# Planet Octave

## August 27, 2015

### Asma Afzal

#### Wrapping up

So, the GSoC period has come to an end now.

Project Goals
My project was about creating Matlab compatible wrappers for the optim package.  Here is a brief list of my project goals.

1- lsqnonlin wrapping nonlin_residmin and residmin_stat
2- lsqcurvefit wrapping nonlin_curvefit and curvefit_stat
3- nlinfit wrapping nonlin_curvefit (it was initially decided to wrap leasqr but changed to avoid extra computations)
4- quadprog wrapping __qp__ instead of qp and returning lambda in the form of a structure as in Matlab
5- fmincon wrapping nonlin_min
6- Test and demos for the above functions
7- Stretch goals: previously decided to create other missing functions or perhaps additional backends but before midterm I decided instead to include optimoptions in my to do list.

Achievements
The functions lsqnonlinlsqcurvefit and nlinfit are complete with tests and demos and integrated in the optim package. Since nlinfit is from the statistics package in Matlab, additional functions such as statset, statget were required for handling options. These functions are implemented with minor modifications in optimset, optimget and __all_opts__ as statset, statget and __all_stat_opts__ and are now a part of optim package.

The function quadprog required directly wrapping __qp__ instead of qp for the ordering of lambda. It is in the final stages of review and will soon be integrated.

fmincon has not been thoroughly reviewed yet. I will send it to Olaf after quadprog is committed to optim.

Hiccups in the stretch goal
I couldn't create optimoptions in the GSoC time frame because it was a bit open ended and I had to come up with an object oriented design for the function. I was trying to understand how Matlab implements it for quite some time. Anyway, I didn't pursue it further and shifted my focus on the refinement of my almost complete functions to get them integrated in optim.

Interesting Takeaways
This is my first experience of working with any open source organization and it's definitely a pleasant one. It's delightful to see people using my functions and possibly benefiting from my work [2-3]. :)

I think I have managed to meet all the goals I set before the start of GSoC. (Regrets? Well, I could have saved more time for optimoptions and it would've been better to discuss it way before than being stuck for a while.)

I'm extremely grateful to the Octave community, especially my mentors Olaf Till and Nir Krakauer for their unrelenting support. GSoC wouldn't have been possible without their constructive feedback. I have learned a lot from this experience.

[1] https://github.com/AsmaAfzal/octave_workspace/issues/1
[2] http://octave.1599824.n4.nabble.com/GSoC-2015-Optimization-Package-Non-linear-and-constrained-least-squares-lsqcurvefit-lsqlin-lsqnonlin-tp4668777p4670940.html

## August 20, 2015

### Piotr Held

#### Summary of project

I have not written a post in a long while because I have had family issues. Although my progress was hampered, it was not completely halted. I was able to add some more features and clean some of the old code up. I have added two new sections with five new functions:
• Testing for nonlinearity:
• surrogates
• endtoend
• timerev
• Spike trains:
• spikeauto
• spikespec
They are complete with documentation and some demos. The wiki has also been updated with a demo of surrogates. Thus the original plan has been completed apart from some of the functions in the final section Tutorial and randomize. There have been some additional functions: endtoend, spikeauto, spikespec and av_d2. These were added as a logical addition to the existing function.

#### What happened to randomize?

The first time I read the documentation and tried to plan the project I did not realize how big randomize is. Therefore, once I realized that it was a toolbox and not a function I needed to design how this toolbox should work with/in Octave. This took about a week of my time. I have not had any experience designing anything similar to randomize. I considered it an interesting challenge. I tried to create a hybrid that could use C++ and Octave functions to run simulated annealing. I wanted the main loop to be in C++ because this algorithm cannot be vectorized, so it will run slowly on Octave.

My plan was to create some "runner" that can call some abstract methods of the base cost function class. I was able to accomplish this only partly, even though I introduced some polymorphism I only created one cost function (I did not finish the second one). This one cost function and the runner worked properly (as tested against the results from TISEAN). I was even able to do a type of pausing (like in other functions in the package), which allows the user to break the execution without having to kill Octave. All of this worked correctly with the one cost function. My plan was to modify the code to ensure the runner could manage any type of cost functions (that inherit from the base cost function class) once the second cost function was completed

Afterwards, I planned to create a base abstract class in Octave using classdef and similar keywords. Then I would have created an instance of the C++ class that would run the Octave methods using polymorphism. This was not possible however because classdef (and similar keywords) are yet not fully supported by Octave. They are not parsed properly by help, are not documented and not all of the Matlab functionality is present as of now.

Looking back at this attempt of porting the randomize toolbox I believe that with a good plan it could have taken more than half of my programming time this summer to create a complete toolbox with good tests and documentation. I do not regret trying to port it, but I think I can say that this part was not completed, even though much work was put into it.

All of the code created is placed in 'devel' folder. There are also some tests there that utilize the cgreen (cgreen++ actually) framework. It is almost completely undocumented, as I spent most of my time developing new code.

#### Other thoughts

I am very pleased with the support of the community. I expected they wouldn't have time to help, and at first I had trouble receiving the help I need, but once I understood what time (depending on who was online) to ask for help and which channel to use it became my biggest help in completing the project. My project needed help from people who knew things about how Octave was written in C++ and not just Octave developers. There were plenty of people who could assist me in just that.

One of the most exciting moments of the whole project was when I saw a problem on the mailing list that I encountered a few weeks earlier and could give some ideas on how to solve it.

If I were to do the project all over again and have to choose between the help of the community and the experience I have today after working on this project for over 3 months I would choose the community, because there are so many knowledgeable people there that know the inns and outs of Octave.

## August 19, 2015

### Juan Pablo Carbajal

#### ¿Para qué me sirven los vectores?

En la clase de vectores comentaba como se puede crear un vector en GNU Octave, pero quedó en el tintero la motivación para el uso de vectores ¿Para qué me pueden servir esas listas de numeritos que llamamos vectores?

#### Cartas de amor...

Supongo que alguna vez han mandado una carta por correo postal. Para que la carta llegue a destino necesitamos proveer cierta información además del nombre del destinatario. Supongamos que la carta la enviamos dentro de Argentina. Necesitamos indicar la provincia a donde va la carta, la cuidad, la calle y el número de casa. En principio podríamos organizar la información en de esta forma

carta = [ "Buenos Aires","Lomas de Zamora", "Ceferino Namuncura", "150" ];

Ok, esto ya se parece a un vector pero las componentes son strings en vez de números. No hay drama, simplemente podemos indicar la provincia con un número del 1 al 23 (Argentina tiene 23 provincias) y hacer algo parecido con las ciudades y las calles. El número de casa lo podemos usar directamente. Es decir que el destino de nuestra carta podría representarse con el vector

carta = [ 23, 1352, 12345, 150 ];

donde cada componente fue reemplazada por un número según una tabla especificada. La primera componente del vector nos indica la provincia, la segunda nos muestra las ciudad, la cuarta nos indica la calle y la quinta componente la altura de la calle ¡Estas cartas viven en un espacio de 5 dimensiones!

