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 |  موضوع: كتاب Numerical Methods for Engineers الإثنين 02 يناير 2023, 5:20 pm | |
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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
و المحتوى كما يلي :
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 Round-Off Errors
. Significant Figures
. Accuracy and Precision
. Error Definitions
. Round-Off 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 . Trade-Offs
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 False-Position Method
. Incremental Searches and Determining Initial Guesses
Problems
CHAPTER
Open Methods
. Simple Fixed-Point Iteration
. The Newton-Raphson 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 . Trade-Offs
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
. Gauss-Jordan
. 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 Gauss-Seidel
. Special Matrices
. Gauss-Seidel
. Linear Algebraic Equations with Software Packages
Problems
CHAPTER
Case Studies: Linear Algebraic Equations
. Steady-State 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)
. Spring-Mass Systems (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART THREE
PT . Trade-Offs
PT . Important Relationships and Formulas
PT . Advanced Methods and Additional References
PART FOUR
OPTIMIZATION PT . Motivation
PT . Mathematical Background
PT . Orientation
CHAPTER
One-Dimensional Unconstrained Optimization
. Golden-Section 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
. Least-Cost Design of a Tank (Chemical/Bio Engineering)
. Least-Cost 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 . Trade-Offs
PT . Additional References
PART FIVE
CURVE FITTING PT . Motivation
PT . Mathematical Background
PT . Orientation
CHAPTER
Least-Squares Regression
. Linear Regression
. Polynomial Regression
. Multiple Linear Regression
. General Linear Least Squares
. Nonlinear Regression
Problems
CHAPTER
Interpolation
. Newton’s Divided-Difference 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 . Trade-Offs
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
Newton-Cotes Integration Formulas
. The Trapezoidal Rule
. Simpson’s Rules
. Integration with Unequal Segments
. Open Integration Formulas
. Multiple Integrals
Problems CONTENTS xi
CHAPTER
Integration of Equations
. Newton-Cotes Algorithms for Equations
. Romberg Integration
. Adaptive Quadrature
. Gauss Quadrature
. Improper Integrals
. Monte Carlo Integration
Problems
CHAPTER
Numerical Differentiation
. High-Accuracy 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)
. Root-Mean-Square Current by Numerical Integration (Electrical
Engineering)
. Numerical Integration to Compute Work (Mechanical/Aerospace
Engineering)
Problems
EPILOGUE: PART SIX
PT . Trade-Offs
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
Runge-Kutta Methods
. Euler’s Method
. Improvements of Euler’s Method
. Runge-Kutta Methods
. Systems of Equations
. Adaptive Runge-Kutta Methods
Problems
CHAPTER
Stiffness and Multistep Methods
. Stiffness
. Multistep Methods
Problems
CHAPTER
Boundary-Value and Eigenvalue Problems
. General Methods for Boundary-Value 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)
. Predator-Prey 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 . Trade-Offs
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 Control-Volume Approach
. Software to Solve Elliptic Equations
Problems
CHAPTER
Finite Difference: Parabolic Equations
. The Heat-Conduction Equation
. Explicit Methods
. A Simple Implicit Method
. The Crank-Nicolson Method
. Parabolic Equations in Two Spatial Dimensions
Problems
CHAPTER
Finite-Element Method
. The General Approach
. Finite-Element Application in One Dimension
. Two-Dimensional Problems
. Solving PDEs with Software Packages
Problems
CHAPTER
Case Studies: Partial Differential Equations
. One-Dimensional Mass Balance of a Reactor (Chemical/Bio
Engineering)
. Deflections of a Plate (Civil/Environmental Engineering)
. Two-Dimensional Electrostatic Field Problems (Electrical
Engineering)
. Finite-Element Solution of a Series of Springs
(Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART EIGHT
PT . Trade-Offs
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, – ,
