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عدد المساهمات : 18140 التقييم : 33204 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
 موضوع: كتاب Numerical Methods for Engineers الإثنين 02 يناير 2023, 5:20 pm  

أخواني في الله أحضرت لكم كتاب Numerical Methods for Engineers Eighth 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 ABOUT THE AUTHORS iv PREFACE xv 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 vi CONTENTS 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 CONTENTS vii . 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 viii CONTENTS 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 CONTENTS ix 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 x CONTENTS 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 . Fitting Enzyme Kinetics (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 CONTENTS xi CHAPTER Integration of Equations . NewtonCotes Algorithms for Equations . Romberg Integration . Adaptive Quadrature . Gauss Quadrature . Improper Integrals . Monte Carlo Integration 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 xii CONTENTS 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 CONTENTS xiii 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 xiv CONTENTS APPENDIX A: THE FOURIER SERIES APPENDIX B: GETTING STARTED WITH MATLAB APPENDIX C: GETTING STARTED WITH MATHCAD BIBLIOGRAPHY INDEX INDEX A Accuracy, – , AdamsBashforth formula, – , AdamsMoulton formula, – Adaptive integration, Adaptive quadrature, – Adaptive RungeKutta (RK) methods, – , Adaptive stepsize control, – , – Addition, estimated error bounds, large and small number, – matrix operations, smearing, – Advanced methods/additional references, – curve fitting, – linear algebraic equations, – numerical integration, ordinary differential equations (ODEs), – partial differential equations (PDEs), roots of equations, – Advectiondiffusion equation, Air resistance falling parachutist problem, – formulation, Allosteric enzymes, – Alternatingdirection implicit (ADI) method, – , – , , Amplitude, – Analytical/direct approach curve fitting, – falling parachutist problem, – . See also Falling parachutist problem finiteelement methods, – linear algebraic equations, – nature of, numerical differentiation, – , – numerical integration, – optimization, – partial differential equations (PDEs), – roots of equations, – , Angular frequency, Antidifferentiation, Antoine’s equation, Approximations, – . See also Estimation accuracy/inaccuracy, – , algorithm for iterative calculations, – approximate percent relative error, continuous Fourier series, – error calculation, – , error definitions, – finiteelement methods, – functional, polynomial, – precision/imprecision, – , significant figures/digits, – , Taylor series, – , – Archimedes’ principle, – Areal integrals, Arithmetic mean, Arithmetic operations, – , – , – Assemblage property matrix, Associative property, matrix operations, Augmentation, matrix operations, – Auxiliary conditions, B Background information blunders, – computer programming and software, – conservation laws and engineering, – curve fitting, – data uncertainty, – , eigenvalue problems, error propagation, – , Excel, – . See also Excel formulation errors, linear algebraic equations, – Mathcad, – , – . See also Mathcad MATLAB, – , – . See also MATLAB modular programming, – numerical differentiation, – , – numerical integration, – optimization, – ordinary differential equations (ODEs), – root equation, – , – roots of polynomials, – roundoff errors, – , – INDEX simple mathematical model, – structured programming, – Taylor series, – total numerical error, – truncation errors, – , – Back substitution, – , – LU decomposition, Backward deflation, Backward difference approximation, – , Bairstow’s method, – , Banded matrices, – Banded matrix, Base (binary) number system, – Base (octal) number system, Base (decimal) number system, – , – Basic feasible solution, Basic variables, Bernoulli’s equation, – BFGS algorithm, Bias/inaccuracy, – Bilinear interpolation, – Binary chopping. See Bisection method Binary (base ) number system, – Binding constraints, Bisection method, – , – , bisection algorithm, computer methods, – , defined, error estimates, – falseposition method vs., – graphical method, – , incremental search methods vs., minimizing function evaluations, – termination criteria, Blasius formula, Blunders, – Bolzano’s method. See Bisection method Boole’s rule, Boundary conditions, – derivative, – , – , finiteelement methods, – , – , irregular boundaries, – Laplace equation, – , – Boundaryvalue problems, – , eigenvalue, – finitedifference method, initialvalue problems vs., shooting method, – Bracketing methods, – , bisection method, – , – , computer methods, – defined, falseposition method, – , – , graphical method, – incremental searches/determining initial guesses, Break command, Break loops, – Brent’s rootlocation method, – , algorithm, – , – graphical method, inverse quadratic interpolation, – optimization, – , roots of polynomials, BroydenFletcherGoldfarbShanno (BFGS) algorithms, B splines, Butcher’s fifthorder RungeKutta method, – Butterfly network, C C++, Cartesian coordinates, CASE structure, CashKarp RK method, – , – Centered finite divideddifference approximation, , Central Limit Theorem, Chaotic solutions, Characteristic, – Characteristic equation, – Charge, conservation of, Chebyshev economization, Chemical/biological engineering analyzing transient response of reactor, – conservation of