . See also Excel Main diagonal, Maintenance, Manning equation, Manning roughness coefficient, Manning’s formula, Mantissa, Marksmanship, Marquardt’s method, – Mass balance, , , Mass balance of reactor, – Massspring system, – Mathcad, – condition number, , constrained nonlinear optimization, cubic spline interpolation, curve fitting, – eigenvalues, , entering text, FFT, graphs, – help, linear equations, – main menu, math palette, mathematical functions and variables, – mathematical operations, – matrix computations and operations, matrix functions, matrix inverse, multiline procedures/subprograms, nonlinear system of equations, numeric mode, numerical integration/differentiation, – numerical methods function, ODEs, – online help, optimization, , PDEs, – Poisson’s equation, probability distributions, QuickSheets, range variables, – resource center, roots of equations, – roots of polynomial, standard tool bar, – stiff systems, symbolic mathematics, – symbolic mode, ToolTips, trig and logs, units, – what is it, Mathematical model, – Mathematical programming problem, Mathsoft, MATLAB, – array operations, assignment, – builtin functions, condition number, curve fitting, – diary file, differentiation, double precision, editor/debugger, eigenvalues, – , , – Euler’s method, extended precision, factorial, , FFT, graphics, – humps function, integration, interpolation, – iterative calculation, linear equations, – Mfiles, , mathematical operations, – matrix, , matrix analysis, multidimensional optimization, – name, numerical integration/differentiation, – ODEs, – onedimensional optimization, – optimization, – PDEs, – pipe friction, , polynomial manipulation, polynomials, potential energy function, predatorprey equations, – primary features, regression, , root location, roots of polynomials, – roundoff/truncation errors in numerical differentiation, save, spline, – statistical analysis, – stiff systems, – twodimensional function, versions, INDEX Matrix, – addition, augmentation, defined, division, inverse, , – linear least squares, – MATLAB, multiplication, – notation, – representing linear algebraic equations, – special, square, subtraction, trace, transpose, Matrix condition evaluation, – Matrix condition number, – Matrix inverse, – calculating the inverse, – illconditioned systems, – MATLAB, , pseudocode, stimulusresponse computations, – system condition, – Matrix multiplication, – Matrix norms, – Maximum attainable growth rate, Maximum likelihood principle, Maximummagnitude norm, Maximum power transfer for circuit, – Mean of continuous data, Mean of discrete points, Mean value, Method of false position, Method of lines, Method of optimal steepest ascent, – Method of steepest ascent, – Method of undetermined coefficients, – Method of weighted residuals (MWR), , – Microsoft, Midpoint method, , – , , , , Midtest loop, Milne’s method, – , – Minimax principles, , Minimum potential energy, – Minor, Mixed partial derivative, Model error, Modified Euler, Modified false position, – Modified NewtonRaphson, – , Modified NewtonRaphson method, – , Modified secant method, – Modular programming, – Modules, Modulus of toughness, Molal volume, , Moler, Cleve, , , , Müller’s method, – Multidimensional interpolation, – Multidimensional problems, Multidimensional unconstrained optimization, – BFGS, conjugate gradient method, DFP, direct methods, – finitedifference approximations, – gradient, – gradient methods, – Hessian, – Marquardt’s method, – Newton’s method, – pattern searches, – Powell’s method, , quasiNewton methods, random search, – steepest ascent method, – tradeoffs, – univariate search method, Multimodal, Multipleapplication Simpson’s / rule, – Multipleapplication trapezoidal rule, – Multiple integrals, – Multiple linear regression, – Multiple root, , – Multiplication, Multistep methods Adams formulas, – fourthorder Adams method, , – higherorder methods, – integration formulas, – Milne’s method, – , – NewtonCotes formulas, – nonselfstarting Heun method, stability, – step size, Multivariate power equation, MWR, – N Naive Gauss elimination, – Neumann boundary condition, , NewtonCotes closed integration formulas, , NewtonCotes integration formulas, – Adams formulas, contrasted, , closed