Adams-Bashforth formula, – ,
Adams-Moulton formula, –
Adaptive integration,
Adaptive quadrature, –
Adaptive Runge-Kutta (RK) methods, – ,
Adaptive step-size 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, –
Advection-diffusion equation,
Air resistance
falling parachutist problem, –
formulation,
Allosteric enzymes, –
Alternating-direction implicit (ADI) method, – , – ,
,
Amplitude, –
Analytical/direct approach
curve fitting, –
falling parachutist problem, – . See also Falling parachutist
problem
finite-element 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, –
finite-element 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, –
round-off 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, –
false-position 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, – , – ,
finite-element methods, – , – ,
irregular boundaries, –
Laplace equation, – , –
Boundary-value problems, – ,
eigenvalue, –
finite-difference method,
initial-value problems vs.,
shooting method, –
Bracketing methods, – ,
bisection method, – , – ,
computer methods, –
defined,
false-position method, – , – ,
graphical method, –
incremental searches/determining initial guesses,
Break command,
Break loops, –
Brent’s root-location method, – ,
algorithm, – , –
graphical method,
inverse quadratic interpolation, –
optimization, – ,
roots of polynomials,
Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithms,
B splines,
Butcher’s fifth-order Runge-Kutta method, –
Butterfly network,
C
C++,
Cartesian coordinates,
CASE structure,
Cash-Karp RK method, – , –
Centered finite divided-difference 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, –
least-cost design of a tank, –
linear algebraic equations, –
numerical integration, –
one-dimensional mass balance of reactor, –
optimization, –
ordinary differential equations (ODEs), –
partial differential equations (PDEs), –
roots of equations, –
steady-state 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, –
least-cost treatment of wastewater, –
linear algebraic equations, –
numerical integration, –
optimization, –
ordinary differential equations (ODEs), –
partial differential equations (PDEs), –
predator-prey models and chaos, –
roots of equations, –
splines to estimate heat transfer, –
Clamped end condition, –
Classical fourth-order Runge-Kutta method, – ,
Coefficient, method of undetermined, –
Coefficient of determination,
Coefficient of interpolating polynomial,
Coefficient of thermal conductivity,
Coefficient of variation,
Colebrook equation,
Column, defined,
Column-sum 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, –
step-size control,
structured programming, –
Condition numbers, –
matrix, –
Confidence intervals, – , –
Conjugate directions, –
Conjugate gradient,
Conservation laws, –
by field of engineering,
simple models in specific fields,
stimulus-response 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,
Control-volume approach, –
Convergences
defined,
fixed-point iteration, –
Gauss-Seidel (Liebmann) method, –
linear, –
nature of,
Newton-Raphson method, –
of numerical methods of problem solving,
Cooley-Tukey algorithm, –
Corrector equation, –
Corrector modifier, –
Correlation coefficient,
Count-controlled loops, – ,
Cramer’s rule, – ,
Crank-Nicolson 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 mean-value 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, finite-element methods, –
Discriminant,
DISPLAY statements,
Distributed-parameter system,
Distributed-variable 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, – ,
boundary-value problem, –
computer methods, –
eigenvalue, defined,
eigenvalue analysis of axially loaded column, –
eigenvectors, –
mass-spring 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, – ,
–
least-squares regression, –
linear regression, –
mathematical background, –
multidimensional interpolation, –
multiple linear regression, – , –
Newton’s divided-difference 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, – ,
Davidon-Fletcher-Powell (DFP) method of optimization,
Decimal (base- ) number system, – , –
Decimation-in-frequency, –
Decimation-in-time, – ,
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, –
two-pronged approach, –
Entering variables, –
Epilimnion,