Un conjunto de estas cartas se puede organizar en una lista de vectores, una matriz:

cartas = [ 23, 1352, 12345, 150; ...
          5, 130, 4, 756; ...
          12, 7, 2341, 29 ];

Cada vector fila de esa matriz representa una carta. La primera columna de la matriz nos muestra las provincias, la segunda columna nos muestra las ciudades, la cuarta nos indica la calle y la quinta columna la altura de la calle.

#### Panadero, quiero pan!

Ok, quizás es ejemplo fue muy abstracto. El siguiente ejemplo me lo pasaron por Facebook, también es abstracto, pero quizás es más fácil de entender.

Pensemos por un momento en recetas de cocina, en particular en la lista de ingredientes de cada receta. En la tabla puse dos ejemplos de ingredientes para hacer pan

 Pan de cazuela Pan fuerte 1.Harina (gr) 250 300 2.Sal (gr) 5 20 3.Levadura (gr) 5 20 4.Aceite (ml) 0 50 5.Agua (ml) 175 50 6.Miel (ml) 0 100 7.Huevos 0 0

Esta tabla ya nos muestra una representación de estas recetas como vectores. En este caso son dos vectores columna que en GNU Octave podríamos anotar como

recetas = [ 250 300; 5 20; 5 20; 0 50; 175 50; 0 100; 0 0 ];

Los ingredientes del pan de cazuela son la primera columna de esta matriz y los del pan fuerte son la segunda. Noten que estos vectores son de 7 dimensiones (7 ingredientes), pero vale preguntarse si la dimensión Huevos tiene sentido en el espacio de recetas de pan... ¿Hay recetas de pan que utilizan huevos?

Los dos ejemplos anteriores son bastante abstractos y si los has entendido me puedo imaginar que ya tienes muchas ideas de como representar otras cosas como vectores o matrices.

#### Trayectorias espaciales

El siguiente ejemplo es paradigmático en el campo del cálculo numérico, y es muy relevante para este curso: representar la posición de objectos en el espacio!

Imagina una hormiga caminando sobre una hoja de papel A4 (ancho: 21cm, largo: 29.7cm). Para nuestra conveniencia hemos marcado una de las esquinas de la hoja como punto de referencia. En cualquier instante podemos leer la distancia entre la hormiga y la esquina de la hoja en las dos dimensiones del papel: ancho y largo. Ponemos la hormiga en un punto del papel y una vez por segundo miramos donde se encuentra. Podríamos observar algo así:

Figura 1: Posiciones de la hormiga en el papel. Cada punto representa la posición de la hormiga en las coordenadas del papel observadas cada 1 segundo. La línea punteada es la trayectoria de la hormiga observada con más frecuencia.

La posición luego del primer segundo la guardamos en una matriz con una fila y con dos columnas (un vector fila!)

p(1,1:2) = [ 10.7 14.8 ];

Luego de otro segundo obtenemos la nueva posición y la guardamos en la segunda fila de la matriz:

p(2,1:2) = [ 11.1 14.8 ];

Y así sucesivamente. Lego de 10 segundos, la matriz tiene 10 filas y 2 columnas. Esta matriz representa la trayectoria de la hormiga a intervalos de 1 segundo. Cada vector fila nos indica la posición en un dado instante de tiempo.

p = [ 10.7 14.8; ...
      11.1 14.8; ...
      11.4 15.1; ...
      11.2 15.0; ...
      11.0 15.3; ...
      11.8 15.8; ...
      11.6 16.8; ...
      11.8 17.0; ...
      11.0 17.3; ...
      10.8 17.2];

Creo que estos tres ejemplos deberían entusiasmarte para pensar qué cosas pueden representarse utilizando vectores y matrices y cómo hacerlo. Vale la pena preguntarse como podemos utilizar estas representaciones. Por ejemplo, usando la matriz de la hormiga ¿Cuál es la distancia entre la primera y la ultima posición de la hormiga?

¿Se te ocurren otros ejemplos? ¡Anótalos en los comentarios!

Nota: El archivo adjunto es un script de GNU Octave que genera la trayectoria de la hormiga y crea la figura que se muestra en esta clase.

## August 16, 2015

### Asma Afzal

#### Week 11 and 12: Integrating existing work in optim package.

A recap of the progress in two weeks:

• I had to let go of optimoptions (for GSoC) mainly because of
• time constraints
• and also because I don't have much experience with objected oriented programming. For optimoptions, I will have to come up with a design. I started with class implementation using classdef as in Matlab, but it is in its infancy in Octave and it could possibly be a limiting factor.
• I am refining my existing functions and including tests and demos so they can be integrated in the optim package.
• lsqnonlin and lsqcurvefit required additional options documentation and OutputFcn and Display setting. These two functions have now been successfully integrated in the optim package. [1]
• Functions nlinfit and quadprog are under review.
• I am working on fmincon now. Still have to discuss which backend should be used. lm_feasible can return Lagrange multipliers, gradient and hessian, but since it adheres to the constraints in all iterations, it behaves differently (from Matlab's algos) and sometimes less efficiently as octave_sqp, which only respects the constraints for the final result.
[1] http://sourceforge.net/p/octave/optim/ci/85d621b7e31a3546383248431b2f340d43edd6da/

## August 03, 2015

### Asma Afzal

#### Week 10: Preliminary work on optimoptions

Thoroughly checking how optimoptions works in  Matlab.

options = optimoptions (SolverName)

Things to do:

• Identify if Solver name is the right string or function handle

• Cater for multiple Algorithms
A subset of options for different algorithms.
• Transfer relevant options of different solvers to modify/create option.
oldoptions = optimoptions('fmincon','TolX',1e-10)
newoptions = optimoptions('lsqnonlin',oldoptions)
• Using dot notation or optimoptions to modify previously created options.
(Second argument in optimoptions can be old options)
• Display options:
Set by user on top (for the current algo)
Default options
Options set for other algorithms.
Implementation ideas:

In Matlab, these two calls generate the same options object optim.options.Fmincon:
optimoptions('fmincon')

optim.options.Fmincon
What would be more appropriate, IMO, will be to have a function optimoptions of the following format
opts = function optimoptions(SolverName,varargin)
...

obj = optim.options.SolverName(varargin)

opts = struct(obj)

...

This function will
1. Instantiate the relevant class and request for default options from the solver.
2. Compare the user provided options to add relevant options.
3. Display options of the current algorithm.
4. The output can also be returned in the form of a struct to be compatible with optimget.

