Post on 18-May-2017
transcript
Exp. No. 1
Basics of MATLAB
>>a1=sin(cos(exp(log(25))))+100*(55/7-1000*(tan(.23)))-100
a1 =
-2.2728e+004
>> a2=(3^4*log(126))/(7^3+115)+sqrt(920)+273^(1/3)+atan(pi/3)-pi
a2 =
35.3408
Exp. No. 2
Basic mathematical operations on matrices.
a =
22 70 57
30 93 81
38 11 10
>> b=[22 30 38;70 93 11;57 81 10]
b =
22 30 38
70 93 11
57 81 10
>> a1=a+b
a1 =
44 100 95
100 186 92
95 92 20
>> a1=a-b
a1 =
0 40 19
-40 0 70
-19 -70 0
>> a1=a*b
a1 =
8633 11787 2176
11787 16110 2973
2176 2973 1665
>> a1=a/b
a1 =
1.3224 -6.2418 7.5409
1.9125 -8.1537 9.8014
0.3443 6.5751 -7.5409
>> a1=a\b
a1 =
1.3224 1.9125 0.3443
-6.2418 -8.1537 6.5751
7.5409 9.8014 -7.5409
Exp. No.3
Transpose, Determinant and Inverse of a matrix
a =
10 -2 -1 -1
-2 10 -1 -1
-1 -1 10 -2
-1 -1 -2 10
>> a1=a'
a1 =
>> a2=det(a)
a2 =
8640
>> a3=inv(a)
a3 =
0.1083 0.0250 0.0167 0.0167
0.0250 0.1083 0.0167 0.0167
0.0167 0.0167 0.1083 0.0250
0.0167 0.0167 0.0250 0.1083
Exp. No.4
Solution of simultaneous algebraic equations by exact methods
Method 1: Gauss elimination method
% workout the number of equations.a=input('enter the coefficient matrix');b=input('enter the constant matrix');N=length(b);%gauss eliminationfor column=1:(N-1) %work on all the rows below diagonal element for row=(column+1):N %workout the value of d. d=a(row,column)/a(column,column); % do the row operation. a(row,:)=a(row,:)-d*a(column,:); b(row)=b(row)-d*b(column); end %loop through rows.end % loop through columns.% back substitution.for row=N:-1:1 x(row)=b(row); for i=(row+1):N x(row)=x(row)-a(row,i)*x(i); end x(row)=x(row)/a(row,row);end % return the answer.x=x'returnx=x'
Run
Result of Method 1:
enter the coefficient matrix[10 -2 -1 -1; -2 10 -1 -1; -1 -1 10 -2; -1 -1 -2 10]enter the constant matrix[3;15;27;-9]x = 1.0000 2.0000 3.0000 0.0000
Method 2: Matrix Inverse method
>> a=[10 -2 -1 -1; -2 10 -1 -1; -1 -1 10 -2; -1 -1 -2 10]
a =
10 -2 -1 -1
-2 10 -1 -1
-1 -1 10 -2
-1 -1 -2 10
>> b=[3;15;27;-9]
b =
3
15
27
-9
>> a1=inv(a)*b
a1 =
1.0000
2.0000
3.0000
-0.0000
Exp. No. 5
Solution of simultaneous algebraic equations by iteration methods
Method 1: Jacobian Iteration
clear;clc;format compact;A = input(‘input the coefficient matrix:’);C = input(‘input the 7teratio matrix:’);n = length(C);X = zeros(n,1);Error_Value = ones(n,1);for i=1:n; j=1:n;end 7teration = 0;while max(Error_Value) > 0.001 7teration = 7teration+1; Z=X; for i=1:n; j=1:n; j(i)=[]; Xtemp=Z; Xtemp(i) = []; X(i) = (C(i) – sum(A(i,j) * Xtemp)) / A(i,i); end Xsolution (:,7teration) = X Error_Value = sqrt((X – Z).^2);endThe_required_matrix =[X]
Run:
Result of Method 1
input the coefficient matrix:[10 -2 -1 -1; -2 10 -1 -1; -1 -1 10 -2; -1 -1 -2 10]
input the constant matrix:[3;15;27;-9]
Iteration No=1
solution =
0.3000
1.5000
2.7000
-0.9000
Iteration No=2
solution =
0.7800
1.7400
2.7000
-0.1800
Iteration No=3
solution =
0.9000
1.9080
2.9160
-0.1080
Iteration No=4
solution =
0.9624
1.9608
2.9592
-0.0360
