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 |  موضوع: كتاب Numerical Methods for Engineers Sixth Edition الأحد 18 ديسمبر 2022, 10:53 pm | |
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أخواني في الله
أحضرت لكم كتاب
Numerical Methods for Engineers Sixth Edition
Steven C. Chapra
Berger Chair in Computing and Engineering
Tufts University
Raymond P. Canale
Professor Emeritus of Civil Engineering
University of Michigan
و المحتوى كما يلي :
Contents
Preface Xiv
Guided Tour Xvi
About the Authors Xviii
Part One
Modeling, Pt . Motivation
Computers, and Pt . Mathematical Background
Error Analysis Pt . Orientation
Chapter
Mathematical Modeling and Engineering Problem Solving
. A Simple Mathematical Model
. Conservation Laws and Engineering
Problems
CHAPTER
Programming and Software
. Packages and Programming
. Structured Programming
. Modular Programming
. Excel
. MATLAB
. Mathcad
. Other Languages and Libraries
Problems
CHAPTER
Approximations and Round-Off Errors
. Significant Figures
. Accuracy and Precision
. Error Definitions
. Round-Off Errors
Problems
ivCONTENTS v
CHAPTER
Truncation Errors and the Taylor Series
. The Taylor Series
. Error Propagation
. Total Numerical Error
. Blunders, Formulation Errors, and Data Uncertainty
Problems
EPILOGUE: PART ONE
PT . 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 vi CONTENTS
. Müller’s Method
. Bairstow’s Method
. Other Methods
. Root Location with Software Packages
Problems
CHAPTER
Case Studies: Roots of Equations
. Ideal and Nonideal Gas Laws (Chemical/Bio Engineering)
. Greenhouse Gases and Rainwater (Civil/Environmental Engineering)
. Design of an Electric Circuit (Electrical Engineering)
. Pipe Friction (Mechanical/Aerospace Engineering)
Problems
EPILOGUE: PART TWO
PT . 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 CONTENTS vii
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 viii CONTENTS
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 CONTENTS ix
CHAPTER
Fourier Approximation
. Curve Fitting with Sinusoidal Functions
. Continuous Fourier Series
. Frequency and Time Domains
. Fourier Integral and Transform
. Discrete Fourier Transform (DFT)
. Fast Fourier Transform (FFT)
. The Power Spectrum
. Curve Fitting with Software Packages
Problems
CHAPTER
Case Studies: Curve Fitting
. Linear Regression and Population Models (Chemical/Bio
Engineering)
. Use of Splines to Estimate Heat Transfer (Civil/Environmental
Engineering)
. Fourier Analysis (Electrical Engineering)
. Analysis of Experimental Data (Mechanical/Aerospace
Engineering)
Problems
EPILOGUE: PART FIVE
PT . 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 x CONTENTS
CHAPTER
Integration of Equations
. Newton-Cotes Algorithms for Equations
. Romberg Integration
. Adaptive Quadrature
. Gauss Quadrature
. Improper Integrals
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 CONTENTS xi
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 xii CONTENTS
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 APPENDIX A: THE FOURIER SERIES
APPENDIX B: GETTING STARTED WITH MATLAB
APPENDIX C: GETTING STARTED WITH MATHCAD
BIBLIOGRAPHY
INDEX
INDEX
A
Accuracy,
Accuracy and precision, –
Adams-Bashforth formulas, –
Adams formulas
Adams-Bashforth formulas, –
Adams-Moulton formulas, –
closed formulas, –
Newton-Coles, contrasted, ,
open formulas, –
Adams-Moulton formulas, –
Adaptive fourth-order RK method, ,
–
Adaptive quadrature, –
Adaptive Runge-Kutta methods, –
adaptive fourth-order RK method, ,
–
adaptive step-size control,
pseudocode, ,
Runge-Kutta Fehlberg, –
step-size control, –
Adaptive step-size control,
Adding large and small number, –
Addition,
ADI scheme, –
Advanced methods/additional references
background material, –
curve fitting,
linear algebraic equations, –
numerical differentiation and integration,
ODEs, –
optimization,
PDEs,
roots of equations, –
Advection-dispersion equation,
Algorithms,
Alternating-direction implicit (ADI) scheme,
–
Alzheimer’s disease,
Amplitude,
Analysis of experimental data, –
Analytical solution, ,
Andrade’s equation,
Angular frequency,
Antidifferentiation,
Approximate percent relative error,
Approximation function,
Approximations and round-off errors, –
accuracy and precision, –
error definitions, –
iterative calculation, –
round-off error. See Round-off error
significant figures, –
Areal integral,
Arithmetic manipulations of computer numbers,
–
Arithmetic mean,
Artificial neural network,
Ascent methods,
Assemblage property matrix,
Augmentation,
Auxiliary conditions,
Axially-dispersed plug-flow reactor, –
Axially loaded column, – ,
B
B splines,
Back substitution, –
Background material, –
accuracy and figures, –
additional references, –
blunders, –
computer objectives,
computer programming, – . See also
Computer programming
condition,
conservation laws, –
data uncertainty,
error, –
error definitions, –
error propagation, –
formulation errors,
important relationships/formulas,
iterative calculation, –
mathematical model, –
overview, ,
round-off error. See Round-off error
scope/preview, ,
significant figures, –
stability,
study objectives,
Taylor series. See Taylor series
total numerical error, –
truncation errors,
Backward deflation,
Backward difference approximation,
Backward Euler’s method,
Backward finite-divided-difference formula,
Bairstow’s method, –
Banded matrix, ,
Base- number system,
Base- number system,
Base- system,
Basic feasible solution,
Basic variables,
Best fit, –
BFGS, ,
Bias,
Bibliography, –
Bilinear functions,
Bilinear interpolation, –
Binary chopping,
Binary system,
Binding constraints,
Bingham plastics,
Biofilm,
Bisection method, –
algorithm,
alternative names,
error estimates, –
false-position method, compared,
–
minimizing function evaluations,
pseudocode,
termination criteria,
Bit reversal,
Bits,
Blasius formula,
Blood,
Blunders, –
Bolzano’s method,
Book, overview, –
Boole’s rule, ,
Boundary conditions, –
Boundary-value problems, –
eigenvalues, –
