Using MATLAB to solve Poisson matrix equation
$begingroup$
I'm not sure if this is the correct forum to post my question. I'm working on a Poisson-based maths assignment and am stuck as regards finding the solution to the Poisson matrix equation. The matrix I have is rectangular. I've found some MATLAB code online for solving Poisson's equation and am just wondering if you could suggest which might be the best to look into for my particular problem (question 4)?
I'm new to Poisson and MATLAB, so thanks for any advice you can give.
Links to MATLAB code for solving Poisson's equation:
https://uk.mathworks.com/matlabcentral/fileexchange/38090-2d-poisson-equation
https://uk.mathworks.com/help/pde/ug/pde.pdemodel.solvepde.html
https://staff.washington.edu/rjl/fdmbook/matlab/poisson.m
https://www.math.utah.edu/~wright/courses/5620/notes/FD2PoissonEx.pdf
http://people.bu.edu/andasari/courses/Fall2015/LectureNotes/Lecture14_27Oct2015.pdf
https://math.boisestate.edu/~wright/courses/m567/FD2-Poisson.pdf
MATLAB output:
ans =
[ -14203947/3248336, -940473/203021, -382635/141232, 233/203021, 6708477/3248336]
[ -362575/31234, -217820/15617, -16915/1358, -146040/15617, -171485/31234]
[ -18233133/3248336, -1440217/203021, -891885/141232, -905553/203021, -7065717/3248336]
$therefore$
$begin{align}
P_{13} = -4.37qquad P_{23} = -4.63qquad P_{33} = -2.71qquad P_{43} = 0.00qquad,,,,, P_{53} = 2.07
end{align}$
$begin{align}
P_{12} = -11.61,,,,,,,,, P_{22} = -13.95,,,,,,,,, P_{32} = -12.46,,,,,,,,, P_{42} = -9.35qquad P_{52} = -5.49
end{align}$
$begin{align}
P_{11} = -5.61qquad P_{21} = -7.09qquad P_{31} = -6.32qquad P_{41} = -4.46qquad P_{51} = -2.18
end{align}$
matlab matrix-equations poissons-equation
asked Dec 24 '18 at 15:56
somerssomers
2828
$endgroup$
add a comment |
$begingroup$
I'm not sure if this is the correct forum to post my question. I'm working on a Poisson-based maths assignment and am stuck as regards finding the solution to the Poisson matrix equation. The matrix I have is rectangular. I've found some MATLAB code online for solving Poisson's equation and am just wondering if you could suggest which might be the best to look into for my particular problem (question 4)?
I'm new to Poisson and MATLAB, so thanks for any advice you can give.
Links to MATLAB code for solving Poisson's equation:
https://uk.mathworks.com/matlabcentral/fileexchange/38090-2d-poisson-equation
https://uk.mathworks.com/help/pde/ug/pde.pdemodel.solvepde.html
https://staff.washington.edu/rjl/fdmbook/matlab/poisson.m
https://www.math.utah.edu/~wright/courses/5620/notes/FD2PoissonEx.pdf
http://people.bu.edu/andasari/courses/Fall2015/LectureNotes/Lecture14_27Oct2015.pdf
https://math.boisestate.edu/~wright/courses/m567/FD2-Poisson.pdf
MATLAB output:
ans =
[ -14203947/3248336, -940473/203021, -382635/141232, 233/203021, 6708477/3248336]
[ -362575/31234, -217820/15617, -16915/1358, -146040/15617, -171485/31234]
[ -18233133/3248336, -1440217/203021, -891885/141232, -905553/203021, -7065717/3248336]
$therefore$
$begin{align}
P_{13} = -4.37qquad P_{23} = -4.63qquad P_{33} = -2.71qquad P_{43} = 0.00qquad,,,,, P_{53} = 2.07
end{align}$
$begin{align}
P_{12} = -11.61,,,,,,,,, P_{22} = -13.95,,,,,,,,, P_{32} = -12.46,,,,,,,,, P_{42} = -9.35qquad P_{52} = -5.49
end{align}$
$begin{align}
P_{11} = -5.61qquad P_{21} = -7.09qquad P_{31} = -6.32qquad P_{41} = -4.46qquad P_{51} = -2.18
end{align}$
matlab matrix-equations poissons-equation
asked Dec 24 '18 at 15:56
somerssomers
2828
$endgroup$
$begingroup$
Can you show the matrix equation corresponding to the given Poisson equation?
$endgroup$
– Alex Vong
Dec 24 '18 at 16:38
$begingroup$
@AlexVong - Yes, I'll post.
