MATLABExamples

Hans-PetterHalvorsen,M.Sc.

Mathematics

MathematicswithMATLAB

• MATLABisapowerfultoolformathematicalcalculations.• Type“helpelfun”(elementarymathfunctions)intheCommandwindowformoreinformationaboutbasicmathematicalfunctions.

MathematicsTopics

• BasicMathFunctionsandExpressions𝑧 = 3𝑥% + 𝑥% + 𝑦%� + 𝑒+,(.)

• Statistics– mean,median,standarddeviation,minimum,maximumandvariance

• TrigonometricFunctionssin() , cos() , tan()

• ComplexNumbers𝑧 = 𝑎 + 𝑗𝑏

• Polynomials𝑝 𝑥 = 𝑝<𝑥= + 𝑝%𝑥=>< + ⋯+ 𝑝=𝑥 +𝑝=@<

BasicMathFunctions

Createafunctionthatcalculatesthefollowingmathematicalexpression:

𝑧 = 3𝑥% + 𝑥% + 𝑦%� + 𝑒+,(.)

Wewilltestwithdifferentvaluesfor𝑥 and𝑦

function z=calcexpression(x,y)

z=3*x^2 + sqrt(x^2+y^2)+exp(log(x));

>> x=2;>> y=2;>> calcexpression(x,y)ans =

16.8284

Wecreatethefunction:

Testingthefunctiongives:

StatisticsFunctions

• MATLABhaslotsofbuilt-infunctionsforStatistics• Createavectorwithrandomnumbersbetween0and100.Findthefollowingstatistics:mean,median,standarddeviation,minimum,maximumandthevariance.

>> x=rand(100,1)*100;

>> mean(x)>> median(x)>> std(x)>> mean(x)>> min(x)>> max(x)>> var(x)

Trigonometricfunctions

sin(𝑥) cos(𝑥) tan(𝑥)

TrigonometricfunctionsItisquiteeasytoconvertfromradianstodegreesorfromdegreestoradians.Wehavethat:

2𝜋𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 360𝑑𝑒𝑔𝑟𝑒𝑠𝑠Thisgives:

𝑑 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 = 𝑟[𝑟𝑎𝑑𝑖𝑎𝑛𝑠] M180𝜋

𝑟[𝑟𝑎𝑑𝑖𝑎𝑛𝑠] = 𝑑[𝑑𝑒𝑔𝑟𝑒𝑒𝑠] M𝜋180

→Createtwofunctionsthatconvertfromradianstodegrees(r2d(x))andfromdegreestoradians(d2r(x))respectively.Testthefunctionstomakesurethattheyworkasexpected.

function d = r2d(r)

d=r*180/pi;

Thefunctionsareasfollows:

function r = d2r(d)

r=d*pi/180;

>> r2d(2*pi)ans =

360>> d2r(180)ans =

3.1416

Testingthefunctions:

Trigonometricfunctions

Givenrighttriangle:

• CreateafunctionthatfindstheangleA(indegrees)basedoninputarguments(a,c),(b,c)and(a,b)respectively.

• Use,e.g.,athirdinput“type”todefinethedifferenttypesabove.• Useyourpreviousfunctionr2d() tomakesuretheoutputofyourfunctionisindegreesandnotinradians.

• Testthefunctionstomakesureitworksproperly.

Trigonometricfunctions

Wehavethat:

sin 𝐴 =𝑎𝑐 , 𝐴 = 𝑎𝑟𝑐𝑠𝑖𝑛

𝑎𝑐

cos 𝐴 =𝑏𝑐 , 𝐴 = 𝑎𝑟𝑐𝑐𝑜𝑠

𝑏𝑐

tan𝐴 =𝑎𝑏 , 𝐴 = 𝑎𝑟𝑐𝑡𝑎𝑛

𝑎𝑏

ThePythagoras'theorem:𝑐% = 𝑎% + 𝑏%

function angleA = right_triangle(x,y, type)

switch typecase 'sin'

angleA=asin(x/y);case 'cos'

angleA=acos(x/y);case 'tan'

angleA=atan(x/y);end

% Convert from radians to degreesangleA = r2d(angleA);

Thefunctionbecomesasfollows:

Testingthefunction:>> a=5a =

5>> b=8b =

8>> c=sqrt(a^2+b^2)c =

9.4340>> right_triangle(a,c,'sin')ans =

32.0054>> right_triangle(b,c,'cos')ans =

32.0054>> right_triangle(a,b,'tan')ans =

32.0054

Lawofcosines

Given:

Createafunctionwhereyoufindcusingthelawofcosines.