finite-difference methods, –
nonlinear two-point problems, –
other techniques, –
shooting method, –
Boussinesq’s equation,
Bracketing methods, –
bisection method, –
false-position method, –
graphical methods, –
incremental searches,
initial guesses,
modified false position, –
open methods, compared,
Breadth of application,
Break loop, –
Brent, Richard,
Brent’s method
algorithm, –
inverse quadratic interpolation, –
optimization, ,
pseudocode, ,
roots of equations, –
Brent’s root finding method,
Brent’s root-location method,
Broyden-Fletcher-Goldfarb-Shanno (BFGS),
,
Bubble sort, –
Buckling load,
Bulirsch-Stoer method,
Butcher’s fifth-order RK method,
Butterfly network,
C
Calculator,
Calculus, . See also Numerical
differentiation and integration
Canned programs,
CASE structure, ,
Case studies
analysis of experimental data, –
chaos, –
currents and voltages in resistor circuits,
–
curve fitting, –
deflections of a plate, –
electrical circuit design, –
equilibrium and minimum potential energy,
–
INDEX
Fourier analysis, –
greenhouse gases, –
heat calculations, –
heat transfer, –
ideal/nonideal gas laws, –
least-cost design of tank, –
least-cost treatment of wastewater, –
linear algebraic equations, –
linear regression and population models,
–
mass balance of reactor, –
maximum power transfer for circuit,
–
numerical integration/differentiation,
–
ODEs, –
optimization, –
PDEs, –
pipe friction, –
population models, –
predator-prey models, –
rainwater, –
reactor, – , –
reactors, –
root-mean-square current, –
roots of equations, –
sailboat, –
series of springs, –
simulating transient current for electric
current, –
spring-mass system, –
statistically determinate truss, –
steady-state analysis of system of reactors,
–
swinging pendulum, –
transient responses of reactor, –
truss, –
two-dimensional electrostatic field
problems, –
work, calculation of, –
Cash-Karp RK method, ,
Casson region,
Casson relationship,
Catenary cable,
Centered difference approximation,
Centered finite-divided-difference formula,
Central limit theorem,
Chaos, –
Chaotic solutions,
Characteristic,
Characteristic equation,
Characteristic value,
Chebyshev economization,
Cholesky decomposition, –
Chopping,
Classical fourth-order RK method, –
Closed integration formulas, ,
Coefficient of determination,
Coefficient of thermal conductivity,
Coefficient of thermal diffusivity,
Coefficient of variation (c.v.),
Colebrook equation, ,
Collocation approach,
Column,
Column-sum norm,
Column vectors,
Commercial programming libraries,
Comparison of one-dimensional methods,
–
Complete pivoting,
Composite integration formulas,
Computer,
Computer program,
Computer programming, –
algorithms,
CASE structure, ,
commercial programming libraries,
computer program,
count-controlled loop, –
DOEXIT construct, ,
DOFOR loop, –
Excel. See Excel
flowchart,
high-level languages,
IF/THEN/ELSE structure, ,
IF/THEN/ELSEIF structure, ,
IF/THEN structure, ,
logical representation, –
loops, –
Mathcad. See Mathcad
MATLAB. See MATLAB
modular programming, –
numerical library,
other languages,
packages, –
pseudocode, ,
repetition, –
selection, –
sequence,
structured programming, –
Computer representation of numbers, –
Computer software. See Excel; Mathcad;
MATLAB
Condition,
Condition number, , –
Confidence intervals, – Confidence intervals for linear regression,
–
Conjugate directions,
Conjugate gradient method,
Conservation laws, –
Conservation of charge,
Conservation of energy,
Conservation of mass, ,
Conservation of momentum,
Consistency index,
Constant of integration,
Constitutive equation,
Constrained optimization, , –
linear programming, –
nonlinear,
simplex method, –
Continuous Fourier series, –
Control-volume approach, –
Convergence, , ,
Cooley, J. W.,
Cooley-Tukey algorithm, –
Corrector,
Corrector equation,
Corrector modifier,
Correlation coefficient,
Count-controlled loop, –
Covariance, n
Cramer’s rule, – ,
Crank-Nicolson algorithm,
Crank-Nicolson method, –
Creep rate,
Critically damped case,
Crout decomposition, –
Cubic interpolating polynomial,
Cubic splines, , , –
Cumulative normal distribution,
Current balance,
Currents and voltages in resistor circuits,
–
Curvature,
Curve fitting, –
advanced methods/references,
analysis of experimental data, –
approaches,
case studies, –
computer objectives, –
engineering practice,
Excel, –
Fourier analysis, –
Fourier approximation. See Fourier
approximation
heat transfer, –
important relationships/formulas,
INDEX
interpolation. See Interpolation
least-squares regression. See Least-squares
regression
linear regression and population models,
–
Mathcad, –
mathematical background, –
MATLAB, –
methods, compared,
noncomputer methods,
overview, –
population models, –
scope/preview, –
sinusoidal functions, –
statistics, –
study objectives, ,
trade-offs, –
Curvilinear interpolation,
c.v.,
D
Danckwerts boundary condition,
Data distribution,
Data uncertainty,
Davidon-Fletcher-Powell (DFP),
Decimal number system,
Decimation in time, ,
Decimation in frequency,
Decision loop,
Definite integration, n
Deflation,
Deflections of a plate, –
Degrees of freedom,
Dekker, Theodorus,
del f,
Dependent variable, ,
Derivative,
Derivative boundary conditions, – ,
Derivative form,
Derivative mean-value theorem,
Descent methods,
Descriptive models,
Design,
Design of electric circuit, –
Design vector,
Determinant, –
Determinant evaluation,
DFP,
DFT, –
Diagonal matrix,
Diagonally dominant,
Differential calculus,
Differential equation
defined,
ordinary. See Ordinary differential equations
partial. See Partial differential equations
what is it,
Differentiation, – . See also Numerical
differentiation and integration
Dirac delta function,
Direct approach, , –
Direct methods, –
Directional derivative,
Dirichlet boundary condition, , ,
Dirichlet condition, n
Discrete Fourier transform (DFT), –
Discretization errors, ,
Discriminant,
Distributed-parameter system,
Distributed variable problems, ,
