How to use fsolve to solve matrix equations involving matrices of different dimensions?

Question

Smriti Verma 2024-1-10

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2068356-how-to-use-fsolve-to-solve-matrix-equations-involving-matrices-of-different-dimensions

评论： Smriti Verma 2024-1-16

I have the following equation to solve using fsolve: (all elements in matrices A0, A1, A2, A3, B1, B2 are given parameter values)

A0 = [1 0 sigma; -kappa 1 0; -phi_x -phi_pi 1];

A1 = [1 sigma 0; 0 beta 0; 0 0 0];

A2 = [1 0; 0 1; 0 0];

A3 = [0 0 0; 0 0 0; sigma_i 0 0];

B1 = [rho_g 0, 0 rho_u];

B2 = [0 sigma_g 0; 0 0 sigma_u];

F0 = fsolve(@(x) A1*x*B1 - A0*x + A2, zeros(3,2));

However, I get the following error:

Error using *

Incorrect dimensions for matrix multiplication. Check that the number of columns in the first matrix matches

the number of rows in the second matrix. To operate on each element of the matrix individually, use TIMES (.*)

for elementwise multiplication.

Error in Standard_NK_economy>@(x)A1(ir,:)*x*B1(:,ir)-A0(ir,:)*x+A2(ir,:) (line 29)

f0 = fsolve(@(x) A1(ir,:)*x*B1(:,ir) - A0(ir,:)*x + A2(ir,:), zeros(1,2)); % x is 1x2

Error in fsolve (line 270)

fuser = feval(funfcn{3},x,varargin{:});

Error in Standard_NK_economy (line 29)

f0 = fsolve(@(x) A1(ir,:)*x*B1(:,ir) - A0(ir,:)*x + A2(ir,:), zeros(1,2)); % x is 1x2

Caused by:

Failure in initial objective function evaluation. FSOLVE cannot continue.

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Dyuman Joshi 2024-1-10

Please share the mathematical formulation of the equations you are trying to solve.

There are undefined variables in the code above, thus we can not run your code.

Smriti Verma 2024-1-10

% Calibration

sigma = 1;

kappa = 0.2;

beta = 0.99;

phi_pi = 1.5;

phi_x = 0.5;

sigma_i = 0.0025;

sigma_g = 0.0025;

sigma_u = 0.0025;

rho_g = 0.5;

rho_u = 0.3;

% Define A0, A1, A2, A3 for the 3 DSGE equations, where Z = [x; pi; i], S =

% [g; u], and eps = [eps_i; eps_g; eps_u]

A0 = [1 0 sigma; -kappa 1 0; -phi_x -phi_pi 1];

A1 = [1 sigma 0; 0 beta 0; 0 0 0];

A2 = [1 0; 0 1; 0 0];

A3 = [0 0 0; 0 0 0; sigma_i 0 0];

% Define B1, B2 for the two state equations: S_t = B1*S_t-1 + B2*eps_t

B1 = [rho_g 0, 0 rho_u];

B2 = [0 sigma_g 0; 0 0 sigma_u];

% Guessing Z_t = F0*S_t + F1*eps_t, we calculate F0 and F1:

F0 = fsolve(@(x) A1*x*B1 - A0*x + A2, zeros(3,2));

F1 = A0\A3; % inv(A0)*A3

G1 = F0*B1;

G2 = F0*B2 + F1;

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Answer 1

Torsten 2024-1-10

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2068356-how-to-use-fsolve-to-solve-matrix-equations-involving-matrices-of-different-dimensions#answer_1387046

编辑：Torsten 2024-1-10

在 MATLAB Online 中打开

Your equations are linear in the unknowns - so you should use a linear solver, not a nonlinear one like "fsolve":

% Calibration

sigma = 1;

kappa = 0.2;

beta = 0.99;

phi_pi = 1.5;

phi_x = 0.5;

sigma_i = 0.0025;

sigma_g = 0.0025;

sigma_u = 0.0025;

rho_g = 0.5;

rho_u = 0.3;

% Define A0, A1, A2, A3 for the 3 DSGE equations, where Z = [x; pi; i], S =

% [g; u], and eps = [eps_i; eps_g; eps_u]

A0 = [1 0 sigma; -kappa 1 0; -phi_x -phi_pi 1];

A1 = [1 sigma 0; 0 beta 0; 0 0 0];

A2 = [1 0; 0 1; 0 0];

A3 = [0 0 0; 0 0 0; sigma_i 0 0];

% Define B1, B2 for the two state equations: S_t = B1*S_t-1 + B2*eps_t

B1 = [rho_g 0; 0 rho_u];

B2 = [0 sigma_g 0; 0 0 sigma_u];