## July 24, 2015

### Asma Afzal

#### Week 8 and 9: The ordering of lambda in quadprog

I was trying to dig through __qp__.cc to figure out why the ordering of lambda in [1] does not match that of quadprog in Matlab.
The example shows how the values were different:

C = [0.9501    0.7620    0.6153    0.4057
0.2311    0.4564    0.7919    0.9354
0.6068    0.0185    0.9218    0.9169
0.4859    0.8214    0.7382    0.4102
0.8912    0.4447    0.1762    0.8936];
d = [0.0578
0.3528
0.8131
0.0098
0.1388];
A =[0.2027    0.2721    0.7467    0.4659
0.1987    0.1988    0.4450    0.4186
0.6037    0.0152    0.9318    0.8462];
b =[0.5251
0.2026
0.6721];
Aeq = [3 5 7 9];
beq = 4;
lb = -0.1*ones(4,1);
ub = 1*ones(4,1);
H = C ' * C;

f = -C ' * d;

[x, obj_qp, INFO, lambda] = qp ([],H,f,Aeq,beq,lb,ub,[],A,b);
lambda =
0.01654
0.06743
0.00000
0.24993
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.49819
0.00000

Reordering lambda based on the length of constraints resulted in
lambda.eqlin =
0.01654
lambda.lower =
0.06743
0.00000
0.24993
0.00000

lambda.upper =
0.00000
0.00000
0.00000
0.00000

lambda.ineqlin =
0.00000
0.49819
0.00000

Matlab however,  gave lambda.eqlin= -0.0165 for this example and lambda.lower =
0.0674
0.2499
0
0

There were two issues with the ordering:
1.  the sign for Lagrange multipliers corresponding to linear equality constraint is always different from Matlab's
2. The multipliers corresponding to the bound constraints as underlined above are swapped.
I tried several different examples to understand what is going on. For all the examples, the sign for lambda.eqlin was consistently different. Although, I still can't pinpoint why but for now I am just multiplying lambda.eqlin by -1.

For the swapping issue, I tried the same example with just the bound constraints:

[x, obj_qp, INFO, lambda] = qp ([],H,f,[],[],lb,ub)
lambda=
0.07377
0.00000
0.29791
0.00000
0.00000
0.00000
0.00000
0.00000

but only considering lower bound constraints gave:
[x, obj_qp, INFO, lambda] = qp ([],H,f,[],[],lb,[])
lambda=
0.07377
0.29791
0.00000
0.00000

which is how it is supposed to be. Tracing back, I found out that the ordering of lambda vector in qp.m [2] is not [equality constraints; lower bounds; upper bounds; other inequality constraints] like I previously assumed. From lines 287 and 288 in qp.m [2], the bound constraints are added to the inequality constraint matrix alternatively. So the issue wasn't swapping but understanding how the constraints are passed to __qp__.

In my code in [1], I had to make significant changes in the original qp.m code such as:
- Inequality constraint matrix has the order: [Linear inequality constraints; lower bounds; upper bounds]
- Check for too close bounds forming an equality constraint- This brings indexing issues as now the Lagrange multiplier value corresponding to bounds is in place of multipliers corresponding to linear equality constraints.
Also, Matlab only accepts too close bounds when using a medium scale algorithm and since the lower bound is approximately equal to the upper bound and it is considered as a single equality constraint, the single Lagrange multiplier is placed in the corresponding lambda.upper field while the corresponding lambda.lower value is zero.

Continuing the above example:

lb(4) = 0.3;
ub = 0.3*ones(4,1);

[x, obj_qp, INFO, lambda] = qp ([],H,f,[],[],lb,ub)
lambda =

-0.04828
0.04154
0.00000
0.28615
0.00000
0.00000
0.01546

Here, lb(4) = ub(4) and hence the constraint is treated as an equality constraint so the value for corresponding Lagrange multipliers is present on top (underlined)
I added checks for such cases and now my code in [1] gives same results as Matlab:

[x,obj,flag,op,lambda]=quadprog(H,f,[],[],[],[],lb,ub)

lambda =
scalar structure containing the fields:
lower =
0.04154
0.28615
0.00000
0.00000
upper =

0.000000
0.000000
0.015456
0.048279

- qp.m strips off the -Inf constraints before passing them to __qp__. I am doing the same in quadprog. I have added further checks to make sure the multipliers are placed in the right positions in their respective fields.

Plans for the next weeks:
- Get feedback from my mentor on the changes in quadprog.
- Begin intial work on optimoptions.

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/quadprog.m
[2] http://fossies.org/dox/octave-4.0.0/qp_8m_source.html

## July 20, 2015

### Piotr Held

#### Plans for randomize

This past week I have spent time trying to establish some testing framework for C++ methods and also trying to create a model for what the abstract base classes for cool, cost and permute should look like. I would like them to have the following methods/members:
• Cost (Cost_fcn):
• const Matrix *series-> pointer to the series for which a surrogate is generated
• double cost-> current cost
• void cost_transform (Matrix *)-> initial transformation which is used for better calculation of cost
• Matrix cost_invert () const-> assigns to the input variable the inverse of the transformation performed above (to get the actual surrogate, not just a representation of it in a different form)
• double cost_update (octave_idx_type nn1, octave_idx_type nn2, double cmax, bool &accept)-> perform quick update of cost (for a swap of elements under index nn1 and nn2) and decide if cost is smaller than maximum cost (cmax) if yes, accept new cost and return true otherwise reject new cost and return false
• double cost_full()-> performs a full calculation of the cost, takes longer than cost_update()
• getters/setters for cost
• Cool (Cool_fcn):
• double temp-> holds the current temperature
• double cool (double cost, bool accept, bool &end)-> takes current cost, accept which holds whether the last cost_update() was accepted or not and returns the new temp and sets flag end to indicate if the simulated annealing is over
• Permute (Permute_fcn):
• Matrix *series-> holds the pointer to the series, and modifies the series only when exch() is called
• void permute (octave_idx_type &n1, octave_idx_type &n2) const-> generates two indexes n1 and n2 that can be used to calculate Cost_fcn::cost_update()
•  void exch (octave_idx_type n1, octave_idx_type n2)-> exchange element under n1 with element under n2 in the series
Those are the methods I intend to have in the base/abstract classes which will be called by the Simulated Annealing runner code. I have not decided what that code should look like, but the current version seems to be working as well as the randomize program from TISEAN package.