mass, curve fitting, – determining total quantity of heat, – fitting enzyme kinetics, – ideal gas law, – leastcost design of a tank, – linear algebraic equations, – numerical integration, – onedimensional mass balance of reactor, – optimization, – ordinary differential equations (ODEs), – partial differential equations (PDEs), – roots of equations, – steadystate analysis of system of reactors, – Cholesky decomposition, – Chopping, – , Civil/environmental engineering analysis of statically determinate truss, – conservation of momentum, curve fitting, – deflections of a plate, – INDEX Civil/environmental engineering—Cont. effective force on mast of racing sailboat, – greenhouse gases and rainwater, – leastcost treatment of wastewater, – linear algebraic equations, – numerical integration, – optimization, – ordinary differential equations (ODEs), – partial differential equations (PDEs), – predatorprey models and chaos, – roots of equations, – splines to estimate heat transfer, – Clamped end condition, – Classical fourthorder RungeKutta method, – , Coefficient, method of undetermined, – Coefficient of determination, Coefficient of interpolating polynomial, Coefficient of thermal conductivity, Coefficient of variation, Colebrook equation, Column, defined, Columnsum norms, Column vectors, Commutative property, matrix operations, Complete pivoting, Complex systems, linear algebraic equations, Composite, integration formulas, – Computational error, – , Computer programming and software, – . See also Pseudocode algorithms bisection method, – , bracketing methods, – computer programs, defined, cost comparison, – curve fitting, – , – , – eigenvalues, – Excel. See Excel linear algebraic equations, – , – , – linear programming, – linear regression, – Mathcad. See Mathcad MATLAB. See MATLAB modular programming, – numerical integration/differentiation, – optimization, – , – ordinary differential equations (ODEs), – other languages and libraries, partial differential equations (PDEs), – , – roots of equations, – , – , – software user types, – stepsize control, structured programming, – Condition numbers, – matrix, – Confidence intervals, – , – Conjugate directions, – Conjugate gradient, Conservation laws, – by field of engineering, simple models in specific fields, stimulusresponse computations, – Conservation of charge, Conservation of energy, Conservation of mass, – Conservation of momentum, Constant of integration, – Constant step size, Constitutive equation, – Constrained optimization, – linear programming, – nonlinear, – , Constraints binding/nonbinding, optimization, Continuous Fourier series, – approximation, – determination of coefficients, Controlvolume approach, – Convergences defined, fixedpoint iteration, – GaussSeidel (Liebmann) method, – linear, – nature of, NewtonRaphson method, – of numerical methods of problem solving, CooleyTukey algorithm, – Corrector equation, – Corrector modifier, – Correlation coefficient, Countcontrolled loops, – , Cramer’s rule, – , CrankNicolson technique, – , Critically damped case, Crout decomposition, – Cubic splines, – , – , derivation, – interpolation with Mathcad, – natural, Cumulative normal distribution, – Current balance, Curvature, Curve fitting, – advanced methods and additional references, – INDEX Dependent variables, – , Derivative defined, first, – second, – Derivative boundary conditions, – , – , Derivative meanvalue theorem, Descriptive statistics, – Design, Design variables, Determinants, in Gauss elimination, – , – Determination, coefficient of, DFP algorithm, Diagonally dominant systems, – Diagonal matrix, Differential calculus. See Numerical differentiation Differential equations, – , Dilatant (“shear thickening”) fluids, Direct approach. See Analytical/direct approach Directional derivative, Dirichlet boundary condition, – , Discrete Fourier transform (DFT), – Discretization, finiteelement methods, – Discriminant, DISPLAY statements, Distributedparameter system, Distributedvariable systems, Distributive property, matrix operations, Division, estimated error bounds, synthetic, – by zero, DOEXIT loops, DOFOR loops, – Double integrals, – Double roots, Drag coefficient, Dynamic instability, E Eigenvalue problems, – , boundaryvalue problem, – computer methods, – eigenvalue, defined, eigenvalue analysis of axially loaded column, – eigenvectors, – massspring system, – mathematical background, other methods, – physical background, – polynomial method, – , – power method, – case studies, – coefficients of an interpolating polynomial, comparisons of alternative methods, – computer methods, – , – , – defined, engineering applications, – , – estimation of confidence intervals, – extrapolation, fast Fourier transform (FFT), Fourier approximation, – frequency domains, – general linear least squares model, – goals/objectives, – important relationships and formulas, – interpolation, – inverse interpolation, – Lagrange interpolating polynomial, – , – leastsquares regression, – linear regression, – mathematical background, – multidimensional interpolation, – multiple linear regression, – , – Newton’s divideddifference interpolating polynomials, – Newton’s interpolating polynomial, – , – noncomputer methods, – nonlinear regression, – , – , normal distribution, polynomial regression, – , power spectrum, – scope/preview, – simple statistics, – with sinusoidal functions, – spline interpolation, – time domains, – D Dartboard Monte Carlo integration, – Data distribution, Data