NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS
CONTENTS CHAPTER INTRODUCTION 1 1 BACKGROUND 1 2 REPRESENTATION OF NUMBERS ON A COMPUTER 1 3 ERRORS IN NUMERICAL SOLUTIONS 1 3 1 ROUND-OFF ERRORS 1 3 2 TRUNCATION ERRORS 1 3 3 TOTAL ERROR 1 4 COMPUTERS AND PROGRAMMING 1 5 PROBLEMS CHAPTER MATHEMATICAL BACKGROUND 2 1 BACKGROUND 2 2 CONCEPTS FROM PRE-CALCULUS AND CALCULUS 2 3 VECTORS 2 3 1 OPERATIONS WITH VECTORS 2 4 MATRICES AND LINEAR ALGEBRA 2 4 1 OPERATIONS WITH MATRICES 2 4 2 SPECIAL MATRICES 2 4 3 INVERSE OF A MATRIX 2 4 4 PROPERTIES OF MATRICES 2 4 5 DETERMINANT OF A MATRIX 2 4 6 CRAMER’S RULE AND SOLUTION OF A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS 2 4 7 NORMS 2 5 ORDINARY DIFFERENTIAL EQUATIONS (ODE) 2 6 FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES 2 6 1 DEFINITION OF THE PARTIAL DERIVATIVE 2 6 2 CHAIN RULE 2 6 3 THE JACOBIAN 2 7 TAYLOR SERIES EXPANSION OF FUNCTIONS 2 7 1 TAYLOR SERIES FOR A FUNCTION OF ONE VARIABLE 2 7 2 TAYLOR SERIES FOR A FUNCTION OF TWO VARIABLES 2 8 INNER PRODUCT AND ORTHOGONALITY 2 9 PROBLEMS SOLVING NONLINEAR EQUATIONS 3 1 BACKGROUND 3 2 ESTIMATION OF ERRORS IN NUMERICAL SOLUTIONS 3 3 BISECTION METHOD 3 4 REGULA FALSI METHOD 3 5 NEWTON’S METHOD 3 6 SECANT METHOD 3 7 FIXED-POINT ITERATION METHOD 3 8 USE OF MATLAB BUILT-IN FUNCTIONS FOR SOLVING NONLINEAR EQUATIONS 3 8 1 THE FZERO COMMAND 3 8 2 THE ROOTS COMMAND 3 9 EQUATIONS WITH MULTIPLE SOLUTIONS 3 10 SYSTEMS OF NONLINEAR EQUATIONS 3 10 1 NEWTON’S METHOD FOR SOLVING A SYSTEM OF NONLINEAR EQUATIONS 3 10 2 FIXED-POINT ITERATION METHOD FOR SOLVING A SYSTEM OF NONLINEAR EQUATIONS 3 11 PROBLEMS SOLVING A SYSTEM OF LINEAR EQUATIONS 4 1 BACKGROUND 4 1 1 OVERVIEW OF NUMERICAL METHODS FOR SOLVING A SYSTEM OF LINEAR ALGEBRAIC EQUATIONS 4 2 GAUSS ELIMINATION METHOD 4 2 1 POTENTIAL DIFFICULTIES WHEN APPLYING THE GAUSS ELIMINATION METHOD 4 3 GAUSS ELIMINATION WITH PIVOTING 4 4 GAUSS–JORDAN ELIMINATION METHOD 4 5 LU DECOMPOSITION METHOD 4 5 1 LU DECOMPOSITION USING THE GAUSS ELIMINATION PROCEDURE 4 5 2 LU DECOMPOSITION USING CROUT’S METHOD 4 5 3 LU DECOMPOSITION WITH PIVOTING 4 6 INVERSE OF A MATRIX 4 6 1 CALCULATING THE INVERSE WITH THE LU DECOMPOSITION METHOD 4 6 2 CALCULATING THE INVERSE USING THE GAUSS–JORDAN METHOD 4 7 ITERATIVE METHODS 4 7 1 JACOBI ITERATIVE METHOD 4 7 2 GAUSS-SEIDEL ITERATIVE METHOD 4 8 USE OF MATLAB BUILT-IN FUNCTIONS FOR SOLVING A SYSTEM OF LINEAR EQUATIONS 