Distribution coefficient,
Divide and average method,
Divided-difference table,
Division,
DOEXIT construct, ,
DOFOR loop, –
Doolittle decomposition,
Double integral, –
Double precision,
Double root,
Drag coefficient,
Dynamic instability,
E
Ease of application,
Eigenvalue, ,
Eigenvalue problems, –
axially loaded column, – ,
boundary-value problem, –
Given’s method,
Hotelling’s method,
Householder’s method,
intermediate eigenvalue,
Jacobi’s method, –
largest eigenvalue, –
LR method,
mathematical background,
physical background, –
polynomial method, –
power method, –
QR method,
smallest eigenvalue,
Eigenvector,
Electrical circuit design, –
Element property,
Element stiffness matrix, Elimination methods, – ,
Elimination of unknowns, –
Elliptic equations, –
boundary conditions, –
control-volume approach, –
derivative boundary conditions, –
flux distribution for heated plate, –
heated plate with insulated edge, –
heated plate with irregular boundary,
–
irregular boundaries, –
Laplace equation, –
Laplacian difference equation, –
Liebmann’s method, –
secondary variables,
software, –
solution technique, –
temperature of heated plate, –
Embedded RK method,
Energy balance,
Engineering practice. See Case studies
Engineering problem solving
accuracy and precision, –
blunders, –
computer programming, – . See also
Computer programming
condition,
conservation laws, –
data uncertainty,
error, –
error definitions, –
error propagation, –
formulation errors,
iterative calculation, –
mathematical model, –
phases,
process,
round-off error. See Round-off error
significant figures, –
stability,
Taylor series. See Taylor series
total numerical error, –
truncation errors,
two-pronged approach,
Engineering problem-solving process,
Entering variable,
Enzymatic reactions,
Epilimnion,
Equal-area graphical differentiation, ,
Equality constraints,
Equilibrium and minimum potential energy,
–
Ergun equation,
INDEX
Error. See Approximations and round-off errors;
Truncation errors and Taylor series
Error analysis and system condition, –
Error definitions, –
Error estimates for interactive methods, –
Error propagation, –
Estimated mean,
Estimation,
Euclidean norm, –
Euler-Cauchy method,
Euler’s formula,
Euler’s identity, ,
Euler’s method, –
algorithm, –
backward,
basic equation,
C,
computer program, –
equation and basic approach, –
error analysis, –
Excel,
Fortran function,
fundamental source of error,
Heun’s method,
higher-order Taylor series methods,
implicit, –
MATLAB,
pseudocode, ,
solving differential equation,
step size, –
systems of equations,
Taylor series, ,
Even function,
Exact method,
Exact solution,
Excel, –
built-in numerical capabilities,
condition number,
conjugate gradient approach,
curve fitting, –
Data Analysis Toolpack, –
Dirichlet boundary conditions,
double precision,
Euler’s method,
evaluation of ex using infinite series, –
factorial, ,
Goal Seek,
iterative calculation,
least-cost treatment of wastewater, –
Liebmann method,
linear equations, –
linear programming, –
maximum power transfer, –
nonlinear optimization, –
nonlinear regression, , –
nonlinear system of equations, –
optimization, –
parachute optimization problem, –
parachutist problem, ,
PDEs, –
random search,
reaction kinetics, –
regression fits of data,
roots of equations, –
roots of quadratic,
Solver, –
Trendline, –
VBA macro,
visualization tools,
Explicit methods, – ,
Exponent,
Exponential model,
Extended midpoint rule,
Extended precision, –
Extrapolation,
Extreme point,
F
Factored form of polynomial,
Factorial, ,
Factorization,
Falling parachutist problem. See Parachutist
problem
False-position formula,
False-position method, –
bisection, compared, –
derivation,
modified false position, –
name,
pitfalls, –
secant method, compared, –
Fanning friction factor, –
Faraday’s law, , ,
Fast Fourier transform (FFT), –
Feasible extreme points,
Feasible solution space,
FFT, –
Fibonacci numbers,
Fick’s first law,
Fick’s law of diffusion,
Finite-difference approximations, –
Finite difference approximations of higher
derivatives, –
Finite difference methods. See Elliptic
equations; Parabolic equations
Finite-difference methods, – Finite divided difference, , , ,
Finite-divided-difference approximations of
derivatives, –
Finite-element method, –
approximation functions, –
assembly, , – , –
boundary conditions, , , –
direct approach, –
discretization, , ,
element equations, – , – ,
–
fit of the function to solution, –
heated rod, – ,
method of weighted residuals (MWR),
–
one dimension, –
solution, , ,
two-dimensional problems, –
First backward difference,
First derivative,
First finite divided difference, ,
First forward difference,
First-order equation,
First-order error propagation, –
First-order splines, –
Fixed-point iteration, –
algorithm, ,
convergence, –
divergence,
nonlinear equations, –
pseudocode,
two-curve graphical method, –
Fletcher-Reeves conjugate gradient
algorithm,
Floating-point operations, –
Floating-point representation, –
Flops, –
Flow balance,
Flowchart,
Flux distribution for heated plate, –
Force balance, ,
Forcing function,
Formulation errors,
Forward deflation, –
Forward difference approximation,
Forward divided difference,
Forward elimination of unknowns, –
Forward finite-divided-difference formulas,
Fourier analysis, –
Fourier approximation, –
continuous Fourier series, –
Cooley-Tukey algorithm, –
discrete Fourier transform (DFT), –
INDEX
fast Fourier transform (FFT), –
Fourier integral, –
Fourier transform, –
frequency domain, –
power spectrum,
references, –
Sande-Tukey algorithm, –
sinusoidal functions, –
time domain, –
Fourier integral, –
Fourier series, –
Fourier transform, –
Fourier transform pair,
Fourier’s heat law,
Fourier’s law, , ,
Fourier’s law of heat conduction, – ,
Fourth Adams-Bashforth open formula,
Fourth Adams-Moulton closed formula,
Fourth-order Adams method, , –
Fourth-order RK methods, –
Frequency domain, –
Frequency plane, –
Friction factor,
Frobenius norm,