forms, , – closed integration formulas, , higherorder formulas, – , , integration of equations, – integration with unequal segments, – multiple integrals, – open forms, , open integration formulas, Simpson’s rules. See Simpson’s rules trapezoidal rule. See Trapezoidal rule unequal segments, – unequally spaced data, – NewtonCotes open integration formulas, NewtonGregory backward formula, NewtonGregory central formula, NewtonGregory forward formula, NewtonRaphson formula, NewtonRaphson method, – additional features, algorithm, error estimates, – evaluate function and derivative, formula, Gauss elimination, , graphical depiction, ideal/nonideal gas laws, – modified method, – , multiple roots, – nonlinear equations, – pitfalls, – slowly converging function, – , termination criteria, twoequation approach, , Newtonian fluid, Newton’s divideddifference interpolating polynomial, Newton’s formula, Newton’s interpolating polynomials, – algorithm, computer applications, errors, – , – general form, linear interpolation, – Newton’s divideddifference interpolating polynomial, INDEX pseudocode, quadratic interpolation, – Newton’s law of cooling, Newton’s laws of motion, Newton’s method, – , – Newton’s second law of motion, , Nodal lines, Node, NonNewtonian fluid, Nonselfstarting Heun method, – derivation, – equations, errors, – Heun approach, – modifiers, – perstep truncation error, sequence of formulas, step size, Nonbasic variables, Nonbinding constraints, Noncomputer methods, Nongradient methods, Nonhomogeneous system, Nonideal gas laws, – Nonideal vs. ideal, Nonlinear boundaryvalue problem, – Nonlinear constrained optimization, Nonlinear equations, Nonlinear programming, Nonlinear regression, , – Nonlinear system of equations, – Nonlinear vs. linear, Norm, Normal distribution, , Normal equation, Normalization, Normalized standard deviate, nth finite divided difference, Number systems, Numerical differentiation, – Numerical differentiation and integration, – advanced methods/references, antidifferentiation, case studies, – commonly used derivatives, computer objectives, – derivatives and integrals for data with errors, – derivatives of unequally spaced data, , differentiation/integration, contrasted, , engineering practice, – equalarea graphical differentiation, , heat calculations, – highaccuracy differentiation formulas, – important relationships/formulas, integrals used in this Part, integration of equations. See Integration of equations Mathcad, – mathematical background, – MATLAB, – method of undetermined coefficients, – methods, compared, NewtonCotes. See NewtonCotes integration formulas noncomputer methods, – overview, – partial derivatives, – Richardson extrapolation, – rootmeansquare current, – sailboat, – scope/preview, – simple strip method, study objectives, , tradeoffs, – uncertain data, work, calculation of, – Numerical double integral, – Numerical error, – Numerical integration, Numerical library, Numerical methods accuracy, defined, engineering practice, – error, hyperbolic equations, iterative approach, linear algebraic equations, – noncomputer methods, precision, – rapid growth, tradeoffs, – why studied, Numerical Recipe, Numerical roundoff errors, – Numerically unstable, O Objective function, , Octal number system, Odd function, ODE. See Ordinary differential equations Ohm’s law, , , Onedimensional parabolic PDEs, Onedimensional problems, Onedimensional unconstrained optimization, – bracketing methods, Brent’s method, , global vs. local extremum, goldensection search, – Newton’s method, – open methods, parabolic interpolation, – tradeoffs, Onepoint iteration, Onesided interval, Onestep methods, Open integration formulas, Open methods, – bracketing methods, compared, Brent’s method, – modified NewtonRaphson, – , modified secant method, – multiple roots, – NewtonRaphson method. See NewtonRaphson method secant method, – simple fixedpoint iteration, – systems of nonlinear equations, – Operation counting, – Optimal steepest ascent, – Optimization, – case studies, – computer objectives, constrained. See Constrained optimization dimensionality, engineering practice, equilibrium and minimum potential energy, – Excel, – fundamental elements, historical overview, leastcost design