I'm trying to solve and heat transfer problem using SciLab's ode function. The thing is: one of the parameters changes with time, h(t).
ODE
My question is: how can I pass an argument to ode function that is changing over time?
ode allows extra function's parameters as list :
It may happen that the simulator f needs extra arguments. In this
case, we can use the following feature. The f argument can also be a
list lst=list(f,u1,u2,...un) where f is a Scilab function with
syntax: ydot = f(t,y,u1,u2,...,un) and u1, u2, ..., un are extra
arguments which are automatically passed to the simulator simuf.
Extra parameter is a function of t
function y = f(t,y,h)
// define y here depending on t and h(t),eg y = t + h(t)
endfunction
function y = h(t)
// define here h(t), eg y = t
endfunction
// define y0,t0 and t
y = ode(y0, t0, t, list(f,h)) // this will pass the h function as a parameter
Extra is a vector for which we want to extract the corresponding term.
Since ode only compute the solution y at t. An idea is to look for Ti < t < Tj when ode performs a computation and get Hi < h < Hj.
This is rather ugly but totally works:
function y = h(t,T,H)
res = abs(t - T) // looking for nearest value of t in T
minres = min(res) // getting the smallest distance
lower = find(res==minres) // getting the index : T(lower)
res(res==minres)=%inf // looking for 2nd nearest value of t in T: nearest is set to inf
minres = min(res) // getting the smallest distance
upper = find(minres==res) // getting the index: T(upper)
// Now t is between T(lower) (nearest) and T(upper) (farest) (! T(lower) may be > T(upper))
y = ((T(upper)-t)*H(lower)+(t-T(lower))*H(upper))/(T(upper)-T(lower)) // computing h such as the barycenter with same distance to H(lower) and H(upper)
endfunction
function ydot=f(t, y,h,T,H)
hi = h(t,T,H) // if Ti< t < Tj; Hi<h(t,T,H)<Hj
disp([t,hi]) // with H = T, hi = t
ydot=y^2-y*sin(t)+cos(t) - hi // example of were to use hi
endfunction
// use base example of `ode`
y0=0;
t0=0;
t=0:0.1:%pi;
H = t // simple example
y = ode(y0,t0,t,list(f,h,t,H));
plot(t,y)
Related
Here is a made up set of function calls (I tried to make it complicated but perhaps it is easy).
function main(arg1, arg2) {
do_foo(arg1, arg2)
}
function do_foo(a, b) {
let x = a + b
let y = x * a
let z = x * b
let p = y + z
let q = x + z
let r = do_bar(&p)
let s = do_bar(&q)
}
function do_bar(&p, &q) {
*p += 1
*q += 3
let r = &p * &q
let s = &p + &q
let v = do_baz(&r, &s)
return &v
}
function do_baz(&a, &b) {
return *a + *b
}
How do you generally go about figuring out the liveness of variables and where you can insert instructions for reference counting?
Here is my attempt...
Start at the top function main. It starts with 2 arguments. Assume there is no copying that occurs. It passes the actual mutable values to do_foo.
Then we have x. X owns a and b. Then we see y. y is set to x, so link the previous x to this x. By r, we don't see x anymore, so perhaps it can be freed.... Looking at do_bar by itself, we know basically that p and q can't be garbage collected within this scope.
Basically, I have no idea how to start implementing an algorithm to implement ARC (ideally compile time reference counting, but runtime would be okay for now too to get started).
function main(arg1, arg2) {
let x = do_foo(arg1, arg2)
free(arg1)
free(arg2)
free(x)
}
function do_foo(a, b) {
let x = a + b
let y = x * a
let z = x * b
let p = y + z
free(y)
let q = x + z
free(x)
free(z)
let r = do_bar(&p)
let s = do_bar(&q)
return r + s
}
function do_bar(&p, &q) {
*p += 1
*q += 3
let r = &p * &q
let s = &p + &q
let v = do_baz(&r, &s)
free(r)
free(s)
return &v
}
function do_baz(&a, &b) {
return *a + *b
}
How do I start with implementing such an algorithm. I have searched for every paper on the topic but found no algorithms.
The following rules should do the job for your language.