4 8 1 SOLVING A SYSTEM OF EQUATIONS USING MATLAB’S LEFT AND RIGHT DIVISION 4 8 2 SOLVING A SYSTEM OF EQUATIONS USING MATLAB’S INVERSE OPERATION 4 8 3 MATLAB’S BUILT-IN FUNCTION FOR LU DECOMPOSITION 4 8 4 ADDITIONAL MATLAB BUILT-IN FUNCTIONS 4 9 TRIDIAGONAL SYSTEMS OF EQUATIONS 4 10 ERROR, RESIDUAL, NORMS, AND CONDITION NUMBER 4 10 1 ERROR AND RESIDUAL 4 10 2 NORMS AND CONDITION NUMBER 4 11 ILL-CONDITIONED SYSTEMS 4 12 PROBLEMS EIGENVALUES AND EIGENVECTORS 5 1 BACKGROUND 5 2 THE CHARACTERISTIC EQUATION 5 3 THE BASIC POWER METHOD 5 4 THE INVERSE POWER METHOD 5 5 THE SHIFTED POWER METHOD 5 6 THE QR FACTORIZATION AND ITERATION METHOD 5 7 USE OF MATLAB BUILT-IN FUNCTIONS FOR DETERMINING EIGENVALUES AND EIGENVECTORS 5 8 PROBLEMS CURVE FITTING AND INTERPOLATION 6 1 BACKGROUND 6 2 CURVE FITTING WITH A LINEAR EQUATION 6 2 1 MEASURING HOW GOOD IS A FIT 6 2 2 LINEAR LEAST-SQUARES REGRESSION 6 3 CURVE FITTING WITH NONLINEAR EQUATION BY WRITING THE EQUATION IN A LINEAR FORM 6 4 CURVE FITTING WITH QUADRATIC AND HIGHER-ORDER POLYNOMIALS 6 5 INTERPOLATION USING A SINGLE POLYNOMIAL 6 5 1 LAGRANGE INTERPOLATING POLYNOMIALS 6 5 2 NEWTON’S INTERPOLATING POLYNOMIALS 6 6 PIECEWISE (SPLINE) INTERPOLATION 6 6 1 LINEAR SPLINES 6 6 2 QUADRATIC SPLINES 6 6 3 CUBIC SPLINES 6 7 USE OF MATLAB BUILT-IN FUNCTIONS FOR CURVE FITTING AND INTERPOLATION 6 8 CURVE FITTING WITH A LINEAR COMBINATION OF NONLINEAR FUNCTIONS 6 9 PROBLEMS FOURIER METHODS 7 1 BACKGROUND 7 2 APPROXIMATING A SQUARE WAVE BY A SERIES OF SINE FUNCTIONS 7 3 GENERAL (INFINITE) FOURIER SERIES 7 4 COMPLEX FORM OF THE FOURIER SERIES 7 5 THE DISCRETE FOURIER SERIES AND DISCRETE FOURIER TRANSFORM 7 6 COMPLEX DISCRETE FOURIER TRANSFORM 7 7 POWER (ENERGY) SPECTRUM 7 8 ALIASING AND NYQUIST FREQUENCY 7 9 ALTERNATIVE FORMS OF THE DISCRETE FOURIER TRANSFORM 7 10 USE OF MATLAB BUILT-IN FUNCTIONS FOR CALCULATING DISCRETE FOURIER TRANSFORM 7 11 LEAKAGE AND WINDOWING 7 12 BANDWIDTH AND FILTERS 7 13 THE FAST FOURIER TRANSFORM (FFT) 7 14 PROBLEMS NUMERICAL DIFFERENTIATION 8 1 BACKGROUND 8 2 FINITE DIFFERENCE APPROXIMATION OF THE DERIVATIVE 8 3 FINITE DIFFERENCE FORMULAS USING TAYLOR SERIES EXPANSION 8 3 1 FINITE DIFFERENCE FORMULAS OF FIRST DERIVATIVE 8 3 2 FINITE DIFFERENCE FORMULAS FOR THE SECOND DERIVATIVE 8 4 SUMMARY OF FINITE DIFFERENCE FORMULAS FOR NUMERICAL DIFFERENTIATION 8 5 DIFFERENTIATION FORMULAS USING LAGRANGE POLYNOMIALS 8 6 DIFFERENTIATION