Obtaining a matrix of solutions with fsolve

Question

Olimpia 2023-1-31

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1903850-obtaining-a-matrix-of-solutions-with-fsolve

编辑： Torsten 2023-1-31

Hello everyone !

I am trying to solve a system of 7 non-linear equations and 7 variables.

I have 4 parameters (c1, c2, c3, c4) and I want to solve my system for different (specific) values of these four parameters. Concretely, I am trying to solve the seven equations over my 210 possible vectors (c1i, c2i, c3i, c4i), that I upload from the file « grid_wide_4firms ». Instead of obtaining a vector of solution (1x7), I then wish to obtain a matrix (210x7).

I have therefore created a loop outside that replaces the vector c at each step, and store the results each step. But have the error:

Function definition are not supported in this context. Functions can only be created as local or nested functions in code files

Any idea ?

Thanks in advance !

PS : I attach the file grid_wide_4firms with the different values of c1, c2, c3, c4 and the code that works over one specific vector of ci (so that hopefully it is clearer)

final_matrix=zeros(210,7);
for i=1:210
clearvars c1 c2 c3 c4 x  my_func
func=@f;
x0=[1.99;1.99;1.99;1.99;0.0109;0.0109;0.0109]
x = zeros(1,7);
x=fsolve(func,x0),
display(x)
final_matrix((i),1)=x
function my_func=f(x)
% import the data
opts = delimitedTextImportOptions("NumVariables", 4);
opts.DataLines = [2, Inf];
opts.Delimiter = ",";
opts.VariableNames = ["cost1", "cost2", "cost3", "cost4"];
opts.VariableTypes = ["double", "double", "double", "double"];
opts.ExtraColumnsRule = "ignore";
opts.EmptyLineRule = "read";
gridwide4firms2 = readtable("C:\Users\o.cutinelli\Downloads\grid_wide_4firms.csv", opts);
costs= table2array(gridwide4firms2);
% gen c1, c2, c3, c4 so they are vectors
c1=costs((i),1)
c2=costs((i),2)
c3=costs((i),3)
c4=costs((i),4)
% parameters
rho = 1/2;
y = 20;
pall = 2;
sigma = 10;
eta = 2;
alpha=3/4;
my_func(1)= (sigma*(1 - (x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c1 -x(1);
my_func(2)= (sigma*(1 - (x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c2 -x(2);
my_func(3)= (sigma*(1 - (x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c3 -x(3);
my_func(4)= (sigma*(1 - (x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma)) + eta*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma))/(sigma*(1 - (x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma)) + eta*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma) - 1)*c4 -x(4);
my_func(5)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(1)^(-sigma)*(x(1) - c1)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(5)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(5)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
my_func(6)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(2)^(-sigma)*(x(2) - c2)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(6)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(6)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
my_func(7)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(3)^(-sigma)*(x(3) - c3)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(7)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(7)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
end 
end

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Answer 1

Torsten 2023-1-31

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1903850-obtaining-a-matrix-of-solutions-with-fsolve#answer_1160360

编辑：Torsten 2023-1-31

Don't solve for all x values at once.

Call "fsolve" in a loop and solve for each pair of combinations for (c1,c2,c3,c4) separately.

And something like x(1;i) does not exist. Indexing a 2d array works as x(1,i).

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Torsten 2023-1-31

在 MATLAB Online 中打开

% import the data
opts = delimitedTextImportOptions("NumVariables", 4);
opts.DataLines = [2, Inf];
opts.Delimiter = ",";
opts.VariableNames = ["cost1", "cost2", "cost3", "cost4"];
opts.VariableTypes = ["double", "double", "double", "double"];
opts.ExtraColumnsRule = "ignore";
opts.EmptyLineRule = "read";
gridwide4firms2 = readtable("grid_wide_4firms.csv", opts);
costs= table2array(gridwide4firms2);
final_matrix=zeros(7,210);
x0=[1.99;1.99;1.99;1.99;0.0109;0.0109;0.0109];
for i=1:210
  x=fsolve(@(x)f(x,costs(i,1:4)),x0);
  x = x(:);
  final_matrix(:,i)=x;
  x0 = x;
end
function my_func=f(x,costs)
c1 = costs(1);
c2 = costs(2);
c3 = costs(3);
c4 = costs(4);
% parameters
rho = 1/2;
y = 20;
pall = 2;
sigma = 10;
eta = 2;
alpha=3/4;
my_func(1)= (sigma*(1 - (x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(1)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c1 -x(1);
my_func(2)= (sigma*(1 - (x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(2)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c2 -x(2);
my_func(3)= (sigma*(1 - (x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma))/(sigma*(1 - (x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma)) + eta*(x(3)/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1 - sigma) - 1)*c3 -x(3);
my_func(4)= (sigma*(1 - (x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma)) + eta*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma))/(sigma*(1 - (x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma)) + eta*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^(1 - sigma) - 1)*c4 -x(4);
my_func(5)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(1)^(-sigma)*(x(1) - c1)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(5)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(5)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
my_func(6)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(2)^(-sigma)*(x(2) - c2)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(6)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(6)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
my_func(7)= pall^(eta - 1)*(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma))^(sigma - eta)*y*x(3)^(-sigma)*(x(3) - c3)*(sigma - eta)*(x(4)*(1+(x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))/(x(1)^(1 - sigma) + x(2)^(1 - sigma) + x(3)^(1 - sigma) + (x(4)*(1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho)))^(1 - sigma))^(1/(1 - sigma)))^(1-sigma)*(x(5)^rho + x(6)^rho + x(7)^rho)^((1 - rho)/rho)*x(7)^(rho - 1) - (1 + (x(5)^rho + x(6)^rho + x(7)^rho)^(alpha/rho))^sigma/(x(7)^(rho - 1)*(x(5)^rho + x(6)^rho + x(7)^rho)^((rho - 1)/rho)*alpha*(x(5)^rho + x(6)^rho + x(7)^rho)^(1/rho))^(alpha-1);
end 

Olimpia 2023-1-31

Thank you very much. I have already checked the solvability.

The initial guess is good for some vectors of c (for instance for i=207,208), but hoping to work on a much bigger set of vectors c, but I cannot find a good initial guess for all of them at the same time...

Torsten 2023-1-31

编辑：Torsten 2023-1-31

There is no general rule that generates good initial guesses.

If your inputs for c don't change much from call to call, you can use the result of a call as initial guess for the next call. If this doesn't work: tough luck.

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Obtaining a matrix of solutions with fsolve

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Obtaining a matrix of solutions with fsolve

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