Problems in finding a set of global minima

Hi all, I have some problems in finding a set of local minimum of some complicated function. I use fminbnd but it only returns one single minium from the range of -50 to 50, but in fact there are 17 minima. How can I find the correct set of minima and save them into an array?

clear;
L=1;
L1=L/6;
omega0 = 0;
omegaR = 10;
tR=  (2*pi)/omegaR;
te = 0.8;
t1 = 0;
t_i = te/50;
gamma_i = 1;
%alpha = (t_i^2)/(2*L);
alpha = (2*pi*gamma_i)/(omegaR*L) ;
beta = sqrt(1-te^2);
delta = sqrt(1-t1^2);


figure(1);
k = -54:0.01:54

k = 1×10801

-54.0000 -53.9900 -53.9800 -53.9700 -53.9600 -53.9500 -53.9400 -53.9300 -53.9200 -53.9100 -53.9000 -53.8900 -53.8800 -53.8700 -53.8600 -53.8500 -53.8400 -53.8300 -53.8200 -53.8100 -53.8000 -53.7900 -53.7800 -53.7700 -53.7600 -53.7500 -53.7400 -53.7300 -53.7200 -53.7100

a3 =  (sqrt(1-te^2) - (te^2) ./ (1./(delta*exp(i*(k*L + i*alpha*L)) - (t1^2)*exp(i*( (k*L + i*alpha*L) + (k*L1 + i*alpha*L1) ))./(1- (sqrt(1-t1^2))*exp(i*(k*L1 + i*alpha*L1)))) - sqrt(1-te^2)));
a3norm = abs(a3);
plot(k,a3norm, 'LineWidth',1)
axis([-54 54 0 0.9 ])
xlabel('unnormalized m')
ylabel('T')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% Finding Minimum %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
syms k;



k = fminbnd(@(k)abs((sqrt(1-te^2) - (te^2) ./ (1./(delta*exp(i*(k*L + i*alpha*L)) - (t1^2)*exp(i*( (k*L + i*alpha*L) + (k*L1 + i*alpha*L1) ))./(1- (sqrt(1-t1^2))*exp(i*(k*L1 + i*alpha*L1)))) - sqrt(1-te^2)))),-50,50)

k = -3.5172e-13

Answer 1

[Dyuman Joshi]

Functions like fminbnd (and fzero, etc) work differently than what you are expecting them to work like.

It would be better to use islocalmin in this case.

clear;
L=1;
L1=L/6;
omega0 = 0;
omegaR = 10;
tR=  (2*pi)/omegaR;
te = 0.8;
t1 = 0;
t_i = te/50;
gamma_i = 1;
%alpha = (t_i^2)/(2*L);
alpha = (2*pi*gamma_i)/(omegaR*L) ;
beta = sqrt(1-te^2);
delta = sqrt(1-t1^2);


figure(1);
k = -54:0.01:54;
a3 =  (sqrt(1-te^2) - (te^2) ./ (1./(delta*exp(i*(k*L + i*alpha*L)) - (t1^2)*exp(i*( (k*L + i*alpha*L) + (k*L1 + i*alpha*L1) ))./(1- (sqrt(1-t1^2))*exp(i*(k*L1 + i*alpha*L1)))) - sqrt(1-te^2)));
a3norm = abs(a3);
plot(k,a3norm, 'LineWidth',1)
axis([-54 54 0 0.9 ])
xlabel('unnormalized m')
ylabel('T')

%Get the indices corresponding to local minimas
idx = islocalmin(a3norm);

%To save them in an array, simply use logical indexing -
k_min = k(idx);
a3norm_min = a3norm(idx);

%Highlight the local minimas on the plot
hold on
plot(k_min,a3norm_min,'r*')

[Tsz Tsun]

Thanks a lot! It is exactly I want!

Answer 2

[Torsten]

L=1;
L1=L/6;
omega0 = 0;
omegaR = 10;
tR=  (2*pi)/omegaR;
te = 0.8;
t1 = 0;
t_i = te/50;
gamma_i = 1;
%alpha = (t_i^2)/(2*L);
alpha = (2*pi*gamma_i)/(omegaR*L) ;
beta = sqrt(1-te^2);
delta = sqrt(1-t1^2);
syms k real
a3 =  abs((sqrt(1-te^2) - (te^2) ./ (1./(delta*exp(i*(k*L + i*alpha*L)) - (t1^2)*exp(i*( (k*L + i*alpha*L) + (k*L1 + i*alpha*L1) ))./(1- (sqrt(1-t1^2))*exp(i*(k*L1 + i*alpha*L1)))) - sqrt(1-te^2))));
da3dk = diff(a3,k);
a3 = matlabFunction(a3);
da3dk = matlabFunction(da3dk);
k0 = -20*pi:2*pi:20*pi;
for i = 1:numel(k0)
  kzero(i) = fzero(da3dk,[k0(i)-0.1 k0(i)+0.1]);
end

hold on
k = k0(1):0.01:k0(end);
%plot(k,da3dk(k))
%plot(kzero,da3dk(kzero),'o')
plot(k,a3(k))
plot(kzero,a3(kzero),'o')
hold off