Equal-area 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, –
Newton-Raphson estimation method, –
Newton’s divided-difference interpolating polynomial estimation,
–
numerical differentiation, – , –
numerical integration, –
predictor-corrector approach, – , –
quantizing, – ,
relative,
residual, –
round-off. See Round-off 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), –
root-mean-square current, –
roots of equations, –
simulating transient current for electric circuit, –
two-dimensional electrostatic field problems, –
Element properties, finite-element methods,
Element stiffness matrix,
Elimination of unknowns, –
back substitution, – , –
forward, – ,
Elliptic partial differential equations (PDEs), – , – ,
boundary conditions, –
computer software solutions, –
control-volume approach, –
Gauss-Seidel (Liebmann) method, – , –
Laplace equation, – , – , –
Laplacian difference equation, –
solution technique, –
Embedded Runge-Kutta (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, – , –
Newton-Raphson estimation method, –
parameter,
standard error of the estimate,
standard normal estimate, –
Euclidean norms, –
Euler-Cauchy 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, round-off 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, – , –
False-position method, – , – ,
bisection method vs., –
false-position formula, – ,
graphical method,
modified false position, – ,
pitfalls, –
secant method vs., –
Fanning friction factor,
Faraday’s law,
Fast Fourier transform (FFT), – ,
Cooley-Tukey algorithm, –
Sande-Tukey algorithm, –
Feasible extreme points,
Feasible solution space, –
Fibonacci numbers, –
Fick’s law of diffusion,
Finish, –
Finite-difference methods, – , –
elliptic partial differential equations (PDEs), – , – ,
,
high-accuracy differentiation formulas, –
optimization, –
ordinary differential equations (ODEs), –
parabolic partial differential equations (PDEs), – ,
– ,
Finite-divided-difference approximations of derivatives, – , – ,
–
Finite-element 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
Fourth-order methods
Adams, – ,
Runge-Kutta, – , – , – , – ,
– , –
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, –
Gauss-Jordan 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, –
Gauss-Jordan method, – , –
Gauss-Legendre formulas, – ,
higher-point, –
two-point, –
Gauss-Newton method, – ,
Gauss quadrature, – , –
error analysis, –
Gauss-Legendre formulas, – ,
method of undetermined coefficients, –
Gauss-Seidel (Liebmann) method, – , – , – ,
,
algorithm, –
First finite divided difference,
First forward difference,
First forward finite divided difference,
First-order approximation, –
First-order equations, – ,
First-order splines, –
Fixed (Dirichlet) boundary condition, – ,
Fixed-point iteration, – ,
algorithm, –
convergences, –
graphical method, –
nonlinear equations, –
Fletcher-Reeves conjugate gradient algorithm,
Floating-point operations/flops, –
Floating-point 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 least-squares model, –
confidence intervals for linear regression, –
general matrix formulation, –
statistical aspects of least-squares theory, –
General solution,
Genetic algorithm,
Given’s method,
Global truncation error,
Golden ratio, –
Golden-section search optimization, – , – ,
extremum, –
golden ratio, –
single-variable optimization,
unimodal, –
Gradient, defined,
Gradient methods of optimization, –
conjugate gradient method (Fletcher-Reeves),
finite-difference approximation, –
gradients, –
Hessian, – ,
Marquardt’s method, – ,
Newton’s method, – , – , –
path of steepest ascent/descent, – , – ,
quasi-Newton methods,
Greenhouse gases, –
H
Hagen-Poiseulle law,
Half-saturation constant,
Hamming’s method,
Harmonics,
Hazen-Williams equation,
Heat balance,
Heat-conduction equation, – , – . See also Parabolic
partial differential equations (PDEs)
Hessenberg form,
Hessian, – ,
Heun’s method, – , – ,
non-self-starting, – ,
High-accuracy 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,
Ill-conditioned 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, –
Initial-value problems,
boundary-value problems vs.,
defined,
Inner products,
In place implementation,
INPUT statements,
Integer part, –
Integer representation, –
Integral calculus. See Numerical integration INDEX