Fully augmented version,
Functional approximation,
Functions,
Fundamental frequency,
Fundamental theorem,
G
Galerkin’s method, ,
Gauss elimination, –
algorithm,
back substitution, –
complex systems,
determinant evaluation,
division by zero,
falling parachutist problem, –
floating-point operations, –
forward elimination of unknowns, –
Gauss-Jordan, –
ill-conditioned systems, –
LU decomposition, –
naive, –
names,
Newton-Raphson method, ,
nonlinear system of equations, –
operation counting, –
pitfalls, –
pivoting, –
pseudocode, ,
round-off errors,
scaling, , –
significant figures,
singular systems,
Gauss-Jordan, –
Gauss-Legendre formulas, , –
Gauss-Newton method, , –
Gauss quadrature, –
error analysis, –
falling parachutist problem,
higher-point formulas, –
method of undetermined coefficients,
–
three-point Gauss-Legendre formula,
two-point Gauss-Legendre formula,
–
Gauss Seidel, –
algorithm, –
convergence, –
divergence,
formula/equations, – ,
Jacobi iteration, compared, ,
PDEs,
problem contexts, –
pseudocode,
relaxation,
trade-offs,
General linear least squares, –
General solution,
Generalized reduced gradient (GRG),
Genetic algorithm,
Given’s method,
Glaucoma,
Global truncation error,
Global warming,
Goal Seek,
Golden ratio,
Golden-section search, –
algorithm,
example,
golden ratio,
initial step, –
minimizing number of function evaluations,
,
pseudocode, ,
second step,
Gossett, W. S.,
Gradient, – ,
Gradient methods, –
Graphical methods
linear equations, –
roots of equations, –
Graphical solution,
Greenhouse gases, – GRG,
Grid search,
Gross error,
H
Half-saturation constant,
Halving-doubling strategy, ,
Hamming’s method,
Harmonics,
Hazen-Williams equation,
Heat balance,
Heat calculations, –
Heat conduction equation, –
Heat transfer, –
Heated plate with insulated edge, –
Heated plate with irregular boundary, –
Heated rod, – ,
Henry’s constant,
Hessenberg form,
Hessian, –
Heun’s method, – , , ,
–
High-accuracy differentiation formulas,
–
High-level languages,
Higher-order multistep methods, –
Higher-order Newton-Cotes closed formulas,
–
Higher-order Newton-Cotes formulas, ,
Higher-order RK methods,
Higher-order Taylor series methods,
Higher-order temporal approximations, –
Higher-point formulas, –
Hilbert matrix, ,
Histogram, ,
Hooke’s law, , , , ,
Hotelling’s method,
Householder’s method,
Human blood,
Humps function,
Hyperbolic equations, ,
Hypolimnion,
Hypothesis testing,
I
Ideal gas law,
Ideal/nonideal gas laws, –
Ideal vs. nonideal,
Identity matrix,
IF/THEN/ELSE structure, ,
IF/THEN/ELSEIF structure, ,
IF/THEN structure, ,
Ill-conditioned, ,
INDEX
Ill-conditioned systems, – , –
Implicit Euler’s method, –
Implicit method, , – ,
Imprecision,
Improper integrals, –
Improved polygon method,
Inaccuracy,
Increment function,
Incremental search methods, ,
Indefinite integral,
Indefinite integration, n
Independent variable, ,
Index, –
Indoor air pollution,
Inequality constraints,
Initial guess,
Initial value,
Initial-value problem, ,
Inner products,
Instability, ,
Integer representation, –
Integral calculus,
Integral form,
Integrand,
Integration, . See also Numerical
differentiation and integration
Integration of equations, –
adaptive quadrature, –
Gauss quadrature. See Gauss quadrature
higher-order error correction of integral
estimates, –
improper integrals, –
Newton-Coles algorithm, –
Richardson’s extrapolation, –
Romberg integration, , –
Interactive methods, –
Interpolation, –
bilinear, –
coefficients of interpolating polynomial,
equally spared data, ,
extrapolation,
ill-conditioning,
inverse, –
Lagrange interpolating polynomials,
linear, –
multidimensional, –
Newton’s interpolating polynomials. See
Newton’s interpolating polynomials
quadratic, –
spline. See Spline interpolation
Interpolation function,
Interval estimator,
Interval halving,
Intraocular pressure,
Inverse, , –
Inverse Fourier transform,
Inverse interpolation, –
Irregular boundaries, –
Iterative approach,
Iterative calculation, –
Iterative Heun method,
Iterative refinement, –
J
Jacobi iteration, ,
Jacobian,
Jacobi’s method, –
Jenkins-Traub method,
K
Kirchhoff’s current law,
Kirchhoff’s laws, , ,
Kirchhoff’s second law,
Kirchhoff’s voltage law, , , ,
Knot,
L
Lagging phase angle,
Lagrange form,
Lagrange interpolating polynomials,
Lagrange multiplier, ,
Lagrange polynomial,
Laguerre’s method,
Laplace equation, , – , ,
Laplacian difference equation, –
Large computations, –
Large vs. small systems,
Leading phase angle,
Least-cost design of tank, –
Least-cost treatment of wastewater, –
Least-squares,
Least-squares fit of sinusoid, –
Least-squares fit of straight line, –
Least-squares regression, –
Gauss-Newton method, , –
general linear least squares, –
linear regression. See Linear regression
multiple linear regression, –
nonlinear regression, –
polynomial regression, –
statistical aspects, –
Leaving variable,
Liebmann’s method, –
Line spectra, ,
Linear algebraic equations, –
advanced methods, –
approximate technique, , case studies, –
Cholesky decomposition, –
computer objectives, –
Cramer’s rule, –
Crout decomposition, –
distributed variable problems, ,
elimination methods, – ,
elimination of unknowns, –
engineering practice, –
exact methods,
Excel, –
Gauss elimination. See Gauss elimination
Gauss-Jordan, –
Gauss-Seidel. See Gauss-Seidel
general form,
graphical method, –
important relationships/formulas,
iterative refinement, –
LU decomposition. See LU decomposition
lumped variable problems, ,
Mathcad, –
mathematical background, –
MATLAB, –
matrix. See Matrix
matrix condition number, –
matrix inverse. See Matrix inverse
matrix norms, –
methods, compared,
noncomputer methods, –
numerical methods, –
overview, –
reactors, –
references, –
resistor circuits, –
scope/preview, –
spring-mass system, –
study objectives,
Thomas algorithm,
trade-offs, –
tridiagonal system, –
truss, –
vector norms, –
Linear convergence,
Linear equation,
Linear interpolation, , , –
Linear-interpolation formula,