$endgroup$
– somers
Dec 24 '18 at 16:48
$begingroup$
@AlexVong - Should I post the set of equations for part 3?
$endgroup$
– somers
Dec 24 '18 at 17:03
$begingroup$
Yes! I think it will help people understand the problem.
$endgroup$
– Alex Vong
Dec 24 '18 at 17:05
$begingroup$
@AlexVong - Ok no worries, I'll post them.
$endgroup$
– somers
Dec 24 '18 at 18:08
add a comment |
$begingroup$
I'm not sure if this is the correct forum to post my question. I'm working on a Poisson-based maths assignment and am stuck as regards finding the solution to the Poisson matrix equation. The matrix I have is rectangular. I've found some MATLAB code online for solving Poisson's equation and am just wondering if you could suggest which might be the best to look into for my particular problem (question 4)?
I'm new to Poisson and MATLAB, so thanks for any advice you can give.
Links to MATLAB code for solving Poisson's equation:
https://uk.mathworks.com/matlabcentral/fileexchange/38090-2d-poisson-equation
https://uk.mathworks.com/help/pde/ug/pde.pdemodel.solvepde.html
https://staff.washington.edu/rjl/fdmbook/matlab/poisson.m
https://www.math.utah.edu/~wright/courses/5620/notes/FD2PoissonEx.pdf
http://people.bu.edu/andasari/courses/Fall2015/LectureNotes/Lecture14_27Oct2015.pdf
https://math.boisestate.edu/~wright/courses/m567/FD2-Poisson.pdf
MATLAB output:
ans =
[ -14203947/3248336, -940473/203021, -382635/141232, 233/203021, 6708477/3248336]
[ -362575/31234, -217820/15617, -16915/1358, -146040/15617, -171485/31234]
[ -18233133/3248336, -1440217/203021, -891885/141232, -905553/203021, -7065717/3248336]
$therefore$
$begin{align}
P_{13} = -4.37qquad P_{23} = -4.63qquad P_{33} = -2.71qquad P_{43} = 0.00qquad,,,,, P_{53} = 2.07
end{align}$
$begin{align}
P_{12} = -11.61,,,,,,,,, P_{22} = -13.95,,,,,,,,, P_{32} = -12.46,,,,,,,,, P_{42} = -9.35qquad P_{52} = -5.49
end{align}$
$begin{align}
P_{11} = -5.61qquad P_{21} = -7.09qquad P_{31} = -6.32qquad P_{41} = -4.46qquad P_{51} = -2.18
end{align}$
matlab matrix-equations poissons-equation
asked Dec 24 '18 at 15:56
somerssomers
2828
$endgroup$
I'm not sure if this is the correct forum to post my question. I'm working on a Poisson-based maths assignment and am stuck as regards finding the solution to the Poisson matrix equation. The matrix I have is rectangular. I've found some MATLAB code online for solving Poisson's equation and am just wondering if you could suggest which might be the best to look into for my particular problem (question 4)?
I'm new to Poisson and MATLAB, so thanks for any advice you can give.
Links to MATLAB code for solving Poisson's equation:
https://uk.mathworks.com/matlabcentral/fileexchange/38090-2d-poisson-equation
https://uk.mathworks.com/help/pde/ug/pde.pdemodel.solvepde.html
https://staff.washington.edu/rjl/fdmbook/matlab/poisson.m
https://www.math.utah.edu/~wright/courses/5620/notes/FD2PoissonEx.pdf
http://people.bu.edu/andasari/courses/Fall2015/LectureNotes/Lecture14_27Oct2015.pdf
https://math.boisestate.edu/~wright/courses/m567/FD2-Poisson.pdf
MATLAB output:
ans =
[ -14203947/3248336, -940473/203021, -382635/141232, 233/203021, 6708477/3248336]
[ -362575/31234, -217820/15617, -16915/1358, -146040/15617, -171485/31234]
[ -18233133/3248336, -1440217/203021, -891885/141232, -905553/203021, -7065717/3248336]
$therefore$
$begin{align}
P_{13} = -4.37qquad P_{23} = -4.63qquad P_{33} = -2.71qquad P_{43} = 0.00qquad,,,,, P_{53} = 2.07
end{align}$
$begin{align}
P_{12} = -11.61,,,,,,,,, P_{22} = -13.95,,,,,,,,, P_{32} = -12.46,,,,,,,,, P_{42} = -9.35qquad P_{52} = -5.49
end{align}$
$begin{align}
P_{11} = -5.61qquad P_{21} = -7.09qquad P_{31} = -6.32qquad P_{41} = -4.46qquad P_{51} = -2.18
end{align}$
matlab matrix-equations poissons-equation
matlab matrix-equations poissons-equation
asked Dec 24 '18 at 15:56
somerssomers
2828
asked Dec 24 '18 at 15:56
somerssomers
2828
edited Dec 25 '18 at 2:50
somers
asked Dec 24 '18 at 15:56
somerssomers
2828
asked Dec 24 '18 at 15:56
somerssomers
2828
asked Dec 24 '18 at 15:56
somerssomers
2828
2828
$begingroup$
Can you show the matrix equation corresponding to the given Poisson equation?