Iteration No=5
solution =
0.9845
1.9848
2.9851
-0.0158
Iteration No=6
solution =
0.9939
1.9938
2.9938
-0.0060
Iteration No=7
solution =
0.9975
1.9975
2.9976
-0.0025
Iteration No=8
solution =
0.9990
1.9990
2.9990
-0.0010
Iteration No=9
solution =
0.9996
1.9996
2.9996
-0.0004
The_required_matrix =
0.9996
1.9996
2.9996
-0.0004
Method 2: Gauss Seidal Iteration
clear;clc;format compact;A = input('input the coefficient matrix:');C = input('input the connstant matrix:');n = length(C);X = zeros(n,1);Error_Value = ones(n,1);for i=1:n; j=1:n;end itteration = 0;while max(Error_Value) > 0.001 itteration = itteration+1; Z=X; for i=1:n; j=1:n; j(i)=[]; Xtemp=X; Xtemp(i) = []; X(i) = (C(i) - sum(A(i,j) * Xtemp)) / A(i,i); end Xsolution (:,itteration) = X Error_Value = sqrt((X - Z).^2);endThe_required_matrix =[X]
Run:Result of Method 2:
input the coefficient matrix:[10 -2 -1 -1; -2 10 -1 -1; -1 -1 10 -2; -1 -1 -2 10]
input the connstant matrix:[3;15;27;-9]
Iteration No=1
solution =
0.3000
1.5600
2.8860
-0.1368
Iteration No=2
solution =
0.8869
1.9523
2.9566
-0.0248
Iteration No=3
solution =
0.9836
1.9899
2.9924
-0.0042
Iteration No=4
solution =
0.9968
1.9982
2.9987
-0.0008
Iteration No=5
solution =
0.9994
1.9997
2.9998
-0.0001
Iteration No=6
solution =
0.9999
1.9999
3.0000
-0.0000
The_required_matrix =
0.9999
1.9999
3.0000
-0.0000
Exp. No. 6Solution of Ordinary Differential Equations
using Finite Difference Methodclear;clc;format compact;% Solution of ODEs using Finite Difference Method% Example of 1-D steady state Heat conduction problem with Temp BCsn=input('Enter the Number of nodes :');a=zeros(n);b=zeros(n,1); % Governing Equation: Ti-1 - 2Ti + Ti+1 = 0% Boundary conditions i) T1=100 ii) Tn=20T1=100; Tn=20; j=0;for i=2:(n-1) j=j+1; a(i,j)=1; a(i,j+1)=-2; a(i,j+2)=1; b(i,1)=-a(i,1)*T1-a(i,n)*Tn;endfor i=1:(n-2) for j=1:(n-2) af(i,j)=a(i+1,j+1); bf(i,1)=b(i+1,1); endendt=af\bf;% Display Resultafbft
Run
Result of method :1
Enter the Number of nodes :5
af =
-2 1 0
1 -2 1
0 1 -2
bf =
-100
0
-20
t =
80.0000
60.0000
40.0000
Result of method 2
Enter the Number of nodes :10
af =
-2 1 0 0 0 0 0 0
1 -2 1 0 0 0 0 0
0 1 -2 1 0 0 0 0
0 0 1 -2 1 0 0 0
0 0 0 1 -2 1 0 0
0 0 0 0 1 -2 1 0
0 0 0 0 0 1 -2 1
0 0 0 0 0 0 1 -2
bf =
-100
0
0
0
0
0
0
-10
t =
90.0000
80.0000
70.0000
60.0000
50.0000
40.0000
30.0000
20.0000
Exp. No. 7
Solution of Partial Differential Equations
using Finite Difference Method
clear;clc;format long%material=copper%Thermal conductivity : K=4450w/m.k%Density : row=8940kg/cubic m.%Specific heat : cp=300J/kg.k%diffusivity : Alpha = K/(row*Cp)=1.659 e-3alpha=1.659*10^-3;n=input('enter the number of nodes:');T=zeros(n);T(1,:)=100; T(n,:)=20; T(2,1)=0; T(3,1)=0;T(4,1)=0; deltat=1; deltax=0.25;for i=1:deltat:(n-1) for j=2:(n-1) T(j,i+deltat)=alpha*(deltat/((deltax)^2))*(T(j-1,i)-2*T(j,i)+T(j+1,i))+T(j,i); end % for L=2:(n-1) % T(L,i+1) %endEnd
run
Exp. No. 7 result for first 5 iterations
enter the number of nodes:5
>> T
T =
1.0e+002 *