𝑐% = 𝑎% + 𝑏% − 2𝑎𝑏𝑐𝑜𝑠𝐶

function c = law_of_cosines(a,b,C)

c = sqrt(a^2 + b^2 - 2*a*b*cos(C));

Thefunctionbecomesasfollows:

Testingthefunction:>> a=2;, b=3;, C=pi;,

>>law_of_cosines(a,b,C)ans =

5

PlottingTrigometricfunctions

• Plot𝑠𝑖𝑛(𝜃)and𝑐𝑜𝑠(𝜃) for0 ≤ 𝜃 ≤ 2𝜋 inthesameplot.• Makesuretoaddlabelsandalegend,andusedifferentlinestylesandcolorsfortheplots.

clf

x=0:0.01:2*pi;

plot(x, sin(x), 'c+:')hold on

plot(x, cos(x), 'r:')hold off

legend('sin', 'cos')xlabel('x')ylabel('f(x)')

% Define x-valuesx=0:0.01:2*pi;

% subplot 1subplot(2,1,1)plot(x, sin(x))title('Plotting sin(x)')xlabel('x')ylabel('sin(x)')

% Subplot 2subplot(2,1,2)plot(x, cos(x))title('Plotting cos(x)')xlabel('x')ylabel('cos(x)')

OrwecanuseSubplots:

clf

x=0:0.01:2*pi;

plot(x, sin(x), x, cos(x))

AComplexNumberisgivenby:

𝑧 = 𝑎 + 𝑗𝑏Where

𝑗 = −1�

Wehavethat:𝑎 = 𝑅𝑒 𝑧

𝑏 = 𝐼𝑚(𝑧)

ComplexNumbers

𝑧 = 𝑎 + 𝑗𝑏

0 𝑎

𝑏

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦𝐴𝑥𝑖𝑠(𝐼𝑚)

𝑅𝑒𝑎𝑙𝐴𝑥𝑖𝑠(𝑅𝑒)

ComplexNumbersPolarform:

𝑧 = 𝑟𝑒\]

Where:𝑟 = 𝑧 = 𝑎%+𝑏%�

𝜃 = 𝑎𝑡𝑎𝑛𝑏𝑎

Note!

𝑎 = 𝑟 cos 𝜃

𝑏 = 𝑟 sin 𝜃

𝑧 = 𝑟𝑒\]

0 𝑎

𝑏

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦𝐴𝑥𝑖𝑠(𝐼𝑚)

𝑅𝑒𝑎𝑙𝐴𝑥𝑖𝑠(𝑅𝑒)

𝜃𝑟

𝑗 = −1�

ComplexNumbersRectangularform ofacomplexnumber Exponential/polarform ofacomplexnumber

Length(“Gain”): Angle(“Phase”):

1 2

𝑧 = 𝑎 + 𝑗𝑏

0 𝑎

𝑏

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦𝐴𝑥𝑖𝑠(𝐼𝑚)

𝑅𝑒𝑎𝑙𝐴𝑥𝑖𝑠(𝑅𝑒)

𝑧 = 𝑟𝑒\]

0 𝑎

𝑏

𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦𝐴𝑥𝑖𝑠(𝐼𝑚)

𝑅𝑒𝑎𝑙𝐴𝑥𝑖𝑠(𝑅𝑒)

𝜃𝑟

𝑟 = 𝑧 = 𝑎% + 𝑏%� 𝜃 = 𝑎𝑡𝑎𝑛𝑏𝑎

𝑗 = −1�

Rectangularform→Exponential/polarform

Giventhecomplexnumbers(Rectangularform):𝑧 = 𝑎 + 𝑗𝑏

Exponential/polarform:𝑧 = 𝑟𝑒\]

Where𝑟 = 𝑧 = 𝑎% + 𝑏%�

𝜃 = 𝑎𝑡𝑎𝑛𝑏𝑎

z = a + bjr = sqrt(a^2 + b^2)

z = a + bjtheta = atan(b/a)

z = a + bjr = abs(z)

z = a + bjtheta = angle(z)

or:

or:

Rectangularform→Exponential/polarformclearclc

a = 5;b = 3; % Rectangular Form:z = a + b*i % Polar Form:r = sqrt(a^2 + b^2)r = abs(z) theta = atan(b/a)theta = angle(z)

z = r*exp(j*theta)

𝑧 = 5 + 𝑗3

𝑧 = 5 + 𝑗3

𝑧 = 𝑟𝑒\] = 5.83𝑒\`.ab

ComplexNumbers

Toaddorsubtracttwocomplexnumbers,wesimplyadd(orsubtract)theirrealpartsandtheirimaginaryparts.Giventhecomplexnumbers:

𝑧< = 𝑎< + 𝑗𝑏<and𝑧% = 𝑎% + 𝑗𝑏%Addition:

𝑧c = 𝑧< + 𝑧% = 𝑎< + 𝑎% + 𝑗 𝑏< +𝑏%Subtraction:

𝑧c = 𝑧< −𝑧% = 𝑎< − 𝑎% + 𝑗 𝑏< −𝑏%

ComplexNumbers

InDivisionandmultiplication,weusethepolarform.