$endgroup$
– Alex Vong
Dec 24 '18 at 16:38
$begingroup$
@AlexVong - Yes, I'll post.
$endgroup$
– somers
Dec 24 '18 at 16:48
$begingroup$
@AlexVong - Should I post the set of equations for part 3?
$endgroup$
– somers
Dec 24 '18 at 17:03
$begingroup$
Yes! I think it will help people understand the problem.
$endgroup$
– Alex Vong
Dec 24 '18 at 17:05
$begingroup$
@AlexVong - Ok no worries, I'll post them.
$endgroup$
– somers
Dec 24 '18 at 18:08
add a comment |
$begingroup$
Can you show the matrix equation corresponding to the given Poisson equation?
$endgroup$
– Alex Vong
Dec 24 '18 at 16:38
$begingroup$
@AlexVong - Yes, I'll post.
$endgroup$
– somers
Dec 24 '18 at 16:48
$begingroup$
@AlexVong - Should I post the set of equations for part 3?
$endgroup$
– somers
Dec 24 '18 at 17:03
$begingroup$
Yes! I think it will help people understand the problem.
$endgroup$
– Alex Vong
Dec 24 '18 at 17:05
$begingroup$
@AlexVong - Ok no worries, I'll post them.
$endgroup$
– somers
Dec 24 '18 at 18:08
$begingroup$
Can you show the matrix equation corresponding to the given Poisson equation?
$endgroup$
– Alex Vong
Dec 24 '18 at 16:38
$begingroup$
Can you show the matrix equation corresponding to the given Poisson equation?
$endgroup$
– Alex Vong
Dec 24 '18 at 16:38
$begingroup$
@AlexVong - Yes, I'll post.
$endgroup$
– somers
Dec 24 '18 at 16:48
$begingroup$
@AlexVong - Yes, I'll post.
$endgroup$
– somers
Dec 24 '18 at 16:48
$begingroup$
@AlexVong - Should I post the set of equations for part 3?
$endgroup$
– somers
Dec 24 '18 at 17:03
$begingroup$
@AlexVong - Should I post the set of equations for part 3?
$endgroup$
– somers
Dec 24 '18 at 17:03
$begingroup$
Yes! I think it will help people understand the problem.
$endgroup$
– Alex Vong
Dec 24 '18 at 17:05
$begingroup$
Yes! I think it will help people understand the problem.
$endgroup$
– Alex Vong
Dec 24 '18 at 17:05
$begingroup$
@AlexVong - Ok no worries, I'll post them.
$endgroup$
– somers
Dec 24 '18 at 18:08
$begingroup$
@AlexVong - Ok no worries, I'll post them.
$endgroup$
– somers
Dec 24 '18 at 18:08
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since you have shown your work, I decide to write an answer. (Although I think your question is more on-topic for stackoverflow. Also, please write in latex instead of posting images in the future since images are impossible to search (our OCR technology is not that good yet))
Actually, you have already solved the harder part of the problem. Assuming you've done it right, the only thing left to do is to solve the linear system given by the system of equations: $$P_{i, j} = frac{P_{i + 1, j} + P_{i - 1, j} + P_{i, j + 1} + P_{i, j - 1} - 25 + 5 frac{i}{2} frac{j}{2}}{4}$$
where $1le i le 5$ and $1 le j le 3$ with the convention that
$P_{i, j} =
begin{cases}
20 &text{ if } j < 1 text{ or } j > 3 \
0 &text{ if (} i < 1 text{ or } i > 5 text{) and } 1 le j le n
end{cases}$
(Thank god we don't have buffer overflow!)
The following program does exactly that. I have tested it in octave with octsympy. Please check if it works in matlab as well.
