chapter 7. symbolic math toolbox

1 out of 27Engineering Computations and Modeling in MATLAB/Simulink

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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 Plotting Symbolic Expressions

• 7.10 Using Symbolic Expressions in Numerical Operations

• 7.11 LaTeX Representation of Symbolic Expression

• 7.12 Symbolic Calculators

• 7.13 Units of Measurements in Symbolic Math Toolbox (optional)

Engineering Computations and Modeling in MATLAB/Simulink





SMT History

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 SMT

Subject Areas Supported

Linear AlgebraInverses, 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

Formula Manipulation and

Simplification

Simplification or modification of symbolic expressions, substitution of parts

of expressions

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

Polynomials Characteristic and minimal polynomials, coefficients of polynomials

Numbers and Precision Variable-precision arithmetic and elementary number-theoretic operations

Number Theory Modular arithmetic operations and sequences of numbers

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 2-D and 3-D plotting of symbolic expressions





SMT Examples





Basic Operations

a=sym('a');

syms x y

Y=(x^a+sqrt(x))/y

pretty(Y)

symvar(Y)

Y =

(x^a + x^(1/2))/y

a

x + sqrt(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

… and other functions





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/factorial(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

gradient Gradient vector of a scalar function

hessian Hessian matrix of a scalar function

jacobian Jacobian matrix

laplacian Laplacian of a scalar function

limit Limit of symbolic expression

symsum Symbolic summation of series

taylor Taylor series expansion


( )

0

( )( ) ( )

!

nn

n

f af x x a

n

=

= −

2(cos( ))d x

dx

2

0

cos ( )x dx

0

sin( )limx

x

x→

0 !

k

k

x

k

=





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(0)=2','D2x(0)=0')

Key Functions

solve Symbolic solution of algebraic equations

dsolve Symbolic solution of ordinary differential equations

syms x(t); D1=diff(x); D2=diff(x,2);

dsolve(diff(x,3)==x-1,[x(0)==0,D1(0)==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

ob

so

lete

syn

tax

syms x y, [A,b]=equationsToMatrix([x+2*y-3,2x+3*y]); z=A\b alternative approach





syms y1(t) y2(t)

% Define system parameters (in kg and kg/s/s)

m1=1; m2=1; k1=3; k2=2;

% Define equations

eqn1 = m1*diff(y1,t,2) == -k1*y1+k2*(y2-y1);

eqn2 = m2*diff(y2,t,2) == -k2*(y2-y1);

% Define initial conditions (in m and m/s)

Dy1 = diff(y1,t); Dy2 = diff(y2,t);

cond = [y1(0)==1, Dy1(0)==-2*sqrt(6),...

y2(0)==2, Dy2(0)==sqrt(6)];

% Find a solution

Sol = dsolve([eqn1 eqn2],cond);

% Plot the results within t=[0;10]s

subplot(221), ezplot(Sol.y1,[0 10]), ylabel('y_1(t), m')

subplot(223), ezplot(Sol.y2,[0 10]), ylabel('y_2(t), m')

subplot(222), ezplot(diff(Sol.y1),[0 10]), ylabel('$$\dot{y}_1(t)$$, m/s','interpreter','latex')

subplot(224), ezplot(diff(Sol.y2),[0 10]), ylabel('$$\dot{y}_2(t)$$, m/s','interpreter','latex')

Dynamic Systems Modeling

m2mm22m1mm11

yy22((tt)) yy11((tt))

kk11kk22

1 1 1 1 2 2 1

2 2 2 2 1

( )

( )

m y k y k y y

m y k y y

= − + −

= − −

A system of differential equations governing an

oscillatory motion of two-body mass–spring system on a

horizontal frictionless surface

1 1

2 2

1 2 6 /

2 6 /

y m y m s

y m y m s

= = −

= =

2 2

1 2 1 21 3 / 2 /m m kg k kg s k kg s= = = =





Linear and Nonlinear Equations and Systems

equationsToMatrix Converts a set of linear equations to the matrix form

finverse Performs a functional inverse

linsolve Solves a linear system of equations given in the matrix form

poles Finds the poles of an expression or function

solve Solves an equation or system of equations symbolically

vpasolve Solves an equation or system of equations numerically

Ordinary Differential Equations and Systems

dsolve Solves an ordinary differential equation or system of equations

massMatrixForm Extracts the mass matrix and right side of semi-linear system of differential algebraic equations (DAEs)

odeFunctionConverts a system of symbolic algebraic expressions to a MATLAB function handle suitable for ode45, ode15s, and

other ODE solvers

odeToVectorField Converts higher-order differential equations to systems of the 1st-order differential equations

