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Steven T. Karris
التاريخ
4 يناير 2021
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Numerical Analysis Using MATLAB and Excel
Third Edition
Steven T. Karris
1 Introduction to MATLAB 1−1
1.1 Command Window .1−1
1.2 Roots of Polynomials .1−3
1.3 Polynomial Construction from Known Roots 1−4
1.4 Evaluation of a Polynomial at Specified Values .1−5
1.5 Rational Polynomials 1−8
1.6 Using MATLAB to Make Plots 1−9
1.7 Subplots .1−18
1.8 Multiplication, Division and Exponentiation .1−19
1.9 Script and Function Files 1−26
1.10 Display Formats 1−31
1.11 Summary .1−33
1.12 Exercises 1−37
1.13 Solutions to End−of−Chapter Exercises .1−38
MATLAB Computations: Entire chapter
2 Root Approximations 2−1
2.1 Newton’s Method for Root Approximation .2−1
2.3 The Bisection Method for Root Approximation .2−19
2.4 Summary .2−27
2.5 Exercises .2−28
2.6 Solutions to End−of−Chapter Exercises .2−29
MATLAB Computations: Pages 2−2 through 2−7, 2−14, 2−21 through 2−23,
2−29 through 2−34
Excel Computations: Pages 2−8 through 2−19, 2−24 through 2−26
3 Sinusoids and Phasors 3−1
3.1 Alternating Voltages and Currents 3−1
3.2 Characteristics of Sinusoids 3−2
3.3 Inverse Trigonometric Functions .3−10
3.4 Phasors 3−10
3.5 Addition and Subtraction of Phasors .3−11
3.6 Multiplication of Phasors 3−12
3.7 Division of Phasors .3−13ii Numerical Analysis Using MATLAB and Excel , Third Edition
3.8 Exponential and Polar Forms of Phasors 3−13
3.9 Summary .3−24
3.10 Exercises 3−27
3.11 Solutions to End−of−Chapter Exercises 3−28
MATLAB Computations: Pages 3−15 through 3−23, 3−28 through 3−31
Simulink Modeling: Pages 3−16 through 3−23
4 Matrices and Determinants 4−1
4.1 Matrix Definition .4−1
4.2 Matrix Operations .4−2
4.3 Special Forms of Matrices 4−5
4.4 Determinants .4−9
4.5 Minors and Cofactors 4−13
4.6 Cramer’s Rule 4−18
4.7 Gaussian Elimination Method .4−20
4.8 The Adjoint of a Matrix 4−22
4.9 Singular and Non−Singular Matrices 4−22
4.10 The Inverse of a Matrix .4−23
4.11 Solution of Simultaneous Equations with Matrices 4−25
4.12 Summary 4−32
4.13 Exercises 4−36
4.14 Solutions to End−of−Chapter Exercises 4−38
MATLAB Computations: Pages 4−3, 4−5 through 4−8, 4−10, 4−12, 4−3, 4−5, 4−19
through 4−20, 4−24, 4−26, 4−28, 4−30, 4−38, 4−41, 4−43
Excel Computations: Pages 4−28 through 4−29, 4−42 through 4−43
5 Differential Equations, State Variables, and State Equations 5−1
5.1 Simple Differential Equations 5−1
5.2 Classification 5−2
5.3 Solutions of Ordinary Differential Equations (ODE) .5−6
5.4 Solution of the Homogeneous ODE 5−8
5.5 Using the Method of Undetermined Coefficients for the Forced Response 5−10
5.6 Using the Method of Variation of Parameters for the Forced Response . 5−20
5.7 Expressing Differential Equations in State Equation Form 5−24
5.8 Solution of Single State Equations . 5−27
5.9 The State Transition Matrix 5−28
5.10 Computation of the State Transition Matrix 5−30
5.11 Eigenvectors 5−38
5.12 Summary 5−42Numerical Analysis Using MATLAB and Excel , Third Edition iii
5.13 Exercises . 5−47
5.14 Solutions to End−of−Chapter Exercises . 5−49
MATLAB Computations: Pages 5−11, 5−13 through 5−14, 5−16 through 5−17,
