How to perform summation with double subscript notation?

Hello everyone, I have to perform some analysis based on the following equation, which contains summation operator with double subscript notation.

I have written a code. Can anyone please look at it and confim whether I am wrong or right?

    alpha =0;
    
    for i=1:2
        
        for j=1:2
            
            alpha=alpha +(C(i)*A(j)*((((E-Eg(j)+Ep(i))^2)/(exp(Ep(i)/(k*T))-1))+(((E-Eg(j)-Ep(i))^2)/(1-exp(-Ep(i)/(k*T))))));
            
        end
        
    end
    
    Alpha=(alpha+(Ad*((E-Egd)^(1/2))));

Answer 1

[Star Strider]

First, provide the ‘C’ and the other missing vectors, then save ‘alpha’ as a matrix, then use the sum function to sum its elements.

    alpha =0;
    
    for i=1:2
        
        for j=1:2
            
            alpha(i,j) = C(i)*A(j)*((((E-Eg(j)+Ep(i))^2)/(exp(Ep(i)/(k*T))-1)));
            
        end
        
    end

Unrecognized function or variable 'C'.

    
    Alpha=(sum(alpha(:))+(Ad*((E-Egd).^(1/2))))

Try that with the vectors to see if the result is as desired.

.

[Md. Golam Zakaria]

@Star Strider actually I only uploaded part of the code. Just wanted to know if my summation formating is right or wrong. I am giving the full code. Can you please comment on this on? I am getting slight different result than expected.

clc
clear all
close all

h=4.136*10^-15;                         % Planck's Constant
k=8.617*10^-5;                          % Boltzmann's Constant
c=3*10^8;                               % speed of light
T=300;                                  % Ambient Temparature

beta=7.021*10^-4;
gamma=1108;
Eg0_1=1.1557;
Eg0_2=2.5;
Egd0=3.2;

Eg1=Eg0_1-((beta*(T^2))/(T+gamma));
Eg2=Eg0_2-((beta*(T^2))/(T+gamma));
Egd=Egd0-((beta*(T^2))/(T+gamma));

Eg=[Eg1 Eg2];
Ep=[1.827*10^-2 5.773*10^-2];

C=[5.5 4.0];
A=[3.231*10^2 7.237*10^3];
Ad=1.052*10^6;

walenength=(.2*10^-6):(.0001*10^-6):(1.2*10^-6);
num=numel(walenength);
Alpha=nan(1,num);


for t=1:num
    
    lambda=walenength(t);
    E=((h*c)/lambda);
    alpha =0;
    
    for i=1:2
        
        for j=1:2
            
            alpha=alpha +(C(i)*A(j)*((((E-Eg(j)+Ep(i))^2)./(exp(Ep(i)/(k.*T))-1))+(((E-Eg(j)-Ep(i))^2)/(1-exp(-Ep(i)/(k*T))))));
            
        end
        
    end
    
    Alpha(t)=(alpha+(Ad*((E-Egd)^(1/2))));
end


plot((walenength./10^-6),Alpha)
set(gca,'YScale','log')
ylim([(10^0) (10^9)])
xlabel('Wavelength \lambda ,(\mum)'),ylabel('Absorption Coefficient, \alpha(m^{-1})')

[Star Strider]

h=4.136*10^-15;                         % Planck's Constant
k=8.617*10^-5;                          % Boltzmann's Constant
c=3*10^8;                               % speed of light
T=300;                                  % Ambient Temparature

beta=7.021*10^-4;
gamma=1108;
Eg0_1=1.1557;
Eg0_2=2.5;
Egd0=3.2;

Eg1=Eg0_1-((beta*(T^2))/(T+gamma));
Eg2=Eg0_2-((beta*(T^2))/(T+gamma));
Egd=Egd0-((beta*(T^2))/(T+gamma));

Eg=[Eg1 Eg2];
Ep=[1.827*10^-2 5.773*10^-2];