Giventhecomplexnumbers:𝑧< = 𝑟<𝑒\]d and𝑧% = 𝑟%𝑒\]e

Multiplication:

𝑧c = 𝑧<𝑧% = 𝑟<𝑟%𝑒\(]d@]e)Division:

𝑧c =𝑧<𝑧%=𝑟<𝑒\]d𝑟%𝑒\]e

=𝑟<𝑟%𝑒\(]d>]e)

ComplexNumbers– MATLABFunctionsFunction Description Example

i,j Imaginaryunit.AsthebasicimaginaryunitSQRT(-1),iandjareusedtoentercomplexnumbers.Forexample,theexpressions3+2i,3+2*i,3+2j,3+2*jand3+2*sqrt(-1)allhavethesamevalue.

>>z=2+4i

>>z=2+4j

abs abs(x)istheabsolutevalueoftheelementsofx.Whenxiscomplex,abs(x)isthecomplexmodulus(magnitude)oftheelementsofX.

>>z=2+4i

>>abs(z)

angle Phaseangle.angle(z)returnsthephaseangles,inradians >>z=2+4i

>>angle(z)

imag Compleximaginarypart.imag(z)istheimaginarypartofz. >>z=2+4i

>>b=imag(z)

real Complexrealpart.real(z)istherealpartofz. >>z=2+4i

>>a=real(z)

conj Complexconjugate.conj(x)isthecomplexconjugateofx. >>z=2+4i

>>z_con=conj(z)

complex Constructcomplexresultfromrealandimaginaryparts.c=complex(a,b)returnsthecomplexresultA+Bi

>>a=2;

>>b=3;

>>z=complex(a,b)

ComplexNumbers

Giventwocomplexnumbers𝑐 = 4 + 𝑗3, 𝑑 = 1 − 𝑗

FindtherealandimaginarypartofcanddinMATLAB.→UseMATLABtofind𝑐 + 𝑑, 𝑐 − 𝑑, 𝑐𝑑𝑎𝑛𝑑𝑐/𝑑.UsethedirectmethodsupportedbyMATLABandthespecificcomplexfunctionsabs,angle,imag,real,conj,complex,etc.togetherwiththeformulasforcomplexnumbers.→Findalso𝑟 and𝜃.Findalsothecomplexconjugate.

c=5+3i;d=1-i;

disp('c+d')%Directly-----------------z = c + d%Manually-----------------z_real = real(c) + real(d);z_imag = imag(c) + imag(d);z = complex(z_real,z_imag)

% r and angle + complex conungater=abs(z)theta=angle(z)complconj=conj(z)

disp('c-d')%Directly--------------z = c - d%Manually--------------z_real = real(c) - real(d);z_imag = imag(c) - imag(d);z = complex(z_real,z_imag)%or: z = z_real + z_imag*i

disp('c*d')%Directly-------------z = c*d%Manually-------------z_abs = abs(c)*abs(d);z_angle = angle(c) + angle(d);z_real = z_abs*cos(z_angle);z_imag = z_abs*sin(z_angle);z = complex(z_real,z_imag)

disp('c/d')%Directly-------------z = c/d%Manually-------------z_abs = abs(c)/abs(d);z_angle = angle(c) - angle(d);z_real = z_abs*cos(z_angle);z_imag = z_abs*sin(z_angle);z = complex(z_real,z_imag)

c+dz =

6.0000 + 2.0000iz =

6.0000 + 2.0000ir =

6.3246theta =

0.3218complconj =

6.0000 - 2.0000ic-dz =

4.0000 + 4.0000iz =

4.0000 + 4.0000ic*dz =

8.0000 - 2.0000iz =

8.0000 - 2.0000ic/dz =

1.0000 + 4.0000iz =

1.0000 + 4.0000i

Thisgives:

ComplexRoots

Findtherootsoftheequation:𝑥% + 4𝑥 + 13

Therootsaregivenby:𝑥% + 4𝑥 + 13 = 0

FindalsotheSumandDifferenceoftheroots.

a=1;b=4;c=13;

solveeq(a,b,c)

Wecane.g.,usethesolveeq functionwecreatedinapreviousexample:function x = solveeq(a,b,c)if a~=0

x = zeros(2,1);x(1,1)=(-b+sqrt(b^2-4*a*c))/(2*a);x(2,1)=(-b-sqrt(b^2-4*a*c))/(2*a);

elseif b~=0x=-c/b;

elseif c~=0disp('No solution')

elsedisp('Any complex number is a solution')

end

a=1;b=4;c=13;

p=[a,b,c]roots(p)

Orwecanusetheroot function:

Note!thesolutioniscomplexconjugate.

