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Engineering Computations and Modeling in MATLAB/Simulink

Chapter 7. Symbolic Math Toolbox



Outline • 7.1 Introduction • 7.2 Symbolic Math Toolbox Overview • 7.3 Getting Started • 7.4 Basic Operations • 7.5 Linear Algebra and Calculus • 7.6 Solving Algebraic and Differential Equations • 7.7 Simplification and Substitution • 7.8 Other Functions • 7.9 MuPAD Development Environment • 7.10 Plotting Symbolic Expressions • 7.11 Using Symbolic Expressions in Numerical Operations • 7.12 LaTeX Representation of Symbolic Expression • 7.13 Symbolic Calculators




Toolbox Overview

Originally, the computational engine underlying the Symbolic Math toolbox was the Maple kernel, developed at the University of Waterloo in Canada. Before R2007b, the Basic Symbolic Math toolbox included in the Student Version was a collection of more than a hundred MATLAB functions allowing access to the functions in Maple’s linear algebra package. The Extended Symbolic Math toolbox augmented the basic toolbox by including access to non-graphics packages, programming features and user defined procedures in Maple.

As of R2007b+, Symbolic Math Toolbox - now a single product - uses the MuPAD computational engine, originally developed by the MuPAD research group at the University of Paderborn, Germany. The functionality of the MuPAD language, together with included libraries, goes far beyond that of the previous Symbolic Math Toolbox software. The functionality of Extended Symbolic Math Toolbox has been taken over by MuPad environment (MuPAD Notebook).



Areas Supported by Toolbox

Subject Areas Supported

Linear Algebra Inverses, determinants, eigenvalues, singular value decomposition, canonical forms of symbolic matrices

Calculus Differentiation, integration, limits, summation, and Taylor series

Solving (Systems of) Equations Symbolic and numerical solutions to algebraic and differential equations

Simplification and Substitution Methods of simplifying algebraic expressions

Integral Transforms Fourier, Laplace, z- transforms and their corresponding inverse transforms

Conversions and Variable Precision Arithmetic

Conversion and numerical evaluation of mathematical expressions to any specified accuracy

Special Functions Special functions in applied mathematics such as Airy functions, binomial

coefficients, error functions, Bessel functions, Dirac delta function, Legendre polynomials, Jacobian, and Chebyschev functions

Plotting Symbolic Expressions Easy 2D and 3D plotting of symbolic expressions



Symbolic Math Demos



Basic Operations

sym('a') syms x y y=(x^a+sqrt(x))/y pretty(y) symvar(y)

y = (x^a + x^(1/2))/y a 1/2 x + x --------- y ans = [ a, x, y]

Arithmetic Operations + Addition - Subtraction * Multiplication .* Array multiplication \ Left division .\ Array left division / Right division ./ Array right division ^ Matrix or scalar raised to a power .^ Array raised to a power ’ Complex conjugate transpose .’ Real transpose

Basic Operations ceil Round symbolic matrix toward positive infinity conj Symbolic complex conjugate eq Symbolic equality test fix Round toward zero floor Round symbolic matrix toward negative infinity frac Symbolic matrix elementwise fractional parts imag Imaginary part of complex number latex LaTeX representation of symbolic expression log10 Logarithm base 10 of entries of symbolic matrix Iog2 Logarithm base 2 of entries of symbolic matrix mod Symbolic matrix elementwise modulus pretty Pretty-print symbolic expressions quorem Symbolic matrix elementwise quotient and remainder real Real part of imaginary number round Symbolic matrix elementwise round size Symbolic matrix dimensions sort Sort symbolic vectors or polynomials sym Symbolic numbers, variables, and objects syms Shortcut for constructing symbolic objects symvar Find symbolic variables in symbolic expression or matrix



Linear Algebra & Calculus

syms a11 a22 a21 a22 inv([a11 a22; a21 a22])

syms x k a diff(cos(x^2)) int(cos(x)^2,0,pi) limit(sin(x)/x) symsum(x^k/sym('k!'),k,0,inf) taylor(sin(x-a),x,a,'Order',10)