Gauss-Seidel (Liebmann) method, – ,
secondary variables, –
solution technique, –
Laplacian difference equation, –
Large computations, interdependent computations, –
Large versus small systems,
Law of mass action,
LC networks/circuits, –
Least-squares fit of a sinusoid, –
Least-squares regression, –
general linear least-squares model, –
least-squares fit of a straight line, –
linear regression, – , –
multiple linear regression, – , –
nonlinear, – , – ,
polynomial regression, – ,
Leaving variables, –
Levenberg-Marquardt method,
Liebmann method. See Gauss-Seidel (Liebmann) method
Linear algebraic equations, –
advanced methods and additional references, –
case studies, –
comparisons of methods, –
complex systems,
computer methods, – , – , –
Cramer’s rule, – ,
determinants, –
distributed-variable systems,
division by zero,
elimination of unknowns, –
engineering applications, – , –
error analysis, –
Gauss elimination. See Gauss elimination
Gauss-Jordan method, – , –
Gauss-Seidel (Liebmann) method. See Gauss-Seidel (Liebmann)
method
general form,
goals/objectives, –
graphical method, – , – ,
ill-conditioned systems, –
important relationships and formulas,
Liebmann method. See Gauss-Seidel (Liebmann) method
LU decomposition methods, – ,
lumped-variable 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,
finite-element methods, –
interpolation functions, –
inverse, –
inverse quadratic interpolation method, –
Lagrange interpolating polynomials, – , –
linear interpolation method, –
multidimensional, –
Newton’s divided-difference 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, –
Gauss-Seidel (Liebmann) method, – , – , – ,
,
iterative refinement, –
J
Jacobian,
Jacobi iteration,
Jacobi’s method, –
Jenkins-Traub 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
round-off errors,
scaling, – , –
scope/preview, –
singular systems, –
special matrices, –
system condition, –
Linear convergences, –
Linear interpolation method. See also False-position method; Secant
method
defined, –
linear-interpolation 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 least-squares model, –
least-squares 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, –
Lotka-Volterra equations, –
Lower triangular matrix,
LR method (Rutishauser),
LU decomposition methods, – ,
algorithm, – , –
Crout decomposition, –
defined,
LU decomposition step, – ,
overview, –
substitution step, –
version of Gauss elimination, –
Lumped-parameter systems,
Lumped-variable 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 problem-solving process,
simple model, – INDEX
conservation of momentum,
curve fitting, –
equilibrium and minimum potential energy, –
finite-element 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, –
spring-mass systems, –
swinging pendulum, –
Method of false position. See False-position method
Method of lines, –
Method of undetermined coefficients, –
Method of weighted residuals (MWR), finite-element methods, –
M-files (MATLAB), – . See also MATLAB
Michaelis-Menten 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 fixed-point method,
Modified Newton-Raphson 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, –
Multiple-application trapezoidal rule, – ,
Multiple integrals, –
Mathematical programming. See Optimization
Mathsoft Inc.,
MathWorks, Inc., The,
MATLAB, – , – , –
assignment of values to variable names, –
built-in 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, –
M-files, – , – ,
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, –
stimulus-response 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, –
Maximum-magnitude norms,
Mean value, –
confidence interval on the mean, –
derivative mean-value theorem,
determining mean of discrete points, –
spread around,
Mechanical/aerospace engineering
analysis of experimental data, – INDEX
Multiple linear regression, – , – ,
Multiple roots, –
double roots,
modified Newton-Raphson method for multiple roots, – ,
Newton-Raphson method, –
secant method, –
triple roots,
Multiplication,
estimated error bounds,
inner products,
matrix operations, –
Multistep methods, – ,
defined,
higher-order methods, –
integration formulas, –
non-self-starting Heun, – ,
step-size control,
N
Naive Gauss elimination, –
back substitution, – , –