%{ see https://stackoverflow.com/questions/35115039 %}
if__ = @(varargin) varargin{3 - varargin{1}}();
if_ = @(pred, if_clause, else_clause) if__(pred, if_clause, else_clause);
%{ size of the matrix %}
m = 5;
n = 3;
%{ if j goes out of range, return 20 %}
%{ else if i goes out of range, return 0 %}
%{ otherwise, return the symbol Pij %}
P = @(i, j) if_(j < 1 || j > n, ...
@() 20, ...
@() if_(i < 1 || i > m, ...
@() 0, ...
@() sym(sprintf('P%d%d', i, j))));
%{ our equation with P(i, j) moved to the RHS %}
eqn = @(i, j) ...
(P(i + 1, j) + P(i - 1, j) + P(i, j + 1) + P(i, j - 1) ...
- 25 ...
+ sym(5) * i / 2 * j / 2) ...
/ 4 ...
- P(i, j);
%{ system of equations obtained by substituting the above equation %}
%{ with 1 <= i <= m and 1 <= j <= n %}
eqns = arrayfun(@(i) arrayfun(@(j) eqn(i, j), ...
1:n, ...
'UniformOutput', false), ...
1:m, ...
'UniformOutput', false);
eqns2 = vertcat(eqns{:});
eqns3 = vertcat(eqns2{:});
%{ solve the system of equations %}
solns = solve(eqns3);
%{ display the solutions %}
[solns.P13 solns.P23 solns.P33 solns.P43 solns.P53; ...
solns.P12 solns.P22 solns.P32 solns.P42 solns.P52; ...
solns.P11 solns.P21 solns.P31 solns.P41 solns.P51]
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
$endgroup$
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
add a comment |
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Since you have shown your work, I decide to write an answer. (Although I think your question is more on-topic for stackoverflow. Also, please write in latex instead of posting images in the future since images are impossible to search (our OCR technology is not that good yet))
Actually, you have already solved the harder part of the problem. Assuming you've done it right, the only thing left to do is to solve the linear system given by the system of equations: $$P_{i, j} = frac{P_{i + 1, j} + P_{i - 1, j} + P_{i, j + 1} + P_{i, j - 1} - 25 + 5 frac{i}{2} frac{j}{2}}{4}$$
where $1le i le 5$ and $1 le j le 3$ with the convention that
$P_{i, j} =
begin{cases}
20 &text{ if } j < 1 text{ or } j > 3 \
0 &text{ if (} i < 1 text{ or } i > 5 text{) and } 1 le j le n
end{cases}$
(Thank god we don't have buffer overflow!)
The following program does exactly that. I have tested it in octave with octsympy. Please check if it works in matlab as well.
%{ see https://stackoverflow.com/questions/35115039 %}
if__ = @(varargin) varargin{3 - varargin{1}}();
if_ = @(pred, if_clause, else_clause) if__(pred, if_clause, else_clause);
%{ size of the matrix %}
m = 5;
n = 3;
%{ if j goes out of range, return 20 %}
%{ else if i goes out of range, return 0 %}
%{ otherwise, return the symbol Pij %}
P = @(i, j) if_(j < 1 || j > n, ...
@() 20, ...
@() if_(i < 1 || i > m, ...
@() 0, ...
@() sym(sprintf('P%d%d', i, j))));
%{ our equation with P(i, j) moved to the RHS %}
eqn = @(i, j) ...
(P(i + 1, j) + P(i - 1, j) + P(i, j + 1) + P(i, j - 1) ...
- 25 ...
+ sym(5) * i / 2 * j / 2) ...
/ 4 ...
- P(i, j);
%{ system of equations obtained by substituting the above equation %}
%{ with 1 <= i <= m and 1 <= j <= n %}
eqns = arrayfun(@(i) arrayfun(@(j) eqn(i, j), ...
1:n, ...
'UniformOutput', false), ...
1:m, ...
'UniformOutput', false);
eqns2 = vertcat(eqns{:});
eqns3 = vertcat(eqns2{:});
%{ solve the system of equations %}
solns = solve(eqns3);
%{ display the solutions %}
[solns.P13 solns.P23 solns.P33 solns.P43 solns.P53; ...
solns.P12 solns.P22 solns.P32 solns.P42 solns.P52; ...