Linear Algebra colspace Basis for column space det Symbolic matrix determinant diag Create or extract symbolic diagonals eig Symbolic eigenvalues and eigenvectors expm Symbolic matrix exponential inv Symbolic matrix inverse jordan Jordan canonical form null Basis for null space poly Characteristic polynomial of matrix rank Symbolic matrix rank rref Reduced row echelon form svd Symbolic singular value decomposition tril Symbolic lower triangle triu Symbolic upper triangle

Calculus diff Differentiate symbolic expression int Integrate symbolic expression jacobian Jacobian matrix limit Limit of symbolic expression symsum Symbolic summation of series taylor Taylor series expansion

( )

0

( )( ) ( )!

nn

n

f af x x an

∞

=

= −∑

2(cos( ))d xdx2

0

cos ( )x dxπ

∫

0

sin( )limx

xx→

0 !

k

k

xk

∞

=∑



Solving Equations

syms a b c x, solve(a*x^2+b*x+c,x)

syms x y, [x,y]=solve([5*x+6*y-8,2*x+3*y-1])

syms a x y, f=solve([a*x^2+y^2-1,x+sqrt(y)-1],[x,y]); [f.x f.y]

dsolve('Dx-a*x=sin(t)','x(0)=1')

g=dsolve('Dx=y','Dy=cos(t)','x(0)=1','y(0)=0'); [g.x g.y]

dsolve('D3x=x-1','x(0)=0','Dx(1)=2','D2x(0)=0')

Solution of Equations compose Functional composition dsolve Symbolic solution of ordinary differential equations finverse Functional inverse solve Symbolic solution of algebraic equations

syms x(t); D1=diff(x); D2=diff(x,2); dsolve(diff(x,3)==x-1,[x(0)==0,D1(1)==2,D2(0)==0])

Dx – the 1st-order derivative

D2x – the 2nd-order derivative

D3x – the 3rd-order derivative

solving a single equation with respect to x

solving two equations with respect to x and y

solving two equations with respect to x and y



Simplification & Substitution

syms x y=cos(2*x) simple(y)

Simplification and Substitution coeffs Coefficients of multivariate polynomial collect Collect coefficients expand Symbolic expansion of polynomials and elementary functions factor Factorization horner Horner nested polynomial representation numden Numerator and denominator simple Search for simplest form of symbolic expression simplify Symbolic simplification subexpr Rewrite symbolic expression in terms of common subexpressions subs Symbolic substitution in symbolic expression of matrix

syms x y rho phi R a zdc=(x-a)^2+y^2-R^2; zpc=subs(zdc,{x,y},{rho*cos(phi),rho*sin(phi)}) simplify(zpc)

Double strikethrough functions should be removed if using R2014b+

syms x y=cos(2*x) expand(y)

ans = 2*cos(x)^2 - 1

ans = - R^2 + a^2 - 2*cos(phi)*a*rho + rho^2



Integral & z-Transforms

syms a t laplace(t*exp(-a*t))

syms w x y ilaplace(w/(y^2 + w^2),y)

Integral Transforms fourier Fourier integral transform ifourier Inverse Fourier integral transform ztrans z-transform transform iztrans Inverse z-transform laplace Laplace transform ilaplace Inverse Laplace transform

1( ) ( )2

i tf t F e dωω ωπ

∞

−∞

= ∫( ) ( ) i tF f t e dtωω

∞−

−∞

= ∫

11( ) ( ) , 1, 2,...2

n

z R

f n F z z dz niπ

−

=

= =∫0

( )( ) nn

f nF zz

∞

=

=∑

0

( ) ( ) stF s f t e dt∞

−= ∫( ) ( )

c ist

c i

f t F s e ds+ ∞

− ∞

= ∫

ans = 1/(a + s)^2

ans = sin(w*y)



Other Functions syms x f=taylor(sin(x),'Order',4) t=inline(char(f)); t(5)

f = - x^3/6 + x ans = -15.8333

Conversions ccode C code representation of symbolic expression char Convert symbolic objects to strings double Convert symbolic matrix to MATLAB numeric form emlBlock Convert symbolic expression to MATLAB Function block fortran Fortran representation of symbolic expression int8, int16, int32, int64 Convert symbolic matrix to signed integers latex LaTeX representation of symbolic expression matlabFunction Convert symbolic expression to function handle or file poly2sym Polynomial coefficient vector to symbolic polynomial single Convert symbolic matrix to single precision sym2poly Symbolic-to-numeric polynomial conversion uint8, uint16, uint32, uint64 Convert symbolic matrix to unsigned integers

