FUNDAMENTAL NUMERICAL METHODS AND DATA ANALYSIS
I TABLE OF CONTENTS 1 INTRODUCTION AND FUNDAMENTAL CONCEPTS BASIC PROPERTIES OF SETS AND GROUPS SCALARS, VECTORS, AND MATRICES COORDINATE SYSTEMS AND COORDINATE TRANSFORMATIONS TENSORS AND TRANSFORMATIONS OPERATORS CHAPTER 1 EXERCISES CHAPTER 1 REFERENCES AND ADDITIONAL READING 2 THE NUMERICAL METHODS FOR LINEAR EQUATIONS AND MATRICES ERRORS AND THEIR PROPAGATION DIRECT METHODS FOR THE SOLUTION OF LINEAR ALGEBRAIC EQUATIONS A SOLUTION BY CRAMER'S RULE B SOLUTION BY GAUSSIAN ELIMINATION C SOLUTION BY GAUSS JORDAN ELIMINATION D SOLUTION BY MATRIX FACTORIZATION: THE CROUT METHOD E THE SOLUTION OF TRI-DIAGONAL SYSTEMS OF LINEAR EQUATIONS SOLUTION OF LINEAR EQUATIONS BY ITERATIVE METHODS A SOLUTION BY THE GAUSS AND GAUSS-SEIDEL ITERATION METHODS B THE METHOD OF HOTELLING AND BODEWIG C RELAXATION METHODS FOR THE SOLUTION OF LINEAR EQUATIONS D CONVERGENCE AND FIXED-POINT ITERATION THEORY THE SIMILARITY TRANSFORMATIONS AND THE EIGENVALUES AND VECTORS OF A MATRIX I CHAPTER 2 EXERCISES CHAPTER 2 REFERENCES AND SUPPLEMENTAL READING 3 POLYNOMIAL APPROXIMATION, INTERPOLATION, AND ORTHOGONAL POLYNOMIALS POLYNOMIALS AND THEIR ROOTS A SOME CONSTRAINTS ON THE ROOTS OF POLYNOMIALS B SYNTHETIC DIVISION C THE GRAFFE ROOT-SQUARING PROCESS D ITERATIVE METHODS CURVE FITTING AND INTERPOLATION A LAGRANGE INTERPOLATION B HERMITE INTERPOLATION C SPLINES D EXTRAPOLATION AND INTERPOLATION CRITERIA ORTHOGONAL POLYNOMIALS A THE LEGENDRE POLYNOMIALS B THE LAGUERRE POLYNOMIALS C THE HERMITE POLYNOMIALS D ADDITIONAL ORTHOGONAL POLYNOMIALS E THE ORTHOGONALITY OF THE TRIGONOMETRIC FUNCTIONS CHAPTER 3 EXERCISES CHAPTER 3 REFERENCES AND SUPPLEMENTAL READING 4 NUMERICAL EVALUATION OF DERIVATIVES AND INTEGRALS NUMERICAL DIFFERENTIATION A CLASSICAL DIFFERENCE FORMULAE B RICHARDSON EXTRAPOLATION FOR DERIVATIVES NUMERICAL EVALUATION OF INTEGRALS: QUADRATURE A THE TRAPEZOID RULE B SIMPSON'S RULE C QUADRATURE SCHEMES FOR ARBITRARILY SPACED FUNCTIONS D GAUSSIAN QUADRATURE SCHEMES E ROMBERG QUADRATURE AND RICHARDSON EXTRAPOLATION F MULTIPLE INTEGRALS MONTE CARLO INTEGRATION SCHEMES AND OTHER TRICKS A MONTE CARLO EVALUATION OF INTEGRALS B THE GENERAL APPLICATION OF QUADRATURE FORMULAE TO INTEGRALS CHAPTER 4 EXERCISES CHAPTER 4 REFERENCES AND SUPPLEMENTAL READING 5 NUMERICAL SOLUTION OF DIFFERENTIAL AND INTEGRAL EQUATIONS THE NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS A ONE STEP METHODS OF THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS B ERROR ESTIMATE AND STEP SIZE CONTROL C MULTI-STEP AND PREDICTOR-CORRECTOR METHODS D SYSTEMS OF DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS E PARTIAL DIFFERENTIAL EQUATIONS THE NUMERICAL SOLUTION OF INTEGRAL EQUATIONS A