C=[5.5 4.0];
A=[3.231*10^2 7.237*10^3];
Ad=1.052*10^6;

walenength=(.2*10^-6):(.0001*10^-6):(1.2*10^-6);
num=numel(walenength);
Alpha=nan(1,num);


for t=1:num
    
    lambda=walenength(t);
    E=((h*c)/lambda);
    alpha =0;
    
    for i=1:2
        
        for j=1:2
            
            alpha=alpha +(C(i)*A(j)*((((E-Eg(j)+Ep(i))^2)./(exp(Ep(i)/(k.*T))-1))+(((E-Eg(j)-Ep(i))^2)/(1-exp(-Ep(i)/(k*T))))));
            
        end
        
    end
    
    Alpha(t)=(alpha+(Ad*((E-Egd)^(1/2))));
end

figure
plot((walenength./10^-6),real(Alpha))
hold on
plot((walenength./10^-6),imag(Alpha))
plot((walenength./10^-6),abs(Alpha),'--')
hold off
set(gca,'YScale','log')
ylim([(10^0) (10^9)])
xlabel('Wavelength \lambda ,(\mum)')
ylabel('Absorption Coefficient, \alpha(m^{-1})')
legend('Re(\alpha(T))','Im(\alpha(T))','|\alpha(T)|', 'Location','best')

This is difficult to follow. I do not understand taking the square root in the second term of the ‘Alpha(t)’ assignment, since the square root is not in the posted symbolic code.

It would likely be best for you to go through this to be certain there are no coding errors, since this is not an area of my expertise. I have no idea what the ‘expected’ result is, so I have nothing to compare it to.

.

[Md. Golam Zakaria]

@Star Strider actually the square root is there, I mistyped the equation. You just tell me if there is any coding error regarding the summation.

[Star Strider]

I do not see anything wrong with it. The easiest way to troubleshoot it is to see what the individual terms evaluate to, and then see if those are correct —

h=4.136*10^-15;                         % Planck's Constant
k=8.617*10^-5;                          % Boltzmann's Constant
c=3*10^8;                               % speed of light
T=300;                                  % Ambient Temparature

beta=7.021*10^-4;
gamma=1108;
Eg0_1=1.1557;
Eg0_2=2.5;
Egd0=3.2;

Eg1=Eg0_1-((beta*(T^2))/(T+gamma));
Eg2=Eg0_2-((beta*(T^2))/(T+gamma));
Egd=Egd0-((beta*(T^2))/(T+gamma));

Eg=[Eg1 Eg2];
Ep=[1.827*10^-2 5.773*10^-2];

C=[5.5 4.0];
A=[3.231*10^2 7.237*10^3];
Ad=1.052*10^6;

walenength=(.2*10^-6):(.0001*10^-6):(1.2*10^-6);
num=numel(walenength);
Alpha=nan(1,num);


for t=1:(num/1000)
    
    lambda=walenength(t);
    E=((h*c)/lambda);
    alpha =0;
    
    for i=1:2
        
        for j=1:2
            
            fprintf([repmat('—',1, 20) '\nt = %4d\ti = %d\tj = %d\n'],t,i,j)
            Term_1(i,j) = (((E-Eg(j)+Ep(i))^2)./(exp(Ep(i)/(k.*T))-1))
            Term_2(i,j) = (((E-Eg(j)-Ep(i))^2)/(1-exp(-Ep(i)/(k*T))))
            
            alpha=alpha +(C(i)*A(j)*((((E-Eg(j)+Ep(i))^2)./(exp(Ep(i)/(k.*T))-1))+(((E-Eg(j)-Ep(i))^2)/(1-exp(-Ep(i)/(k*T))))));
            