Linear interpolation method,
Linear least squares, –
Linear least-squares fit, – ,
Linear ordinary differential equation,
Linear programming, , –
graphical solution, –
possible outcomes, –
INDEX
simplex method, –
standard form, –
Linear regression, –
algorithm,
best fit, –
computer program, –
general comments,
least-squares fit of straight line, –
linearization of nonlinear relationships,
–
pseudocode,
quantification of error, –
statistical assumptions,
Linear regression and population models,
–
Linear splines, –
Linear vs. nonlinear,
Linearization,
Linearization of nonlinear relationships,
–
Linearization of power equation, –
Little, John N.,
Local truncation error,
Logical representation, –
Loops, –
Lorenz, Edward,
Lorenz equation,
Lotka, Alfred J.,
Lotka-Volterra equation, ,
Lower Colorado River,
Lower triangular matrix,
LR method,
LU decomposition, –
algorithm, ,
Crout decomposition, –
decomposition phase,
forward-substitution steps,
Gauss elimination, –
overview, ,
pseudocode, , ,
steps in process, ,
substitution steps, –
Lumped-parameter systems,
Lumped variable problems, ,
M
M-files, ,
MacCormack’s method,
Machine epsilon,
Maclaurin series expansion, ,
Macro, . See also Excel
Main diagonal,
Maintenance,
Manning equation,
Manning roughness coefficient,
Manning’s formula,
Mantissa,
Marksmanship,
Marquardt’s method, –
Mass balance, , ,
Mass balance of reactor, –
Mass-spring system, –
Mathcad, –
condition number, ,
constrained nonlinear optimization,
cubic spline interpolation,
curve fitting, –
eigenvalues, ,
entering text,
FFT,
graphs, –
help,
linear equations, –
main menu,
math palette,
mathematical functions and variables,
–
mathematical operations, –
matrix computations and operations,
matrix functions,
matrix inverse,
multiline procedures/subprograms,
nonlinear system of equations,
numeric mode,
numerical integration/differentiation,
–
numerical methods function,
ODEs, –
online help,
optimization, ,
PDEs, –
Poisson’s equation,
probability distributions,
QuickSheets,
range variables, –
resource center,
roots of equations, –
roots of polynomial,
standard tool bar, –
stiff systems,
symbolic mathematics, –
symbolic mode,
ToolTips,
trig and logs,
units, –
what is it, Mathematical model, –
Mathematical programming problem,
Mathsoft,
MATLAB, –
array operations,
assignment, –
built-in functions,
condition number,
curve fitting, –
diary file,
differentiation,
double precision,
editor/debugger,
eigenvalues, – , , –
Euler’s method,
extended precision,
factorial, ,
FFT,
graphics, –
humps function,
integration,
interpolation, –
iterative calculation,
linear equations, –
M-files, ,
mathematical operations, –
matrix, ,
matrix analysis,
multidimensional optimization, –
name,
numerical integration/differentiation,
–
ODEs, –
one-dimensional optimization, –
optimization, –
PDEs, –
pipe friction, ,
polynomial manipulation,
polynomials,
potential energy function,
predator-prey equations, –
primary features,
regression, ,
root location,
roots of polynomials, –
round-off/truncation errors in numerical
differentiation,
save,
spline, –
statistical analysis, –
stiff systems, –
two-dimensional function,
versions,
INDEX
Matrix, –
addition,
augmentation,
defined,
division,
inverse, , –
linear least squares, –
MATLAB,
multiplication, –
notation, –
representing linear algebraic equations,
–
special,
square,
subtraction,
trace,
transpose,
Matrix condition evaluation, –
Matrix condition number, –
Matrix inverse, –
calculating the inverse, –
ill-conditioned systems, –
MATLAB, ,
pseudocode,
stimulus-response computations, –
system condition, –
Matrix multiplication, –
Matrix norms, –
Maximum attainable growth rate,
Maximum likelihood principle,
Maximum-magnitude norm,
Maximum power transfer for circuit, –
Mean of continuous data,
Mean of discrete points,
Mean value,
Method of false position,
Method of lines,
Method of optimal steepest ascent, –
Method of steepest ascent, –
Method of undetermined coefficients, –
Method of weighted residuals (MWR), ,
–
Microsoft,
Midpoint method, , – , , ,
,
Midtest loop,
Milne’s method, – , –
Minimax principles, ,
Minimum potential energy, –
Minor,
Mixed partial derivative,
Model error,
Modified Euler,
Modified false position, –
Modified Newton-Raphson, – ,
Modified Newton-Raphson method,
– ,
Modified secant method, –
Modular programming, –
Modules,
Modulus of toughness,
Molal volume, ,
Moler, Cleve, , , ,
Müller’s method, –
Multidimensional interpolation, –
Multidimensional problems,
Multidimensional unconstrained optimization,
–
BFGS,
conjugate gradient method,
DFP,
direct methods, –
finite-difference approximations, –
gradient, –
gradient methods, –
Hessian, –
Marquardt’s method, –
Newton’s method, –
pattern searches, –
Powell’s method, ,
quasi-Newton methods,
random search, –
steepest ascent method, –
trade-offs, –
univariate search method,
Multimodal,
Multiple-application Simpson’s / rule,
–
Multiple-application trapezoidal rule, –
Multiple integrals, –
Multiple linear regression, –
Multiple root, , –
Multiplication,
Multistep methods
Adams formulas, –
fourth-order Adams method, ,
–
higher-order methods, –
integration formulas, –
Milne’s method, – , –
Newton-Cotes formulas, –
non-self-starting Heun method,
stability, –
step size,
Multivariate power equation,
MWR, – N
Naive Gauss elimination, –
Neumann boundary condition, ,
Newton-Cotes closed integration formulas,
,
Newton-Cotes integration formulas, –
Adams formulas, contrasted, ,
closed forms, , –
closed integration formulas, ,
higher-order formulas, – , ,
integration of equations, –
integration with unequal segments, –
multiple integrals, –
open forms, ,
open integration formulas,
Simpson’s rules. See Simpson’s rules
trapezoidal rule. See Trapezoidal rule
unequal segments, –
unequally spaced data, –
Newton-Cotes open integration formulas,
Newton-Gregory backward formula,
Newton-Gregory central formula,
Newton-Gregory forward formula,
Newton-Raphson formula,
Newton-Raphson method, –
