Resonance in moonpool type structures

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Javeria il 2 Ago 2025

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https://it.mathworks.com/matlabcentral/answers/2178916-resonance-in-moonpool-type-structures

Commentato: Javeria il 4 Ago 2025

1st_file.pdf

I have the following code for my problem where i apply the match eigen function technique to find the unknown coefficients and then plotting one result wave motion inside a moonpool structure. However the expected plot for me should be decreasing after the resonance peak. But in my code it is decrease but then again blows up. I did not find any source of error. The analytical expressions which i used for my problem is write in pdf file at the end. Here is my code . I just want to figure out my source of error to fix it but i couldn't. So I need help in acheiving my desired results. I also attached the refrence plot.

clear all;

close all;

clc;

tic;

% Parameters from your setup

N = 5; % Number of evanescent modes

M = 3; % Number of azimuthal modes

RE = 1.0; % Outer radius (m)

ratio = 2.0; % Ratio R_E / R_I

RI = RE / ratio; % Inner radius (m)

h = 4.0 * RE; % Water depth (m)

b = 2.0; % Distance from cylinder bottom to seabed (m)

d = h - b; % Interface depth (m)

g = 9.81; % Gravity (m/s^2)

l = 0; % Azimuthal order

% I_given_wavenumber = 1;

X_kRE = linspace(0.01, 4, 200);

% Initialize eta0 array before loop

eta0 = zeros(size(X_kRE)); % Preallocate

for idx = 1:length(X_kRE)

% if I_given_wavenumber == 1

wk = X_kRE(idx) / RE; % dimensional wavenumber

omega = sqrt(g * wk * tanh(wk * h)); % dispersion relation

fun_alpha = @(x) omega^2 + g*x.*tan(x*h); %dispersion equation for evanescent modes

for n = 1:N

if n == 1

k(n) = -1i * wk;

else

x0_left = (2*n - 3) * pi / (2*h);

x0_right = (2*n - 1) * pi / (2*h);

guess = (x0_left + x0_right)/2;

% Use lsqnonlin for better convergence

k(n) = lsqnonlin(fun_alpha, guess, x0_left, x0_right);

end

% % find evanescent modes k1…k_{N-1} stored in k_all(2)…k_all(N)

% for n = 2 : N

% m = n - 1; % mode index for evanescent

% a = ((2*m-1)*pi)/(2*h) + eps;

% b = (m*pi)/h - eps;

% % at a: tan→ -∞ so f(a)<0; at b: tan→0 so f(b)=+omega^2>0

% k_all(n) = fzero(@(k) g*k.*tan(k*h) + omega^2, [a, b]);

% end

% % % end

%% finding the root \lemda_m =m*pi/b%%%%%%

for j=1:M

m(j)=(pi*(j-1))/b;

end

% %%%%%%%%%%%%%%%%%%%%% Define matrix A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A = zeros(M,N);

A(1,1) = (cosh(wk*h)*sinh(wk*b))/(wk*b*(2*wk*h+sinh(2*wk*h)))*(besselh(l, wk*RE)) /(dbesselh(l, wk*RE)); % Set A_00

for j = 2:N

A(1,j) =(cos(k(j)*h)*sin(k(j)*b) )/ (k(j)*b*(2*k(j)*h + sin(2*k(j)*h)))*( besselk(l, k(j)*RE) )...

/(dbesselk(l, k(j)*RE)); % Set A_0n

end

for i = 2:M

A(i,1) =(2 * cosh(wk*h) * (-1)^(i-1) * wk*sinh(wk*b)) / (b * (2*wk*h + sinh(2*wk*h))*(wk^2 + m(i)^2))...

* (besselh(l, wk*RE))/ (dbesselh(l, wk*RE)); % Set A_m0

end

for i = 2:M

for j = 2:N

A(i,j) = (2*cos(k(j)*h)*(-1)^(i-1) *k(j)* sin(k(j)*b))/ (b * (2*k(j)*h + sin(2*k(j)*h))*(k(j)^2-m(i)^2))...