forward elimination of unknowns, – ,
operation counting, –
n-dimensional vector,
Nelder-Mead method,
Newmann boundary condition, – , –
Newton-Cotes integration formulas, – , – , – ,
Boole’s rule,
closed formulas, –
comparisons, –
defined,
higher-order, – , – , –
open formulas, –
ordinary differential equations (ODEs), –
Simpson’s / rule, – , – , – , – , – ,
, – ,
Simpson’s / rule, – , – , – , – ,
trapezoidal rule, – , – , – , – , – ,
,
Newton-Gregory forward formula,
Newtonian fluid,
Newton-Raphson method, – , – , – ,
algorithm, – ,
error estimates, –
graphical method,
modified method for multiple roots, – ,
multiple roots, –
Newton-Raphson formula,
nonlinear equations, –
pitfalls, –
roots of polynomials,
slowly converging function, –
Taylor series derivation, –
Taylor series expansion,
termination criteria, –
Newton’s divided-difference 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,
fixed-point iteration, –
linear equations vs.,
Newton-Raphson method, –
roots of equations, –
shooting method for boundary-value problems, –
systems of equations, – , –
Nonlinear programming optimization,
Nonlinear regression, – , – ,
Non-self-starting Heun, – ,
Normal distribution,
Normalization,
Normalized standard deviate, –
Norms
defined,
matrix, –
vector, –
“Not-a-knot” 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,
One-dimensional unconstrained optimization, –
Brent’s root-location method, – ,
golden-section search, – , – ,
MATLAB, –
multimodal, –
Newton’s method, – , – , –
parabolic interpolation, – ,
One-point iteration, . See also Fixed-point iteration
One-sided interval,
One-step methods, – , – ,
Open methods, – , – ,
Brent’s root-location method, – ,
defined, –
fixed-point iteration, – ,
graphical method,
multiple roots, –
Newton-Raphson method, – , – , – ,
,
secant method, – ,
simple fixed-point iteration, –
systems of nonlinear equations, –
Optimal steepest ascent, – ,
Optimization, –
additional references,
Brent’s root-location method, – ,
case studies, –
computer methods, – , –
defined,
engineering applications, – , –
goals/objectives, –
golden-section 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, – , –
finite-divided-difference approximations, – , –
first derivative, –
forward difference approximations,
goals/objectives, –
high-accuracy 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, – , – , –
round-off 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, –
Newton-Cotes formulas, – , – , – , – ,
, –
noncomputer methods, –
open forms, – , –
Richardson extrapolation, – , – , –
Romberg integration, – ,
scope/preview, – INDEX
mathematical background, –
multidimensional unconstrained, –
Newton’s method, – , – , –
noncomputer methods,
nonlinear constrained optimization, – ,
one-dimensional unconstrained, –
parabolic interpolation, – ,
problem classification, –
random search method, –
scope/preview, –
Order of polynomials,
Ordinary differential equations (ODEs), – , –
advanced methods and additional references, –
algorithms, – , – , – , –
boundary-value problems, – , – ,
case studies, –
components,
computer programming and software, –
defined,
eigenvalue problems, – ,
engineering applications, – , –
Euler’s method, – , – , – , –
explicit solution technique, –
falling parachutist problem, –
finite-difference methods, –
first-order equations, – ,
fourth-order Adams, – , –
fourth-order RK, – , – , – , – ,
– , –
goals/objectives, –
Heun’s method, – , – , – ,
higher-order equations, – , –
implicit solution technique, –
important relationships and formulas,
initial-value problems,
mathematical background, –
midpoint (improved polygon) method, – , – ,
– ,
Milne’s method, – , – ,
multistep methods, – ,
Newton-Cotes integration formulas,
noncomputer methods, –
one-step methods, – , – ,
power methods, –
Ralston’s method, – ,
Runge-Kutta (RK) methods, – ,
scope/preview, –
second-order equations, – , –
shooting method, –
stiff systems, – , – ,
systems of equations, –
third-order RK, –
Orthogonal,
Orthogonal polynomials, –
Overconstrained optimization,
Overdamped case,
Overdetermined equations,
Overflow error, –
Overrelaxation,
P
Parabolic interpolation optimization, – ,
Parabolic partial differential equations (PDEs), –
alternating-direction implicit (ADI) method, – , – ,