        end
        
    end
    
    Alpha(t)=(alpha+(Ad*((E-Egd)^(1/2))));
end

————————————————————
t =    1	i = 1	j = 1

Term_1 = 25.4307

Term_2 = 50.8231

————————————————————
t =    1	i = 1	j = 2

Term_1 = 1×2

25.4307 13.8133

Term_2 = 1×2

50.8231 27.4640

————————————————————
t =    1	i = 2	j = 1

Term_1 = 2×2

25.4307 13.8133 3.1853 0

Term_2 = 2×2

50.8231 27.4640 28.3998 0

————————————————————
t =    1	i = 2	j = 2

Term_1 = 2×2

25.4307 13.8133 3.1853 1.7396

Term_2 = 2×2

50.8231 27.4640 28.3998 15.2603

————————————————————
t =    2	i = 1	j = 1

Term_1 = 2×2

25.3999 13.8133 3.1853 1.7396

Term_2 = 2×2

50.7610 27.4640 28.3998 15.2603

————————————————————
t =    2	i = 1	j = 2

Term_1 = 2×2

25.3999 13.7905 3.1853 1.7396

Term_2 = 2×2

50.7610 27.4184 28.3998 15.2603

————————————————————
t =    2	i = 2	j = 1

Term_1 = 2×2

25.3999 13.7905 3.1815 1.7396

Term_2 = 2×2

50.7610 27.4184 28.3649 15.2603

————————————————————
t =    2	i = 2	j = 2

Term_1 = 2×2

25.3999 13.7905 3.1815 1.7368

Term_2 = 2×2

50.7610 27.4184 28.3649 15.2347

————————————————————
t =    3	i = 1	j = 1

Term_1 = 2×2

25.3691 13.7905 3.1815 1.7368

Term_2 = 2×2

50.6991 27.4184 28.3649 15.2347

————————————————————
t =    3	i = 1	j = 2

Term_1 = 2×2

25.3691 13.7678 3.1815 1.7368

Term_2 = 2×2

50.6991 27.3728 28.3649 15.2347

————————————————————
t =    3	i = 2	j = 1

Term_1 = 2×2

25.3691 13.7678 3.1776 1.7368

Term_2 = 2×2

50.6991 27.3728 28.3300 15.2347

————————————————————
t =    3	i = 2	j = 2

Term_1 = 2×2

25.3691 13.7678 3.1776 1.7340

Term_2 = 2×2

50.6991 27.3728 28.3300 15.2091

————————————————————
t =    4	i = 1	j = 1

Term_1 = 2×2

25.3383 13.7678 3.1776 1.7340

Term_2 = 2×2

50.6372 27.3728 28.3300 15.2091

————————————————————
t =    4	i = 1	j = 2

Term_1 = 2×2

25.3383 13.7452 3.1776 1.7340

Term_2 = 2×2

50.6372 27.3274 28.3300 15.2091

————————————————————
t =    4	i = 2	j = 1

Term_1 = 2×2

25.3383 13.7452 3.1738 1.7340

Term_2 = 2×2

50.6372 27.3274 28.2951 15.2091

————————————————————
t =    4	i = 2	j = 2

Term_1 = 2×2

25.3383 13.7452 3.1738 1.7311

Term_2 = 2×2

50.6372 27.3274 28.2951 15.1835

————————————————————
t =    5	i = 1	j = 1

Term_1 = 2×2

25.3076 13.7452 3.1738 1.7311

Term_2 = 2×2

50.5754 27.3274 28.2951 15.1835

————————————————————
t =    5	i = 1	j = 2

Term_1 = 2×2

25.3076 13.7226 3.1738 1.7311

Term_2 = 2×2

50.5754 27.2820 28.2951 15.1835

————————————————————
t =    5	i = 2	j = 1

Term_1 = 2×2

25.3076 13.7226 3.1700 1.7311

Term_2 = 2×2

50.5754 27.2820 28.2603 15.1835

————————————————————
t =    5	i = 2	j = 2

Term_1 = 2×2

25.3076 13.7226 3.1700 1.7283

Term_2 = 2×2

50.5754 27.2820 28.2603 15.1581

————————————————————
t =    6	i = 1	j = 1

Term_1 = 2×2

25.2770 13.7226 3.1700 1.7283

Term_2 = 2×2

50.5137 27.2820 28.2603 15.1581

————————————————————
t =    6	i = 1	j = 2

Term_1 = 2×2

25.2770 13.7000 3.1700 1.7283