*(besselk(l, k(j)*RE)) / (dbesselk(l, k(j)*RE));% Set A_mn

end

%%%%%%%%%%%%%%%%%%%%%%%% DefineMatrix B %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

B=zeros(N,M);

B(1,1) = (4*sinh(wk*b)) / (RE * wk*log(RE/RI) *cosh(wk*h)); %set B_00

%%%%%%%%%%%B_0m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for j = 2:M

Rml_prime_RE = m(j)*(besselk(l, m(j)*RI) * dbesseli(l, m(j)*RE) ...

- besseli(l, m(j)*RI) * dbesselk(l, m(j)*RE))...

/(besselk(l, m(j)*RI) * besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI));

B(1,j) = Rml_prime_RE * (4 * wk * (-1)^(j-1) * sinh(wk*b)) / (cosh(wk*h) * (wk^2 + m(j)^2));

end

%%%%%%%%%%%B_n0%%%%%%%%%%%%%%%%%%%%%%%%%%%

for i=2:N

B(i,1)=(4*sin(k(i)*b))/(RE*k(i)*log(RE/RI)*cos(k(i)*h));% Set B_n0

end

%%%%%%%%%%%%%%%%%%%%%%%%%%B_nm%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for i=2:N

for j=2:M

Rml_prime_RE = m(j)*(besselk(l, m(j)*RI) * dbesseli(l, m(j)*RE) ...

- besseli(l, m(j)*RI) * dbesselk(l, m(j)*RE))...

/(besselk(l, m(j)*RI) * besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI));

B(i,j) = Rml_prime_RE * (4 * k(i) * (-1)^(j-1) * sin(k(i)*b)) / (cos(k(i)*h) * (k(i)^2 - m(j)^2));

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Define Matrix C %%%%%%%%%%%%%%%%%%%%%%%%

C = zeros(N,M);

C(1,1) = -4 * sinh(wk*b) / (RE * wk*log(RE/RI) * cosh(wk*h));

% %%%%% DEfine C0m%%%%%%

for j = 2:M

Rml_prime_star_RE = m(j)*(besseli(l, m(j)*RE) * dbesselk(l, m(j)*RE) ...

- besselk(l, m(j)*RE) * dbesseli(l, m(j)*RE))...

/(besselk(l, m(j)*RI) * besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI));

C(1,j) = Rml_prime_star_RE * (4 * wk * (-1)^(j-1) * sinh(wk*b)) / (cosh(wk*h) * (wk^2 + m(j)^2));

end

% %%%%%%% Define Cn0%%%%%%

for i = 2:N

C(i,1) = -4 * sin(k(i)*b) / (RE *k(i)* log(RE/RI) * cos(k(i)*h));

end

% % %%%%%% Define Cnm%%%%%%

for i = 2:N

for j =2:M

Rml_prime_star_RE = m(j)*(besseli(l, m(j)*RE) * dbesselk(l, m(j)*RE) ...

- besselk(l, m(j)*RE) * dbesseli(l, m(j)*RE))...

/(besselk(l, m(j)*RI) * besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI));

C(i,j) = Rml_prime_star_RE * (4 * k(i) * (-1)^(j-1) * sin(k(i)*b)) / (cos(k(i)*h) * (k(i)^2 - m(j)^2));

end

%%%%%%% write Matrix D %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

D = zeros(M,N);

%%% write D00 %%%%%%

D(1,1) = (cosh(wk*h) * sinh(wk*b)) / (wk*b * (2*wk*h + sinh(2*wk*h))) * (besselj(l, wk*RI))/ (dbesselj(l, wk*RI));

%%%%% write D0n %%%%%%%%%%

for j =2:N

D(1,j) = (cos(k(j)*h) * sin(k(j)*b)) / (k(j)*b * (2*k(j)*h + sin(2*k(j)*h))) * (besseli(l, k(j)*RI) )/ (dbesseli(l, k(j)*RI));

end

%%%%%% write Dm0 %%%%%%%%%

for i = 2:M

D(i,1) = (2 * cosh(wk*h) * (-1)^(i-1) * wk * sinh(wk*b)) /(b * (2*wk*h + sinh(2*wk*h)) * (wk^2 + m(i)^2))...