of tank, – leastcost treatment of wastewater, – Mathcad, – mathematical background, – MATLAB, – maximum power transfer for circuit, – multidimensional. See Multidimensional unconstrained optimization noncomputer methods, INDEX onedimensional. See Onedimensional unconstrained optimization overview, , parachute, – references, root location, contrasted, scope/preview, , study objectives, , tradeoffs, – Optimum, Ordinary differential equations, – boundaryvalue problems. See Boundaryvalue problems case studies, – chaos, – computer objectives, eigenvalue problems. See Eigenvalue problems engineering practice, – important relationships/formulas, – Mathcad, – mathematical background, – MATLAB, – methods, compared, multistep methods. See Multistep methods noncomputer methods, – overview, – predatorprey models, – reactor, – RK methods. See RungeKutta methods scope/preview, – simulating transient current for electric circuit, – stiffness, – study objectives, , swinging pendulum, – tradeoffs, – transient response of reactor, – Orthogonal polynomials, Overconstrained, Overdamped case, Overdetermined, Overflow error, Overrelaxation, Overview of book, – Pp norm, Parabola, Parabolic equations, – ADI scheme, – comparison of onedimensional methods, – convergence, CrankNicolson method, – derivative boundary conditions, explicit methods, – , heat conduction equation, – higherorder temporal approximations, – MacCormack’s method, method of lines, simple implicit method, – , stability, two spatial dimensions, – Parabolic interpolation, – Parachutist problem air resistance, algorithm, analytical solution, – error, , evaluating integrals, – Excel, – Gauss quadrature, gravity, numerical/analytical solution, compared, numerical solution, schematic diagram, Parameter estimation, Parameters, Parametric Technology Corporation (PTC), Parthenon, Partial derivative, Partial derivatives, – Partial differential equations, – area of focus, case studies, – classification, computer objectives, – deflections of a plate, – elliptic equations. See Elliptic equations engineering practice, – Excel, – finite difference methods, – finiteelement method. See Finiteelement method important relationships/formulas, linear equations, mass balance of reactor, – Mathcad, – MATLAB, – overview, – parabolic equations. See Parabolic equations precomputer methods, reactor, – references, scope/preview, – series of springs, – study objectives, , tradeoffs, twodimensional electrostatic field problems, – Partial pivoting, – Pattern directions, Pattern searches, – PDE. See Partial differential equations Penalty functions, Pentadiagonal system, Percent relative error, Perfection, Period, Periodic function, Phase line spectra, , Phaseplane representation, Phase shift, Phases of engineering problem solving, Piecewise functions, Pipe friction, – Pivot coefficient, Pivot element, Pivot equation, Pivoting, – Place value, Plane, Pointslope method, Poisson equation, , , Polynomial. See Roots of polynomials Polynomial deflation, – Polynomial evaluation and differentiation, – Polynomial method, – Polynomial regression, – Polynomials, Population, Population models, – Positional notation, Positive definite matrix, n Posttest loop, Potential energy, Potentiometers, Powell’s method, , Power equation, – Power method, – Power spectrum, Practical issues, Practice applications. See Case studies Precision, – , Predatorprey equations, – Predatorprey models, – INDEX Predictor, Predictorcorrector approach, Predictor equation, – Predictor modifier, Prescriptive models, Pretest loop, Principal diagonal, Problem solving. See Engineering problem solving Problemsolving process, Program development cost, Programming. See Computer programming Programming effort required, Programming languages. See Excel; Mathcad; MATLAB Propagated truncation error, Propagation problems, , Proportionality, Pseudocode, , Pseudoplastics, PTC, Q QD algorithm, QR factorization, QR method, Quadratic convergence, Quadratic interpolation, – Quadratic polynomial, Quadratic programming, Quadratic splines, – Quadrature, Quantizing errors, – QuasiNewton methods, Quotient difference (QD) algorithm, R r, r , Rainwater, – Ralston’s method, Random search, – Rate equation, Rate of