Columns 1 through 4
1.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000
0 0.026544000000000 0.051678832128000 0.075501749259185
0 0 0.000845500723200 0.002446730284814
0 0.005308800000000 0.010335766425600 0.015118304228794
0.200000000000000 0.200000000000000 0.200000000000000 0.200000000000000
Column 5
1.000000000000000
0.098102458403194
0.004722256967238
0.019689449702576
0.200000000000000
Exp. No. 8
Numerical Integration
clear;clc;format compact%Problem 1: Find Integration of the function (2+5*Zeta) with in the limits% -1 to +1 %Problem 2: Find Integration of the function (2+5*Zeta-3*Zeta*Zeta) with in % the limits -1 to +1a=input(‘enter the example number:’);b=input (‘Enter the Rule to be used for Numerical Integration :’);if a==1 if b==1 w1=2; zeta1=0; I11=w1*(2+5*zeta1) else w1=1; w2=1; zeta1=-1/3^(1/2); zeta2=1/3^(1/2); I12=w1*(2+5*zeta1)+w2*(2+5*zeta2) endelse if b==1 w1=2; zeta1=0; I21=w1*(2+5*zeta1-3*zeta1*zeta1) else w1=1; w2=1; zeta1=-1/3^(1/2); zeta2=1/3^(1/2); I22=w1*(2+5*zeta1-3*zeta1*zeta1)+w2*(2+5*zeta2-3*zeta2*zeta2) endend
Run
Exp. No. 8 result
enter the example number:1
Enter the Rule to be used for Numerical Integration :1
I11 =
4
enter the example number:1
Enter the Rule to be used for Numerical Integration :2
I12 =
4
enter the example number:2
Enter the Rule to be used for Numerical Integration :1
I21 =
4
enter the example number:2
Enter the Rule to be used for Numerical Integration :2
I22 =
2.000000000000000
Exp. No. 9
Line graphs and Surface plots
Line graphs
x =
Columns 1 through 4
0 0.104719755119660 0.209439510239320 0.314159265358979
Columns 5 through 8
0.418879020478639 0.523598775598299 0.628318530717959 0.733038285837618
Columns 9 through 12
0.837758040957278 0.942477796076938 1.047197551196598 1.151917306316257
Columns 13 through 16
1.256637061435917 1.361356816555577 1.466076571675237 1.570796326794897
Columns 17 through 20
1.675516081914556 1.780235837034216 1.884955592153876 1.989675347273536
Columns 21 through 24
2.094395102393195 2.199114857512855 2.303834612632515 2.408554367752175
Columns 25 through 28
2.513274122871835 2.617993877991494 2.722713633111154 2.827433388230814
Columns 29 through 32
2.932153143350473 3.036872898470133 3.141592653589793 3.246312408709453
Columns 33 through 36
3.351032163829113 3.455751918948772 3.560471674068432 3.665191429188092
Columns 37 through 40
3.769911184307752 3.874630939427412 3.979350694547072 4.084070449666731
Columns 41 through 44
4.188790204786391 4.293509959906051 4.398229715025710 4.502949470145371
Columns 45 through 48
4.607669225265030 4.712388980384690 4.817108735504350 4.921828490624010
Columns 49 through 52
5.026548245743669 5.131268000863329 5.235987755982989 5.340707511102648
Columns 53 through 56
5.445427266222308 5.550147021341968 5.654866776461628 5.759586531581287
Columns 57 through 60
5.864306286700947 5.969026041820607 6.073745796940266 6.178465552059927
Column 61
6.283185307179586
>> y1=sin(x)
y1 =