I was also hoping to create a subclass of each of those abstract classes built to call GNU Octave code. This would allow the user to create their own functions without having to write anything in C++. However, this idea might not be practical for the following reasons:
1. The example of Simulated Annealing provided in the TISEAN package (ver. 3.0.1), takes about 0.7 seconds to run using only C++ code and performs on average 900,000 calls to Cost_fcn::cost_update() and calling a simple function in Octave that many times (using the for loop) took 16 seconds
2. I have trouble deciding how to neatly pass these functions/classes to randomize along with some parameters the user might want to include. I originally thought of using classdef - the new keyword introduced in Octave 4.0.0. I hoped to create an abstract Octave class and then let anyone subclass it to create their own cost, cool and permute classes. The problem is that classdef and all of the associated keywords are not documented, moreover according to Carnë Draug the help function will not recognize this new type of Octave class. So even if all of the needed functionality was available in Octave I might not be able to document it for the user
If obstacle 2. can be overcome it might still be beneficial for the package to create this type of functionality, regardless of how long the code will execute.

This week I plan to refine the design of the abstract classes as well as port more of the cost function options from TISEAN.

[Update]: I modified the design a bit and updated this post to fit the new code.

## July 14, 2015

### Asma Afzal

#### Week 7: quadprog wrapping __qp__

I have wrapped quadprog on __qp__ instead of qp.m in [1].

Main differences between quadprog in [1] and qp.m.

- Input argument placement
quadprog(H, f, A, b, Aeq, beq, lb, ub, x0, options)  =  qp (x0, H, f, Aeq, beq, lb, ub, [], A, b, options)

- Check for empty inputs A and b
qp ([],H,f,Aeq,beq,lb,ub,[],A,[])

This works. qp simply ignores inequality constraints due to if checks
in lines  258, 266 and 275 of qp.m. Matlab gives an error if A is empty and b is not and vice versa.

Example:
quadprog (H, f, A, [], Aeq, beq, lb, ub)
Error: The number of rows in A must be the same as the length of b. I have added this check in line 181 in [1].

- Lambda output as a structure instead of a vector as in qp.m.

Ordering of lambda:
• The order of lambda vector output (qp_lambda) from __qp__(in my code) is [equality constraints; inequality constraints; lower bounds; upper bounds].
• The multipliers are present if the constraints are given as inputs so the size of qp_lambda depends on the size of constraints.
• Variables idx_ineq, idx_lb and idx_ub make sure I pick the right values.
Example:

H = diag([1; 0]);
f = [3; 4];
A = [-1 -3; 2 5; 3 4];
b = [-15; 100; 80];
l = zeros(2,1);
[x, obj, info, qp_lambda] = qp ([], H, f, [], [],l,[],[], A, b)
[x,fval,exitflag,output,lambda] = quadprog (H, f, A, b,[],[],l,[])

qp_lambda =

1.66667
0.00000
1.33333
0.00000
0.00000

lambda =
scalar structure containing the fields:

lower =

1.66667
0.00000

upper = [](0x0)
eqlin = [](0x0)
ineqlin =

1.33333
0.00000
0.00000

Things to do:
• Check the sign issue for lambda.eqlin (qp gives values -1* Matlab's)
• Check if __qp__ changes the order of constraints. The values of lambda from qp.m in the last example in [2] are there but not coinciding with the respective constraints.
• Move on to optimoptions.

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/quadprog.m
[2] https://github.com/AsmaAfzal/octave_workspace/blob/master/runquadprog.m

## July 09, 2015

### Piotr Held

#### Progress report and plans

So far my progress has been as planned. Before the end of the midterm evaluation I was able to publish on my repository version 0.2.0 of the package, which included all of the functions from section Dimensions and entropies from the TISEAN documentation. As I mentioned in my previous post the functions that needed to be ported in this section are slightly different from what I wrote in my outline. The ported functions are:
• d2
• d1
• boxcount
• c2t
• c2g
• c2d
• av_d2
I also wrote demos for most of those functions and updated the tutorial on the wiki page.

The first part of this week I spent improving on the build process. The function __c2g__ relies on C++ lambdas to work, therefore a configure script needed to be introduced to ensure the compiler has this capability. As was suggested by John Eaton, I tried to make the impact of not having that capability as small as possible. Currently if the compiler does not recognize C++ lambdas simply __c2g__ is not compiled and the function c2g does not work.

### The plans

I was hoping to port all of the functions in the next section, Testing for Nonlinearity, by the end of the week. This might not be possible as randomize turned out to be a bigger function than I anticipated. It is actually not a function at all but, as the author of the TISEAN documentation puts it, "a toolbox". It generates surrogate data using simulated annealing. It needs to be supplied with three functions:
1. the cost function
2. the cooling function -- how the temperature decreases
3. the permutation function -- what to change every step
So currently if the user wants their own version of any of the functions above the user needs to write it in FORTRAN. My goal for this project would be to allow the user to write (use) their own octave function. The SA algorithm is an iterative method so using Octave code is not a good idea (as each line must be parsed when using for or while loops). As far as I understand the samin routine from the optim package will not suffice as it does not generate surrogate data, and has fewer options. Due to the size of this function it might take me some time to complete it.

I plan to tackle this problem as follows: I will rewrite in C++ the equivalent function to randomize_auto_exp_random and then try to refactor and modify the code to accept other functions. I plan to include all of the functions that are available in TISEAN in the Octave package, either through rewriting them or through linking to them. And I would like to make it easy for new functions to be added.

Further reading on randomize is available on the TISEAN documentation in the General Constrained RandomizationSurrogates paper Appendixrandomize description and randomize function extension description.

## July 02, 2015

### Asma Afzal

#### Week 5 and 6: Refining fmincon

So my fmincon implementation is coming in shape [1].

[x,fval,exitflag,output,lambda,grad,hessian] =
fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

I came across a few issues which turned out to be bugs. Olaf pushed fixes in the central repository. Listing the issues for the record:
Setting gradc (the gradient of general equality/inequality functions)A bug in nonlin_min.m (and __nonlin_residmin__.m)
Example:
objective_function = @ (p) p(1)^2 + p(2)^2;
pin = [-2; 5];
constraint_function = @ (p) p(1)^2 + 1 - p(2);
gradc = @(p) [2*p(1);-1];
[p, objf, cvg, outp] = nonlin_min (objective_function, pin, optimset
("equc", {constraint_function, gradc}))

error: function handle type invalid as index value
- Giving linear inequality/ equality constraints to lm_feasible. A bug in nonlin_min.m
Example:
f = @(x) -x(1) * x(2) * x(3);
S = [1  -1;   2  -2;   2  -2]
b = [0;72];
x0 = [10;10;10];
[x,fval] = nonlin_min( f, x0, optimset ("inequc",{S,b}) )
error: __lm_feasible__: operator -: nonconformant arguments (op1 is
3x1, op2 is 3x0)

-Any zero value in initial guess vector for nonin_residmin/nonlin_min gave an error. Required a minor change of sign(0)==0 in __dfdp__.m.
Example:
k = 1:10;
func = @(x) 2 + 2 * k - exp (k * x(1)) - exp (k * x(2));
x0 = [0;0.5];
x = nonlin_residmin(func,x0)

warning: division by zero
warning: called from
__dfdp__ at line 367 column 21
__nonlin_residmin__> at line -1 column -1
__lm_svd__ at line 191 column 9
__nonlin_residmin__ at line 1125 column 21
nonlin_residmin at line 98 column 21
runlsqnonlin at line 9 column 3
error: svd: cannot take SVD of matrix containing Inf or NaN values

Functionality for Returning Hessian and Gradient
New options "ret_objf_grad" and "ret_hessian" to be introduced in nonlin_min (by Olaf). If anyone of these options is set to true, the 'outp' structure output argument of nonlin_min will contain additional fields .objf_grad and .hessian. My code currently checks this.

