MATLAB code False Position Method (root)

%The sequence of points that converges to the root is
%generated via c=[a*f(b)-b*f(a)]/[f(b)-f(a)]

f = @(x)(x*cos(x)+1);

%
% False position uses the regula falsi method to approximate a root of f(x) = 0
% in the interval [a,b].
%
%       f is an anonymous function representing f(x),
%       a and b are the limits of interval [a,b],
%       kmax is the maximum number of iterations (default 20),
%       tol is the scalar tolerance for convergence (default 1e-4),
%
%       r is the approximate root of f(x) = 0,
%       k is the number of iterations needed for convergence.
%
a=-2;
b=4;
tol = 1e-2;
kmax = 20;
c = zeros(1,kmax);      % Pre-allocate
if f(a)*f(b) > 0
     r = 'failure';
     return
end
disp(' k      a           b')
for k = 1:kmax,
    c(k) = (a*f(b)-b*f(a))/(f(b)-f(a));      % Find the x-intercept
    if f(c(k)) == 0      % Stop if a root has been found
        return
    end
   fprintf('%2i      %11.6f%11.6f\n',k,a,b)
    if f(b)*f(c(k)) > 0      % Check sign changes
        b = c(k);          % Adjust the endpoint of interval
    else a = c(k);
    end
    c(k+1) = (a*f(b)-b*f(a))/(f(b)-f(a));   % Find the next x-intercept
    if abs(c(k+1)-c(k)) < tol,     % Stop if tolerance is met
        r = c(k+1);
        break
    end
end

fx = f(r);
fprintf('r =  %11.6f\n', r);
fprintf('k = %11.1f\n', k);

mp = linspace(-2,4);
fp = arrayfun(f,mp,'UniformOutput',false); %func(mp);
fp = cat(1, fp{:});
hold on
plot(mp,fp),grid
plot(r,fx,'.','MarkerSize',20)
hold off

k      a           b
 1        -2.000000   4.000000
 2         1.189493   4.000000
 3         1.189493   2.515720
 4         1.960504   2.515720
r =     2.073843
k =         4.0

Published with MATLAB® R2017a

MATLAB code Bisection Method (root)

%Bisection method
%Step 1: Choose lower xl and upper xu guesses for
%the root such that the function changes sign
%over the interval. This can be checked by ensuring
%that f(xl)*f(xu) < 0.
%Step 2: An estimate of the root xr is determined
%by xr = (xl + xu)/2
%Step 3: Make the following evaluations to determine
%in which subinterval the root lies:
%(a) If f(xl)f(xr) < 0, the root lies in the lower
%subinterval. Therefore, set xu 5 xr and return to step 2.
%(b) If f(xl)f(xr) > 0, the root lies in the upper
%subinterval. Therefore, set xl 5 xr and return to step 2.
%(c) If f(xl)f(xr) = 0, the root equals xr; terminate the computation.

%Example

Pipes and connections between functions in R

A recent addition to R is the pipe-forwarding mechanism (%>%) within the magrittr package. This is extremely useful when using the dplyr, ggvis, and tidyr packages, among others. Pipe forwarding is an alternative to nesting that yields code that can be read from top to bottom. Here we demonstrate an example that compares traditional (nested) dplyr function calls to the new pipe operator.

Summary of Matrix Operators in R

Summary of Matrix Operators in R

Parallel with R. Example with snow

Snow provides support for easily executing R functions in parallel. Most of the parallel execution functions in snow are variations of the standard lapply() function, making snow fairly easy to learn. To implement these parallel operations, snow uses a master/ worker architecture, where the master sends tasks to the workers, and the workers execute the tasks and return the results to the master.

The basic cluster creation function is makeCluster() which can create any type of cluster. snow includes a number of functions that we could use, including clusterApply(), clusterApplyLB(), and parLapply(). For this example, we’ll use clusterApply(). You call it exactly the same as lapply(), except that it takes a snow cluster object as the first argument. We also need to load MASS on the workers, rather than on the master, since it’s the workers that use the “Boston” dataset.

We’ll use snow.time() to gather timing information about the overall execution. We will also use snow.time()’s plotting capability to visualize the task execution on the workers. 

ARS (Cartesian)

ARS is a basic reservoir simulation spreadsheet. With ARS you can do examples 

considering the follow features:

    -One dimension
    -One phase (oil)
    -Incompressible or Slightly compressible fluid
    -Rock porosity constant or depending of pressure
    -5 different types of boundary conditions
    -One well (specified production rate or specified flowing pressure)

Reservoir Engineering Tools Android App Tutorial

Reservoir Engineering tools

Android App with Equation of state, Fluid flow in porous media, Reservoir Simulation (One phase-1D) and Waterflooding

Download it in Playstore