uncertainty, – , DavidonFletcherPowell (DFP) method of optimization, Decimal (base ) number system, – , – Decimationinfrequency, – Decimationintime, – , Decision loops, Definite integration, n Deflation, backward, forward, polynomial, – Degrees of freedom, INDEX numerical integration, – , – , – optimization, – , – ordinary differential equations (ODEs), – , – parameters, – , partial differential equations (PDEs), – , – , – practical issues, roots of equations, – , – , – roots of polynomials, – twopronged approach, – Entering variables, – Epilimnion, Equalarea graphical differentiation, Equality constraint optimization, Error(s) approximations. See Approximations bisection method, – blunders, – calculation, – , data uncertainty, – , defined, – estimates for iterative methods, – estimates in multistep method, – estimation, – , estimation for Euler’s method, – falling parachutist problem, formulation, Gauss quadrature, – linear algebraic equations, – NewtonRaphson estimation method, – Newton’s divideddifference interpolating polynomial estimation, – numerical differentiation, – , – numerical integration, – predictorcorrector approach, – , – quantizing, – , relative, residual, – roundoff. See Roundoff errors Simpson’s / rule estimation, total numerical, – trapezoidal rule, – , – , – true, true fractional relative error, – truncation. See Truncation errors Error definitions, – approximate percent relative error, stopping criterion, – , true error, true fractional relative error, – true percent relative error, – , Error propagation, – , condition, – functions of more than one variable, – Eigenvectors, – Electrical engineering conservation of charge, conservation of energy, currents and voltages in resistor circuits, – curve fitting, – design of electric circuit, – Fourier analysis, – linear algebraic equations, – maximum power transfer for a circuit, – numerical integration, – optimization, – ordinary differential equations (ODEs), – partial differential equations (PDEs), – rootmeansquare current, – roots of equations, – simulating transient current for electric circuit, – twodimensional electrostatic field problems, – Element properties, finiteelement methods, Element stiffness matrix, Elimination of unknowns, – back substitution, – , – forward, – , Elliptic partial differential equations (PDEs), – , – , boundary conditions, – computer software solutions, – controlvolume approach, – GaussSeidel (Liebmann) method, – , – Laplace equation, – , – , – Laplacian difference equation, – solution technique, – Embedded RungeKutta (RK) method, – ENDDO statement, – End statement, Energy conservation of, equilibrium and minimum potential, – Energy balance, Engineering problem solving chemical engineering. See Chemical/biological engineering civil engineering. See Civil/environmental engineering conservation laws, – curve fitting, – , – dependent variables, – , electrical engineering. See Electrical engineering falling parachutist problem. See Falling parachutist problem forcing functions, – fundamental principles, independent variables, – , linear algebraic equations, – , – mechanical engineering. See Mechanical/aerospace engineering Newton’s laws of motion, – , – numerical differentiation, – , – INDEX functions of single variable, – stability, – Estimated mean, Estimation. See also Approximations confidence interval, – , – defined, errors, – , – NewtonRaphson estimation method, – parameter, standard error of the estimate, standard normal estimate, – Euclidean norms, – EulerCauchy method. See Euler’s method Euler’s method, – , – algorithm, – backward/implicit, – effect of reduced step size, – error analysis, – Euler’s formula, improvements, – ordinary differential equations (ODEs), – , – , – , – as predictor, – systems of equations, Excel, – , – , – computer implementation of iterative calculation, – curve fitting, – Data Analysis Toolpack, – described, double precision to represent numerical quantities, Goal Seek, infinite series evaluation, linear algebraic equations, – linear programming, – linear regression, nonlinear constrained optimization, – optimization, – , – , – , – ordinary differential equations (ODEs), – partial differential equations (PDEs), – pseudocode vs., roots of equations, – , – , – Solver, – , – , – , – , – standard use, – Trendline command, – VBA macros, – Explicit solution technique, ordinary differential equations (ODEs), – parabolic partial differential equations (PDEs), – , Exponent, – Exponential model of linear regression, – Extended midpoint rule, Extended precision, roundoff error, – Extrapolation, Extreme points, Extremum, – F Factors, polynomial, Falling parachutist problem, – , – algorithm, – error, Gauss elimination, – Gauss quadrature application, optimization of parachute drop cost, – , – schematic diagram, velocity of the parachutist, – , – Falseposition method, – , – , bisection method vs., – falseposition formula, – , graphical method, modified false position, – , pitfalls, – secant method vs., – Fanning friction factor, Faraday’s law, Fast Fourier transform (FFT), – , CooleyTukey algorithm, – SandeTukey algorithm, – Feasible extreme points, Feasible solution space, – Fibonacci numbers, – Fick’s law of diffusion, Finish, – Finitedifference methods, – , – elliptic partial differential equations (PDEs), – , – , , highaccuracy differentiation formulas, – optimization, – ordinary differential equations (ODEs), – parabolic partial differential equations (PDEs), – , – , Finitedivideddifference approximations of