[PDF] Introduction to MATLAB - Numerical Mathematics (TMA4215)

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IntroductiontoMATLAB

NumericalMathematics(TMA4215)

EirikHoelHøiseth

21August, 2014

www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 2

WhatisMA TLAB?

- Highle velprogramminglanguage. - Simpletouse ,but significantlyslowerthan lowerlevel languageslike C++andJava. - Initiallya toolfor matrixcomputations (MATrixLABorator y), thatev olvedintoaplatformforscientificcomputing. - Hugescientificlibr ary. - Expensivelicense. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 3

Numbers

MATLABsupportsall theusualnumbers:

- Integers - Realnumbers - Complexnumbers Canmixintegers ,complex numbers,andreal numbersaswelik e.

Example:

>>3.47*4-(5+4i) ans=

8.8800-4.0000i

Note:Bothiandjcanbeused forthe imaginaryunit.

www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 4

Storageofnumbers

- Bydefault anintegeristreatedandstored asareal number. - Realnumbers are,bydef ault,storedwithdoubleprecision (roughly16digits accuracy). - Acomplex numberisstoredas2 realnumbers(realand imaginarypart). www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 5

Variables

Definition:V ariable

Avariableisareferencetoanobject. Theassignmentoper ator,=, isused toassigna valueto avar iable. Note:Variablesneednotbedeclaredbeforethe yareused.

Example:Assignment

>>x= 2; >>y= x%xandyrefertodifferentobjects y= 2 www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 6

Arrays

Definition:Arr ays

Anarrayisadatastructure .Itconsists ofacollection ofelements. Inanarr ayof dimensionnthepositionof anelement inmemory is givenbyan orderedsetofnindices.

Note:InMATLAB allvariablesref ertoarr ays.

www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 7

Definition:Scalars

Definition:Scalar

AscalarinMATLAB isa0-dimensionalarr ayholdingasingle value.

Example:Creatingn umerical scalars

>>x= 3.4; >>y= 5+4i; www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 8

Vectors

Definition:V ector

AvectorinMATLAB isa1-dimensionaln umericalarray.

Arbitraryvectorscanbecreatedusing thesquarebrac ket[]

Wedistinguishbetweentw otypes:

- Rowvector:Elementssepar atedby acommaorspace - Columnvector :Elementssepar atedby asemicolon. Note:Acolumnv ectorisoften madebytransposingarow vector.

Example:Creatingv ectors

>>x= [3 ,4, -1];%Rowvectoroflength3 >>y= [3+i;2-2i];%Columnvectoroflength2; www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 9

GeneratingEqually spacedvectors

Thecolonoperator ,:,isuseful forcreating equallyspacedreal vectors: - a:bforb!acreates[a,a+1,...,a+m],wherem="b#a$. - a:d:bforrealdcreates[a,a+d,...,a+md],where m="(b#a)/d$,assuming(b#a)/d!0. Analternativ eisthebuiltinMATLAB functionlinspace: - linspace(a,b,n)createsaro wvector ofnequallyspaced pointsbetween aandb. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 10

Matrices

Definition:Matrix

AmatrixinMATLAB isa2-dimensionaln umericalarray.

Matricescanbecreatedsimilar lytov ectorsusingsquare brack ets: - Elementsina roware separatedb yacommaorspace - Rowsareseparatedb yasemicolon. Note:Vectorsandnumeric scalarscanbe thoughtofasmatrices.

Example:Creating matrices

>>A= [3 21; 47-1];%Matrixhas2rowsand3columns www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 11

Generatingcommonmatrices

BuiltinMA TLABfunctionsused togeneratesomefrequentlyused matrices: - zeros(n,m)generatesann%mmatrixofzeros. - ones(n,m)generatesann%mmatrixofones. - eye(n)generatesann%nidentitymatrix. - rand(n,m)generatesann%mrandommatrix. Note:Onlyneedto statethelength ofonedimension ifthematr ix issquare, e.g.zeros(n). www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 12

Basicmatrixoperations

Themostcommon operationsare:

- +and#additionandsubtraction. matrixmultiplication. - ˆmatrixpowerforsquarematrices. - /and\matrixrightandleftdivision x.Similarly b/Asolvesthelinearsystem xA=bforx. •Preferabletomultiplyingbytheinv erse,e .g.(Aˆ(-1))*band b*(Aˆ(-1))respectively. - Puttingaperiodbeforetheoperatorgiv esthecorresponding elementwiseoperations:. ,.ˆ,./and.\. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 13