TYPES OF LINEAR INTEGRAL EQUATIONS B THE NUMERICAL SOLUTION OF FREDHOLM EQUATIONS C THE NUMERICAL SOLUTION OF VOLTERRA EQUATIONS D THE INFLUENCE OF THE KERNEL ON THE SOLUTION CHAPTER 5 EXERCISES CHAPTER 5 REFERENCES AND SUPPLEMENTAL READING 6 LEAST SQUARES, FOURIER ANALYSIS, AND RELATED APPROXIMATION NORMS LEGENDRE'S PRINCIPLE OF LEAST SQUARES A THE NORMAL EQUATIONS OF LEAST SQUARES B LINEAR LEAST SQUARES C THE LEGENDRE APPROXIMATION LEAST SQUARES, FOURIER SERIES, AND FOURIER TRANSFORMS A LEAST SQUARES, THE LEGENDRE APPROXIMATION, AND FOURIER SERIES B THE FOURIER INTEGRAL C THE FOURIER TRANSFORM D THE FAST FOURIER TRANSFORM ALGORITHM ERROR ANALYSIS FOR LINEAR LEAST-SQUARES A ERRORS OF THE LEAST SQUARE COEFFICIENTS B THE RELATION OF THE WEIGHTED MEAN SQUARE OBSERVATIONAL ERROR TO THE WEIGHTED MEAN SQUARE RESIDUAL C DETERMINING THE WEIGHTED MEAN SQUARE RESIDUAL D THE EFFECTS OF ERRORS IN THE INDEPENDENT VARIABLE NON-LINEAR LEAST SQUARES A THE METHOD OF STEEPEST DESCENT B LINEAR APPROXIMATION OF F(AJ,X) C ERRORS OF THE LEAST SQUARES COEFFICIENTS OTHER APPROXIMATION NORMS A THE CHEBYSCHEV NORM AND POLYNOMIAL APPROXIMATION B THE CHEBYSCHEV NORM, LINEAR PROGRAMMING, AND THE SIMPLEX METHOD C THE CHEBYSCHEV NORM AND LEAST SQUARES CHAPTER 6 EXERCISES CHAPTER 6 REFERENCES AND SUPPLEMENTARY READING 7 PROBABILITY THEORY AND STATISTICS BASIC ASPECTS OF PROBABILITY THEORY A THE PROBABILITY OF COMBINATIONS OF EVENTS B PROBABILITIES AND RANDOM VARIABLES C DISTRIBUTIONS OF RANDOM VARIABLES COMMON DISTRIBUTION FUNCTIONS A PERMUTATIONS AND COMBINATIONS B THE BINOMIAL PROBABILITY DISTRIBUTION C THE POISSON DISTRIBUTION D THE NORMAL CURVE E SOME DISTRIBUTION FUNCTIONS OF THE PHYSICAL WORLD MOMENTS OF DISTRIBUTION FUNCTIONS THE FOUNDATIONS OF STATISTICAL ANALYSIS A MOMENTS OF THE BINOMIAL DISTRIBUTION B MULTIPLE VARIABLES, VARIANCE, AND COVARIANCE C MAXIMUM LIKELIHOOD IV CHAPTER 7 EXERCISES CHAPTER 7 REFERENCES AND SUPPLEMENTAL READING 8 SAMPLING DISTRIBUTIONS OF MOMENTS, STATISTICAL TESTS, AND PROCEDURES THE T, ?2 , AND F STATISTICAL DISTRIBUTION FUNCTIONS A THE T-DENSITY DISTRIBUTION FUNCTION B THE ?2 -DENSITY DISTRIBUTION FUNCTION C THE F-DENSITY DISTRIBUTION FUNCTION THE LEVEL OF SIGNIFICANCE AND STATISTICAL TESTS A THE "STUDENTS" T-TEST B THE ?2-TEST C THE F-TEST D KOLMOGOROV-SMIRNOV TESTS LINEAR REGRESSION, AND CORRELATION ANALYSIS A THE SEPARATION OF VARIANCES AND THE TWO-VARIABLE CORRELATION COEFFICIENT B THE MEANING AND SIGNIFICANCE OF THE CORRELATION COEFFICIENT C CORRELATIONS OF MANY VARIABLES AND LINEAR REGRESSION D ANALYSIS OF VARIANCE THE DESIGN OF EXPERIMENTS A THE TERMINOLOGY OF EXPERIMENT DESIGN B BLOCKED DESIGNS C FACTORIAL DESIGNS CHAPTER 8 EXERCISES CHAPTER 8 REFERENCES AND SUPPLEMENTAL READING