derivatives, – , – , – Finiteelement methods, – assembly, – , – boundary conditions, – , – , defined, – discretization, – element equations, – , – , – , – general approach, – partial differential equations (PDEs), – , single dimension, – solution and postprocessing, – , two dimensions, – First backward difference, First derivative, – INDEX Fourthorder methods Adams, – , RungeKutta, – , – , – , – , – , – Fractional parts, – Frequency domain, – Frequency plane, – Friction factor, – Frobenius norms, Fully augmented version, FUNCTION, Function(s) error propagation, – forcing, – interpolation, – mathematical behavior, modular programming, – penalty, sinusoidal, – spline, Functional approximation, Fundamental frequency, Fundamental theorem of integral calculus, G Gauss elimination, – , algorithm, – Cramer’s rule, – , determinants, – , – elimination of unknowns, – GaussJordan method, – , – graphical method, – improving solutions, – LU decomposition version, – more significant figures, naive approach, – operation counting, – pitfalls of elimination methods, – pivoting, – , – , solving small numbers of equations, – GaussJordan method, – , – GaussLegendre formulas, – , higherpoint, – twopoint, – GaussNewton method, – , Gauss quadrature, – , – error analysis, – GaussLegendre formulas, – , method of undetermined coefficients, – GaussSeidel (Liebmann) method, – , – , – , , algorithm, – First finite divided difference, First forward difference, First forward finite divided difference, Firstorder approximation, – Firstorder equations, – , Firstorder splines, – Fixed (Dirichlet) boundary condition, – , Fixedpoint iteration, – , algorithm, – convergences, – graphical method, – nonlinear equations, – FletcherReeves conjugate gradient algorithm, Floatingpoint operations/flops, – Floatingpoint representation chopping, – , fractional part/mantissa/significand, – integer part/exponent/characteristic, – machine epsilon, – quantizing errors, – , Flowcharts, – defined, sequence structure, simple selection constructs, symbols, Force balance, Forcing functions, – Formulation errors, Fortran , – Forward deflation, Forward difference approximation, Forward elimination of unknowns, – , Forward substitution, LU decomposition, – , Fourier approximation, – continuous Fourier series, – curve fitting with sinusoidal functions, – defined, – discrete Fourier transform (DFT), – engineering applications, – fast Fourier transform (FFT), – , Fourier integral and transform, – frequency domain, – power spectrum, – time domain, – Fourier integral, – Fourier series, – , – Fourier’s law of heat conduction, – , Fourier transform, – discrete Fourier transform (DFT), – fast Fourier transform (FFT), – , Fourier transform pair, Fourth derivative, – INDEX convergence criterion, – elliptic partial differential equations (PDEs), – , – graphical method, – iteration cobwebs, – problem contexts, – relaxation, Generalized reduced gradient (GRG), General linear leastsquares model, – confidence intervals for linear regression, – general matrix formulation, – statistical aspects of leastsquares theory, – General solution, Genetic algorithm, Given’s method, Global truncation error, Golden ratio, – Goldensection search optimization, – , – , extremum, – golden ratio, – singlevariable optimization, unimodal, – Gradient, defined, Gradient methods of optimization, – conjugate gradient method (FletcherReeves), finitedifference approximation, – gradients, – Hessian, – , Marquardt’s method, – , Newton’s method, – , – , – path of steepest ascent/descent, – , – , quasiNewton methods, Greenhouse gases, – H HagenPoiseulle law, Halfsaturation constant, Hamming’s method, Harmonics, HazenWilliams equation, Heat balance, Heatconduction equation, – , – . See also Parabolic partial differential equations (PDEs) Hessenberg form, Hessian, – , Heun’s method, – , – , nonselfstarting, – , Highaccuracy differentiation formulas, – Hilbert matrix, – , – Histograms, – Hooke’s law, – Hotelling’s method, Householder’s method, Humps function, Hyperbolic partial differential equations (PDEs), Hypolimnion, Hypothesis testing, I Ideal gas law, – Identity matrix, IEEE format, IF/THEN structure, – , IF/THEN/ELSE structure, – , IF/THEN/ELSE/IF structure, Illconditioned systems, – effect of scale on determinant, – elements of matrix inverse as measure of, singular systems, – Implicit solution technique, ordinary differential equations (ODEs), – parabolic partial differential equations (PDEs), – , – , Imprecision, – , Improper integrals, – cumulative normal distribution, – extended midpoint rule, normalized standard deviate, – Improved polygon (midpoint) method, – , – , – , Inaccuracy, – Incremental search methods bisection method vs., defined, determining initial guesses, Increment function, – Indefinite integral, Indefinite integration, n Independent variables, – , Indexes, – Inequality constraint optimization, Inferential statistics, Infinite series computation, smearing, – Initial value, – Initialvalue problems, boundaryvalue problems vs., defined, Inner products, In place implementation, INPUT statements, Integer part, – Integer representation, – Integral calculus. See Numerical integration INDEX GaussSeidel (Liebmann) method, – , secondary variables, – solution technique, – Laplacian difference equation, – Large computations, interdependent computations, – Large versus small systems, Law of mass action, LC networks/circuits, – Leastsquares fit of a sinusoid, – Leastsquares regression, – general linear