5−19, 5−23, 5−33 through 5−35, 5−37,
5−49 through 5−53, 5−55
6 Fourier, Taylor, and Maclaurin Series 6−1
6.1 Wave Analysis 6−1
6.2 Evaluation of the Coefficients .6−2
6.3 Symmetry .6−7
6.4 Waveforms in Trigonometric Form of Fourier Series .6−12
6.5 Alternate Forms of the Trigonometric Fourier Series .6−25
6.6 The Exponential Form of the Fourier Series .6−29
6.7 Line Spectra .6−33
6.8 Numerical Evaluation of Fourier Coefficients .6−36
6.9 Power Series Expansion of Functions 6−40
6.10 Taylor and Maclaurin Series 6−41
6.11 Summary 6−48
6.12 Exercises 6−51
6.13 Solutions to End−of−Chapter Exercises 6−53
MATLAB Computations: Pages 6−35, 6−45, 6−58 through 6−61
Excel Computations: Pages 6−37 through 6−39
7 Finite Differences and Interpolation 7−1
7.1 Divided Differences .7−1
7.2 Factorial Polynomials .7−6
7.3 Antidifferences .7−12
7.4 Newton’s Divided Difference Interpolation Method .7−15
7.5 Lagrange’s Interpolation Method 7−17
7.6 Gregory−Newton Forward Interpolation Method 7−19
7.7 Gregory−Newton Backward Interpolation Method .7−21
7.8 Interpolation with MATLAB .7−24
7.9 Summary .7−39
7.10 Exercises .7−44
7.11 Solutions to End−of−Chapter Exercises .7−45
MATLAB Computations: Pages 7−8 through 7−9, 7−13 through 7−15,
7−26 through 7−38, 7−45 through 7−46,
7−48, 7−50, 7−52
Excel Computations: Pages 7−17 through 7−19, 7−22 through 7−25, 7−49, 7−52iv Numerical Analysis Using MATLAB and Excel , Third Edition
8 Linear and Parabolic Regression 8−1
8.1 Curve Fitting 8−1
8.2 Linear Regression .8−2
8.3 Parabolic Regression 8−7
8.4 Regression with Power Series Approximations 8−14
8.5 Summary 8−24
8.6 Exercises .8−26
8.7 Solutions to End−of−Chapter Exercises .8−28
MATLAB Computations: Pages 8−11 through 8−14, 8−17 through 8−23,
8−30 through 8−34
Excel Computations: Pages 8−5 through 8−10, 8−15 through 8−19, 8−28 through 8−32
9 Solution of Differential Equations by Numerical Methods 9−1
9.1 Taylor Series Method 9−1
9.2 Runge−Kutta Method 9−5
9.4 Milne’s Method 9−15
9.5 Summary 9−17
9.6 Exercises 9−20
9.7 Solutions to End−of−Chapter Exercises 9−21
MATLAB Computations: Pages 9−5, 9−9 through 9−12, 9−21 through 9−23
Excel Computations: Page 9−2, 9−14, 9−22 through 9−26
10 Integration by Numerical Methods 10−1
10.1 The Trapezoidal Rule 10−1
10.2 Simpson’s Rule . 10−6
10.3 Summary 10−14
10.4 Exercises 10−15
10.5 Solution to End−of−Chapter Exercises 10−16
MATLAB Computations: Pages 10−3 through 10−6, 10−9 through 10−13,
10−16, 10−18 through 10−21
Excel Computations: Pages 10−10, 10−19 through 10−21
11 Difference Equations 11−1
11.1 Introduction . 11−1
11.2 Definition, Solutions, and Applications 11−1
11.3 Fibonacci Numbers 11−7Numerical Analysis Using MATLAB and Excel , Third Edition v
11.4 Summary .11−11
11.5 Exercises . 11−13
11.6 Solutions to End−of−Chapter Exercises .11−14
12 Partial Fraction Expansion 12−1
12.1 Partial Fraction Expansion 12−1
12.2 Alternate Method of Partial Fraction Expansion 12−13
12.3 Summary 12−19
12.4 Exercises 12−22
12.5 Solutions to End−of−Chapter Exercises 12−23
MATLAB Computations: Pages 12−3 through 12−5, 12−9 through 12−12,
12−16 through 12-18, 12−23 through 12−28
13 The Gamma and Beta Functions and Distributions 13−1