Usefulvectoroperations

Foravectorv

- sum(v)andprod(v)computesthesumandproduct respectivelyoftheelementsin v. - length(v)givesthelengthofv. - max(v)andmin(v)findsthemaximumandminimumofthe elementsinv. - normcomputesnorms ,e.g.norm(v)forthevector2-nor m. - diff(v)computesav ectorofdiff erencesofneighbouring elementsinv. - sort(v)sortstheelementsin vinascendingorder .

Note:Mostofthese arealsoapplicab letomatr ices.

www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 14

Usefulmatrixoperations

ForamatrixA

- A'andA.'givetheconjugatetranspose andtranspose respectivelyofA. - diagcanbeused togenerate diagonalmatrices andextr act diagonalsfrommatr ices. - size(A)givesthesizeof A. - eig(A)computestheeigenvaluesandeigenvectorsofA - cond(A)computestheconditionnumber ofA Note:Manyothersexist,and thelistedones oftenhaveadditional optionalf eatures. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 15

Indexing

Definition:Inde xing

Indexingintoanarr ayis ameansofselectingasubset ofthe elements. MATLABusesone-basedindexing, whichmeans theindicesstar t at1. Note:Indexingisanimportant programming toolinMA TLAB. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 16

Indexing:Vectors

- Togetthen-thelementinav ectorb,write b(n) - Letabeav ectorcontainingv alidindicesofthevector b.Then b(a)andahavethesamelength,andelementiofthev ector b(a)iselementa(i)ofb.Specialcases: •b(1:n)givesthenfirstelementsofb. •b(m:n)withm&nisthesub vectorfrom b(m)andupto b(n).

Example:Indexing vectors

>>x= [1 020 30];%Vectortobeindexed >>x( [3,1])%Newvector ans= 3010
www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 17

Indexing:Matrices

Definition:Subscr iptindexing

Withsubscriptindexingformatrices,w egiveanindex(orv ectorof indices)for eachofthetwodimensions ,separated bycomma, insideparentheses. Thefirstindexrefersto therows,andthe secondtothecolumns.

Note:Thisisthe commonwa yofinde xingmatrices .

Somespesificcases:

- Theelementin them-thro wandn-thcolumn ofamatr ixAis indexedasA(m,n). - :isshorthand for1:end,andcan beused todenoteall rows orcolumns. E.g.A(1:n,:)extractsthefirstnrowsofA. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 18

Indexing:Matrices

Definition:Linear indexing

Withlinearindexing weindexasif thematrixwasa vector. The matrixbeingindex edistreated asifitselementsarestrungoutin a longcolumnvector . Note:Theresultingsub vectorhas thesameformastheindex vector. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 19

Example:Indexing

>>A= [13 5;7 911] A= 135
7911
>>A( 2,2:3)%Subscriptindexing ans= 911
>>A( 3:5)%Linearindexing ans= 395
www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 20

Charactersandstrings

Definition:Char acter

Acharacterisaninter pretednonnegative integer.

Acharacter iswrittenbyenclosing itin"single" quotationmarks, e.g.'a'.

Definition:Str ing

Astringisav ectorofchar acters.

Astring iswritteninthesame way asasinglecharacter, e.g. 'Thisis astring',andcan beindex edjustlik eanor malvector. www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 21

Logicals

Definition:Logical

Alogicalisa datatypewhich canonlyha vethe valuetrueor false. - true(logical1)and false(logical0)e valuateto 1and0 respectivelywhenusedincomputations . - Anyrealnonzeron umberbecomes truewhile0becomes false,whencon verted toorusedasalogical.

Logicaloperators:

- &(and)and|(or)combinelogical values. - '(not)gives thelogicalnegationofthelogical valuethat follows.

Note:Alsowor kelementwisewithmatrices.

www.ntnu.noEirikHoelHøiseth, IntroductiontoMA TLAB 22

Logicalexpression

Logicalsfrequentlyar iseasthe resultofalogicalexpression

Definition:Logicale xpression

Alogicalexpression isane xpressionthatcan onlyevaluatetotr uequotesdbs_dbs14.pdfusesText_20

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