additional features,
algorithm,
error estimates, –
evaluate function and derivative,
formula,
Gauss elimination, ,
graphical depiction,
ideal/nonideal gas laws, –
modified method, – ,
multiple roots, –
nonlinear equations, –
pitfalls, –
slowly converging function, – ,
termination criteria,
two-equation approach, ,
Newtonian fluid,
Newton’s divided-difference interpolating
polynomial,
Newton’s formula,
Newton’s interpolating polynomials, –
algorithm,
computer applications,
errors, – , –
general form,
linear interpolation, –
Newton’s divided-difference interpolating
polynomial,
INDEX
pseudocode,
quadratic interpolation, –
Newton’s law of cooling,
Newton’s laws of motion,
Newton’s method, – , –
Newton’s second law of motion, ,
Nodal lines,
Node,
Non-Newtonian fluid,
Non-self-starting Heun method, –
derivation, –
equations,
errors, –
Heun approach, –
modifiers, –
per-step truncation error,
sequence of formulas,
step size,
Nonbasic variables,
Nonbinding constraints,
Noncomputer methods,
Nongradient methods,
Nonhomogeneous system,
Nonideal gas laws, –
Nonideal vs. ideal,
Nonlinear boundary-value problem, –
Nonlinear constrained optimization,
Nonlinear equations,
Nonlinear programming,
Nonlinear regression, , –
Nonlinear system of equations, –
Nonlinear vs. linear,
Norm,
Normal distribution, ,
Normal equation,
Normalization,
Normalized standard deviate,
nth finite divided difference,
Number systems,
Numerical differentiation, –
Numerical differentiation and integration,
–
advanced methods/references,
antidifferentiation,
case studies, –
commonly used derivatives,
computer objectives, –
derivatives and integrals for data with errors,
–
derivatives of unequally spaced data, ,
differentiation/integration, contrasted,
,
engineering practice, –
equal-area graphical differentiation, ,
heat calculations, –
high-accuracy differentiation formulas,
–
important relationships/formulas,
integrals used in this Part,
integration of equations. See Integration of
equations
Mathcad, –
mathematical background, –
MATLAB, –
method of undetermined coefficients,
–
methods, compared,
Newton-Cotes. See Newton-Cotes
integration formulas
noncomputer methods, –
overview, –
partial derivatives, –
Richardson extrapolation, –
root-mean-square current, –
sailboat, –
scope/preview, –
simple strip method,
study objectives, ,
trade-offs, –
uncertain data,
work, calculation of, –
Numerical double integral, –
Numerical error, –
Numerical integration,
Numerical library,
Numerical methods
accuracy,
defined,
engineering practice, –
error,
hyperbolic equations,
iterative approach,
linear algebraic equations, –
noncomputer methods,
precision, –
rapid growth,
trade-offs, –
why studied,
Numerical Recipe,
Numerical round-off errors, –
Numerically unstable,
O
Objective function, ,
Octal number system,
Odd function, ODE. See Ordinary differential equations
Ohm’s law, , ,
One-dimensional parabolic PDEs,
One-dimensional problems,
One-dimensional unconstrained optimization,
–
bracketing methods,
Brent’s method, ,
global vs. local extremum,
golden-section search, –
Newton’s method, –
open methods,
parabolic interpolation, –
trade-offs,
One-point iteration,
One-sided interval,
One-step methods,
Open integration formulas,
Open methods, –
bracketing methods, compared,
Brent’s method, –
modified Newton-Raphson, – ,
modified secant method, –
multiple roots, –
Newton-Raphson method. See NewtonRaphson method
secant method, –
simple fixed-point iteration, –
systems of nonlinear equations, –
Operation counting, –
Optimal steepest ascent, –
Optimization, –
case studies, –
computer objectives,
constrained. See Constrained optimization
dimensionality,
engineering practice,
equilibrium and minimum potential energy,
–
Excel, –
fundamental elements,
historical overview,
least-cost design of tank, –
least-cost treatment of wastewater,
–
Mathcad, –
mathematical background, –
MATLAB, –
maximum power transfer for circuit,
–
multidimensional. See Multidimensional
unconstrained optimization
noncomputer methods,
INDEX
one-dimensional. See One-dimensional
unconstrained optimization
overview, ,
parachute, –
references,
root location, contrasted,
scope/preview, ,
study objectives, ,
trade-offs, –
Optimum,
Ordinary differential equations, –
boundary-value problems. See Boundaryvalue problems
case studies, –
chaos, –
computer objectives,
eigenvalue problems. See Eigenvalue
problems
engineering practice, –
important relationships/formulas, –
Mathcad, –
mathematical background, –
MATLAB, –
methods, compared,
multistep methods. See Multistep methods
noncomputer methods, –
overview, –
predator-prey models, –
reactor, –
RK methods. See Runge-Kutta methods
scope/preview, –
simulating transient current for electric
circuit, –
stiffness, –
study objectives, ,
swinging pendulum, –
trade-offs, –
transient response of reactor, –
Orthogonal polynomials,
Overconstrained,
Overdamped case,
Overdetermined,
Overflow error,
Overrelaxation,
Overview of book, –
Pp
norm,
Parabola,
Parabolic equations, –
ADI scheme, –
comparison of one-dimensional methods,
–
convergence,
Crank-Nicolson method, –
derivative boundary conditions,
explicit methods, – ,
heat conduction equation, –
higher-order temporal approximations,
–
MacCormack’s method,
method of lines,
simple implicit method, – ,
stability,
two spatial dimensions, –
Parabolic interpolation, –
Parachutist problem
air resistance,
algorithm,
analytical solution, –
error, ,
evaluating integrals, –
Excel, –
Gauss quadrature,
gravity,
numerical/analytical solution, compared,
numerical solution,
schematic diagram,
Parameter estimation,
Parameters,
Parametric Technology Corporation (PTC),
Parthenon,
Partial derivative,
Partial derivatives, –
Partial differential equations, –
area of focus,
case studies, –
classification,
computer objectives, –
deflections of a plate, –
elliptic equations. See Elliptic equations
engineering practice, –
Excel, –
finite difference methods, –