When a variable is declared, increment its refcount
When a variable goes out of scope, decrement its refcount
When a reference-to-variable is assigned to a variable, adjust the reference counts for the variable(s):
increment the refcount for the variable whose reference is being assigned
decrement the refcount for the variable whose references was previously in the variable being assigned to (if it was not null)
When a variable containing a non-null reference-to-variable goes out of scope, decrement the refcount for the variable it referred to.
Note:
If your language allows reference-to-variable types to be used in data structures, "static" variables, etcetera, the rules abouve need to be extended ... in the obvious fashion.
An optimizing compiler may be able to eliminate some refcount increments and decrements.
Compile time reference counting:
There isn't really any such thing. Reference counting is done at runtime. It doesn't make sense to do it at compile time.
You are probably talking about analyzing the code to determine if runtime reference counting can be optimized or entirely eliminated.
I alluded to the former above. It is really a kind of peephole optimization.
The latter entails checking whether a reference-to-variable can ever escape; i.e. whether it could be used after the variable goes out of scope. (Try Googling for "escape analysis". This is kind of analogous to the "escape analysis" that a compiler could do to decide if an object could be allocated on the stack rather than in the heap.)
I have the following function which calculates statistic parameters. I would like to pass this function to nlfilter to do the calculation for a whole image. But the output of nlfilter must be a scalar.
How can I convert this to a function handle suitable for use with nlfilter so I can save the output of the function getStatistics2?
The getStatistics2 function's output is an struct array.
function [out] = getStatistics2(D)
D = double(D);
% out.MAX = max(D);%maximum
% out.MIN = min(D);%minimum
out.MEA = mean(D);%mean
out.MAD = mad(D);% mean absolute deviation y=mean(abs(X-mean(x)))
out.MED = median(D);%median
out.RAN = max(D) - min(D);%range
out.RMS = rms(D);%root mean square
out.STD = std(D);%stardard deviation
out.VAR= var(D);%variance
This is an interesting question. What's interesting is that your approach is almost perfect. The only reason it fails is because struct cannot be constructed using a numeric scalar input (i.e. struct(3)). The reason I mention this is because somewhere during the execution of nlfilter (specifically in mkconstarray.m), it calls the the following code:
repmat(feval(class, value), size);
Where:
class is 'struct'.
value is 0.
size is the size() of the input image, e.g. [100,100].
... and this fails because feval('struct', 0), which is equivalent to struct(0) - and this we already know to be invalid.
So what do we do? Create a custom class that can be constructed this way!
Here's an example of one such class:
classdef MyStatsClass % Value class
properties (GetAccess = public, SetAccess = private)
MAX#double scalar = NaN; % Maximum
MIN#double scalar = NaN; % Minimum
MEA#double scalar = NaN; % Mean
MAD#double scalar = NaN; % Mean absolute deviation y = mean(abs(X-mean(x)))
MED#double scalar = NaN; % Median
RMS#double scalar = NaN; % Root mean square
STD#double scalar = NaN; % Stardard deviation
VAR#double scalar = NaN; % Variance
RAN#double scalar = NaN; % Range
end % properties
methods (Access = public)
%% Constructor:
function obj = MyStatsClass(vec)
%% Special case:
if (nargin == 0) || (numel(vec) == 1) && (vec == 0)
% This happens during nlfilter allocation
return
end
%% Regular case:
obj.MAX = max(vec(:));
obj.MIN = min(vec(:));
obj.MEA = mean(vec(:));
obj.MAD = mad(vec(:));
obj.MED = median(vec(:));
obj.RMS = rms(vec(:));
obj.STD = std(vec(:));
obj.VAR = var(vec(:));
obj.RAN = obj.MAX - obj.MIN;
end % default constructor
end % public methods
end % classdef
And here's how you can use it:
function imF = q35693068(outputAsStruct)
if nargin == 0 || ~islogical(outputAsStruct) || ~isscalar(outputAsStruct)
outputAsStruct = false;
end
rng(35693068); % Set the random seed, for repeatability
WINDOW_SZ = 3;
im = randn(100);
imF = nlfilter(im, [WINDOW_SZ WINDOW_SZ], #MyStatsClass);
% If output is strictly needed as a struct:
if outputAsStruct
warning off MATLAB:structOnObject
imF = arrayfun(#struct,imF);
warning on MATLAB:structOnObject
end
Notice that I have added an optional input (outputAsStruct) that can force the output to be a struct array (and not an array of the type of our custom class, which is functionally identical to a read-only struct).
