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Cambridge2010Numerical and Statistical Methods for Bioengineering
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Contents Preface page ix 1 Types and sources of numerical error 1 1.1 Introduction 1 1.2 Representation of floating-point numbers 4 1.2.1 How computers store numbers 7 1.2.2 Binary to decimal system 7 1.2.3 Decimal to binary system 9 1.2.4 Binary representation of floating-point numbers 10 1.3 Methods used to measure error 16 1.4 Significant digits 18 1.5 Round-off errors generated by floating-point operations 20 1.6 Taylor series and truncation error 26 1.6.1 Order of magnitude estimation of truncation error 28 1.6.2 Convergence of a series 32 1.6.3 Finite difference formulas for numerical differentiation 33 1.7 Criteria for convergence 39 1.8 End of Chapter 1: key points to consider 40 1.9 Problems 40 References 46 2 Systems of linear equations 47 2.1 Introduction 47 2.2 Fundamentals of linear algebra 53 2.2.1 Vectors and matrices 53 2.2.2 Matrix operations 56 2.2.3 Vector and matrix norms 64 2.2.4 Linear combinations of vectors 66 2.2.5 Vector spaces and basis vectors 69 2.2.6 Rank, determinant, and inverse of matrices 71 2.3 Matrix representation of a system of linear equations 75 2.4 Gaussian elimination with backward substitution 76 2.4.1 Gaussian elimination without pivoting 76 2.4.2 Gaussian elimination with pivoting 84 2.5 LU factorization 87 2.5.1 LU factorization without pivoting 88 2.5.2 LU factorization with pivoting 93 2.5.3 The MATLAB lu function 95 2.6 The MATLAB backslash (\) operator 96 2.7 III-conditioned problems and the condition number 97 2.8 Linear regression 101 2.9 Curve fitting using linear least-squares approximation 107 2.9.1 The normal equations 109 2.9.2 Coefficient of determination and quality of fit 115 2.10 Linear least-squares approximation of transformed equations 118 2.11 Multivariable linear least-squares regression 123 2.12 The MATLAB function polyfit 124 2.13 End of Chapter 2: key points to consider 125 2.14 Problems 127 References 139 3 Probability and statistics 141 3.1 Introduction 141 3.2 Characterizing a population: descriptive statistics 144 3.2.1 Measures of central tendency 145 3.2.2 Measures of dispersion 146 3.3 Concepts from probability 147 3.3.1 Random sampling and probability 149 3.3.2 Combinatorics: permutations and combinations 154 3.4 Discrete probability distributions 157 3.4.1 Binomial distribution 159 3.4.2 Poisson distribution 163 3.5 Normal distribution 166 3.5.1 Continuous probability distributions 167 3.5.2 Normal probability density 169 3.5.3 Expectations of sample-derived statistics 171 3.5.4 Standard normal distribution and the z statistic 175 3.5.5 Confidence intervals using the z statistic and the t statistic 177 3.5.6 Non-normal samples and the central-limit theorem 183 3.6 Propagation of error 186 3.6.1 Addition/subtraction of random variables 187 3.6.2 Multiplication/division of random variables 188 3.6.3 General functional relationship between two random variables 190 3.7 Linear regression error 191 3.7.1 Error in model parameters 193 3.7.2 Error in model predictions 196 3.8 End of Chapter 3: key points to consider 199 3.9 Problems 202 References 208 4 Hypothesis testing 209 4.1 Introduction 209 4.2 Formulating a hypothesis 210 4.2.1 Designing a scientific study 211 4.2.2 Null and alternate hypotheses 217 4.3 Testing a hypothesis 219 4.3.1 The p value and assessing statistical significance 220 4.3.2 Type I and type II errors 226 4.3.3 Types of variables 228 4.3.4 Choosing a hypothesis test 230 vi Contents 4.4 Parametric tests and assessing normality 231 4.5 The z test 235 4.5.1 One-sample z test 235 4.5.2 Two-sample z test 241 4.6 The t test 244 4.6.1 One-sample and paired sample t tests 244 4.6.2 Independent two-sample t test 249 4.7 Hypothesis testing for population proportions 251 4.7.1 Hypothesis testing for a single population proportion 256 4.7.2 Hypothesis testing for two population proportions 257 4.8 One-way ANOVA 260 4.9 Chi-square tests for nominal scale data 274 4.9.1 Goodness-of-fit test 276 4.9.2 Test of independence 281 4.9.3 Test of homogeneity 285 4.10 More on non-parametric (distribution-free) tests 288 4.10.1 Sign test 289 4.10.2 Wilcoxon signed-rank test 292 4.10.3 Wilcoxon rank-sum test 296 4.11 End of Chapter 4: key points to consider 299 4.12 Problems 299 References 308 5 Root-finding techniques for nonlinear equations 310 5.1 Introduction 310 5.2 Bisection method 312 5.3 Regula-falsi method 319 5.4 Fixed-point iteration 320 5.5 Newton’s method 327 5.5.1 Convergence issues 329 5.6 Secant method 336 5.7 Solving systems of nonlinear equations 338 5.8 MATLAB function fzero 346 5.9 End of Chapter 5: key points to consider 348 5.10 Problems 349 References 353 6 Numerical quadrature 354 6.1 Introduction 354 6.2 Polynomial interpolation 361 6.3 Newton–Cotes formulas 371 6.3.1 Trapezoidal rule 372 6.3.2 Simpson’s 1/3 rule 380 6.3.3 Simpson’s 3/8 rule 384 6.4 Richardson’s extrapolation and Romberg integration 387 6.5 Gaussian quadrature 391 6.6 End of Chapter 6: key points to consider 402 6.7 Problems 403 References 408 vii Contents 7 Numerical integration of ordinary differential equations 409 7.1 Introduction 409 7.2 Euler’s methods 416 7.2.1 Euler’s forward method 417 7.2.2 Euler’s backward method 428 7.2.3 Modified Euler’s method 431 7.3 Runge–Kutta (RK) methods 434 7.3.1 Second-order RK methods 434 7.3.2 Fourth-order RK methods 438 7.4 Adaptive step size methods 440 7.5 Multistep ODE solvers 451 7.5.1 Adams methods 452 7.5.2 Predictor–corrector methods 454 7.6 Stability and stiff equations 456 7.7 Shooting method for boundary-value problems 461 7.7.1 Linear ODEs 463 7.7.2 Nonlinear ODEs 464 7.8 End of Chapter 7: key points to consider 472 7.9 Problems 473 References 478 8 Nonlinear model regression and optimization 480 8.1 Introduction 480 8.2 Unconstrained single-variable optimization 487 8.2.1 Newton’s method 488 8.2.2 Successive parabolic interpolation 492 8.2.3 Golden section search method 495 8.3 Unconstrained multivariable optimization 500 8.3.1 Steepest descent or gradient method 502 8.3.2 Multidimensional Newton’s method 509 8.3.3 Simplex method 513 8.4 Constrained nonlinear optimization 523 8.5 Nonlinear error analysis 530 8.6 End of Chapter 8: key points to consider 533 8.7 Problems 534 References 538 9 Basic algorithms of bioinformatics 539 9.1 Introduction 539 9.2 Sequence alignment and database searches 540 9.3 Phylogenetic trees using distance-based methods 554 9.4 End of Chapter 9: key points to consider 557 9.5 Problems 558 References 558 Appendix A Introduction to MATLAB 560 Appendix B Location of nodes for Gauss–Legendre quadrature 576 Index for MATLAB commands 578 Index 579 |
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