finite-element method. See Finite-element
method
important relationships/formulas,
linear equations,
mass balance of reactor, –
Mathcad, –
MATLAB, –
overview, –
parabolic equations. See Parabolic equations
precomputer methods,
reactor, –
references, scope/preview, –
series of springs, –
study objectives, ,
trade-offs,
two-dimensional electrostatic field
problems, –
Partial pivoting, –
Pattern directions,
Pattern searches, –
PDE. See Partial differential equations
Penalty functions,
Pentadiagonal system,
Percent relative error,
Perfection,
Period,
Periodic function,
Phase line spectra, ,
Phase-plane representation,
Phase shift,
Phases of engineering problem solving,
Piecewise functions,
Pipe friction, –
Pivot coefficient,
Pivot element,
Pivot equation,
Pivoting, –
Place value,
Plane,
Point-slope method,
Poisson equation, , ,
Polynomial. See Roots of polynomials
Polynomial deflation, –
Polynomial evaluation and differentiation,
–
Polynomial method, –
Polynomial regression, –
Polynomials,
Population,
Population models, –
Positional notation,
Positive definite matrix, n
Posttest loop,
Potential energy,
Potentiometers,
Powell’s method, ,
Power equation, –
Power method, –
Power spectrum,
Practical issues,
Practice applications. See Case studies
Precision, – ,
Predator-prey equations, –
Predator-prey models, –
INDEX
Predictor,
Predictor-corrector approach,
Predictor equation, –
Predictor modifier,
Prescriptive models,
Pretest loop,
Principal diagonal,
Problem solving. See Engineering
problem solving
Problem-solving process,
Program development cost,
Programming. See Computer programming
Programming effort required,
Programming languages. See Excel; Mathcad;
MATLAB
Propagated truncation error,
Propagation problems, ,
Proportionality,
Pseudocode, ,
Pseudoplastics,
PTC,
Q
QD algorithm,
QR factorization,
QR method,
Quadratic convergence,
Quadratic interpolation, –
Quadratic polynomial,
Quadratic programming,
Quadratic splines, –
Quadrature,
Quantizing errors, –
Quasi-Newton methods,
Quotient difference (QD) algorithm,
R
r,
r ,
Rainwater, –
Ralston’s method,
Random search, –
Rate equation,
Rate of convergence,
Razdow, Allen,
Reaction kinetics,
Reactor, – , –
Reactors, –
Redlich-Kwong equation of state,
References. See Advanced methods/additional
references
References (bibliography), –
Relative error,
Relaxation,
Repetition, –
Residual,
Resistor circuits, –
Reynolds number, , ,
Richardson extrapolation, –
Richardson’s extrapolation, –
RK methods. See Runge-Kutta methods
Romberg integration, , –
Root-mean-square current, –
Root polishing,
Roots of equations, –
advanced methods/additional references,
–
bracketing methods. See Bracketing
methods
bracketing/open methods, compared,
case studies, –
computer objectives,
electrical circuit design, –
engineering practice, –
Excel, –
graphical methods, – ,
greenhouse gases, –
ideal/nonideal gas laws, –
important relationships/formulas,
Mathcad, –
mathematical background, –
MATLAB, –
methods, compared,
noncomputer methods,
open methods. See Open methods
optimization, contrasted,
overview, –
pipe friction, –
problem areas,
QD algorithm,
rainwater, –
root polishing,
roots of polynomials. See Roots of
polynomials
scope/preview, –
study objectives,
trade-offs, –
Roots of polynomials, –
Bairstow’s method, –
characteristic equation,
conventional methods, –
Excel, –
factored form of polynomial,
general solution,
Jenkins-Traub method,
Laguerre’s method, Roots of polynomials—Cont.
Mathcad, –
MATLAB, –
Müller’s method, –
ODE,
overdamped/critically
damped/underdamped, –
polynomial deflation, –
polynomial evaluation and differentiation,
–
root polishing,
Roots of quadratic algorithm,
Rosenbrock method,
Rosin-Rammler-Bennet (RRB) equation,
Round-off error, –
adding large and small number, –
addition,
arithmetic manipulations of computer
numbers, –
chopping,
computer representation of numbers, –
division,
extended precision, –
floating-point representation, –
Gauss elimination,
inner products,
integer representation, –
large computations, –
multiplication,
normalization,
number systems,
overflow error,
quantizing errors, –
rounding, ,
smearing,
subtraction,
subtractive cancellation, –
underflow “hole,”
Rounding, ,
Row,
Row-sum norm, ,
Row vectors,
RRB equation,
Run-time cost,
Runge-Kutta Fehlberg, –
Runge-Kutta methods, –
adaptive. See Adaptive Runge-Kutta
methods
Euler’s method. See Euler’s method
fourth-order RK methods, –
Heun’s method, – , , ,
–
higher-order RK methods,
INDEX
methods, compared, –
midpoint method, – , ,
pseudocode,
Ralston’s method,
Runge-Kutta Fehlberg, –
second-order RK methods, –
step-size control, –
systems of equations, –
third-order RK methods, –
Runge’s function,
S
Saddle,
Sailboat, –
Sample, ,
Sample mean,
Sande-Tukey algorithm, –
Saturation-growth-rate equation, ,
Saturation-growth-rate model, –
Scaling, , –
Secant formula,
Secant method, –
additional features, ,
algorithm,
false-position method, compared, –
formula,
graphical depiction,
modified method, –
multiple roots, ,
Second Adams-Bashforth formula,
Second Adams-Moulton formula,
Second derivative,
Second finite divided difference,
Second forward finite divided difference,
Second-order closed Adams formula,
Second-order equation,
Second-order open Adams formula,
Second-order Ralston RK method,
Second-order RK methods, –
Secondary variables,
Selection, –
Sensitivity analysis,
Sequence,
Series of springs, –
Shadow price,
Shape function,
Shooting method, –
Signed magnitude method,
Significance level,
Significand,
Significant digits,
Significant figures, –
Simple fixed-point iteration, –
Simple implicit method, – ,
Simple strip method,
Simplex method, –
Simplex procedure,
Simpson’s / rule, –
Simpson’s / rule, –
Simpson’s rules, –
algorithms, ,
multiple-application Simpson’s / rule,
–
pseudocode,
Simpson’s / rule, –
Simpson’s / rule, –
uneven data,
Simulated annealing,
Simulating transient current for electric current,
–
Simultaneous linear algebraic equations.