leastsquares model, – leastsquares fit of a straight line, – linear regression, – , – multiple linear regression, – , – nonlinear, – , – , polynomial regression, – , Leaving variables, – LevenbergMarquardt method, Liebmann method. See GaussSeidel (Liebmann) method Linear algebraic equations, – advanced methods and additional references, – case studies, – comparisons of methods, – complex systems, computer methods, – , – , – Cramer’s rule, – , determinants, – distributedvariable systems, division by zero, elimination of unknowns, – engineering applications, – , – error analysis, – Gauss elimination. See Gauss elimination GaussJordan method, – , – GaussSeidel (Liebmann) method. See GaussSeidel (Liebmann) method general form, goals/objectives, – graphical method, – , – , illconditioned systems, – important relationships and formulas, Liebmann method. See GaussSeidel (Liebmann) method LU decomposition methods, – , lumpedvariable systems, mathematical background, – matrix inverse, – , – matrix notation, – matrix operating rules, – more significant figures, noncomputer methods, – nonlinear systems of equations, – pivoting, – , – representing in matrix form, – Integral form, Integrand, Interdependent computations, – Interpolation, – coefficients of interpolating polynomial, computers in, – , – curve fitting, with equally spaced data, finiteelement methods, – interpolation functions, – inverse, – inverse quadratic interpolation method, – Lagrange interpolating polynomials, – , – linear interpolation method, – multidimensional, – Newton’s divideddifference interpolating polynomials, – , – polynomial, – quadratic, – spline, – Interval estimator, Interval halving. See Bisection method Inverse Fourier transform, – Inverse interpolation, – Inverse quadratic interpolation, – Irregular boundaries, – Iterative approach to computation algorithms, – defined, – error estimates, – GaussSeidel (Liebmann) method, – , – , – , , iterative refinement, – J Jacobian, Jacobi iteration, Jacobi’s method, – JenkinsTraub method, K Kirchhoff’s laws, – , – , – L Lagging phase angle, Lagrange interpolating polynomials, – , – Lagrange multiplier, Lagrange polynomial, Laguerre’s method, Laplace equation, – , – boundary conditions, – , – described, – flux distribution of heated plate, – INDEX roundoff errors, scaling, – , – scope/preview, – singular systems, – special matrices, – system condition, – Linear convergences, – Linear interpolation method. See also Falseposition method; Secant method defined, – linearinterpolation formula, – Linearization, Linear programming (LP) computer solutions, – defined, feasible solution space, – graphical solution, – optimization, – possible outcomes, – setting up LP problem, – simplex method, – standard form, – Linear regression, – computer programs, – confidence intervals, – criteria for “best” fit, – curve fitting, engineering applications, – estimation errors, exponential model, – general comments, general linear leastsquares model, – leastsquares fit of straight line, – linearization of nonlinear relationships, – linearization of power equation, – minimax criterion, multiple, – , – quantification of error, – residual error, – spread around the regression line, standard error of the estimate, Linear splines, – Linear trend, – Line spectra, – Local truncation error, Logical loops, – Logical representation, – algorithm for roots of a quadratic equation, – repetition, – selection, – sequence, Loops, – , Lorenz equations, – LotkaVolterra equations, – Lower triangular matrix, LR method (Rutishauser), LU decomposition methods, – , algorithm, – , – Crout decomposition, – defined, LU decomposition step, – , overview, – substitution step, – version of Gauss elimination, – Lumpedparameter systems, Lumpedvariable systems, M MacCormack’s method, – Machine epsilon, – Maclaurin series expansion, – , – Manning equation, Mantissa, storage, n Maple V, Marquardt’s method, – , Mass, conservation of, – Mass balance, Mathcad, – , – basics, – curve fitting, – double precision to represent numerical quantities, entering text, – graphics, – linear algebraic equations, – mathematical functions and variables, – mathematical operations, – Minerr, multigrid function, – multiline procedures/subprograms, numerical integration/differentiation, – numerical methods function, online help, optimization, – ordinary differential equations (ODEs), – partial differential equations (PDEs), – QuickSheets, relax function, – resource center, roots of equations, – , – symbolic mathematics, – ToolTips, Mathematical laws, Mathematical models defined, – overview of problemsolving process, simple model, – INDEX conservation of momentum, curve fitting, – equilibrium and minimum potential energy, – finiteelement solution of series of springs, – linear algebraic equations, – numerical integration to compute work, – optimization, – ordinary differential equations (ODEs), – partial differential equations (PDEs), – pipe friction, – roots of equations, – springmass systems, – swinging pendulum, – Method of false position. See Falseposition method Method of lines, – Method of undetermined coefficients, – Method of weighted residuals (MWR), finiteelement methods, – Mfiles (MATLAB), – . See also MATLAB MichaelisMenten model, – , Microsoft, Inc., Midpoint (improved polygon) method, – , – , – , Midtest loops, Milne’s method, – , – , Minimax criterion, MINPACK algorithms, Mixed partial derivatives, Model errors, Modified Euler. See Midpoint (improved polygon) method