Columns 1 through 4
0 0.104528463267653 0.207911690817759 0.309016994374947
Columns 5 through 8
0.406736643075800 0.500000000000000 0.587785252292473 0.669130606358858
Columns 9 through 12
0.743144825477394 0.809016994374947 0.866025403784439 0.913545457642601
Columns 13 through 16
0.951056516295154 0.978147600733806 0.994521895368273 1.000000000000000
Columns 17 through 20
0.994521895368273 0.978147600733806 0.951056516295154 0.913545457642601
Columns 21 through 24
0.866025403784439 0.809016994374947 0.743144825477394 0.669130606358858
Columns 25 through 28
0.587785252292473 0.500000000000000 0.406736643075800 0.309016994374948
Columns 29 through 32
0.207911690817760 0.104528463267654 0.000000000000000 -0.104528463267654
Columns 33 through 36
-0.207911690817760 -0.309016994374947 -0.406736643075800 -0.500000000000000
Columns 37 through 40
-0.587785252292473 -0.669130606358858 -0.743144825477394 -0.809016994374947
Columns 41 through 44
-0.866025403784439 -0.913545457642601 -0.951056516295154 -0.978147600733806
Columns 45 through 48
-0.994521895368273 -1.000000000000000 -0.994521895368273 -0.978147600733806
Columns 49 through 52
-0.951056516295154 -0.913545457642601 -0.866025403784439 -0.809016994374948
Columns 53 through 56
-0.743144825477395 -0.669130606358858 -0.587785252292473 -0.500000000000000
Columns 57 through 60
-0.406736643075800 -0.309016994374948 -0.207911690817760 -0.104528463267653
Column 61
-0.000000000000000
>> y2=cos(x)
y2 =
Columns 1 through 4
1.000000000000000 0.994521895368273 0.978147600733806 0.951056516295154
Columns 5 through 8
0.913545457642601 0.866025403784439 0.809016994374947 0.743144825477394
Columns 9 through 12
0.669130606358858 0.587785252292473 0.500000000000000 0.406736643075800
Columns 13 through 16
0.309016994374947 0.207911690817759 0.104528463267654 0.000000000000000
Columns 17 through 20
-0.104528463267653 -0.207911690817759 -0.309016994374947 -0.406736643075800
Columns 21 through 24
-0.500000000000000 -0.587785252292473 -0.669130606358858 -0.743144825477394
Columns 25 through 28
-0.809016994374947 -0.866025403784438 -0.913545457642601 -0.951056516295154
Columns 29 through 32
-0.978147600733806 -0.994521895368273 -1.000000000000000 -0.994521895368273
Columns 33 through 36
-0.978147600733806 -0.951056516295154 -0.913545457642601 -0.866025403784439
Columns 37 through 40
-0.809016994374947 -0.743144825477394 -0.669130606358858 -0.587785252292473
Columns 41 through 44
-0.500000000000000 -0.406736643075800 -0.309016994374948 -0.207911690817759
Columns 45 through 48
-0.104528463267653 -0.000000000000000 0.104528463267654 0.207911690817759
Columns 49 through 52
0.309016994374947 0.406736643075801 0.500000000000000 0.587785252292473
Columns 53 through 56
0.669130606358858 0.743144825477394 0.809016994374947 0.866025403784438
Columns 57 through 60
0.913545457642601 0.951056516295154 0.978147600733806 0.994521895368273
Column 61
1.000000000000000
>> plot(y1,y2,'r --')
Surface Plots: