Using Jacobi method and Gauss-Seidel iterative methods to solve the following system
The required precision is =0.00001, and the maximum iteration number N=25. Compare the number of iterations and the convergence of these two methods
The MDP toolbox proposes functions related to the resolution of discrete-time Markov Decision Process : finite horizon, value iteration, policy iteration, linear programming algorithms with some variants.
The functions (m-functions) were developped with MATLAB v6.0 (one of the functions requires the Mathworks Optimization Toolbox) by the decision team of the Biometry and Artificial Intelligence Unit of INRA Toulouse (France).
The version 2.0 (February 2005) handles sparse matrices and contains an example
％ EM algorithm for k multidimensional Gaussian mixture estimation
％
％ Inputs:
％ X(n,d) - input data, n=number of observations, d=dimension of variable
％ k - maximum number of Gaussian components allowed
％ ltol - percentage of the log likelihood difference between 2 iterations ([] for none)
％ maxiter - maximum number of iteration allowed ([] for none)
％ pflag - 1 for plotting GM for 1D or 2D cases only, 0 otherwise ([] for none)
％ Init - structure of initial W, M, V: Init.W, Init.M, Init.V ([] for none)
％
％ Ouputs:
％ W(1,k) - estimated weights of GM
％ M(d,k) - estimated mean vectors of GM
％ V(d,d,k) - estimated covariance matrices of GM
％ L - log likelihood of estimates
％
function [U,center,result,w,obj_fcn]= fenlei(data)
[data_n,in_n] = size(data)
m= 2 ％ Exponent for U
max_iter = 100 ％ Max. iteration
min_impro =1e-5 ％ Min. improvement
c=3
[center, U, obj_fcn] = fcm(data, c)
for i=1:max_iter
if F(U)>0.98
break
else
w_new=eye(in_n,in_n)
center1=sum(center)/c
a=center1(1)./center1
deta=center-center1(ones(c,1),:)
w=sqrt(sum(deta.^2)).*a
for j=1:in_n
w_new(j,j)=w(j)
end
data1=data*w_new
[center, U, obj_fcn] = fcm(data1, c)
center=center./w(ones(c,1),:)
obj_fcn=obj_fcn/sum(w.^2)
end
end
display(i)
result=zeros(1,data_n) U_=max(U)
for i=1:data_n
for j=1:c
if U(j,i)==U_(i)
result(i)=j continue
end
end
end
％ Train a two layer neural network with the Levenberg-Marquardt
％ method.
％
％ If desired, it is possible to use regularization by
％ weight decay. Also pruned (ie. not fully connected) networks can
％ be trained.
％
％ Given a set of corresponding input-output pairs and an initial
％ network,
％ [W1,W2,critvec,iteration,lambda]=marq(NetDef,W1,W2,PHI,Y,trparms)
％ trains the network with the Levenberg-Marquardt method.
％
％ The activation functions can be either linear or tanh. The
％ network architecture is defined by the matrix NetDef which
％ has two rows. The first row specifies the hidden layer and the
％ second row specifies the output layer.
program to solve a finite difference discretization of Helmholtz equation :
(d2/dx2)u + (d2/dy2)u - alpha u = f using Jacobi iterative method.
COMMENTS: OpenMP version 3: 1 PR outside the iteration loop, 4 Barriers
Directives are used in this code to achieve paralleism.
All do loops are parallized with default static scheduling.
A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. The term was coined by Benoî t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2]