Modified false position, – , Modified fixedpoint method, Modified NewtonRaphson method, – , Modified secant method, – , Modular programming, – advantages, defined, Momentum, conservation of, Monte Carlo (MC) integration, – , m surplus variables, – Müller’s method, – , – , Multidimensional interpolation, – Multidimensional unconstrained optimization, – direct methods (nongradient), – gradient methods (descent/ascent), – MATLAB, – pattern searches/directions, – Powell’s method, – , random search method, – univariate search method, Multimodal optimization, – Multipleapplication trapezoidal rule, – , Multiple integrals, – Mathematical programming. See Optimization Mathsoft Inc., MathWorks, Inc., The, MATLAB, – , – , – assignment of values to variable names, – builtin functions, computer implementation of iterative calculation, – curve fitting, – described, double precision to represent numerical quantities, Fourier analysis, – graphics, – linear algebraic equations, – linear regression, mathematical operations, – Mfiles, – , – , numerical differentiation errors, – numerical integration/differentiation, – optimization, – , – , ordinary differential equations (ODEs), – , – , – partial differential equations (PDEs), – polynomials, roots of equations, – , – , – statistical analysis, – Matrix condition number, – Matrix inverse, – , – calculating, – stimulusresponse computations, – Matrix norms, – Matrix operations banded matrices, – Cholesky decomposition, – components, – error analysis and system condition, – matrix, defined, matrix condition number, – matrix inverse, – , – matrix notation, – representing linear algebraic equations in matrix form, – rules, – special matrices, – symmetric matrices, tridiagonal systems, – Maximum likelihood principle, – Maximummagnitude norms, Mean value, – confidence interval on the mean, – derivative meanvalue theorem, determining mean of discrete points, – spread around, Mechanical/aerospace engineering analysis of experimental data, – INDEX Multiple linear regression, – , – , Multiple roots, – double roots, modified NewtonRaphson method for multiple roots, – , NewtonRaphson method, – secant method, – triple roots, Multiplication, estimated error bounds, inner products, matrix operations, – Multistep methods, – , defined, higherorder methods, – integration formulas, – nonselfstarting Heun, – , stepsize control, N Naive Gauss elimination, – back substitution, – , – forward elimination of unknowns, – , operation counting, – ndimensional vector, NelderMead method, Newmann boundary condition, – , – NewtonCotes integration formulas, – , – , – , Boole’s rule, closed formulas, – comparisons, – defined, higherorder, – , – , – open formulas, – ordinary differential equations (ODEs), – Simpson’s / rule, – , – , – , – , – , , – , Simpson’s / rule, – , – , – , – , trapezoidal rule, – , – , – , – , – , , NewtonGregory forward formula, Newtonian fluid, NewtonRaphson method, – , – , – , algorithm, – , error estimates, – graphical method, modified method for multiple roots, – , multiple roots, – NewtonRaphson formula, nonlinear equations, – pitfalls, – roots of polynomials, slowly converging function, – Taylor series derivation, – Taylor series expansion, termination criteria, – Newton’s divideddifference interpolating polynomials, – , algorithm, – defined, derivation of Lagrange interpolating polynomial from, – error estimation, – general form, – quadratic interpolation, – Newton’s formula, Newton’s law of cooling, Newton’s laws of motion, – second law of motion, – , Newton’s method of optimization, – , – , – Nodal lines/planes, Nonbasic variables, Nonbinding constraints, Nonideal versus idealized laws, Nonlinear constrained optimization, Excel, – Mathcad, Nonlinear equations defined, fixedpoint iteration, – linear equations vs., NewtonRaphson method, – roots of equations, – shooting method for boundaryvalue problems, – systems of equations, – , – Nonlinear programming optimization, Nonlinear regression, – , – , Nonselfstarting Heun, – , Normal distribution, Normalization, Normalized standard deviate, – Norms defined, matrix, – vector, – “Notaknot” condition, – n structural variables, – nth finite divided difference, – Number systems, . See also specific number systems Numerical differentiation, – , – . See also Optimization backward difference approximation, centered difference approximation, with computer software, – control of numerical errors, – INDEX Simpson’s / rule, – , – , – , – , – , Simpson’s / rule, – , – , – , – , terminology, – trapezoidal rule, – , – , – , – , – , Numerical methods of problem solving, – , – falling parachutist problem, – nature of, – Numerical Recipe library, Numerical stability, – Nyquist frequency, O Objective function optimization, Octal (base ) number system, ODEs. See Ordinary differential equations (ODEs) Ohm’s law, Onedimensional unconstrained optimization, – Brent’s rootlocation method, – , goldensection search, – , – , MATLAB, – multimodal, – Newton’s method, – , – , – parabolic interpolation, – , Onepoint iteration, . See also Fixedpoint iteration Onesided interval, Onestep methods, – , – , Open methods, – , – , Brent’s rootlocation method, – , defined, – fixedpoint iteration, – , graphical method, multiple roots, – NewtonRaphson method, – , – , – , , secant method, – , simple fixedpoint iteration, – systems of nonlinear equations, – Optimal steepest ascent, – , Optimization, – additional references, Brent’s rootlocation method, – , case studies, – computer