Notice also that by default nlfilter pads your array with zeros, which means that the (1,1) output will operate on an array that looks like this (assuming WINDOW_SZ=3):
[0 0 0
0 1.8096 0.5189
0 -0.3434 0.6586]
and not on im(1:WINDOW_SZ,1:WINDOW_SZ) which is:
[ 1.8096 0.5189 0.2811
-0.3434 0.6586 0.8919
-0.1525 0.7549 0.4497]
the "expected result" for im(1:WINDOW_SZ,1:WINDOW_SZ) will be found further "inside" the output array (in the case of WINDOW_SZ=3 at index (2,2)).
I am implementing a fast optimization algorithm using fixed point method in matlab. The goal of that method is that find optimal value of u. Denote u={u_i,i=1..2}. The optimal value of u can be obtained as following steps:
Sorry about my image because I cannot type mathematics equation in here.
To do that task, I tried to find u follows above steps. However, I don't know how to implement the term \sum_{j!=i} (u_j-1) in equation 25. This is my code. Please see it and could you give me some comment or suggestion about my implementation to correct them. Currently, I tried to run that code but it give an incorrect answer.
function u = compute_u_TV(Im0, N_class)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Initialization
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
theta=0.001;
gamma=0.01;
tau=0.1;
sigma=0.1;
N_class=2; % only have u1 and u2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Iterative segmentation process
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:N_class
v(:,:,i) = Im0/max(Im0(:)); % u between 0 and 1.
qxv(:,:,i) = zeros(size(Im0));
qyv(:,:,i) = zeros(size(Im0));
u(:,:,i) = v(:,:,i);
for iteration=1:10000
u_temp=u;
% Update v
Divqi = ( BackwardX(qxv(:,:,i)) + BackwardY(qyv(:,:,i)) );
Term = Divqi - u(:,:,i)/ (theta*gamma);
TermX = ForwardX(Term);
TermY = ForwardY(Term);
Norm = sqrt(TermX.^2 + TermY.^2);
Denom = 1 + tau*Norm;
%Equation 24
qxv(:,:,i) = (qxv(:,:,i) + tau*TermX)./Denom;
qyv(:,:,i) = (qyv(:,:,i) + tau*TermY)./Denom;
v(:,:,i) = u(:,:,i) - theta*gamma* Divqi; %Equation 23
% Update u
u(:,:,i) = (v(:,:,i) - theta* gamma* Divqi -theta*gamma*sigma*(sum(u(:))-u(:,:,i)-1))./(1+theta* gamma*sigma);
u(:,:,i) = max(u(:,:,i),0);
u(:,:,i) = min(u(:,:,i),1);
check=u_temp(:,:,i)-u(:,:,i);
if(abs(sum(check(:)))<=0.1)
break;
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Sub-functions- X.Berson
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [dx]=BackwardX(u);
[Ny,Nx] = size(u);
dx = u;
dx(2:Ny-1,2:Nx-1)=( u(2:Ny-1,2:Nx-1) - u(2:Ny-1,1:Nx-2) );
dx(:,Nx) = -u(:,Nx-1);
function [dy]=BackwardY(u);
[Ny,Nx] = size(u);
dy = u;
dy(2:Ny-1,2:Nx-1)=( u(2:Ny-1,2:Nx-1) - u(1:Ny-2,2:Nx-1) );
dy(Ny,:) = -u(Ny-1,:);
function [dx]=ForwardX(u);
[Ny,Nx] = size(u);
dx = zeros(Ny,Nx);
dx(1:Ny-1,1:Nx-1)=( u(1:Ny-1,2:Nx) - u(1:Ny-1,1:Nx-1) );
function [dy]=ForwardY(u);
[Ny,Nx] = size(u);
dy = zeros(Ny,Nx);
dy(1:Ny-1,1:Nx-1)=( u(2:Ny,1:Nx-1) - u(1:Ny-1,1:Nx-1) );
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of sub-function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
You should do
u(:,:,i) = (v(:,:,i) - theta* gamma* Divqi -theta*gamma*sigma* ...
(sum(u(:,:,1:size(u,3) ~= i),3) -1))./(1+theta* gamma*sigma);
The part you were searching for is
sum(u(:,:,1:size(u,3) ~= i),3)
Let's decompose this :
1:size(u,3) ~= i
is a vector containing all values from 1 to the max size of u on the third dimension except i.