Thesumoftwocomplexconjugatenumbersisalwaysreal.Inourcase:ans =

-4

Whilethedifferenceisimaginary(norealpart).ans =

0 + 6.0000i

Polynomials

Apolynomialisexpressedas:𝑝 𝑥 = 𝑝<𝑥= + 𝑝%𝑥=>< + ⋯+ 𝑝=𝑥 +𝑝=@<

where𝑝<, 𝑝%, 𝑝c, … arethecoefficientsofthepolynomial.

Exampleofpolynomial:𝑝 𝑥 = −5.45𝑥b + 3.2𝑥% + 8𝑥 + 5.6

PolynomialsinMATLAB

MATLABrepresentspolynomialsasrowarrayscontainingcoefficientsorderedbydescendingpowers.Example:

𝑝 𝑥 = −5.45𝑥b + 3.2𝑥% + 8𝑥 + 5.6

>> p=[-5.45 0 3.2 8 5.8]p =

-5.4500 0 3.2000 8.0000 5.8000

MATLABofferslotsoffunctionsonpolynomials,suchasconv,roots,deconv,polyval,polyint,polyder,polyfit,etc.→YoushouldlookupthesefunctionsintheHelpsysteminMATLAB.

Polynomials

DefinethefollowingpolynomialinMATLAB:

𝑝 𝑥 = −2.1𝑥b + 2𝑥c + 5𝑥 + 11

→Findtherootsofthepolynomial(𝑝 𝑥 = 0)

→Find𝑝 𝑥 = 2

Usethepolynomialfunctionslistedabove.

p = [-2.1, 2, 0, 5, 11]

roots(p)

x = 2;polyval(p,x)

MATLABCode:

p =-2.1000 2.0000 0 5.0000 11.0000

ans =2.0820

-0.0199 + 1.5193i-0.0199 - 1.5193i-1.0898

ans =3.4000

Thisgives:

𝑝 𝑥 = 2 𝑝 𝑥 = 0

Insteadofusingthepolyval()functionwecouldofcoursealsofoundtheanswerlikethis:x = 2;p = -2.1*x^4 + 2*x^3+5*x+11

Wecanusee.g.,thepolyval()functiontocheckiftheanswersarecorrect:>> x = 2.0820;>> polyval(p,x)etc.

Theanswersshallthenofcoursebe0 (oratleastaverysmallnumber).

Polynomials

Giventhefollowingpolynomials:𝑝< 𝑥 = 1 + 𝑥 − 𝑥%𝑝% 𝑥 = 2 + 𝑥c

→Findthepolynomial𝑝(𝑥) = 𝑝<(𝑥) M 𝑝%(𝑥) usingMATLABandfindtheroots→Findtherootsofthepolynomial(𝑝 𝑥 = 0)→Find𝑝 𝑥 = 2→Findthedifferentiation/derivativeof𝑝% 𝑥 ,i.e.,𝑝%i

Wewillusethepolynomialfunctionslistedabove.

Note!Thepolynomialsmayberewrittenas:

𝑝< 𝑥 = −𝑥% + 𝑥 + 1𝑝% 𝑥 = 𝑥c + 0𝑥% + 0𝑥 + 2

P1 = [-1, 1, 1];P2 = [1, 0, 0, 2];

p = conv(p1,p2)

r = roots(p)

polyval(p,2)

TheMATLABcodebecomes:Thisgives:

p =-1 1 1 -2 2 2

r =1.6180 0.6300 + 1.0911i0.6300 - 1.0911i-1.2599 -0.6180

ans =-10

ThePolynomialbecomes:

𝑝 𝑥 = −𝑥a + 𝑥b + 𝑥c − 2𝑥% + 2𝑥 + 2

PolynomialFitting

Findthe6.orderPolynomialthatbestfitsthefollowingfunction:𝑦 = sin(𝑥)

→Plotboththefunctionandthe6.orderPolynomialtocomparetheresults.

x=0:0.1:2*pi;y=sin(x);

figure(1)plot(x,y)

% Finding a 6. order polynomialp=polyfit(x,y,6)

y2=polyval(p,x);

figure(2)plot(x,y2)

Hans-PetterHalvorsen,M.Sc.

UniversityCollegeofSoutheastNorwaywww.usn.no

E-mail:hans.p.halvorsen@hit.noBlog:http://home.hit.no/~hansha/