King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

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Equation (B.1) has a weighting function equal to one. If two continuous functions

are orthogonal to each other, their inner product is equal to zero. (Section 2.2

discusses orthogonality (normality) of vectors in Euclidean vector space.) Let

xðÞbe any orthogonal polynomial function, where the subscript n is the degree of

the polynomial. The inner product of two orthogonal polynomial functions is

deﬁned as

hΦ

ðxÞ; Φ

ðxÞi ¼

ðxÞΦ

ðxÞwðxÞdx ¼ c

; (B:4)

where δ is the identity matrix I such that δ

¼ 1 for m ¼ n and δ

¼ 0 for m 6¼ n;

is a positive constant.

A class of orthogonal polynomials called Legendre

polynomials is deﬁned for a weighting factor wxðÞ¼1. Their properties of

orthogonality exist only on the interval ½1; 1. The inner product of two Legendre

polynomials is given by

hΦ

ðzÞ; Φ

ðzÞi ¼

1

ðzÞΦ

ðzÞdx ¼ c

If the roots of ψ zðÞare made equal to the roots z

(i ¼ 0 ; 2; ...; n) of the (n + 1)th-

degree Legendre polynomial Φ

nþ1

zðÞ, then ψ zðÞis a Legendre polynomial of degree

n + 1. Any polynomial of degree n can be expressed as a linear combination of

Legendre polynomials of degree up to n. Therefore, the polynomial function ω zðÞ

can be written as a linear combination of the ﬁrst n + 1 Leg endre polynomials:

ω zðÞ¼a

zðÞþa

zð Þþþa

zðÞ:

The inner product of a Legendre polynomial of degree n + 1 with a function that

is a linear combinat ion of Legendre polynomials of up to degree n is zero. There is no

term involving the inner product of two orthogonal polynomials of the same degree

as long as fðxÞ is a polynomial of degree ≤ 2n + 1. If the degree of fxðÞis equal to

2n + 2, then ω xðÞwill be of degree n + 1, the same as that of ψ xðÞand according to

Equation (B .3) , a non-zero inner product produces the non-zero truncation error.

If c

¼ 1 for n ¼ m and c

¼ 0 for n 6¼ m, then the polynomials are said to be orthonormal.