convergence, Razdow, Allen, Reaction kinetics, Reactor, – , – Reactors, – RedlichKwong equation of state, References. See Advanced methods/additional references References (bibliography), – Relative error, Relaxation, Repetition, – Residual, Resistor circuits, – Reynolds number, , , Richardson extrapolation, – Richardson’s extrapolation, – RK methods. See RungeKutta methods Romberg integration, , – Rootmeansquare current, – Root polishing, Roots of equations, – advanced methods/additional references, – bracketing methods. See Bracketing methods bracketing/open methods, compared, case studies, – computer objectives, electrical circuit design, – engineering practice, – Excel, – graphical methods, – , greenhouse gases, – ideal/nonideal gas laws, – important relationships/formulas, Mathcad, – mathematical background, – MATLAB, – methods, compared, noncomputer methods, open methods. See Open methods optimization, contrasted, overview, – pipe friction, – problem areas, QD algorithm, rainwater, – root polishing, roots of polynomials. See Roots of polynomials scope/preview, – study objectives, tradeoffs, – Roots of polynomials, – Bairstow’s method, – characteristic equation, conventional methods, – Excel, – factored form of polynomial, general solution, JenkinsTraub method, Laguerre’s method, Roots of polynomials—Cont. Mathcad, – MATLAB, – Müller’s method, – ODE, overdamped/critically damped/underdamped, – polynomial deflation, – polynomial evaluation and differentiation, – root polishing, Roots of quadratic algorithm, Rosenbrock method, RosinRammlerBennet (RRB) equation, Roundoff error, – adding large and small number, – addition, arithmetic manipulations of computer numbers, – chopping, computer representation of numbers, – division, extended precision, – floatingpoint representation, – Gauss elimination, inner products, integer representation, – large computations, – multiplication, normalization, number systems, overflow error, quantizing errors, – rounding, , smearing, subtraction, subtractive cancellation, – underflow “hole,” Rounding, , Row, Rowsum norm, , Row vectors, RRB equation, Runtime cost, RungeKutta Fehlberg, – RungeKutta methods, – adaptive. See Adaptive RungeKutta methods Euler’s method. See Euler’s method fourthorder RK methods, – Heun’s method, – , , , – higherorder RK methods, INDEX methods, compared, – midpoint method, – , , pseudocode, Ralston’s method, RungeKutta Fehlberg, – secondorder RK methods, – stepsize control, – systems of equations, – thirdorder RK methods, – Runge’s function, S Saddle, Sailboat, – Sample, , Sample mean, SandeTukey algorithm, – Saturationgrowthrate equation, , Saturationgrowthrate model, – Scaling, , – Secant formula, Secant method, – additional features, , algorithm, falseposition method, compared, – formula, graphical depiction, modified method, – multiple roots, , Second AdamsBashforth formula, Second AdamsMoulton formula, Second derivative, Second finite divided difference, Second forward finite divided difference, Secondorder closed Adams formula, Secondorder equation, Secondorder open Adams formula, Secondorder Ralston RK method, Secondorder RK methods, – Secondary variables, Selection, – Sensitivity analysis, Sequence, Series of springs, – Shadow price, Shape function, Shooting method, – Signed magnitude method, Significance level, Significand, Significant digits, Significant figures, – Simple fixedpoint iteration, – Simple implicit method, – , Simple strip method, Simplex method, – Simplex procedure, Simpson’s / rule, – Simpson’s / rule, – Simpson’s rules, – algorithms, , multipleapplication Simpson’s / rule, – pseudocode, Simpson’s / rule, – Simpson’s / rule, – uneven data, Simulated annealing, Simulating transient current for electric current, – Simultaneous linear algebraic equations. See Linear algebraic equations Simultaneous nonlinear equations, – Simultaneous overrelaxation, Singlevalue decomposition, Singlevariable optimization, Singular system, Singular systems, Sinusoid, Sinusoidal functions, – Slide rule, Small vs. large systems, Smearing, Software cost, Software packages. See Excel; Mathcad; MATLAB Solution technique, – Solver, – SOR, Special matrix, Specific growth rate, Spectral norm, Spline, Spline functions, Spline