USING CURVE FITTING 8 7 USE OF MATLAB BUILT-IN FUNCTIONS FOR NUMERICAL DIFFERENTIATION 8 8 RICHARDSON’S EXTRAPOLATION 8 9 ERROR IN NUMERICAL DIFFERENTIATION 8 10 NUMERICAL PARTIAL DIFFERENTIATION 8 11 PROBLEMS NUMERICAL INTEGRATION 9 1 BACKGROUND 9 1 1 OVERVIEW OF APPROACHES IN NUMERICAL INTEGRATION 9 2 RECTANGLE AND MIDPOINT METHODS 9 3 TRAPEZOIDAL METHOD 9 3 1 COMPOSITE TRAPEZOIDAL METHOD 9 4 SIMPSON’S METHODS 9 4 1 SIMPSON’S 1/3 METHOD 9 4 2 SIMPSON’S 3/8 METHOD 9 5 GAUSS QUADRATURE 9 6 EVALUATION OF MULTIPLE INTEGRALS 9 7 USE OF MATLAB BUILT-IN FUNCTIONS FOR INTEGRATION 9 8 ESTIMATION OF ERROR IN NUMERICAL INTEGRATION 9 9 RICHARDSON’S EXTRAPOLATION 9 10 ROMBERG INTEGRATION 9 11 IMPROPER INTEGRALS 9 11 1 INTEGRALS WITH SINGULARITIES 9 11 2 INTEGRALS WITH UNBOUNDED LIMITS 9 12 PROBLEMS CHAPTER 10 ORDINARY DIFFERENTIAL EQUATIONS: INITIAL-VALUE PROBLEMS 10 1 BACKGROUND 10 2 EULER’S METHODS 10 2 1 EULER’S EXPLICIT METHOD 10 2 2 ANALYSIS OF TRUNCATION ERROR IN EULER’S EXPLICIT METHOD 10 2 3 EULER’S IMPLICIT METHOD 10 3 MODIFIED EULER’S METHOD 10 4 MIDPOINT METHOD 10 5 RUNGE–KUTTA METHODS 10 5 1 SECOND-ORDER RUNGE–KUTTA METHODS 10 5 2 THIRD-ORDER RUNGE–KUTTA METHODS 10 5 3 FOURTH-ORDER RUNGE–KUTTA METHODS 10 6 MULTISTEP METHODS 10 6 1 ADAMS–BASHFORTH METHOD 10 6 2 ADAMS–MOULTON METHOD 10 7 PREDICTOR–CORRECTOR METHODS 10 8 SYSTEM OF FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS 10 8 1 SOLVING A SYSTEM OF FIRST-ORDER ODES USING EULER’S EXPLICIT METHOD 10 8 2 SOLVING A SYSTEM OF FIRST-ORDER ODES USING SECOND-ORDER RUNGE–KUTTA METHOD (MODIFIED EULER VERSION) 10 8 3 SOLVING A SYSTEM OF FIRST-ORDER ODES USING THE CLASSICAL FOURTH-ORDER RUNGE–KUTTA METHOD 10 9 SOLVING A HIGHER-ORDER INITIAL VALUE PROBLEM 10 10 USE OF MATLAB BUILT-IN FUNCTIONS FOR SOLVING INITIAL-VALUE PROBLEMS 10 10 1 SOLVING A SINGLE FIRST-ORDER ODE USING MATLAB 10 10 2 SOLVING A SYSTEM OF FIRST-ORDER ODES USING MATLAB 10 11 LOCAL TRUNCATION ERROR IN SECOND-ORDER RANGE–KUTTA METHOD 10 12 STEP SIZE FOR DESIRED ACCURACY 10 13 STABILITY 10 14 STIFF ORDINARY DIFFERENTIAL EQUATIONS 10 15 PROBLEMS CHAPTER 11 ORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY-VALUE PROBLEMS 11 1 BACKGROUND 11 2 THE SHOOTING METHOD 11 3 FINITE DIFFERENCE METHOD 11 4 USE OF MATLAB BUILT-IN FUNCTIONS FOR SOLVING BOUNDARY VALUE PROBLEMS 11 5 ERROR AND STABILITY IN NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS 11 6 PROBLEMS