*(besselj(l, wk*RI) )/(dbesselj(l, wk*RI));

end

%%%%% Define Dmn%%%%%%%%

for i = 2:M

for j = 2:N

D(i,j) = (2 * cos(k(j)*h) * (-1)^(i-1) * k(j) * sin(k(j)*b)) /(b * (2*k(j)*h + sin(2*k(j)*h)) * (k(j)^2 - m(i)^2))...

*(besseli(l, k(j)*RI)) / (dbesseli(l, k(j)*RI));

end

%%%%%%%%%%%%%%%%%%%%%%%%%% Define Matrix E %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

E = zeros(N, M); % Preallocate the matrix E

%%%%% Define E00%%%%%%%%%%%%

E(1,1) = (4 * sinh(wk*b)) / (RI *wk* log(RE/RI) * cosh(wk*h));

%%%% Define Eom %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for j = 2:M

Rml_prime_RI = m(j)*(besselk(l, m(j)*RI) * dbesseli(l, m(j)*RI) ...

- besseli(l, m(j)*RI) * dbesselk(l, m(j)*RI))...

/ (besselk(l, m(j)*RI) * besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI));

E(1,j) = Rml_prime_RI * (4 * wk * (-1)^(j-1) * sinh(wk*b)) / (cosh(wk*h) * (wk^2 + m(j)^2));

%

end

%%%%%% Define Eno%%%%%%%%%%%%%%

for i = 2:N

E(i,1) = (4 * sin(k(i)*b)) / (RI *k(i)* log(RE/RI) * cos(k(i)*h));

end

for i = 2:N

for j = 2:M

Rml_prime_RI = m(j)*(besselk(l, m(j)*RI) * dbesseli(l, m(j)*RI) ...

- besseli(l, m(j)*RI) * dbesselk(l, m(j)*RI))...

/ (besselk(l, m(j)*RI) * besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI));

E(i,j) = Rml_prime_RI * (4 * k(i) * (-1)^(j-1) * sin(k(i)*b)) / (cos(k(i)*h) * (k(i)^2 - m(j)^2));

end

%%%%%%%%%%%%%%%%%%%%%%%% Now write matrix F %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

F = zeros(N,M);

%%%%%%%% Define F00 %%%%%%%%%%

F(1,1) = (-4 * sinh(wk*b)) / (RI * wk*log(RE/RI) * cosh(wk*h));

% %%%%%% Define F0m%%%%

for j = 2:M

Rml_star_prime_RI = m(j)*(besseli(l, m(j)*RE) * dbesselk(l, m(j)*RI) ...

- besselk(l, m(j)*RE) * dbesseli(l, m(j)*RI))/((besselk(l, m(j)*RI) ...

* besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI)));

F(1,j) = Rml_star_prime_RI * (4 * wk * (-1)^(j-1) * sinh(wk*b)) / (cosh(wk*h) * (wk^2 + m(j)^2));

end

%%%%% Defin Fn0%%%%%%

for i = 2:N

F(i,1) = (-4 * sin(k(i)*b)) / (RI *k(i)* log(RE/RI) * cos(k(i)*h));

end

%%%%%%% Define Fnm %%%%%%%

for i = 2:N

for j = 2:M

Rml_star_prime_RI = m(j)*(besseli(l, m(j)*RE) * dbesselk(l, m(j)*RI) ...

- besselk(l, m(j)*RE) * dbesseli(l, m(j)*RI))/((besselk(l, m(j)*RI) ...

* besseli(l, m(j)*RE) - besselk(l, m(j)*RE) * besseli(l, m(j)*RI)));

F(i,j) = Rml_star_prime_RI * (4 * k(i)* (-1)^(j-1) * sin(k(i)*b)) / (cos(k(i)*h) * (k(i)^2 - m(j)^2));

end

% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% % Define right-hand side vector G