methods, – , – defined, engineering applications, – , – goals/objectives, – goldensection search, – , – , gradient methods. See Gradient methods of optimization history, linear programming, – Numerical differentiation—Cont. data with errors, – derivatives of unequally spaced data, – differentiate, defined, engineering applications, – , – error analysis, – , – finitedivideddifference approximations, – , – first derivative, – forward difference approximations, goals/objectives, – highaccuracy differentiation formulas, – mathematical background, – , – noncomputer methods, – , – ordinary differential equations. See Ordinary differential equations (ODEs) partial derivatives, – , – partial differential equations. See Partial differential equations (PDEs) polynomials, – Richardson extrapolation, – , – , – roundoff errors, – scope/preview, – second derivative, – terminology, – Numerical integration, – Adams formula, – , – , adaptive integration, adaptive quadrature, – advanced methods and additional references, Boole’s rule, calculation of integrals, – case studies, – closed forms, – , – , – , – comparisons, – with computer software, – data with errors, – engineering applications, – , – , – fundamental theorem, Gauss quadrature, – , goals/objectives, – important relationships and formulas, improper integrals, – integrate, defined, integration with unequal segments, – mathematical background, – Monte Carlo (MC) integration, – , multiple integrals, – NewtonCotes formulas, – , – , – , – , , – noncomputer methods, – open forms, – , – Richardson extrapolation, – , – , – Romberg integration, – , scope/preview, – INDEX mathematical background, – multidimensional unconstrained, – Newton’s method, – , – , – noncomputer methods, nonlinear constrained optimization, – , onedimensional unconstrained, – parabolic interpolation, – , problem classification, – random search method, – scope/preview, – Order of polynomials, Ordinary differential equations (ODEs), – , – advanced methods and additional references, – algorithms, – , – , – , – boundaryvalue problems, – , – , case studies, – components, computer programming and software, – defined, eigenvalue problems, – , engineering applications, – , – Euler’s method, – , – , – , – explicit solution technique, – falling parachutist problem, – finitedifference methods, – firstorder equations, – , fourthorder Adams, – , – fourthorder RK, – , – , – , – , – , – goals/objectives, – Heun’s method, – , – , – , higherorder equations, – , – implicit solution technique, – important relationships and formulas, initialvalue problems, mathematical background, – midpoint (improved polygon) method, – , – , – , Milne’s method, – , – , multistep methods, – , NewtonCotes integration formulas, noncomputer methods, – onestep methods, – , – , power methods, – Ralston’s method, – , RungeKutta (RK) methods, – , scope/preview, – secondorder equations, – , – shooting method, – stiff systems, – , – , systems of equations, – thirdorder RK, – Orthogonal, Orthogonal polynomials, – Overconstrained optimization, Overdamped case, Overdetermined equations, Overflow error, – Overrelaxation, P Parabolic interpolation optimization, – , Parabolic partial differential equations (PDEs), – alternatingdirection implicit (ADI) method, – , – , , CrankNicolson technique, – , explicit methods, – , finitedifference methods, – , – , heatconduction equation, – , – implicit methods, – , – , onedimensional, – , twodimensional, – , Parameter estimation, Parameters, – , distributedparameter system, estimation, lumpedparameter systems, sinusoidal function, – Parametric Technology Corporation (PTC), Partial derivatives, – , – Partial differential equations (PDEs), – advanced methods and additional references, boundary conditions, – , – case studies, – characteristics, – computer solutions, – , – defined, elliptic equations, – , – , engineering applications, – , – , – finitedifference methods, – , – , finiteelement methods, – , goals/objectives, – higherorder temporal approximations, – hyperbolic equations, important relationships and formulas, – order of, parabolic equations, – , – , precomputer methods, – scope/preview, – Partial pivoting, – , Pattern searches/directions, – Penalty functions, Period, sinusoidal function, – Phaseplane representation, – INDEX Phase shift, Pivoting, – , complete, division by zero, effect of scaling, – partial, – , pivot coefficient/element, – Place value, Pointslope method. See Euler’s method Poisson equation, – Polynomial regression, – , algorithm, – fit of secondorder polynomial, – Polynomials computing with, – defined, deflation, – eigenvalue problems, – , – engineering applications, – evaluation and differentiation, – factored form, interpolation, – Lagrange, Lagrange interpolating, – , – Newton’s divideddifference, – , – , , order, ordinary differential equations (ODEs), – , orthogonal, – polynomial approximation, – regression, – , roots. See Roots of polynomials synthetic division, – Polyroots, Populations, estimating properties of, – Positional notation, Positive definite matrix, Postprocessing, finiteelement methods, – , Posttest loops, – Potential energy, – Potentiometers, Powell’s method of optimization, – , Power equations, linear regression of, – Power methods, – defined, determining largest eigenvalue, – determining smallest eigenvalue, – Power spectrum, – Precision, – , Predatorprey models, – Predictorcorrector approach, – , – Predictor equation, – , Predictor modifier, – Pretest loops, – Principal/main