,
Crank-Nicolson technique, – ,
explicit methods, – ,
finite-difference methods, – , – ,
heat-conduction equation, – , –
implicit methods, – , – ,
one-dimensional, – ,
two-dimensional, – ,
Parameter estimation,
Parameters, – ,
distributed-parameter system,
estimation,
lumped-parameter 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, – , – , –
finite-difference methods, – , – ,
finite-element methods, – ,
goals/objectives, –
higher-order 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, –
Phase-plane representation, – INDEX
Phase shift,
Pivoting, – ,
complete,
division by zero,
effect of scaling, –
partial, – ,
pivot coefficient/element, –
Place value,
Point-slope method. See Euler’s method
Poisson equation, –
Polynomial regression, – ,
algorithm, –
fit of second-order polynomial, –
Polynomials
computing with, –
defined,
deflation, –
eigenvalue problems, – , –
engineering applications, –
evaluation and differentiation, –
factored form,
interpolation, –
Lagrange,
Lagrange interpolating, – , –
Newton’s divided-difference, – , – ,
,
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, finite-element 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, – ,
Predator-prey models, –
Predictor-corrector 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 root-location method, – , –
Cholesky decomposition,
computing with polynomials, –
curve fitting,
defined,
discrete Fourier transform (DFT), – , –
Euler’s method, –
Excel VBA vs.,
fast Fourier transform (FFT), –
fixed-point iteration, –
forward elimination,
function that solves differential equations,
Gauss-Jordan method,
Gauss-Seidel (Liebmann) method, –
for generic iterative calculation, –
golden-section-search optimization, –
Lagrange interpolation,
linear regression,
logical representation, –
LU decomposition, – , –
MATLAB vs.,
matrix inverse, –
modified false-position method,
Müller’s method, –
multiple linear regression,
Newton’s divided-difference interpolating polynomials, –
ordinary differential equations (ODEs), – , – , – ,
–
partial pivoting,
polynomial regression,
Romberg integration, –
roots of quadratic equation, – , –
Runge-Kutta (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,
Jenkins-Traub method,
Laguerre’s method,
mathematical background, –
Müller’s method, – , – ,
Newton-Raphson method,
other methods,
overdamped case,
Ridder method,
secant method, –
underdamped case,
Rosin-Rommler-Bennet (RRB) equation,
Rounding,
Round-off 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,
Row-sum norms,
Row vectors,
Runge-Kutta Fehlberg method, – , –
Quadratic interpolation, –
Quadratic programming,
Quadratic splines, –
Quadrature methods,
Quantizing errors, – ,
Quasi-Newton methods of optimization,
Quotient difference (QD) algorithm, –
R
Ralston’s method, – ,
Random search method of optimization, –
Rate equations,
Razdow, Allen,
Reaction kinetics,
Redlich-Kwong 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, – ,
Root-mean-square 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, – , – , –
false-position method, – , – ,
fixed-point iteration, – ,
goals/objectives,
graphical methods, – , – , – ,
important relationships and formulas,
incremental searches/determined incremental guesses,
Jenkins-Traub method,
Laguerre’s method,
mathematical background, – , –
multiple roots, –
nature of “roots,”
Newton-Raphson method, – , – , – ,
,
noncomputer methods, – ,
nonlinear equations, – INDEX
implementation, –
slack variables,
Simpson’s / rule, – , – , – , – ,
algorithms, –
derivation and error estimate,
multiple-application, – ,
single-application, –
with unequal segments, –
Simpson’s / rule, – , – , – ,
algorithms, –
with unequal segments, –
Simultaneous overrelaxation,
Single-value decomposition,
Single-variable optimization,
Singular systems, –
Sinusoidal functions, –
least-squares 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,
least-squares theory, –
Runge-Kutta (RK) methods, – , –
adaptive, – ,
adaptive step-size control, – , –
algorithms, –
Cash-Karp RK method, – , –
comparison, –
embedded, –
fourth-order, – , – , – , – ,
– , –
higher-order, –
Runge-Kutta Fehlberg method, – , –
second-order, –
systems of equations, –
third-order, –
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