Rearranging values of lambda in the fields of a structure.
- For lm_feasible, outp will contain an additional field lambda, a structure which contains Lagrange multipliers in fields separated by constraint type.

I added an additional feature in [1] to cater for the non linear constraint function set using deal();
Example:

c = @(x) [x(1) ^ 2 / 9 + x(2) ^ 2 / 4 - 1;
x(1) ^ 2 - x(2) - 1];
ceq = @(x) tanh (x(1)) - x(2);
nonlinfcn = @(x) deal (c(x), ceq(x));
obj = @(x) cosh (x(1)) + sinh (x(2));
z = fmincon (obj, [0;0], [], [], [], [], [], [], nonlinfcn)

z =
-0.6530
-0.5737

To do:
1- Write test cases/refined examples for for lsqnonlin, lsqcurvefit, nlinfit and fmincon.
2- Start wrapping quadprog to __qp__ instead of qp.m (because of the ordering of the lambda output).

## June 26, 2015

### Piotr Held

#### Progress report

The main goal of this post will be to create a progress report before the coming midterm assessment.
As I mentioned before I planned to complete the Dimensions and entropies section of the TISEAN documentation. This seems to be still a realistic goal.

Currently I have ported d2, av-d2, c2g, c2t along with writing documentation and demos for them. The current state of the tests needs improvement because they rely heavily on external files generated using the corresponding TISEAN programs. Because most of those functions/programs are closely linked I plan to improve on this feature once functions from the entire section are ported.

Currently I am working on c1 which already passes it's test. Once I complete it and write the documentation and demo, the only programs/functions that will need to be ported are boxcount and c2d. Once they are complete I plan to release version 0.2.0.

My elaborated proposal located on the octave wiki states that I planned to also port c2. Although the source code for such a program does exist in the TISEAN package (ver. 3.0.1), it does not seem to be mentioned in the documentation. Furthermore, installing the package on a computer does not give access to this program. Also, it seems to be redundant with other programs in the package. Therefore, I will not port it.

## June 24, 2015

### Asma Afzal

#### Progress Update: Midterm Evaluation

Objective:
Adding functions to the Optim package for Octave using existing back-ends.

Expected deliverables before midterm:
• 'lsqnonlin' using 'nonlin_residmin'
Done in [1]. Differences in backends, nonlin_residmin uses "lm_svd" algorithm for optimization as currently the only backend. However, lsqnonlin in Matlab can choose between "trust-region-reflective" and "Levenberg-Marquardt" (LM) algorithms.
Another difference is in complex inputs. lm_svd does not support complex valued inputs whereas Matlab's LM algorithm can accept complex input arguments. One way of providing complex inputs to lsqnonlin in Octave is to split the real and imaginary parts into separate variables and running the optimization.
• 'lsqcurvefit' using 'nonlin_curvefit', 'nonlin_residmin', or 'lsqnonlin'
Done in [2] using nonlin_curvefit. lsqcurvefit is very similar to lsqnonlin with only a few minor interface differences. lsqcurvefit explicitly takes independent variables and the observations as inputs while these values can be wrapped inside the objective function while using lsqnonlin. Additional bounds for the optimized parameters can be specified.
• 'nlinfit' using 'leasqr',
I wrapped nlinfit on nonlin_curvefit and curvefit_stat as leasqr repeats the optimization to compute the additional statistics (Jacobian and Covariance matrix) while curvefit_stat saves this computation overhead. I have partially implemented nlinfit in [3] (It hasn't been thoroughly reviewed yet). Two missing features are: 1) Error models and Error parameters estimation, and 2) Robust Weight function. Meanwhile, no such functionality exists in the Octave's optimization backends for the missing features. My current implementation supports the input of scalar positive array of weights for robust regression.
Since nlinfit is from the statistics toolbox of Matlab, it uses statset and statget to create and get options respectively. I created additional functions statset, statget and __all_stat_opts__ with minor changes to the code in optimset, optimget and __all_opts__.
• 'fmincon' using 'nonlin_min',
In progress [4].

Future goals:
1. Complete fmincon implementation.
2. Create solver specific options using optimoptions and desirably still be able to use optimget.
3. Arranging lambda output for quadprog by wrapping it on __qp__ instead of qp.m
4. Test cases for all the implemented functions.

[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/lsqnonlin.m
[2] https://github.com/AsmaAfzal/octave_workspace/blob/master/lsqcurvefit.m
[3] https://github.com/AsmaAfzal/octave_workspace/blob/master/nlinfit.m
[4] https://github.com/AsmaAfzal/octave_workspace/blob/master/fmincon.m

### Juan Pablo Carbajal

#### Crea tus propias funciones

A esta altura están en posición de escribir sus propias funciones. Pero ¿Qué es una función?

Una función es una entidad que recibe una serie de argumentos como entradas, opera sobre ellos y luego devuelve ciertos resultados. Las funciones cumplen un rol fundamental: encapsulan tareas. Al encapsular tareas, en especial aquellas que se repiten seguido, nuestros programas son mas fáciles de leer y a veces hasta más eficientes.

Considera la situación de saludar a personas que encuentras durante tu día. Supongamos que tu frase elegida como saludo es "Hola, <nombre>",  donde <nombre> es el nombre de pila de la persona a al que saludamos. Dado que a esto lo vamos a hacer muy seguido, podríamos definir una función (o rutina) que haga lo siguiente:

1. Toma el nombre de la persona a saludar.
2. Adjunta "Hola, ".
3. Imprima el saludo
4. Devuelva un resultado con el saludo.

Esta función en octave sería

function frase = saludar (nombre)
  frase = ["Hola, " nombre];
  disp (frase);
enfunction

En una sesión de octave utilizaríamos esta función de la siguiente manera

x = "María";
y = saludar (x);

En la pantalla veríamos "Hola, María" y el contenido de la variable y sería exactamente esa frase.

Pregunta 1
¿Puedes definir una función que tome dos entradas y devuelva la suma de las mismas?

## Estructura de una función

El siguiente esquema muestra la estructura de una función

function <resultados> = <nombre de función> ( <entradas> )
<código>
endfunction

Las palabras y símbolos en negrita son necesarios y obligatorios.