diagonal of matrix, Product, matrix operations, Programming and software. See Computer programming and software; Pseudocode algorithms Propagated truncation error, Propagation problems, – . See also Hyperbolic partial differential equations (PDEs); Parabolic partial differential equations (PDEs) Proportionality, Pseudocode algorithms, – adaptive quadrature, – Bairstow’s method, – bisection, Brent’s rootlocation method, – , – Cholesky decomposition, computing with polynomials, – curve fitting, defined, discrete Fourier transform (DFT), – , – Euler’s method, – Excel VBA vs., fast Fourier transform (FFT), – fixedpoint iteration, – forward elimination, function that solves differential equations, GaussJordan method, GaussSeidel (Liebmann) method, – for generic iterative calculation, – goldensectionsearch optimization, – Lagrange interpolation, linear regression, logical representation, – LU decomposition, – , – MATLAB vs., matrix inverse, – modified falseposition method, Müller’s method, – multiple linear regression, Newton’s divideddifference interpolating polynomials, – ordinary differential equations (ODEs), – , – , – , – partial pivoting, polynomial regression, Romberg integration, – roots of quadratic equation, – , – RungeKutta (RK) method, – Simpson’s rules, – , – Thomas algorithm, – trapezoidal rule, – Q QR factorization, QR method (Francis), Quadratic equation, algorithm for roots, – INDEX open methods. See Open methods optimization and, polynomials. See Roots of polynomials scope/preview, – secant method, – , – , as zeros of equation, – Roots of polynomials, – . See also Roots of equations Bairstow’s method, – , Brent’s method, characteristic equation, – computer methods, – conventional methods, – critically damped case, discriminant, eigenvalue problems, – engineering applications, – general solution, JenkinsTraub method, Laguerre’s method, mathematical background, – Müller’s method, – , – , NewtonRaphson method, other methods, overdamped case, Ridder method, secant method, – underdamped case, RosinRommlerBennet (RRB) equation, Rounding, Roundoff errors, – adding a large and a small numbers, – arithmetic manipulation of computer numbers, – common arithmetic operations, – computer representation of numbers, – defined, Euler’s method, extended precision, – Gauss elimination, integer representation, – iterative refinement, – large computations, – linear algebraic equations, number systems, numerical differentiation, – polynomial deflation, – significant digits and, – , smearing, – subtractive cancellation, – total numerical error, – Row, defined, Rowsum norms, Row vectors, RungeKutta Fehlberg method, – , – Quadratic interpolation, – Quadratic programming, Quadratic splines, – Quadrature methods, Quantizing errors, – , QuasiNewton methods of optimization, Quotient difference (QD) algorithm, – R Ralston’s method, – , Random search method of optimization, – Rate equations, Razdow, Allen, Reaction kinetics, RedlichKwong equation, Regression. See Linear regression; Polynomial regression Relative error, Relaxation, – Remainder, Taylor series, – , Repetition, in logical representation, – Residual error, – Respiratory quotient, Response, Richardson extrapolation, – , – , – Ridder method, root of polynomials, Romberg integration, – , Rootmeansquare current, – Root polishing, Roots of equations, – advanced methods and additional references, – analytical/direct method, bisection method, – , – , – bracketing methods. See Bracketing methods Brent’s method, – , case studies, – computer methods, – , – engineering applications, – , – , – falseposition method, – , – , fixedpoint iteration, – , goals/objectives, graphical methods, – , – , – , important relationships and formulas, incremental searches/determined incremental guesses, JenkinsTraub method, Laguerre’s method, mathematical background, – , – multiple roots, – nature of “roots,” NewtonRaphson method, – , – , – , , noncomputer methods, – , nonlinear equations, – INDEX implementation, – slack variables, Simpson’s / rule, – , – , – , – , algorithms, – derivation and error estimate, multipleapplication, – , singleapplication, – with unequal segments, – Simpson’s / rule, – , – , – , algorithms, – with unequal segments, – Simultaneous overrelaxation, Singlevalue decomposition, Singlevariable optimization, Singular systems, – Sinusoidal functions, – leastsquares fit of sinusoid, – parameters, – Slack variables, Smearing, – Software. See Computer programming and software Special matrices, – Spectral norms, Spline functions, Spline interpolation, – cubic splines, – , – , – , end conditions, – engineering applications, – linear splines, – quadratic splines, – splines, defined, Spread around the mean, Spread around the regression line, Spreadsheets. See Excel Square matrices, Stability defined, error propagation, – of multistep methods, – , of numerical methods of problem solving, – Standard atmosphere, Standard deviation, Standard error of the estimate, Standard normal estimate, – Standard normal random variable, Start, – Statistical inference, Statistics, – descriptive, – estimation of confidence interval, – , – inferential, leastsquares theory, – RungeKutta (RK) methods, – , – adaptive, – , adaptive stepsize control, – , – algorithms, – CashKarp RK method, – , – comparison, – embedded, – fourthorder, – , – , – , – , – , – higherorder, – RungeKutta Fehlberg method, – , – secondorder, – systems of equations, – thirdorder, –
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