Differential Algebraic Equations

daeFunction Converts a system of DAEs to a MATLAB function handle suitable for the ode15i solver

decic Finds consistent initial conditions for the 1st-order implicit ODE system with algebraic constraints

findDecoupledBlocks Searches for decoupled blocks in systems of equations

incidenceMatrix Finds an incidence matrix of system of equations

isLowIndexDAE Checks if the differential index of system of equations is lower than 2

massMatrixForm Extracts the mass matrix and right side of semi-linear system of DAEs

reduceDAEIndex Converts a system of the 1st-order DAEs to the equivalent system of differential index 1

reduceDAEToODE Converts a system of the 1st-order semi-linear DAEs to equivalent system of differential index 0

reduceDifferentialOrder Reduces a system of higher-order differential equations to equivalent system of the 1st-order differential equations

reduceRedundancies Simplifies a system of the 1st-order DAEs by eliminating redundant equations and variables

The Full List of Equation

Solving Functions

For advanced users





Simplification & Substitution

syms x

y=cos(2*x)

simple(y)

children Extracting subexpressions or terms of symbolic expression

coeffs Determining coefficients of multivariate polynomial

collect Collecting coefficients

combine Combining the terms of an identical algebraic structure

compose Performing functional composition

isolate Isolating a variable or expression in equation

expand Symbolic expansion of polynomials and elementary functions

factor Factorization

horner Horner nested polynomial representation

numden Extracting numerator and denominator

lhs, hrs Extracting the left- and right-hand sides of equation

rewrite Rewriting an expression in terms of another function

simplifyFraction Simplifying symbolic rational expressions

simple Search for simplest form of symbolic expression

simplify Symbolic simplification

subexpr Rewriting a symbolic expression in terms of common subexpressions

subs Symbolic substitution

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)

The simple function was removed after R2014b

syms x

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

ans = 2*cos(x)^2 - 1ans = - R^2 + a^2 - 2*cos(phi)*a*rho + rho^2

syms x

[c,t]=coeffs(16*x^2 + 19*x + 11)

c = [ 16, 19, 11]

t = [ x^2, x, 1]

syms y(x)

f=y^2*x+y*x^2+y*sin(x)+x*y;

coefy=collect(f,y)

coefy(x)=x*y(x)^2+(x+sin(x)+x^2)*y(x)

Simplification and Substitution

syms x

sin2tan=rewrite(sin(x),'tan')

sin2tan=(2*tan(x/2))/(tan(x/2)^2+1)

T = [ -x*y, x^2, -y^2]

syms x y

T=children(x^2-x*y-y^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 d

−

=

( ) ( )i t

F f t e dt

−

−

=

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

2

n

z R

f n F z z dz ni

−

=

= =0

( )( )

nn

f nF z

z

=

=

0

( ) ( )st

F s f t e dt

−=

( ) ( )

c i

st

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






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

Easy Visualization Tools &

Calculators

syms x(t) y(t)

g=dsolve([diff(x)==y,diff(y)==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)

Easy Plotting Functions(may be removed in the future releases)

Graphical Calculatorsfuntool Function calculator

rsums Interactive evaluation of Riemann sums

taylortool Taylor series calculator





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 Character &

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





finl =

Inline function:

finl(t) = exp(t)*co(t)

xmin = 3.9270

Current plot held

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 = exp(t)*cos(t)

Error using fcnchk (line 107)

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

Error in fminsearch (line 173)

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

1

2

1

2

3

3

1

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)





The matlabFunction

Function>> 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-2018 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+04 *

0.2837 0.1680 2.5534





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(0.2,0.25,['$' H1 '$'],'interpreter','latex','fontsize',14)

>> text(0.6,0.2, ['$' H2 '$'],'interpreter','latex','fontsize',16)



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





Using odeToVectorField

Function

%% Integrate numerically within 0<t<2s

initConditions=[0,-3,5,-3]; % set initial conditions

ode23(odefun,[0,2],initConditions)

grid, xlabel('Time, s'), ylabel('Solution')

legend('$x(t)$','$\dot{x}(t)$','$\ddot{x}(t)$','$x^{(iii)}(t)$',...