interpolation, – cubic splines, , , – linear splines, – quadratic splines, – superiority, , Spreadsheet, . See also Excel Springmass system, – Square matrix, Stability, , , – , Stage extraction process, Standard deviation, , Standard error of the estimate, Standard normal estimate, Starting point, Static instability, Statistical inference, Statistically determinate truss, – Statistics, – Steadystate, Steadystate analysis of system of reactors, – Steepest ascent method, – StefanBoltzmann constant, StefanBoltzmann law, Step halving, Stiff system, Stiffness, – Stiffness matrix, Stimulusresponse computations, – Stopping criterion, Straightening, Strange attractors, StreeterPhelps model, Strip method, Structured programming, – Studentt, Subdomain method, Subroutine, Subtraction, Subtractive cancellation, – Successive overrelaxation, Successive substitution, Superposition, SVD method, SwameeJain equation, Swinging pendulum, , – Symmetric form, Symmetric matrix, , Synthetic division, System condition, – Systems of nonlinear equations, – Tt distribution, Table lookup, Tableau, Tabu search, Taylor series, – approximation of function with infinite number of derivatives, approximation of polynomial, backward difference approximation, centered difference approximation, finite difference approximations of higher derivatives, – INDEX finitedivideddifference approximations of derivatives, – firstorder approximation, first theorem of mean for integrals, forward difference approximation, NewtonRaphson method, nonlinearity, – nthorder expansion, numerical differentiation, – remainder, , , , – secondorder approximation, second theorem of mean for integrals, step size, – Taylor series expansion, theorem, truncation errors, what is it, zeroorder approximation, Taylor series expansion, Taylor’s formula, Taylor’s theorem, Temperature of heated plate, – Terminal velocity, Termination criteria, The MathWorks, Inc., Thermocline, Thirdorder RK methods, – Thomas algorithm, Threepoint GaussLegendre formula, Time domain, – Time plane, Timevariable, Total numerical error, – Total sum of the squares, Trace, Tradeoffs curve fitting, – linear algebraic equations, – numerical differentiation/integration, – numerical methods, – ODEs, – optimization, – PDEs, roots of equations, – Transcendental function, Transient, Transient responses of reactor, – Transpose, Trapezoidal rule, – algorithms, area under straight line, – computer program, – conclusions, – derivation, , error, error corrections, – formula, graphical depiction, Heun’s method, multipleapplication rule, – pseudocode, single application, – unequal segments, Trend analysis, Triangular matrix, Tridiagonal matrix, Tridiagonal system, – Triple root, True error, True fractional relative error, True local truncation error, True mean, True solution, True value, Truncation, Truncation error, , Truncation errors and Taylor series, – blunders, – condition, data uncertainty, error propagation, – formulation errors, stability, Taylor series. See Taylor series total numerical error, – truncation errors, Truss, – Tukey, J. W., Twiddle factors, Twodimensional electrostatic field problems, – Twodimensional interpolation, – Twodimensional parabolic PDEs, Twoequation NewtonRaphson approach, , Twopoint GaussLegendre formula, – Twosegment trapezoidal rule, Twosided interval, – U Uncertain data, Uncertainty, Unconditionally stable, Unconstrained optimization, Underdamped case, Underdetermined, , Underflow “hole,” Underrelaxation, Underspecified, Unexplained sum of the squares, Uniformmatrix norm, , Uniformvector norm, Unimodal, Univariate search method, Unstable, Upper triangular matrix, V Van der Pol’s equation, Van der Waals equation, Vandermonde matrix, INDEX Variable metric methods, Variance, Variational approach, VBA macro, . See also Excel Vector norms, – Vibrating string, Videoangiography, Voltage balance, Volterra, Vito, Volume integral, Volumeintegral approach, Von Karman equation, W Waste minimization, Wastewater treatment, – Wave equation, , Wellconditioned systems, WHILE, Wolf, Johann Rudolph, Wolf sunspot number, Word, Work, calculation of, – Y Yield stress, Young’s modulus, Z Zeroorder approximation,
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