Variable Precision Arithmetic digits Variable precision accuracy

vpa Variable precisian arithmetic

Special Functions cosint Cosine integral dirac Dirac delta gamma Gamma function heaviside Heaviside step function hyperqeom Generalized hypergeometric lambertw Lambert's W function mfun Numeric evaluation of special mathematical function mfunlist List special functions for use with mfun sinint Sine integral zeta Riemann Zeta



Pedagogical and Graphical Applications ezcontour Contour plotter ezcontourf Filled contour plotter ezmesh 3D mesh plotter ezmeshc Combined mesh and contour plotter ezplot 2D parametric curve plotter ezplot3 3D parametric curve plotter ezpolar Polar coordinate plotter ezsurf 3D colored surface plotter ezsurfc Combined surface and contour plot funtool Function calculator rsums Interactive evaluation of Riemann sums taylortool Taylor series calculator

Calculators & Plotting g=dsolve('Dx=y','Dy=cos(t)',... 'x(0)=1','y(0)=0'); ezplot(g.x,g.y)

syms r phi x=r*cos(phi); y=r*sin(phi); z=1+besselj(1,3*r)*cos(phi); ezmesh(x,y,z)



MuPAD Functions

MuPAD clear all Remove items from MATLAB workspace and reset MuPAD engine doc Get help for MuPAD functions evalin Evaluate MuPAD expressions feval Evaluate MuPAD expressions getVar Get variable from MuPAD notebook mupad Start MuPAD notebook mupadwelcome Start MuPAD interfaces openmn Open MuPAD notebook openmu Open MuPAD program file openmuphlp Open MuPAD help file openxvc Open MuPAD XVC graphics file openxvz Open MuPAD XVZ graphics file reset Close MuPAD engine setVar Assign variable in MuPAD notebook symengine Return symbolic engine trace Enable and disable tracing of MuPAD commands



MuPAD >> mupadwelcome or >> mupad



MuPAD Notebook

Insert Calculation Insert Text Paragraph Evaluate Stop



MuPAD Syntax MATLAB MuPAD

Assignment = =:

List variables whos anames(All,User)

Numerical value of an expression double(expression) float(expression)

Suppress output ; :

Enter a matrix [a, b, c; d, e, f] matrix([a, b ,c], [d, e, f])

{a,b,c} cell array set

Linear algebra commands nothing extra is needed linalg:: prefix or use(linalg)

Autocompletion <Tab> <Ctrl>+<Space>

Equality and inequality comparison ==, ~= =, <>

MATLAB MuPAD

besselj, bessely, besseli, besselk besselJ, besselY, besselI, besselK

lambertw lambertW

sinint, cosint Si, Ci

eilergamma, catalan EULER, CATALAN

conj conjugate

laplace, ilaplace transform::laplace, transform::ilaplace

ztrans, iztrans transform::ztrans, transform::iztrans

MATLAB MuPAD

Inf infinity

pi PI

i I

NaN undefined

fix trunc

log ln

MATLAB MuPAD

asin acos atan asinh acosh atanh acsc asec acot

acsch asech acoth

arcsin arccos arctan arcsinh arccosh arctanh arcsc arcsec arccot

arccsch arcsech arccoth



MuPAD Computations1

1 2 3



MuPAD Computations2

4 5 6



MuPAD Computations3

7 8



Quick MuPAD/MATLAB Interchange

Copy and paste to the MATLAB Workspace to receive

sin(x)^2/x



The getVar and setVar Functions

>> g=sqrt(f) g = (exp(y)*asin(x))^(1/2) >> setVar(h_ntb,'w',g)

>> h_ntb=mupad; >> f = getVar(h_ntb,'f') f = exp(y)*asin(x)

MuPAD => MATLAB MATLAB => MuPAD



The matlabFunction Functions >> syms ps M

>> q = 0.7*ps*M^2; >> hq = matlabFunction(q) hq = @(M,ps)M.^2.*ps.*(7.0./1.0e1)

syms lat lon Rlon = [cos(lon) sin(lon) 0; -sin(lon) cos(lon) 0; 0 0 1]; Rlat = [cos(lat) 0 sin(lat); 0 1 0; -sin(lat) 0 cos(lat)]; R = Rlat*Rlon; F = matlabFunction(R,'file','RotationMatrix.m')

function R = RotationMatrix(lat,lon) %ROTATIONMATRIX % R = ROTATIONMATRIX(LAT,LON) % 13-Nov-2017 19:59:41 t2 = cos(lat); t3 = sin(lon); t4 = cos(lon); t5 = sin(lat); R = reshape([t2.*t4,-t3,-t4.*t5,t2.*t3,t4,-t3.*t5,t5,0.0,t2],[3,3]);