'location','ne','interpreter','latex','fontsize',13)

( )2 10 0

ivy y yy+ − =

odefun =

@(t,Y)[Y(2);Y(3);Y(4);Y(3).*-2.0+Y(1).*Y(2).*1.0e1]

1 2

2 3

3 4

4 3 1 2

( )

( )

( )

2 10 ( 2 10 )

Y Y y

Y Y y

Y Y y

Y Y Y Y y yy

= =

= =

= =

= − + = − +

%% Rewrite equation into state space

[V,S]=odeToVectorField(eqn)

odefun=matlabFunction(V,'vars',{'t','Y'})

%% Define the 4th-order ODE

syms y(t)

eqn=diff(y,4)+2*diff(y,2)-10*y*diff(y);

V = Y[2]

Y[3]

Y[4]

10*Y[1]*Y[2] - 2*Y[3]

S = y

Dy

D2y

D3y

For advanced users





Using odeFunction

Function to Create a Handle

%%

%% Define the 4th-order ODE

syms y(t)

eqn=diff(y,4)+2*diff(y,2)-10*y*diff(y);

%% Rewrite ODE as a system of 1st-order

ODEs

[eqs,vars]=reduceDifferentialOrder(eqn,y)

%% Find the mass matrix M and the right-

hand sides of equations

% $M(t,x(t),\dot{x}(t)=F(t,x(t))$

[M,F]=massMatrixForm(eqs,vars)

%% Rewrite the system in the standard form

% $\dot{x}(t)=f(t,x(t))$

f=M\F

%% Convert to a MATLAB function handle

odefun=odeFunction(f,vars);

For advanced users





Riemann Sums Calculator>> syms x

>> rsums(sin(pi*x))





Function Calculator>> funtool





Taylor Series Calculator>> taylortool

>> taylortool('f')





%%

%% Define units

u=symunit; % loading units collection

d=5*u.fps % using foot per second

w=50*u.lbm % using pound mass

s=10*u.km/u.hr, % using length and time units

s=simplify(d) % simplifying to (default) speed unit

%% Combine the different-unit values

d=500*u.m+0.9*u.mi+70*u.ft+157*u.in

d=simplify(d) % simplifying to (default) single unit

d=separateUnits(d); % extracting expression with no units

d=double(d) % converting symbolic variable to a double-

precision number

%% Unit conversions

len=1*u.mi; % define 1 statute mile

len=unitConvert(len,u.inm) % convert to international nautical miles

len=unitConvert(len,u.nmile) % convert to British imperial nautical

miles

len=unitConvert(len,u.km) % convert to kilometers

[len,un]=separateUnits(len); % separate units from expression

len=double(len) % convert expression to double type

un % show units

%% Convert to SI or U.S. unit system

m=5*u.kg;

a=2*u.m/u.s^2;

F=m*a

F=unitConvert(F,'SI','Derived')

F=unitConvert(F,'US')

F=unitConvert(F,'US','Derived')

%% Use units in differential equations

syms V(t) a v0

T=t*u.s; % time in seconds

A=a*u.m/u.s^2; % acceleration in meters per second

eqn1=A==diff(V,T) % differential equation

cond=V(0)==v0*u.m/u.s; % initial condition

eqn2=V==dsolve(eqn1,cond)

SMT Units of Measurements

For advanced users

Introduced in R2017a





Unit Manipulations

checkUnits Check for compatible dimensions and consistent units

findUnits Find units in input

isUnit Determine if input is a symbolic unit

mixedUnits Split unit into sum of units

newUnit Define new unit

removeUnit Remove unit

rewrite Rewrite expression in terms of another function

separateUnits Separate units from expression

simplify Algebraic simplification

str2symunit Convert character vector or string to unit

symunit Units of measurement

symunit2str Convert unit to character vector

unitConversionFactor Conversion factor between units

unitConvert Convert units to other units of measurement

unitInfo Information on units of measurement

baseUnits Base units of unit system

derivedUnits Derived units of unit system

newUnitSystem Define unit system

removeUnitSystem Remove unit system

unitSystems List available unit systems

Unit Systems

Define, Create, Convert Units

For advanced users





The End of Chapter 7

Questions?

chapter 7. symbolic math toolbox

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