Las palabra clave function es necesaria (y obligatoria) para definir una función. Un archivo que contiene una función debe siempre tener como primer comando ejecutable esta palabra clave. El archivo donde guardamos la función siempre tiene que llamarse  <nombre de funcion>.m , es decir el nombre del archivo debe coincidir con le nombre de a función que allí se define.
Las variables que se pasarán al ámbito desde donde se ha llamado la función (p. ej. la sesión de Octave) se devuelven como un vector en <resultados>. En el caso de la función saludar.m de más arriba, devolvíamos solo una variable. Pero una función puede devolver un montón de cosas o nada. Ejemplos

function [x, y] = ceroyuno ()
 x = 0;
 y = 1;
endfunction

function diceHola ()
  disp ("Hola");
endfunction

De manera similar, las variables de entrada  se pasan en una lista en <entradas>. Las entradas pueden ser muchas o ninguna (como en los ejemplos anteriores). Ejemplo

function z = resta (x,y)
  z = x - y;
endfunction

Pregunta 2:
¿Cual es el nombre del archivo .m donde debemos guardar la función resta definida en el ejemplo anterior?

Pregunta 3:
¿Como se define una función que devuelva la cantidad de letras "a" en un string?

## Ámbitos

Cuando discutimos como guardar nuestro trabajo, vimos que desde Octave podemos ejecutar el contenido de un archivo de texto. Los comandos en ese archivo tienen acceso a las variables que existen en el ámbito (en inglés scope) de la sesión de Octave; esto quiere decir que los scripts pueden "ver" y "editar" las variables que creamos directamente en la sesión, y que si estos scripts crean nuevas variables estas quedaran en nuestra sesión luego de que el script haya terminado de ejecutarse.

Una función, en cambio, crea su propio ámbito de variables. Las variables que son creadas dentro de una función no son accesibles desde la sesión de Octave, o por lo menos no directamente. Excepto en casos particulares, las variables creadas dentro de una función son eliminadas cuando la función termina.

Con estas ayuditas ya puedes empezar crear tus propias funciones. El mundo de las funciones es mucho más amplio de lo que he descrito aquí ¡No te olvides de explorar!. Espero tus dudas y preguntas en el foro.

## June 22, 2015

### Asma Afzal

#### Week4: fmincon wrapping nonlin_min

Time flies.. A third of the way through already..

fmincon/nonlin_min is the most elaborate function of all that I have previously
implemented so before actual coding I would like to thoroughly check the the
mapping of arguments and options.

[x,fval,exitflag,output,lambda,grad,hessian] = fmincon (fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
[x,fval,exitflag,output] = nonlin_min (fun,x0,settings)

A,b Aeq, beq- Linear inequality and equality constraints
Inequality constraints:
Matlab standard: A * x - b <= 0 ,
Octave standard: m.' * x  + v >= 0.
This implies: m=-A.' , v=b.
Set in Octave using
optimset ("inequc", {m,v})
Similar for equality constraints:
optimset ("equc", {m,v})

lb, ub- Lower and upper bounds
Set in Octave using optimset
optimset ("lbound", ..., "ubound",...)

nonlcon- Nonliner constraint function handle
In Matlab, nonlinear constraints are given in a function with the following format

function [c,ceq,gradc,gradceq] = mycon(x)
c = ...     % Compute nonlinear inequalities at x.
ceq = ...   % Compute nonlinear equalities at x.
% Optional output arguments:
gradc = ...   % Gradient of c(x).
gradceq = ...   % Gradient of ceq(x).
options = optimoptions('fmincon','GradConstr','on')

Alternative to nonlcon in nonlin_min
optimset ("equc", {constraint_function})

Options- Options common to both fmincon and nonlin_min
User-supplied Gradient
In Matlab, the objective function must return the second output when the
GradObj option is set:
optimset('GradObj','on')
Octave:
optimset("objf_grad",@fun)
User-supplied Hessian
In Matlab, the objective function must return the third output when the Hessian
option is set:
optimset('Hessian','user-supplied')
Octave:
optimset("objf_hessian",@fun)

Lambda- returned as a structure in Matlab but as a vector field of the "output"
ouput argument in Octave.

Things to investigate:
1. Algorithm mapping.
2. Exitflag mapping.
3. Setting gradc (the gradient of general equality/inequality functions). The second
entry for the equc/inequc setting implements this feature in Octave as stated by
optim_doc but I was unable to make it work properly.
4. Returning additional statistics (Hessian, Gradient)- I used
residmin_stat/curvefit_stat previously for lsqnonlin and lsqcurvefit. No such function for nonlin_min.
5. Rearranging values of lambda in the fields of a structure.

P.s. this week I spent quite some time with Mercurial and how to possibly work in the optim package. There was a slight miscommunication/confusion but it is clear now and I will continue to publish my code on github..

## June 18, 2015

### Juan Pablo Carbajal

#### Si, si en cambio, sino: IF-ELSEIF-ELSE

En la clase sobre lógica discutimos un poco la forma de generar valores verdadero/false, usualmente llamados valores booleanos (se lee bu-leanos y viene del nombre de un inglés llamado George Boole).

Casi todos los lenguajes de programación ofrecen una forma de controlar la ejecución de código utilizando como criterio valores booleanos. Es decir, cierta parte del código se ejecuta si una condición lógica es verdadera. Una forma ejemplar de realizar esta ejecución selectiva es utilizando IF-ELSE (si, sino) o IF-ELSEIF (si, si en cambio) o una combinación de ambos.

## IF

En Octave, todos aquellos comandos dentro de un bloque if se ejecutan solo si a condición dada es verdadera.

if (x == 0)
  disp ("Equis es igual a cero");
endif

El bloque if se empieza en la linea donde llamamos a esta función y termina con endif (o simplemente end, es lo mismo). En este caso, se imprime a la pantalla la oración "Equis es igual a cero" cuando la comparación x == 0 es verdadera. La condición puede ser tan compleja como sea necesario (aunque esto puede hacer el código difícil de entender), por ejemplo

if (x == 0 || x == 5)
  disp ("Equis es igual a cero o igual a cinco");
endif

Si la condición fuese falsa, el código dentro del bloque nunca se ejecuta. En cualquier caso, una vez evaluado el bloque (con o sin ejecución del código correspondiente), el programa continua. El siguiente ejemplo imprime "Equis es igual a cinco" dependiendo del valor de x, pero siempre imprime "Terminé".

if (x == 5)
  disp ("Equis es igual a cinco");
endif
disp ("Terminé.");

Pregunta 1:
¿En que situación se imprime a pantalla el mensaje "¡Muy loco!"?

if (x == 0 && x == 5)
  disp ("¡Muy loco!");
endif

¿Veremos alguna vez el mensaje en pantalla?

