An introduction to MATLAB

ByS. Ziaei Rad

About MATLABMATLAB is an interactive softwareMATLAB is easy to learn and easy to develop a program inside it.Numerical calculation in MATLAB uses collections of well written scientific/mathematical subroutines such as LINPACK and EISPACK.In general, MATLAB is a useful tool for vector and matrix manipulations.The FEM method is a well defined candidate for which MATLAB can be very useful as a solution tool.

Vector and Matrix Manipulations

Once we get into Matlab, we meet a prompt >> called Matlabprompt.

Define a matrix/vector Transpose of a matrix

>> A‘

ans =

1 2 03 7 36 8 9

>> A=[1 3 6; 2 7 8; 0 3 9]

A =

1 3 6

2 7 8

0 3 9

>> size(A)

ans =

3 3

Vector and Matrix Manipulations

Column or row components>> A(:,3)ans =

689

>> A(1,:)

ans =

1 3 6We can try

>> A(1,:)+A(3,:)

ans =

1 6 15

Matrix AdditionLet’s define another matrix>> B=[3 4 5; 6 7 2; 8 1 10];>> BB =

3 4 56 7 28 1 10

>> C=A+B

C =4 7 118 14 108 4 19

Matrix Subtraction>> C=A-B

C =-2 -1 1-4 0 6-8 2 -1

Vector and Matrix Manipulations

Matrix Multiplication>> C=A*B

C =69 31 71

112 65 10490 30 96

>> C=A*B‘;>> C=A’*B;Matrix Function>> inv(A)

ans =1.8571 -0.4286 -0.8571

-0.8571 0.4286 0.19050.2857 -0.1429 0.0476

>> A*inv(A)

ans =1.0000 0.0000 00.0000 1.0000 0

0 0 1.0000

Basic matrix Functions

Symbol Explanation----------------------------------inv Inverse of a matrixdet Determinate of a matrixrank Rank of a matrixcond Condition number of a matrixeye(n) The n by n identity matrixtrace Summation of diagonal

elements of a matrixzeros(n,m) The n by m matrix

consistings of all zerosones(n,m) The n by m matrix

consistings of all onesExamples:inv(A);det(A);rank(A);eye(5);ones(6,7);

Basic Matrix Function

Matrix of random number

>> A=rand(3,3)

A =

0.3529 0.1389 0.60380.8132 0.2028 0.27220.0099 0.1987 0.1988

Matrix exponential

>> expm(A)

ans =

1.5267 0.2680 0.84721.1116 1.3533 0.66510.1191 0.2521 1.2783

EigenvalusThe eigenvalus problem of a matrix is defined

>> A=[5 3 2; 1 4 6; 9 7 2]

A =5 3 21 4 69 7 2

>> e=eig(A)

e =

12.53611.7486

-3.2847

λφφ =A

Basic Matrix Function

Eigenvalues and Eignvectors

>> [V,D]=eig(A)

V =

-0.4127 -0.5992 0.0459-0.5557 0.7773 -0.6388-0.7217 -0.1918 0.7680

D =

12.5361 0 00 1.7486 00 0 -3.2847

LU decomposition

>> A=[1 3 5; 2 4 8; 4 7 3];>> [L,U]=lu(A)

L =0.2500 1.0000 00.5000 0.4000 1.00001.0000 0 0

U =4.0000 7.0000 3.0000

0 1.2500 4.25000 0 4.8000

>> L*Uans =

1 3 52 4 84 7 3

Basic Matrix Function

QR Decomposition

Q is a matrix with orthonormalColumn and R is the upper triangularMatrix.>>A=[1 3 5; 2 4 8; 4 7 3];>> [Q,R]=qr(A)

Q =-0.2182 0.9117 0.3482-0.4364 0.2279 -0.8704-0.8729 -0.3419 0.3482

R =-4.5826 -8.5105 -7.2012

0 1.2536 5.35610 0 -4.1779

Singular Value Decomposition (SVD)

>> D=[1 3 7; 2 9 5; 2 8 5];>> [U,sigma,V]=svd(D)

U =-0.4295 0.8998 -0.0775-0.6629 -0.3723 -0.6495-0.6133 -0.2276 0.7564

sigma =15.6492 0 0

0 4.1333 00 0 0.1391

V =-0.1905 -0.0726 0.9790-0.7771 -0.5982 -0.1956-0.5999 0.7980 -0.0576

'VUA Σ= QRA =

Basic Matrix Function

Solution of linear equations

Ax=y

>> A=[1 3 5; 2 4 8; 4 7 3];>> A=[1 3 4; 5 7 8; 2 3 5];>> y=[10; 9; 8];>> x=inv(A)*y

x =

-4.25001.75002.2500

OrAx=y

>> x=A\y

x =

-4.25001.75002.2500

Data Analysis FunctionsSymbol Explanations-------------------------------------------------------------

min (max) minimum (maximum) of a vectorMIN(X) is a row vector containing the minimum element from each[Y,I] = MIN(X) returns the indices of the minimum values in vector I.