solns.P11 solns.P21 solns.P31 solns.P41 solns.P51]
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
$endgroup$
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
add a comment |
$begingroup$
Since you have shown your work, I decide to write an answer. (Although I think your question is more on-topic for stackoverflow. Also, please write in latex instead of posting images in the future since images are impossible to search (our OCR technology is not that good yet))
Actually, you have already solved the harder part of the problem. Assuming you've done it right, the only thing left to do is to solve the linear system given by the system of equations: $$P_{i, j} = frac{P_{i + 1, j} + P_{i - 1, j} + P_{i, j + 1} + P_{i, j - 1} - 25 + 5 frac{i}{2} frac{j}{2}}{4}$$
where $1le i le 5$ and $1 le j le 3$ with the convention that
$P_{i, j} =
begin{cases}
20 &text{ if } j < 1 text{ or } j > 3 \
0 &text{ if (} i < 1 text{ or } i > 5 text{) and } 1 le j le n
end{cases}$
(Thank god we don't have buffer overflow!)
The following program does exactly that. I have tested it in octave with octsympy. Please check if it works in matlab as well.
%{ see https://stackoverflow.com/questions/35115039 %}
if__ = @(varargin) varargin{3 - varargin{1}}();
if_ = @(pred, if_clause, else_clause) if__(pred, if_clause, else_clause);
%{ size of the matrix %}
m = 5;
n = 3;
%{ if j goes out of range, return 20 %}
%{ else if i goes out of range, return 0 %}
%{ otherwise, return the symbol Pij %}
P = @(i, j) if_(j < 1 || j > n, ...
@() 20, ...
@() if_(i < 1 || i > m, ...
@() 0, ...
@() sym(sprintf('P%d%d', i, j))));
%{ our equation with P(i, j) moved to the RHS %}
eqn = @(i, j) ...
(P(i + 1, j) + P(i - 1, j) + P(i, j + 1) + P(i, j - 1) ...
- 25 ...
+ sym(5) * i / 2 * j / 2) ...
/ 4 ...
- P(i, j);
%{ system of equations obtained by substituting the above equation %}
%{ with 1 <= i <= m and 1 <= j <= n %}
eqns = arrayfun(@(i) arrayfun(@(j) eqn(i, j), ...
1:n, ...
'UniformOutput', false), ...
1:m, ...
'UniformOutput', false);
eqns2 = vertcat(eqns{:});
eqns3 = vertcat(eqns2{:});
%{ solve the system of equations %}
solns = solve(eqns3);
%{ display the solutions %}
[solns.P13 solns.P23 solns.P33 solns.P43 solns.P53; ...
solns.P12 solns.P22 solns.P32 solns.P42 solns.P52; ...
solns.P11 solns.P21 solns.P31 solns.P41 solns.P51]
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
$endgroup$
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
add a comment |
$begingroup$
Since you have shown your work, I decide to write an answer. (Although I think your question is more on-topic for stackoverflow. Also, please write in latex instead of posting images in the future since images are impossible to search (our OCR technology is not that good yet))
Actually, you have already solved the harder part of the problem. Assuming you've done it right, the only thing left to do is to solve the linear system given by the system of equations: $$P_{i, j} = frac{P_{i + 1, j} + P_{i - 1, j} + P_{i, j + 1} + P_{i, j - 1} - 25 + 5 frac{i}{2} frac{j}{2}}{4}$$
where $1le i le 5$ and $1 le j le 3$ with the convention that
$P_{i, j} =
begin{cases}
20 &text{ if } j < 1 text{ or } j > 3 \
0 &text{ if (} i < 1 text{ or } i > 5 text{) and } 1 le j le n
end{cases}$
(Thank god we don't have buffer overflow!)
The following program does exactly that. I have tested it in octave with octsympy. Please check if it works in matlab as well.
%{ see https://stackoverflow.com/questions/35115039 %}
if__ = @(varargin) varargin{3 - varargin{1}}();
if_ = @(pred, if_clause, else_clause) if__(pred, if_clause, else_clause);
%{ size of the matrix %}
m = 5;
n = 3;
%{ if j goes out of range, return 20 %}
%{ else if i goes out of range, return 0 %}
%{ otherwise, return the symbol Pij %}
P = @(i, j) if_(j < 1 || j > n, ...
@() 20, ...
@() if_(i < 1 || i > m, ...
@() 0, ...
@() sym(sprintf('P%d%d', i, j))));
%{ our equation with P(i, j) moved to the RHS %}
eqn = @(i, j) ...
(P(i + 1, j) + P(i - 1, j) + P(i, j + 1) + P(i, j - 1) ...
- 25 ...
+ sym(5) * i / 2 * j / 2) ...
/ 4 ...
- P(i, j);
%{ system of equations obtained by substituting the above equation %}
%{ with 1 <= i <= m and 1 <= j <= n %}
eqns = arrayfun(@(i) arrayfun(@(j) eqn(i, j), ...
1:n, ...
'UniformOutput', false), ...
1:m, ...
'UniformOutput', false);
eqns2 = vertcat(eqns{:});
eqns3 = vertcat(eqns2{:});
%{ solve the system of equations %}
solns = solve(eqns3);
%{ display the solutions %}
[solns.P13 solns.P23 solns.P33 solns.P43 solns.P53; ...
solns.P12 solns.P22 solns.P32 solns.P42 solns.P52; ...
solns.P11 solns.P21 solns.P31 solns.P41 solns.P51]
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
$endgroup$
Since you have shown your work, I decide to write an answer. (Although I think your question is more on-topic for stackoverflow. Also, please write in latex instead of posting images in the future since images are impossible to search (our OCR technology is not that good yet))
Actually, you have already solved the harder part of the problem. Assuming you've done it right, the only thing left to do is to solve the linear system given by the system of equations: $$P_{i, j} = frac{P_{i + 1, j} + P_{i - 1, j} + P_{i, j + 1} + P_{i, j - 1} - 25 + 5 frac{i}{2} frac{j}{2}}{4}$$
where $1le i le 5$ and $1 le j le 3$ with the convention that
$P_{i, j} =
begin{cases}
20 &text{ if } j < 1 text{ or } j > 3 \
0 &text{ if (} i < 1 text{ or } i > 5 text{) and } 1 le j le n
end{cases}$
(Thank god we don't have buffer overflow!)
The following program does exactly that. I have tested it in octave with octsympy. Please check if it works in matlab as well.
%{ see https://stackoverflow.com/questions/35115039 %}
if__ = @(varargin) varargin{3 - varargin{1}}();
if_ = @(pred, if_clause, else_clause) if__(pred, if_clause, else_clause);
%{ size of the matrix %}
m = 5;
n = 3;
%{ if j goes out of range, return 20 %}
%{ else if i goes out of range, return 0 %}
%{ otherwise, return the symbol Pij %}
P = @(i, j) if_(j < 1 || j > n, ...
@() 20, ...
@() if_(i < 1 || i > m, ...
@() 0, ...
@() sym(sprintf('P%d%d', i, j))));
%{ our equation with P(i, j) moved to the RHS %}
eqn = @(i, j) ...
(P(i + 1, j) + P(i - 1, j) + P(i, j + 1) + P(i, j - 1) ...
- 25 ...
+ sym(5) * i / 2 * j / 2) ...
/ 4 ...
- P(i, j);
%{ system of equations obtained by substituting the above equation %}
%{ with 1 <= i <= m and 1 <= j <= n %}
eqns = arrayfun(@(i) arrayfun(@(j) eqn(i, j), ...
1:n, ...
'UniformOutput', false), ...
1:m, ...
'UniformOutput', false);
eqns2 = vertcat(eqns{:});
eqns3 = vertcat(eqns2{:});
%{ solve the system of equations %}
solns = solve(eqns3);
%{ display the solutions %}
[solns.P13 solns.P23 solns.P33 solns.P43 solns.P53; ...
solns.P12 solns.P22 solns.P32 solns.P42 solns.P52; ...
solns.P11 solns.P21 solns.P31 solns.P41 solns.P51]
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
edited Dec 24 '18 at 22:13
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
answered Dec 24 '18 at 20:08
Alex VongAlex Vong
1,342819
1,342819
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
add a comment |
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
My god, thank you very much Alex! I will test in MATLAB and let you know.
$endgroup$
– somers
Dec 24 '18 at 22:15
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
$begingroup$
Your code works fine in MATLAB R2018b, I've posted the output.
$endgroup$
– somers
Dec 25 '18 at 3:22
add a comment |
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$begingroup$
Can you show the matrix equation corresponding to the given Poisson equation?
$endgroup$
– Alex Vong
Dec 24 '18 at 16:38
$begingroup$
@AlexVong - Yes, I'll post.
$endgroup$
– somers
Dec 24 '18 at 16:48
$begingroup$
@AlexVong - Should I post the set of equations for part 3?
$endgroup$
– somers
Dec 24 '18 at 17:03
$begingroup$
Yes! I think it will help people understand the problem.
$endgroup$
– Alex Vong
Dec 24 '18 at 17:05
$begingroup$
@AlexVong - Ok no worries, I'll post them.
$endgroup$
– somers
Dec 24 '18 at 18:08