## IF-ELSE

El bloque if se puede extender para que cierto código se ejecute solo cuando la condición es falsa. Considera el siguiente código

disp ("Eqis es: ")

if (x > 0)
    disp ("positivo.");
endif

if (x <= 0)
    disp ("negativo o cero.");
endif

Dado que la segunda condición es lo opuesto a la primera podemos simplificar usando else

disp ("Eqis es: ")
if (x > 0)
    disp ("positivo.");
else
    disp ("negativo o cero.");
endif

Pregunta 2:
¿Cómo harías para separar en "positivo", "cero" o "negativo" el ejemplo anterior?

## IF-ELSEIF

Una respuesta a la pregunta anterior viene de la mano de elseif (si en cambio), que nos permite evaluar una condición adicional.

disp ("Eqis es: ")
if (x > 0)
    disp ("positivo.");
elseif (x == 0)
    disp ("cero.");
else
    disp ("negativo.");
endif

Pregunta 3:
¿Podés dar otros ejemplos que produzcan lo mismo que el ejemplo anterior?

Debe notarse que el código dentro de un elseif se ejecutara solo y solo si la condición del if es falsa. En el siguiente ejemplo jamás veremos el mensaje "¡Ahá!" en pantalla aunque la condición del elseif es siempre verdadera.

if (true)
 disp ("Sí, es verdad.");
elseif (true)
 disp ("¡Ahá!");
endif

Esto quiere decir que las condiciones en un bloque IF-ELSEIF deberían que ser mutuamente exclusivas (nunca ambas pueden ser verdaderas al mismo tiempo). Esto no es necesario, como se muestra en el siguiente ejemplo, pero la lógica resultante es confusa

if (x > 0 && x <=1)
    disp ("Entre cero y uno (incluido).");
elseif (x > 0.5)
    disp ("Más que un medio.");
endif

Pregunta 4:
¿Podés simplificar la lógica del ejemplo anterior?

Espero recibir sus preguntas y dudas en el foro!

## June 14, 2015

### Piotr Held

#### Improving the code

TISEAN was originally written as a command line set of programs. Because of this all the code is not very portable and many variables are global ones. So far this has been dealt with by creating local variables and extending the number of variables in function calls (in some cases up to 11). This is not optimal for code clarity, ease of maintaince and because many of the variables are passed as values (they are parameters) it also caused a slight slowdown in execution speed.

Due to all of these downsides I have contemplated possible solutions which I will attempt to describe.

### Using structs

One idea that came to mind is to pack all of the global variables into a struct and pass the struct to all of the functions and obtain the global variables from this struct. This solution certainly solves the problem of passing so many parameters to functions. However, it does not improve code portability because every *.cc oct-file function needs its own struct. This solution is also problematic because all of the names of global variables have to be referenced now through the struct so center[i][j] would become parameter_struct.center[i][j] (obviously the name of the struct could be as short as p).

### Using classes

Another quite simple solution would be to create a class. This class would have data members that were previously the global variables and function members that were the old functions called from old main(). As there are similarities between different TISEAN programs, it could be possible to even create a prototype class and inherit from it.

There are however downsides to this option as well. First of all, Octave code guidelines specifies that classes should be in separate files. This would mean creating 2 more files for each program that was ported using the C++ wrapper. Apart from that, the memory might have to allocated using new/delete, because the preferred method of using the macro OCTAVE_LOCAL_BUFFER might be difficult (or impossible) to apply to this case. This objection can be worked around in other ways, such as using Array classes to allocate the data and then get a pointer to them using fortran_vec().

### Summary

Performing the aforementioned code improvement, although helpful, is not critical. Therefore any attempts to implement it will be deferred until after the functions outlined at the beginning of the project are complete.

#### Timeline update

So far I have been giving progress reports on the TISEAN porting project. This time, however, I would like to also compare the outlined schedule for the project with the actual progress made.

Since the last post I have additionally ported:
• xzero
• lyap_r
• lyap_k
• lyap_spec
In one of my first posts I stated that I would like to finish Dimensions and Entropies before the midterm assessment. Currently I have finished up Lyapunov Exponents and I plan to start working on Dimensions and Entropies this week. Since there are 2+ weeks to the Midterm Assessment I believe it is possible to complete all of the goals for this section of the project as planned.

### Asma Afzal

#### Week3: nlinfit, statset, stat_get and __all_stat_opts__

So this week I achieved the following milestones:

Wrapping nlinfit on nonlin_curvefit

[beta,R,J,CovB,MSE,ErrorModelInfo] = nlinfit(X,Y,modelfun,beta0,options,Name,Value)

Implementation [1]:

I chose not to wrap nlinfit on lsqcurvefit because
1. We might end up wrapping lsqcurvefit on lsqnonlin eventually so it is undecided.
2. The default options for lsqnonlin/lsqcurvefit are different from nlinfit.
Missing features:
1. RobustWgtFun - The field RobustWgtFun in options can be provided with a function handle which computes the residuals at every iteration. The backend optimization algorithm in Octave currently does not support this functionality.
2. Name-Value pairs. Currently the only implemented one is "weights",  which takes an array of weights for the weighted optimization. "ErrorModelInfo" and "ErrorParameters" are not implemented. The possible error models include, "constant", "proportional" and "combined". The error model also translates to a weight function which helps in reducing the effect of outliers.
3. ErrorModelInfo- output field which gives information about the error variance, and estimates the parameters of Error models.
Setting options using statset

In Matlab, options for statistics toolbox are set using statset [2].  The functionality is almost identical to optimset but the different functions in Matlab are because of different toolboxes (statset for statistics and optimset for optimization toolbox).

Added functions:
statset.m, statget.m and __all_stat_opts__.m [3]-[5]
Creating these functions was pretty straight forward.

Still to do:
1. Have to check if the weighted residual and weighted Jacobian output in octave is consistent with Matlab and further refine the functions with the feedback from my mentors.
2. Move on to fmincon wrapping nonlin_min.
[1] https://github.com/AsmaAfzal/octave_workspace/blob/master/nlinfit.m
[2] http://uk.mathworks.com/help/stats/statset.html
[3] https://github.com/AsmaAfzal/octave_workspace/blob/master/statset.m
[4] https://github.com/AsmaAfzal/octave_workspace/blob/master/statget.m
[5] https://github.com/AsmaAfzal/octave_workspace/blob/master/__all_stat_opts__.m

## June 09, 2015

### Asma Afzal

#### Week 2: lsqnonlin and lsqcurvefit

A bit late blogging about week 2.

Almost completed functions lsqnonlin and lsqcurvefit.
Successfully mapped user-specified Jacobian.
In Matlab, if the Jacobian option is set to "on",  the model function must return
a second output which is the Jacobian function.
In Octave, the Jacobian function handle is given to the dfdp option using optimset.