See Linear algebraic equations
Simultaneous nonlinear equations, –
Simultaneous overrelaxation,
Single-value decomposition,
Single-variable optimization,
Singular system,
Singular systems,
Sinusoid,
Sinusoidal functions, –
Slide rule,
Small vs. large systems,
Smearing,
Software cost,
Software packages. See Excel; Mathcad;
MATLAB
Solution technique, –
Solver, –
SOR,
Special matrix,
Specific growth rate,
Spectral norm,
Spline,
Spline functions,
Spline interpolation, –
cubic splines, , , –
linear splines, –
quadratic splines, –
superiority, ,
Spreadsheet, . See also Excel
Spring-mass system, –
Square matrix,
Stability, , , – ,
Stage extraction process,
Standard deviation, ,
Standard error of the estimate, Standard normal estimate,
Starting point,
Static instability,
Statistical inference,
Statistically determinate truss, –
Statistics, –
Steady-state,
Steady-state analysis of system of reactors,
–
Steepest ascent method, –
Stefan-Boltzmann constant,
Stefan-Boltzmann law,
Step halving,
Stiff system,
Stiffness, –
Stiffness matrix,
Stimulus-response computations, –
Stopping criterion,
Straightening,
Strange attractors,
Streeter-Phelps model,
Strip method,
Structured programming, –
Student-t,
Subdomain method,
Subroutine,
Subtraction,
Subtractive cancellation, –
Successive overrelaxation,
Successive substitution,
Superposition,
SVD method,
Swamee-Jain equation,
Swinging pendulum, , –
Symmetric form,
Symmetric matrix, ,
Synthetic division,
System condition, –
Systems of nonlinear equations, –
Tt
distribution,
Table look-up,
Tableau,
Tabu search,
Taylor series, –
approximation of function with infinite
number of derivatives,
approximation of polynomial,
backward difference approximation,
centered difference approximation,
finite difference approximations of higher
derivatives, –
INDEX
finite-divided-difference approximations of
derivatives, –
first-order approximation,
first theorem of mean for integrals,
forward difference approximation,
Newton-Raphson method,
nonlinearity, –
nth-order expansion,
numerical differentiation, –
remainder, , , , –
second-order approximation,
second theorem of mean for integrals,
step size, –
Taylor series expansion,
theorem,
truncation errors,
what is it,
zero-order approximation,
Taylor series expansion,
Taylor’s formula,
Taylor’s theorem,
Temperature of heated plate, –
Terminal velocity,
Termination criteria,
The MathWorks, Inc.,
Thermocline,
Third-order RK methods, –
Thomas algorithm,
Three-point Gauss-Legendre formula,
Time domain, –
Time plane,
Time-variable,
Total numerical error, –
Total sum of the squares,
Trace,
Trade-offs
curve fitting, –
linear algebraic equations, –
numerical differentiation/integration,
–
numerical methods, –
ODEs, –
optimization, –
PDEs,
roots of equations, –
Transcendental function,
Transient,
Transient responses of reactor, –
Transpose,
Trapezoidal rule, –
algorithms,
area under straight line, –
computer program, –
conclusions, –
derivation, ,
error,
error corrections, –
formula,
graphical depiction,
Heun’s method,
multiple-application rule, –
pseudocode,
single application, –
unequal segments,
Trend analysis,
Triangular matrix,
Tridiagonal matrix,
Tridiagonal system, –
Triple root,
True error,
True fractional relative error,
True local truncation error,
True mean,
True solution,
True value,
Truncation,
Truncation error, ,
Truncation errors and Taylor series, –
blunders, –
condition,
data uncertainty,
error propagation, –
formulation errors,
stability,
Taylor series. See Taylor series
total numerical error, –
truncation errors,
Truss, –
Tukey, J. W.,
Twiddle factors,
Two-dimensional electrostatic field problems,
–
Two-dimensional interpolation, –
Two-dimensional parabolic PDEs,
Two-equation Newton-Raphson
approach, ,
Two-point Gauss-Legendre formula, –
Two-segment trapezoidal rule,
Two-sided interval, –
U
Uncertain data,
Uncertainty,
Unconditionally stable,
Unconstrained optimization,
Underdamped case, Underdetermined, ,
Underflow “hole,”
Underrelaxation,
Underspecified,
Unexplained sum of the squares,
Uniform-matrix norm, ,
Uniform-vector norm,
Unimodal,
Univariate search method,
Unstable,
Upper triangular matrix,
V
Van der Pol’s equation,
Van der Waals equation,
Vandermonde matrix,
INDEX
Variable metric methods,
Variance,
Variational approach,
VBA macro, . See also Excel
Vector norms, –
Vibrating string,
Videoangiography,
Voltage balance,
Volterra, Vito,
Volume integral,
Volume-integral approach,
Von Karman equation,
W
Waste minimization,
Wastewater treatment, –
Wave equation, ,
Well-conditioned systems,
WHILE,
Wolf, Johann Rudolph,
Wolf sunspot number,
Word,
Work, calculation of, –
Y
Yield stress,
Young’s modulus,
Z
Zero-order approximation,
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