Then
u(:,:,1:size(u,3) ~= i)
is all the matrix of the third dimension of u except for j = i
Finally,
sum(...,3)
is the sum of all the matrix by the thrid dimension.
Let me know if it does help!
I write in variable 'O' some values using
for i = 1:size(I,1)
for j = 1:size(1,I)
h = i * j;
O{h} = I(i, j) * theta(h);
end
end
I - double, theta - double.
I need to sum()all 'O' values, but when I do it its give me error: sum: wrong type argument 'cell'.
How can I sum() it?
P.s. when I want to see O(), its give me
O =
{
[1,1] = 0.0079764
[1,2] = 0.0035291
[1,3] = 0.0027539
[1,4] = 0.0034392
[1,5] = 0.017066
[1,6] = 0.0082958
[1,7] = 1.4764e-04
[1,8] = 0.0024597
[1,9] = 1.1155e-04
[1,10] = 0.0010342
[1,11] = 0.0039654
[1,12] = 0.0047713
[1,13] = 0.0054305
[1,14] = 3.3794e-04
[1,15] = 0.014323
[1,16] = 0.0026826
[1,17] = 0.013864
[1,18] = 0.0097778
[1,19] = 0.0058029
[1,20] = 0.0020726
[1,21] = 0.0016430
etc...
The exact answer to your question is to use cell2mat
sum (cell2mat (your_cell_o))
However, this is the very wrong way to solve your problem. The thing is that you should not have created a cell array in first place. You should have created a numeric array:
O = zeros (size (I), class (I));
for i = 1:rows (I)
for j = 1:columns (I)
h = i * j;
O(h) = I(i, j) * theta(h);
endfor
endfor
but even this is just really bad and slow. Octave is a language to vectorize operations. Instead, you should have:
h = (1:rows (I))' .* (1:columns (I)); # automatic broadcasting
O = I .* theta (h);
which assumes your function theta behaves properly and if givena matrix will compute the value for each of the element of h and return something of the same size.
If you get an error about wrong sizes, I will guess you have an old version of Octave that does not perform automatic broadcasting. If so, update Octave. If you really can't, then:
h = bsxfun (#times, (1:rows (I))', 1:columns (I));
:) I'm trying to code a Least Squares algorithm and I've come up with this:
function [y] = ex1_Least_Squares(xValues,yValues,x) % a + b*x + c*x^2 = y
points = size(xValues,1);
A = ones(points,3);
b = zeros(points,1);
for i=1:points
A(i,1) = 1;
A(i,2) = xValues(i);
A(i,3) = xValues(i)^2;
b(i) = yValues(i);
end
constants = (A'*A)\(A'*b);
y = constants(1) + constants(2)*x + constants(3)*x^2;
When I use this matlab script for linear functions, it works fine I think. However, when I'm passing 12 points of the sin(x) function I get really bad results.
These are the points I pass to the function:
xValues = [ -180; -144; -108; -72; -36; 0; 36; 72; 108; 144; 160; 180];
yValues = [sind(-180); sind(-144); sind(-108); sind(-72); sind(-36); sind(0); sind(36); sind(72); sind(108); sind(144); sind(160); sind(180) ];
And the result is sin(165°) = 0.559935259380508, when it should be sin(165°) = 0.258819
There is no reason why fitting a parabola to a full period of a sinusoid should give good results. These two curves are unrelated.
MATLAB already contains a least square polynomial fitting function, polyfit and a complementary function, polyval. Although you are probably supposed to write your own, trying out something like the following will be educational:
xValues = [ -180; -144; -108; -72; -36; 0; 36; 72; 108; 144; 160; 180];
% you may want to experiment with different ranges of xValues
yValues = sind(xValues);
% try this with different values of n, say 2, 3, and 4
p = polyfit(xValues,yValues,n);
x = -180:36:180;
y = polyval(p,x);
plot(xValues,yValues);
hold on
plot(x,y,'r');
Also, more generically, you should avoid using loops as you have in your code. This should be equivalent:
points = size(xValues,1);
A = ones(points,3);
A(:,2) = xValues;
A(:,3) = xValues.^2; % .^ and ^ are different
The part of the loop involving b is equivalent to doing b = yValues; either name the incoming variable b or just use the variable yValues, there's no need to make a copy of it.