Term_2 = 2×2

50.5137 27.2367 28.2603 15.1581

————————————————————
t =    6	i = 2	j = 1

Term_1 = 2×2

25.2770 13.7000 3.1662 1.7283

Term_2 = 2×2

50.5137 27.2367 28.2256 15.1581

————————————————————
t =    6	i = 2	j = 2

Term_1 = 2×2

25.2770 13.7000 3.1662 1.7255

Term_2 = 2×2

50.5137 27.2367 28.2256 15.1326

————————————————————
t =    7	i = 1	j = 1

Term_1 = 2×2

25.2464 13.7000 3.1662 1.7255

Term_2 = 2×2

50.4521 27.2367 28.2256 15.1326

————————————————————
t =    7	i = 1	j = 2

Term_1 = 2×2

25.2464 13.6775 3.1662 1.7255

Term_2 = 2×2

50.4521 27.1915 28.2256 15.1326

————————————————————
t =    7	i = 2	j = 1

Term_1 = 2×2

25.2464 13.6775 3.1624 1.7255

Term_2 = 2×2

50.4521 27.1915 28.1909 15.1326

————————————————————
t =    7	i = 2	j = 2

Term_1 = 2×2

25.2464 13.6775 3.1624 1.7227

Term_2 = 2×2

50.4521 27.1915 28.1909 15.1072

————————————————————
t =    8	i = 1	j = 1

Term_1 = 2×2

25.2159 13.6775 3.1624 1.7227

Term_2 = 2×2

50.3906 27.1915 28.1909 15.1072

————————————————————
t =    8	i = 1	j = 2

Term_1 = 2×2

25.2159 13.6550 3.1624 1.7227

Term_2 = 2×2

50.3906 27.1464 28.1909 15.1072

————————————————————
t =    8	i = 2	j = 1

Term_1 = 2×2

25.2159 13.6550 3.1586 1.7227

Term_2 = 2×2

50.3906 27.1464 28.1563 15.1072

————————————————————
t =    8	i = 2	j = 2

Term_1 = 2×2

25.2159 13.6550 3.1586 1.7199

Term_2 = 2×2

50.3906 27.1464 28.1563 15.0819

————————————————————
t =    9	i = 1	j = 1

Term_1 = 2×2

25.1854 13.6550 3.1586 1.7199

Term_2 = 2×2

50.3293 27.1464 28.1563 15.0819

————————————————————
t =    9	i = 1	j = 2

Term_1 = 2×2

25.1854 13.6326 3.1586 1.7199

Term_2 = 2×2

50.3293 27.1013 28.1563 15.0819

————————————————————
t =    9	i = 2	j = 1

Term_1 = 2×2

25.1854 13.6326 3.1548 1.7199

Term_2 = 2×2

50.3293 27.1013 28.1217 15.0819

————————————————————
t =    9	i = 2	j = 2

Term_1 = 2×2

25.1854 13.6326 3.1548 1.7171

Term_2 = 2×2

50.3293 27.1013 28.1217 15.0566

————————————————————
t =   10	i = 1	j = 1

Term_1 = 2×2

25.1549 13.6326 3.1548 1.7171

Term_2 = 2×2

50.2680 27.1013 28.1217 15.0566

————————————————————
t =   10	i = 1	j = 2

Term_1 = 2×2

25.1549 13.6102 3.1548 1.7171

Term_2 = 2×2

50.2680 27.0563 28.1217 15.0566

————————————————————
t =   10	i = 2	j = 1

Term_1 = 2×2

25.1549 13.6102 3.1510 1.7171

Term_2 = 2×2

50.2680 27.0563 28.0872 15.0566

————————————————————
t =   10	i = 2	j = 2

Term_1 = 2×2

25.1549 13.6102 3.1510 1.7143

Term_2 = 2×2

50.2680 27.0563 28.0872 15.0313

figure
plot((walenength./10^-6),real(Alpha))
hold on
plot((walenength./10^-6),imag(Alpha))
plot((walenength./10^-6),abs(Alpha),'--')
hold off
set(gca,'YScale','log')
ylim([(10^0) (10^9)])
xlabel('Wavelength \lambda ,(\mum)')
ylabel('Absorption Coefficient, \alpha(m^{-1})')
legend('Re(\alpha(T))','Im(\alpha(T))','|\alpha(T)|', 'Location','best')

See if the intermediate values appear to be correct.

.