13.1 The Gamma Function .13−1
13.2 The Gamma Distribution 13−16
13.3 The Beta Function .13−17
13.4 The Beta Distribution 13−20
13.5 Summary 13−22
13.6 Exercises 13−24
13.7 Solutions to End−of−Chapter Exercises 13−25
MATLAB Computations: Pages 13−3, 13−5, 13−10, 13−19, 13−25
Excel Computations: Pages 13−5, 13−10, 13−16 through 13−17, 13−19, 13−21
14 Orthogonal Functions and Matrix Factorizations 14−1
14.1 Orthogonal Functions 14−1
14.2 Orthogonal Trajectories .14−2
14.3 Orthogonal Vectors 14−4
14.4 The Gram−Schmidt Orthogonalization Procedure 14−7
14.5 The LU Factorization .14−9
14.6 The Cholesky Factorization 14−23
14.7 The QR Factorization .14−25
14.8 Singular Value Decomposition .14−28
14.9 Summary .14−30
14.10 Exercises .14−32
14.11 Solutions to End−of−Chapter Exercises .14−34
MATLAB Computations: Pages 14−8 through 14−9, 14−11 through 14−29,
14−36, 14−38 through 14−39vi Numerical Analysis Using MATLAB and Excel , Third Edition
15 Bessel, Legendre, and Chebyshev Functions 15−1
15.1 The Bessel Function 15−1
15.2 Legendre Functions .15−10
15.3 Laguerre Polynomials .15−21
15.4 Chebyshev Polynomials .15−22
15.5 Summary 15−27
15.6 Exercises .15−32
15.7 Solutions to End−of−Chapter Exercises 15−33
MATLAB Computations: Pages 15−3 through 15−4, 15−6, 15−9,
14−19 through 15−22, 15−25, 15−33, 15−35 through 15−37
Excel Computations: Pages 15−5, 15−9
16 Optimization Methods 16−1
16.1 Linear Programming . 16−1
16.2 Dynamic Programming 16−4
16.3 Network Analysis .16−14
16.4 Summary 16−19
16.5 Exercises .15−20
16.6 Solutions to End−of−Chapter Exercises 15−22
MATLAB Computations: Pages 16−3
Excel Computations: Pages 16−4, 16−23, 16−25 through 16−27
A Difference Equations in Discrete−Time Systems A−1
A.1 Recursive Method for Solving Difference Equations . A−1
A.2 Method of Undetermined Coefficients A−1
MATLAB Computations: Pages A−4, A−7, A−9
B.1 Simulink and its Relation to MATLAB .B−1
MATLAB Computations and Simulink Modeling: Entire Appendix B
C Ill-Conditioned Matrices C−1
C.1 The Norm of a Matrix .C−1
C.2 Condition Number of a Matrix .C−2
C.3 Hilbert Matrices C−3Numerical Analysis Using MATLAB and Excel , Third Edition vii
MATLAB Computations: Pages C−1, C−4 through C−5
References R−1
Index
Symbols Chebyshev polynomials of the disp(x) MATLAB function 2-5
% (percent) symbol 1-2 second kind 15-22 Display block in Simulink B-18
%d MATLAB symbol 2-5 Cholesky factorization 14-23 display formats in MATLAB 1-31
%e MATLAB symbol 2-21 classification of differential equations 5-2 distinct poles 12-2
%s MATLAB symbol 2-21 clc MATLAB command 1-2 distinct roots of characteristic
%u MATLAB symbol 2-21 clear MATLAB command 1-2 equation 5-9
coefficient matrix – see matrix divided differences 7-1
A cofactors – see matrix division of complex numbers 3-13
collect(s) MATLAB function 7-15, 12-11 dot division operator in MATLAB 1-21
abs(z) MATLAB function 1-24 column vector 1-19 dot exponentation operator
absolute cell in Excel 2-19 command screen in MATLAB 1-1 in MATLAB 1-21
absolutely convergent 15-10 command window in MATLAB 1-1 dot multiplication operator
Adams’ method 9-13 commas in MATLAB 1-7 in MATLAB 1-21
Add Trendline Excel feature 8-6 comment line in MATLAB 1-2 dsolve(s) MATLAB function 9-5
adjoint of a matrix – see matrix Commonly Used Blocks in Simulink B-7 dynamic programming 16-4
algebraic constrain block in Simulink B-18 complementary solution A-1
algebraic form of the Legendre complex conjugate 1-4 E
polynomials 15-17 complex numbers 1-2, 3-11
alternate form of the trigonometric complex poles 12-5 Editor window in MATLAB 1-1, 1-27
Fourier series 6-25 complex roots of characteristic Editor/Debugger in MATLAB 1-1, 1-2
alternating current 3-1 equation 5-9 eigenvalues 5-30
angle(z) MATLAB function 1-24 cond(A) MATLAB function C-2 eigenvectors 5-39
angular velocity 3-2 condition number of a matrix – see matrix element-by-element division
antidifference 7-12 configuration parameters in Simulink B-12 in MATLAB 1-21
approximations with Excel 2-7 congugate of a matrix – see matrix element-by-element exponentiation
associated Legendre differential conj(A) MATLAB function 4-8 in MATLAB 1-21
equation 15-18 conj(x) MATLAB function 4-8 element-by-element multiplication
associated Legendre functions of the conjugate of a complex number 3-12 in MATLAB 1-19, 1-21
first and second kind 15-18 connected network 16-15 elements of a matrix – see matrix
AutoFill Excel feature 2-9, 2-15 constraints in linear programming 16-1 eps in MATLAB 1-22, 1-27
autoscale icon in Simulink B-12 Contents Pane in Simulink B-7 equating the numerators procedure
axis MATLAB command 1-17, 1-22 conv(a,b) MATLAB function 1-6 in partial fraction expansion 12-13
Cramer’s rule 4-18, 8-4 Erlang distribution 13-16
B cubic interpolarion in MATLAB 7-25 error function 10-4
cubic spline interpolarion in MATLAB 7-25 error in method of least squares 8-3
backward substitution. 14-10 curve fitting 8-1 Euclidean norm C-1
Bessel function of negative order n 15-2 curved regression 8-7, 8-14 Euler’s identities 3-14
Bessel function of n 15-1 cycle 3-2 even functions 6-7, 6-31
Bessel functions 15-1 cyclic frequency 3-3 even symmetry 6-7
Bessel functions of the first kind 15-7 exit MATLAB command 2, 33
Bessel functions of the second kind 15-7 D EXP(GAMMALN(n)) Excel function 13-5
besselj(n,x) MATLAB function 15-3 expand(s) MATLAB function 7-13, 12-9
BESSELJ(x,n) Excel function 15-3 Data Analysis Toolpack in Excel 8-7 exponential form of complex
beta distribution 13-20 data points in MATLAB 1-14 numbers 3-14
beta function 13-17 decibel 12 exponential form of the
beta(m,n) MATLAB function 13-19 decimal format 2-21 Fourier series 6-29
BETADIST Excel function 13-21 deconv(p,q) MATLAB function 1-6, 7-8 eye(n) MATLAB function 4-6, 14-24
bisection method for root default color in MATLAB 1-14 eye(size(A)) MATLAB function 4-7
approximation 2-19 default line in MATLAB 1-15
box MATLAB command 1-12 default marker in MATLAB 1-14 F
default values in MATLAB 1-11
C degree of differential equation 5-3 factor(p) MATLAB function 12-3
demo in MATLAB 1-2 factorial polynomials 7-6
Casorati’s determinant 11-2 determinant – see matrix Fibonacci numbers 11-7
Cayley-Hamilton theorem 5-30 diag(v,k) MATLAB function 14-24 figure window in MATLAB 1-14
characteristic (auxiliary) equation 5-8 diagonal elements of a matrix – see matrix finite differences 7-1
characteristic equation of a second diagonal of a matrix – see matrix first divided difference 7-1
order difference equation 11-3 diff(s) MATLAB function 2-6 first harmonic 6-1
chart type in Excel 2-10 difference equations A-1 fixed point format 2-21
Chart Wizard Excel feature 2-10, 8-6 difference operator 7-4, 11-1 Flip Block command in Simulink B-11
Chebyshev polynomials 15-22 differences – see finite differences forced response 5-8
Chebyshev polynomials of the dipole 15-12 forcing function A-1
first kind 15-22 direct terms in MATLAB 12-1 format MATLAB command 1-31
directed network 16-15 format specifiers in MATLAB 2-21
IN1forward substitution 14-10 improper integral 13-1 Math Operations Library in Simulink B-11
Fourier analysis 3-2 improper rational function 12-1 MATLAB Demos 1-2
Fourier series 6-1 increments between points matrix (matrices) 4-1
fplot MATLAB command 1-27 in MATLAB 1-14 adjoint of 4-22
fprintf MATLAB command 7-29 in-phase sinusoids 3-3 coefficient C-3
frequency 3-2 input(‘string’) MATLAB command 2-2 cofactors of 4-11
Frequency Modulation (FM) 15-4 int(f,a,b) MATLAB function 10-5 condition number of C-2
frequency response 1-11 interp1(x,y,xi) MATLAB function 7-24 conformable for addition 4-2
full-wave rectifier with even interp1(x,y,xi,’method’) MATLAB conformable for multiplication 4-4
symmetry 6-24 function 7-25 conformable for subtraction 4-2
function block parameters in Simulink B-10 interp2(x,y,z,xi,yi) MATLAB congugate of 4-8
function files in MATLAB 1-26, 1-27, 2-4 function 7-25 determinant of 4-9
fundamental frequency 6-1 interp2(x,y,z,xi,yi,’method’) MATLAB diagonal elements of 4-1
fundamental theorem of function 7-25 diagonal of 4-6
integral calculus 7-12 interpolation 7-1 elements of 4-1
fundamental theorem of interpolation with MATLAB 7-24 Hermitian 4-9
sum calculus 7-13 interval halving 2-19 Hilbert C-3
fzero MATLAB function 1-27, 1-29 inverse of a matrix – see matrix identity 4-6
fzero(‘function’,x0) MATLAB isoprofit line 16-2 ill-conditioned 4-22, C-2
function 12-18 inverse of 4-23
fzero(f,x) MATLAB function 2-7 J lower triangular 4-6
main diagonal elements of 4-1
G j operator 3-11 main diagonal of 4-1
minor of determinant 4-13
Gain block in Simulink B-18 L non-singular 4-22
gamma distribution 13-16 scalar 4-6
gamma function 13-1 L’ Hôpital’s rule 13-2 singular 4-22
gamma(n) MATLAB function 13-3, 13-10 lagging sinudoid 3-3 size of 4-7
GAMMADIST Excel function 13-16 Lagrange’s interpolation method 7-17 skew-Hermitian 4-9
GAMMALN Excel function 13-5 Laguerre polynomials 15-21 skew-symmetric 4-8
Gaussian elimination method 4-20, 14-9 leading sinusoid 3-3 square 4-1
Genenbauer 15-22 least squares 8-2, 8-3 symmetric 4-8
general solution of a differential least squares line 8-2 trace of 4-2
equation 5-6 least-squares curve 8-2 transpose of 4-7
generalized factorial function 13-1 least-squares parabola 8-2, 8-7 unitary 14-25
generating function for Bessel functions Legendre functions 15-10 upper triangular 4-5
of the first kind of integer order 15-9 Legendre functions of the well-conditioned C-2
generating function for Legendre second kind 15-11 zero matrix – see matrix
polynomials 15-12 Legendre polynomials 15-11 matrix left division in MATLAB 4-26
Goal Seek Excel feature 2-16 Legendre polynomials in matrix multiplication in MATLAB 1-19, 1-20
Gram-Schmidt orthogonalization trigonometric form 15-18 mesh(x,y,z) MATLAB command 1-17
procedure 5-39, 14-7 legendre(n,x) MATLAB function 15-19 mesh(Z) MATLAB function 7-32
Gregory-Newton Backward Legendre’s equation 15-10 meshgrid(x,y) MATLAB command 1-17
Interpolation Formula 7-21 lims = MATLAB function 1-27 method of Frobenius 15-1
Gregory-Newton Forward line graph 16-4 method of least squares 8-2, 8-3
Interpolation Formula 7-19 line spectrum 6-33 method of Picard 15-1
grid MATLAB command 1-11 linear factor 1-9 method of undetermined
gtext MATLAB command 1-14, 15-25 linear interpolarion 7-25 coefficients 5-10, 11-2, A-1
linear programming 16-1 method of variation of parameters 5-20
H linear regression 8-2 m-file in MATLAB 1-1 1-26, 1-27
linspace MATLAB command 1-14 Milne’s method 9-15
half-wave rectifier 6-19, 6-22 ln (natural logarithm) 1-12 minimum span problems 16-15
half-wave symmetry 6-7, 6-31 log (common logarithm) 1-12 minor of determinant – see matrix
help in MATLAB 1-2, 1-15 log(x) MATLAB function 1-12 MINVERSE Excel function 4-28
Hermitian matrix – see matrix log10(x) MATLAB function 1-12 MMULT Excel function 4-28, 4-29
hilb(n) MATLAB function C-4 log2(x) MATLAB function 1-12 modified Bessel functions 15-7
Hilbert matrix – see matrix loglog MATLAB command 1-12, 34 modulation index 15-4
homogeneous difference equation 11-2 lower triangular matrix – see matrix multiple poles 12-6
Hz (Hertz) 3-3 lsqnonneg MATLAB function 14-28 multiplication of complex numbers 3-12
LU factorization method 14-9
I lu(A) MATLAB function 14-12, 14-14 N
identity matrix – see matrix M NaN in MATLAB 1-27
ill-conditioned matrix – see matrix natural response 5-8, A-1
imag(z) MATLAB function 1-24 Maclaurin power series 6-41, 7-6 nearest neighbor interpolarion 7-25
imaginary axis 3-10 main diagonal elements – see matrix network analysis 16-14
imaginary number 3-10 main diagonal of a matrix – see matrix Neumann functions 15-7
IN2Newton’s divided difference quadratic factor 1-9 Stirling’s asymptotic series
interpolation method 7-15 quit MATLAB command 1-2 for the G(n) function 13-9
Newton-Cotes 8-panel rule 10-10 string in MATLAB 1-17
non-homogeneous difference R subplot MATLAB command 1-18
equation 11-2 surface zonal harmonics 15-11
non-homogeneous ODE 5-6 radian frequency 3-2 svd(A) MATLAB function 14-28
non-singular matrix – see matrix rational polynomial 1-8 sym, syms MATLAB symbolic
norm C-1 rationalization of the quotient 3-13 expressions 12-4
norm(A) MATLAB function C-1 real axis 3-10 symbolic expressions in MATLAB 12-3
numeric expressions in MATLAB 12-4 real number 3-11 Symbolic Math Toolbox in MATLAB 12-4
numerical evaluation of Fourier real(z) MATLAB function 1-24 symmetric matrix – see matrix
coefficients 6-36 recursion A-1 symmetry 6-7, 6-14, 6-31
recursive method A-1
O regression 8-1 T
regression analysis 8-7
odd functions 6-8, 6-31 relative cell in Excel 2-19 Taylor series 5-24, 6-41, 6-44
odd symmetry 6-7 repeated poles 5-9, 12- 6 Taylor series expansion method 9-1