sum sum of elements of a vectorstd standard deviation of a data collectionsort sort of element of a vector

[Y,I] = SORT(X) also returns an index matrix I. If X is a vector, then Y = X(I).mean mean value of a vector

Vector componentwise operation>> v1=[1 5 6 7]; v2=[0 2 3 5];>> v3=v1.*v2

v3 =0 10 18 35

>> v4=v2./v1

v4 =0 0.4000 0.5000 0.7143

Polynomial FunctionsSymbol Explanations-------------------------------------------------------------poly convert collection of roots into a polynomial equationroots finds the roots of a polynomial equationpolyval evaluate a polynomial for a given valueconv multiply two polynomialdeconv decompose a polynomial into a dividend and a residualpolyfit curve fitting of a given polynomial

Example:If C has N+1 components, the polynomial is C(1)*X^N + ... + C(N)*X + C(N+1).>> C=[1 15 136 498 968 592];>> roots(C)ans =

-5.0000 + 7.0000i-5.0000 - 7.0000i-2.0000 + 2.0000i-2.0000 - 2.0000i-1.0000

Making Complex NumberIn order to make a complex number For example 2+3*I>> 2+3*i

ans =2.0000 + 3.0000i

Or>> 2+3*jans =

2.0000 + 3.0000iNote: Matlab takes i and j as a pure complex number>> i=sqrt(-1)

i = 0 + 1.0000i>> k=sqrt(-1)

k =0 + 1.0000i

Commands for complex numberabs the magnitude of a numberangle the phase anglereal the real part of a complex numberimag the imaginary partconj the complex conjugate

>> c=-1+i

c =-1.0000 + 1.0000i

>> [ abs(c) angle(c) real(c) imag(c)]

ans =1.4142 2.3562 -1.0000 1.0000

>> conj(c)

ans =-1.0000 - 1.0000i

Some Numerical Techniquesfminbnd find minimum of a function of one variablefzero solves a nonlinear algebric equation of one variable

>> fminbnd('x*cos(x)',-2,2)

ans =-0.8603

>> x=fzero('tan(x)',2)

x =1.5708

ode23 solution using the 2nd/3rd order Runge-Kutta algorithmode45 solution using the 4th/5th order Runge-Kutta algorithm

[t,y]=ode45(‘func’,[t0 tf],[x0,v0]);>> [t,y]=ode45('vdp1',[0 20],[2 0]); >> [size(t) size(y)]ans =

237 1 237 2

Plotting ToolsAssume that t , y are the solution of ODE of previous slide>> plot(t,y) >> xlabel('Time (s)')>> plot(t,y(:,1)) >> ylabel('Displacement and velocity')>> plot(t,y(:,1),t,y(:,2)) >> legend('Displacement','Velocity')

Loop and logical statement Symbol Explanations-------------------------------------------------------------for loop command similar to other languagewhile used for a loop combined with conditional statementif produce a conditional statementelseif, else used in conjugate with if commandbreak breaks a loop when a condition is satisfied

== two conditions are equal~= two conditions are not equal<=(>=) one is less (greater) than the other

& and operator| or operator~ not operatorA(i,j) element i and j of matrix AV(i) element I of vector V A(:,i) all elements of matrix A in column i A(i,:) all elements of matrix A in row i

Writing Function SubroutineFunction(ov1,ov2,…]=func1(iv1,iv2,…)iv1, iv2, … are input variablesov1, ov2, … are output variablesthe file then will save as func1.m and can be called inside Matlab by >> [ov1,ov2, …]=func1(iv1,iv2,…)

Example: ax^2+b*x+c=0Function[r1,r2]=secroot(a,b,c)Det=b^2-4*a*cif (Det<0)R1=(-b+j*sqrt(-Det))/2/a; R2=(-b-j*sqrt(-Det))/2/a;elseif(Det==0)R1=-b/2/a; R2=-b/2/a;else(Det>0)R1=(-b+sqrt(Det))/2/a; R2=(-b-sqrt(Det))/2/a;end

Writing Function Subroutine

Then save the function as secroot.mIn order to call it inside Matlab:

>> [r1,r2]=secroot(3,4,5)r1=

-0.6667 + 1.1055ir2=

-0.6667 – 1.1055i

>> a=3; b=0; c=-5;>> [p1,p2]=secroot(a,b,c)

p1=1.2910

p2=-1.2910