G = zeros(2*M+2*N, 1); % Total length = M + N+ M+ N = 2M+2N

%

% Block 1: G (size M = 4)

for i = 1:M

if i == 1

G(i, 1) = (sinh(wk*b))/((b*wk)*cosh(wk*h))*besselj(l, wk*RE) ; % Z_0^l

else

G(i, 1) = (2 *wk*(-1)^(i-1)*sinh(wk*b))/(b *(wk^2 +m(i)^2))*besselj(l, wk*RE); %Z_m^l

end

%

% Block 2: Y (size N = 3)

for j = 1:N

if j == 1

G(j + M, 1) = -dbesselj(l, wk*RE) * (2*wk*h + sinh(2*wk*h)) /(cosh(wk*h)^2); % Y_0^l

else

G(j + M, 1) = 0; % Y_n^l

end

% Block 3: X (size M = 4)

for i = 1:M

G(i + M + N, 1) = 0; % X_0^l, X_m^l

end

% Block 4: W (size N = 3)

for j = 1:N

G(j + M + N + M, 1) = 0; % W_0^l, W_n^l

end

% Identity matrices

I_M = eye(M);

I_N = eye(N);

% Zero matrices

ZMM = zeros(M, M);

ZMN = zeros(M, N);

ZNM = zeros(N, M);

ZNN = zeros(N, N);

% Construct the full block matrix H

H = [ I_M, -A, ZMM, ZMN; % Block row 1

-B, I_N, -C, ZNN; % Block row 2

ZMM, ZMN, I_M, -D; % Block row 3

-E, ZNN, -F, I_N]; % Block row 4

% X = H \ G; % Solve the linear system

X = pinv(H) * G; % Use pseudoinverse for stability

%

b_vec = X(1:M); % Coefficients b^l

a_vec = X(M+1 : M+N); % Coefficients a^l

c_vec = X(M+N+1 : 2*M+N); % Coefficients c^l

d_vec = X(2*M+N+1 : end); % Coefficients d^l

%%%%%%%%% Wave motion%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

term1 = (d_vec(1)*cosh(wk*h)^2) / ((2*wk*h + sinh(2*wk*h)) * dbesselj(l, wk*RI));

sum1 = 0;

for n =2:N

sum1 = sum1 + d_vec(n)*cos(k(n)*h)^2/ ((2*k(n)*h + sin(2*k(n)*h))* dbesseli(l, k(n)*RI));

end

eta0(idx) = abs(term1+sum1);

% % disp(k.') % Should all be real and positive for n ≥ 2

% %

end

Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance.

plot(X_kRE, eta0, 'k', 'LineWidth', 1.5);

xlabel('$k_0 R_E$', 'Interpreter', 'latex');

ylabel('$\eta / (iA)$', 'Interpreter', 'latex');

title('Wave motion amplitude for RE/RI = 4.0');

grid on;

xlim([0 4]); % Match the reference plot

ylim([0 6]); % Optional: based on expected peak height

elapsedTime = toc;

disp(['Time consuming = ', num2str(elapsedTime), ' s']);

Time consuming = 3.9218 s

function out = dbesselh(l, z)
    if l == 0
        out = -besselh(1, 1, z);
    else
        out = 0.5 * (besselh(l - 1, 1, z) - besselh(l + 1, 1, z));
    end
end

function out = dbesseli(l, z)
    if l == 0
        out = besseli(1, z);
    else
        out = 0.5 * (besseli(l - 1, z) + besseli(l + 1, z));
    end
end

function out = dbesselj(l, z)
    if l == 0
        out = -besselj(1, z);
    else
        out = 0.5 * (besselj(l - 1, z) - besselj(l + 1, z));
    end
end

function out = dbesselk(l, z)
    if l == 0
        out = -besselk(1, z);
    else
        out = -0.5 * (besselk(l - 1, z) + besselk(l + 1, z));
    end
end

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Javeria il 2 Ago 2025

@Walter Roberson but at the end i need to plot this eta0 against k0RE. then how i can plot it?

Javeria il 2 Ago 2025

@Walter Roberson And the mathematical expression for it is

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Answer 1

Torsten il 2 Ago 2025

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https://it.mathworks.com/matlabcentral/answers/2178916-resonance-in-moonpool-type-structures#answer_1568696

Modificato: Torsten il 2 Ago 2025

Apri in MATLAB Online

I'd change

k(n) = lsqnonlin(fun_alpha, guess, x0_left, x0_right);

to

k(n) = lsqnonlin(fun_alpha, guess, x0_left, x0_right,optimset('Display','none'));

to avoid the outputs of success from lsqnonlin.