Creates a function handle F to the

RotationMatrix function saved as

RotationMatrix.m

>> M = [0.2,0.2,0.6]; ps = [101325, 60000, 101325]; >> hq(M,ps) ans = 1.0e+004 * 0.2837 0.1680 2.5534



z=[1 3 4]; x=poly(z);

fs=poly2sym(x)

fc=char(fs)

xz=fzero(fc,1.82)

fa=str2func(['@(x)' fc])

fplot(fa,[0 5]), grid

hold, plot(xz,fa(xz),'rp')

Symbolic to String and Function Handle

fs = x^3-8*x^2+19*x-12

fc = x^3-8*x^2+19*x-12

xz = 1.0000

fa = @(x)19*x-8*x^2+x^3-12



syms t

fsym=cos(t)*exp(t)

ezplot(fsym) % it works! :-)

xmin=fminsearch(fsym,3) % it doesn’t work :-(

finl=inline(char(fsym))

xmin=fminsearch(finl,3) % it works now :-)

hold, plot(xmin,finl(xmin),'rp')

Symbolic to Inline Function

fsym = cos(t)*exp(t)

??? Error using ==> fcnchk at 105

If FUN is a MATLAB object, it must have an feval method.

Error in ==> fminsearch at 178

funfcn = fcnchk(funfcn,length(varargin));

finl =

Inline function:

finl(t) = cos(t)*exp(t)

xmin = 3.9270

Current plot held

1

2

1

2

3

3



inline vs. fcnchk syms t

fsym=cos(t)*exp(t)

ezplot(fsym)

finl=fcnchk(char(fsym)) % fcnchk is equivalent to inline

xmin=fminsearch(finl,3) % it works now :-)

hold, plot(xmin,finl(xmin),'rp')

ezplot('sin(2*x)')

ezplot('sin(2*t)')

syms t

fsym=cos(t)*exp(t);

finl=fcnchk(char(fsym),'vectorized')

finl =

Inline function:

finl(t) = cos(t).*exp(t).*ones(size(t))

What’s the difference?

finl=inline(vectorize(char(fsym)))

fzero('sin(2*x)',3)

fzero('sin(2*t)',3)



LaTeX Representation of Symbolic Expression

>> syms x y t >> F1 = diff(atan(x)) >> F2 = sym(hilb(3)) >> F4 =(x^2+1/x)/sqrt((sin(x)+cos(x))/x^3.5) >> F5 = taylor(sin(x),'Order',8,'ExpansionPoint',pi/2)

>> H1=latex(F1); >> H2=latex(F2); >> H4=latex(F4); >> H5=latex(F5); >> text(.2,.25,['$$' H1 '$$'],'interpreter','latex','fontsize',14) >> text(.6,.2, ['$$' H2 '$$'],'interpreter','latex','fontsize',16) >> text(.3,.5, ['$$' H4 '$$'],'interpreter','latex','fontsize',18) >> text(.1,.8, ['$$' H5 '$$'],'interpreter','latex','fontsize',14)

F1 = 1/(x^2 + 1) F2 = [ 1, 1/2, 1/3] [ 1/2, 1/3, 1/4] [ 1/3, 1/4, 1/5] F4 = (1/x + x^2)/((cos(x) + sin(x))/x^(7/2))^(1/2) F5 = (x - pi/2)^4/24 - (x - pi/2)^2/2 - (x - pi/2)^6/720 + 1



Riemann Sums Calculator >> syms x >> rsums(sin(pi*x))



Function Calculator funtool



Taylor Series Calculator taylortool taylortool('f')



The End of Chapter 7

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Chapter 7. Symbolic Math Toolbox - Naval Postgraduate …faculty.nps.edu/oayakime/AE2440/Slides/Chapter 07 Symbolic Math... · ... error functions, Bessel functions, Dirac delta function,