X = sym('X',[size(A1,2),size(B1,1)]);

size(X)

ans = 1×2

3 2

eqns = A1*X*B1 - A0*X + A2 == 0;

symvar(eqns)

ans =

[A,b] = equationsToMatrix(eqns)

A =

b =

F0 = A\b

F0 =

F0 = [F0(1),F0(2);F0(3),F0(4);F0(5),F0(6)];

F1 = A0\A3 % inv(A0)*A3

F1 = 3×3

-0.0014 0 0 -0.0003 0 0 0.0014 0 0

G1 = F0*B1

G1 =

G2 = F0*B2 + F1

G2 =

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Torsten 2024-1-11

编辑：Torsten 2024-1-11

在 MATLAB Online 中打开

Seems it also works this way:

% Calibration
sigma = 1;
kappa = 0.2;
beta = 0.99;
phi_pi = 1.5;
phi_x = 0.5;
sigma_i = 0.0025;
sigma_g = 0.0025;
sigma_u = 0.0025;
rho_g = 0.5;
rho_u = 0.3;
% Define A0, A1, A2, A3 for the 3 DSGE equations, where Z = [x; pi; i], S =
% [g; u], and eps = [eps_i; eps_g; eps_u]
A0 = [1 0 sigma; -kappa 1 0; -phi_x -phi_pi 1];
A1 = [1 sigma 0; 0 beta 0; 0 0 0];
A2 = [1 0; 0 1; 0 0];
A3 = [0 0 0; 0 0 0; sigma_i 0 0];
% Define B1, B2 for the two state equations: S_t = B1*S_t-1 + B2*eps_t
B1 = [rho_g 0; 0 rho_u];
B2 = [0 sigma_g 0; 0 0 sigma_u];
F0 = fsolve(@(x) A1*x*B1 - A0*x + A2, zeros(3,2));
Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.
F1 = A0\A3 % inv(A0)*A3
F1 = 3×3
   -0.0014         0         0
   -0.0003         0         0
    0.0014         0         0
G1 = F0*B1
G1 = 3×2
    0.3582   -0.3322
    0.1418    0.3322
    0.3918    0.3322
G2 = F0*B2 + F1
G2 = 3×3
   -0.0014    0.0018   -0.0028
   -0.0003    0.0007    0.0028
    0.0014    0.0020    0.0028

The error you made in your code was that you specified B1 as a 1x4 vector instead of a 2x2 matrix:

B1 = [rho_g 0, 0 rho_u];

instead of

B1 = [rho_g 0; 0 rho_u];

Dyuman Joshi 2024-1-12

Best to ask a new question.

Smriti Verma 2024-1-16

I have a posted a question about modeling in Economics, would you have any idea by any chance? I could provide the link to the question.

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Answer 2

Dyuman Joshi 2024-1-10

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2068356-how-to-use-fsolve-to-solve-matrix-equations-involving-matrices-of-different-dimensions#answer_1387071

编辑：Dyuman Joshi 2024-1-10

在 MATLAB Online 中打开

% Calibration
sigma = 1;
kappa = 0.2;
beta = 0.99;
phi_pi = 1.5;
phi_x = 0.5;
sigma_i = 0.0025;
sigma_g = 0.0025;
sigma_u = 0.0025;
rho_g = 0.5;
rho_u = 0.3;
% Define A0, A1, A2, A3 for the 3 DSGE equations, where Z = [x; pi; i], S =
% [g; u], and eps = [eps_i; eps_g; eps_u]
A0 = [1 0 sigma; -kappa 1 0; -phi_x -phi_pi 1];
A1 = [1 sigma 0; 0 beta 0; 0 0 0];
A2 = [1 0; 0 1; 0 0];
A3 = [0 0 0; 0 0 0; sigma_i 0 0];
% Define B1, B2 for the two state equations: S_t = B1*S_t-1 + B2*eps_t
B1 = [rho_g 0; 0 rho_u];
B2 = [0 sigma_g 0; 0 0 sigma_u];
%size of x
r = 3;
c = 2;
F0 = fsolve(@(x) fun(x, A0, A1, A2, B1, r, c), zeros(r, c))
Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.
F0 = 3×2
    0.7163   -1.1074
    0.2837    1.1074
    0.7837    1.1074
F1 = A0\A3 % inv(A0)*A3
F1 = 3×3
   -0.0014         0         0
   -0.0003         0         0
    0.0014         0         0
G1 = F0*B1
G1 = 3×2
    0.3582   -0.3322
    0.1418    0.3322
    0.3918    0.3322
G2 = F0*B2 + F1
G2 = 3×3
   -0.0014    0.0018   -0.0028
   -0.0003    0.0007    0.0028
    0.0014    0.0020    0.0028

%% Function definition
%Define equations to be solved as a function
function y = fun(x, A0, A1, A2, B1, r, c)
x = reshape(x, r, c); 
y = A1*x*B1 - A0*x + A2;
end