ODE (Ordinary Differential Equation) 5-3 residue(r,p,k) MATLAB function 12-1 text MATLAB command 1-14, 1-17
ode23 MATLAB function 9-9 revolutions per second 3-5 third harmonic 6-1
ode45 MATLAB function 9-9 Rodrigues’ formula 15-12, 15-18 title(‘string’) in MATLAB 1-12
one-dimensional wave equation 5-3 roots of polynomials 1-3 trace of a matrix – see matrix
optimum path policy 16-5 roots(p) MATLAB function 1-3, 1-8 transpose of a matrix – see matrix
order of a differential equation 5-3 rotating vector 3-5 trapezoidal rule 10-1
ordinary differential equation 5-3 round(n) MATLAB function 1-24 trapz(x,y) MATLAB function 10-3, 10-5
oriented network 16-15 row vector 1-3, 1-19 Tree Pane in Simulink B-7
orthogonal basis 14-5 Runge-Kutta method 5-24, 9-5 Trendline Excel feature 8-9
orthogonal functions 6-2, 14-1, 14-2 triangular waveform 6-11, 6-19
orthogonal system 15-9 S trigonometric Fourier series 6-1
orthogonal trajectories 14-2 trigonometric relations 3-5
orthogonal unit vectors 14-5 sawtooth waveform 6-10, 6-18 two-dimensional plots 7-32
orthogonal vectors 5-39, 14-4 scalar matrix – see matrix type of a diferential equation 5-2
orthonormal basis 14-5 Scope block in Simulink B-12
out-of-phase sinusoids 3-3 script file in MATLAB 1-26 U
overdetermined system 8-3 second divided difference 7-1
second harmonic 6-1 ultraspherical functions 15-22
P semicolons in MATLAB 1-7 undetermined system 8-3
semilogx MATLAB command 1-12 unit fraction C-3
parabolic curve 8-1 semilogy MATLAB command 1-12 unitary matrix – see matrix
partial differential equation (PDE) 5-3 simple differential equations 5-1 upper triangular matrix – see matrix
partial fraction expansion 12-1 simplex method 16-4
PDE (Partial Differential Equation) 5-3 Simpson’s rule 10-6 V
period 3-2, 3-3 simulation start icon in Simulink B-12 VLOOKUP Excel function 7-23
periodic waveform 3-2 Simulink icon B-7
phasor 3-2 Simulink Library Browser B-8 W
plot area in Excel 6 singular matrix – see matrix
plot MATLAB command 1-9, 1-12, 1-15 Singular Value Decomposition 14-28 Wallis’s formulas 13-16
plot3 MATLAB command 1-16 Sinks library B-18 Weber functions 15-7
polar form of complex numbers 3-15 sinusoids 3-2 well-conditioned matrix – see matrix
polar plot in MATLAB 1-24 size of a matrix – see matrix while end in MATLAB 2-4
polar(theta,r) MATLAB function 1-24 skew-Hermitian matrix – see matrix Wronskian determinant 5-10, 11-2
poles 12-2 skew-symmetric matrix – see matrix
poly MATLAB function 1-4 solution of the homogeneous ODE 5-8 X
polyder MATLAB function 1-7 solutions of ODEs 5-6
polyfit(x,y,n) MATLAB function 8-11 space equations 5-24 xlabel MATLAB command 1-12
polynomial construction from known spectrum analyzer 33 XY (Scatter) in Excel 8-6
roots in MATLAB 1-4 spherical harmonics 15-18
polyval(p,x) MATLAB function 1-5, 8-11 sprintf MATLAB command 2-5 Y
power series 6-40 sqrt MATLAB function 10-12
proper rational function 12-1 square matrix – see matrix ylabel MATLAB command 1-12
square waveform
Q 6-9, 6-14, 6-16, 6-48, 6-49 Z
QR factorization 14-25 state equations 5-24 zero matrix – see matrix
qr(A) MATLAB function 14-25 state transition matrix 5-28 zeros 12-2
quad MATLAB function 10-10 state variables 5-24 zlabel MATLAB command 1-17