Further, you should avoid pinv as long as possible. So I'd use

   X = H \ G;  % Solve the linear system
%X = pinv(H) * G;  % Use pseudoinverse for stability

instead of

  % X = H \ G;  % Solve the linear system
X = pinv(H) * G;  % Use pseudoinverse for stability

as long as you don't get a warning from mldivide (\).

Then I arbitrarily compared two sections of your code with your documentation and found discrepancies. I know debugging is a tedious job and your equations are complicated, but you should hope that in the end, discipline will pay:

You use

% Block 1: G (size M = 4)
for i = 1:M
    if i == 1
        G(i, 1) = (sinh(wk*b))/((b*wk)*cosh(wk*h))*besselj(l, wk*RE) ;  % Z_0^l
    else
        G(i, 1) = (2 *wk*(-1)^(i-1)*sinh(wk*b))/(b *(wk^2 +m(i)^2))*besselj(l, wk*RE); %Z_m^l
    end
end

instead of

		% Block 1: G (size M = 4)
for i = 1:M
    if i == 1
        G(i, 1) = (sinh(wk*b))/((b*wk)*cosh(wk*h))*besselj(l, wk*RE) ;  % Z_0^l
    else
        G(i, 1) = (2 *wk*(-1)^(i-1)*sinh(wk*b))/(b *(wk^2 +m(i)^2)*cosh(wk*h))*besselj(l, wk*RE); %Z_m^l
    end
end

as in your documentation.

Further, in your code you use

sum1 = 0;
for n =2:N
    sum1 = sum1 + d_vec(n)*cos(k(n)*h)^2/ ((2*k(n)*h + sin(2*k(n)*h))* dbesseli(l, k(n)*RI)); 
end 

instead of

sum1 = 0;
for n =2:N
    sum1 = sum1 + d_vec(n)*cos(k(n)*h)^2/ ((2*k(n)*h + sin(2*k(n)*h))* dbesselj(l, k(n)*RI)); 
end 

in the formula of your posting.

You even used

sum1 = 0;
for n =2:N
    sum1 = sum1 + d_vec(n)*cos(k(n)*h)^2/ ((2*k(n)*h + sin(2*k(n)*h))* dbesselj(l, wk*RI)); 
end 

in your posting, but I think this is wrong.

And as @Walter Roberson said: eta0(idx) = abs(term1 + sum1) looks incorrect. From the reference plots you included I'd deduce that eta is purely imaginary such that it has to be normalized by i*A to be real-valued and possible to be plotted. But I don't have the knowledge about the physical background of your work.

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Torsten il 3 Ago 2025

Modificato: Torsten il 3 Ago 2025

Apri in MATLAB Online

Do you think it would help if you included the parameters and equations from which the reference plots were generated ? Or are your method and the method used for the reference plots not comparable ?

I'm still confused about the expression for eta you use in your code:

term1 = (d_vec(1)*cosh(wk*h)^2) / ((2*wk*h + sinh(2*wk*h)) * dbesselj(l, wk*RI));
sum1 = 0;
for n =2:N
    sum1 = sum1 + d_vec(n)*cos(k(n)*h)^2/ ((2*k(n)*h + sin(2*k(n)*h))* dbesseli(l, k(n)*RI)); 
end 
eta0(idx) = abs(term1+sum1);

How does this conform with equation (27) ? In equation (27), 1/besselj and 1/besseli are used instead of 1/dbesselj and 1/dbesseli and either term1 or sum1 should have a minus sign, shouldn't they ?

Javeria il 4 Ago 2025

@Torsten the equation which i write i actually this one in my code d_vec is c and just they write it specifically for l=0 however i used the recurrence formula in matlab to find the derivative and they just put i directly that's why they have minus sign. And we don't need to divided by this 1iA it is already given in normalized form just need to take absoulte because we have imiginary value.

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Resonance in moonpool type structures

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