577

Appendix B: Location of nodes for Gauss–Legendre quadrature

Index for MATLAB commands

anova1, MATLAB function 267

backslash operator, MATLAB function 96, 118

bar, MATLAB function 157

binopdf, MATLAB function 168

chi2cdf, MATLAB function 276

chi2gof, MATLAB function 281

chi2pdf, MATLAB function 276

cond, MATLAB function 100

det, MATLAB function 74

diag, MATLAB function 55

double, MATLAB data type 16

eps, MATLAB function 13

errorbar, MATLAB function 272

eye, MATLAB function 55

fminbnd, MATLAB function 498

fmincon, MATLAB function 529

fminsearch, MATLAB function 515

fminunc, MATLAB function 511

format, MATLAB function 16, 21

fprintf, MATLAB function 572

global, MATLAB keyword 563

hist, MATLAB function 157

inv, MATLAB function 75

lu, MATLAB function 95

mean, MATLAB function 146

mldivide, MATLAB function

see backslash operator

multcompare, MATLAB function 267

norm, MATLAB function 65

normcdf, MATLAB function 176

norminv, MATLAB function 177

normpdf, MATLAB function 169

normplot, MATLAB function 234

ODE solvers, MATLAB function 445

–447

ones, MATLA

B function 54

poisspdf, MATLAB function 169

polyﬁt, MATLAB function 124

polyval, MATLAB function 110

quad, MATLAB function 175, 387

quadl, MATLAB function 402

range, MATLAB function 147

rank, MATLAB function 72

ranksum, MATLAB function 298

realmax, MATLAB function 11

realmin, MATLAB function 13

round, MATLAB function 7

signrank, MATLAB function 295

signtest, MATLAB function 292

single, MATLAB data type 16

single, MATLAB function 17

size, MATLAB function 56

std, MATAB function 147

structure, MATLAB data type 446

tcdf, MATLAB function 182

tinv, MATLAB function 182

ttest, MATLAB function 244, 249

ttest2, MATLAB function 251

var, MATLAB function 147

zeros, MATLAB function 54

ztest, MATLAB function 236

Index

absolute error 16

adaptive quadrature 385

array indexing 561

backward substitution 76, 81,

82, 85

banded matrices 87

base-2 number system 7

basis vectors 69, 70

Bernoulli, Jacob 148, 159

Bernoulli, Jacques

see Bernoulli, Jacob

Bernoulli, James

see Bernoulli, Jacob

Bernoulli trial 159, 253

Bernoulli process 159

bimodal 145

binomial distribution 160

binomial variable 161, 254

Bonferroni adjustment 266

box plot 267

cardinal variable 229

case-control study

see retrospective study

categorical variable

see nominal variable

chemostat 415

chi-square statistic 274

coefﬁcient of determination 115

cofactor 74

cohort study

see prospective study

colon operator 55, 58

column space 69, 70

column vector 53

complementary event 149

completely randomized

design 211

condition number 99–100

confounding variable 213

contingency table 281, 285

control group 211

convergence

of a series, 32

criteria, 39

covariance 187

covariance matrix

model parameters 198

of y 199

of model predictions 199

cubic spline 363

cumulative distribution function

(cdf) 176

density curve

see probability density curve

density function 168

descriptive statistics 142

determinant of a matrix 73

calculation of 74,

direct methods 76

Dirichlet boundary condition 462

dot plot 540

dot product 59

operator, 59

double precision 11

dynamic programming 547

BLAST 548

Needleman and Wunsch 547

eigenvalue problem 458

eigenvalues 458

eigenvectors 458

elementary row transformation 78

empirical model 480

enzymes 334

epsilon

see machine epsilon

equivalent matrix 78

event 149

expectation

see expected value

expected value 162, 170

exponent 6, 11

exposure variable 211

F statistic 264

false negative 156, 227

false positive 156, 226

ﬁnite difference formula 33–36

ﬁrst backward ﬁnite difference, 36

ﬁrst central ﬁnite difference, 36

ﬁrst forward ﬁnite difference, 36

Fisher, R. A. 261,

264

ﬁxed-point 320

ﬂoating-point

numbers 6, 10

discontinuity in ﬂoating-point number

line 13

ﬂops 88

forward substitution method 88

function workspace 562

Gauss, Carl Friedrich 167

Gaussian elimination 76,

using Thomas method 87, 133–134

with complete pivoting 87

with partial pivoting 82, 85,

with scaled partial pivoting 87

without pivoting 80, 82

global minimum 484

Gombauld, Antoine 147

Gosset, William Sealy 181

Hessian matrix 501

histogram 143, 157

ill-conditioning (instability) 26

independent events 151

inferential statistics 142, 144, 178

inﬂection point 487

inline function 347

inner product 575

interval estimate 178

inverse of a matrix 74

ligand 331

linear operator 65

linearly dependent 68

linearly independent 68

linear transformation 65, 66

Lineweaver–Burk analysis 194

local minimum 483, 484

logical indexing 561

lower triangular matrix 78

LU decomposition method

see LU factorization method

LU factorization method 88

using Doolittle method 89

machine epsilon 13

matrix constructor operator 54

matrix–matrix multiplication

61–63

matrix–vector multiplication 61

mean 145

mean value theorem 419

mechanistic model 480

median 145–146

m-ﬁle

function 562

script 561–562

Michaelis–Menten kinetics

334, 413

minor 73

principal 502

leading principal 502

mixed boundary condition 462

mode

146

Moivre, Abraham

De 166

Monod equation 37

multimodal function 484

multiple root 313

mutually exclusive events 149

Neumann boundary condition 462

nominal variable 229

norm 64

matrix norm, 65

p-norm, 64

vector norm, 64

normal equations 109,

112, 114

normal probability plot 232

Ockham, William 101

Ockham’s razor 101

ODE solvers, MATLAB functions

445–447

order of convergence

see rate of convergence

order of magnitude 6, 28

ordinal variables 229

overﬂow error 12

oxygen transport

in the body, 3

in skeletal tissue, 3–4

p value 220

Pascal, Blaise 148

permutation matrices 87, 93–94

pharmacodynamics 104

pharmacokinetic models 48

pharmacokinetics 104

phylogenetic tree 553

pivot column 78

pivot element 78

pivot row 78

point estimate 178

population parameter 145

power of a hypothesis test 227

of a one-sample z test 236–240

of a two sample z test 243

precision 5

probability density curve 167

probability

density function (pdf)

see density function

probability mass function 169

prospective study 215

quantile 232

random sample 149

random variable 157

range 146

rank

deﬁciency of, 71

of a coefﬁcient matrix, 71

of a matrix, 71

rate of convergence 316

receptor 331

relative error 16

representative sample 104, 149