Lsqnonlin function description:

[x,resnorm,residual,exitflag,output,lambda,jacobian] = ...
lsqnonlin(fun,x0,lb,ub,options)

This function maps on:

[x, residual, exitflag, output] = nonlin_residmin (fun, x0, options)

Features of lsqnonlin in Octave:
• Input arguments: Acceptable forms are lsqnonlin(fun,x0), lsqnonlin(fun,x0,lb,ub) and lsqnonlin(fun,x0,lb,ub,options)
• Outputs
• x, exitflag, residual and output currently same as nonlin_residmin.
• resnorm=sum(residual.^2)
• Lambda is computed using the complementary pivoting in __lm_svd__.
It's values differ from Matlab's due to the difference in backends.
• Jacobian is computed using the function residmin_stat ().

Lsqcurvefit function description:

[x,resnorm,residual,exitflag,output,lambda,jacobian] = ...
lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)

This function maps on

[x, fy, exitflag, output] = nonlin_curvefit (fun, x0, xdata, ydata, options)

Features of lsqcurvefit in Octave:
• Input arguments: Acceptable forms are lsqcurvefit (fun,x0,xdata,ydata), lsqcurvefit (fun,x0,xdata,ydata,lb,ub) and lsqcurvefit (fun,x0,lb,ub,xdata,ydata,options)
• Outputs
• x, exitflag, residual and output currently same as nonlin_curvefit.
• residual = fy-ydata, resnorm = sum(residual.^2)
• Lambda and Jacobian same as in lsqnonlin.
There are only minor interface differences between lsqcurvefit and lsqnonlin.

This week's plan:

• Hopefully, with lsqnonlin and lsqcurvefit wrapped up, I'll move on to nlinfit
• Three key challenges need to be addressed when wrapping nlinfit using nonlin_curvefit
and curvefit_stat:
• Weight functions: Currently, no such functionality exists in nonlin_curvefit,
where a user can specify weight functions to perform Robust regression
(weights computed using the specified function in every iteration).
• Error Models and ErrorModelInfo
• Setting options using statset instead of optimset or optimoptions.

## June 07, 2015

### Piotr Held

#### Analyzing lfo-run

I have written tests that compare lfo-run from TISEAN to the ported version lfo_run. The test that uses amplitude.dat works perfectly, but when I analyzed the results both programs/function gave for henon (Henon Maps) I ran into some problems. I will attempt to describe them.

### Input data

The problems occur when analyzing a 1000 element Henon map (henon(1000)). For all of the implementations if I used a simple call with default parameters (m = 2, d= 1) the programs would quit due to a matrix singularity. The problems arose when (m = 4, d= 6) was used. With these parameters the program gave various results for various implementation methods.

It is important to note that the prediction that I was testing tried to predict 1000 future elements (default for all implementations) on the basis of given 1000 elements.

### Implementations

There are 3 implementations I used:
1. The TISEAN implementation (uses lfo-run)
2. The implementation similar to 1. but compiled as c++ and wrapped in enough code to run as m-file (uses __lfo_run__ and invert_matrix())
3. The implementation that uses Matrix::solve() method
I tried to find out if maybe method 1. and 2. do not differ due to a bug that was introduced while porting. I therefore ported it twice (the second time to a very rudimentary stage) and both times the same results were encountered. I do not understand why there is a discrepancy between between these two implementations.

### Discrepencies

Since the goal of this project is to port TISEAN functions I have compare implementation 1. with 2. and 1. with 3, to see what differences I come across.

A comparison between implementation 1. and 2. results in an error from implementation 2. The function generates about 700 elements (of the default 1000) and then throws an error that the forecast has failed.

The comparison between implementation 1. and 3. is much more fruitful as the results are the same for about 150 elements and then they begin to differ (see Fig. 1.)
 Fig. 1 Comparison between the TISEAN implementation and using Matrix::solve() in TISEAN package from Octave
These results can be achieved by cloning the repo, doing make run and running the script:

cd tests/lfo_run/; test_lfo_run

### Analysis

Before I give my suggestions for what is the cause for these discrepancies I would like to discuss another interesting discrepancy. This discrepancy is the maximum difference between the solution of the equation system obtained from implementation 2. and 3. When using implementation 3. for the forecast this value was 8.5e-14, but when using implementation 2. for the forecast this difference was 5e-13.

I believe this is because the computational error is accumulated throughout the program. Each new forecast point is dependent on the previous ones. Moreover the Kahan algorithm (compensated summation) is never used in the TISEAN implementations. Even matrix multiplication (as seen e.g. in multiply_matrix())  uses the simple, but error accumulating for(...) sum += vec[i].

As to why implementation 1. and 2. give different results I have two theories: either there still is a bug which I was unable to detect, or some compilation difference (e.g. a linked library) between the TISEAN program written in C (lfo-run) and the TISEAN package function written in C++(__lfo_run__).

### Summary

The question that poses itself is whether this warrants a rewriting of other TISEAN function that use the simple summation, or if this problem can be ignored. The authors of TISEAN said in their introduction that blindly using the programs they wrote may result in unintended or even wrong results. Trying to predict 1000 elements of a 1000 element Henon Map using first order local linear prediction might be considered a bad use-case.

#### Progress report

Since the last post I wrote a tutorial on http://wiki.octave.org/TISEAN_package. I also ported:
• lzo_gm
• lzo_run
• ikeda
• lfo_run
• lfo_ar
• lfo_test
• rbf
• polynom
During my work I discovered that polynom has similar functions (polynomppolyparpolyback) which provide extra options for performing polynomial fits. I will not include those functions in the project now, but they have a high priority once I finish all of the functions I outlined for this project.

These newly ported functions aren't completely polished (some need demos and documentation) but they pass tests and don't have memory leaks. Once I clean these functions up and port xzero the last function in this section, I intend to create version 0.1.0 of the package. With this version I intend to branch out the repo to have a 'devel' and a 'stable' branch.

Afterwards, I will add more information to the tutorials on the wiki page.

## June 02, 2015

### Asma Afzal

#### optimoptions

Answering some questions to better understand the behavior of optimoptions.
Although the answers have been discussed in [1], I am pasting some examples for clarity.
*I was using the terms solver/algorithm in the wrong context before. Solver names are functions such as lsqnonlin, fmincon, etc. and one solver can have multiple algorithms such as interior-point, lev-mar, etc. *

1) Is there an error or warning if optimoptions is used to assign an option not contained in the specified algorithm?

If the option belongs to a different algorithm of the solver, Matlab stores it as options "not" used by the current algorithm so if we change the algorithm to which the option belongs, we do not need to set it again.

For eg. The default algorithm of lsqnonlin is trust-region-reflective but when I try to set the option 'ScaleProblem' which is specific to 'levenberg-marquardt' algorithm, I get:

opts = optimoptions ('lsqnonlin', 'ScaleProblem', 'Jacobian')

opts =

lsqnonlin options:

Options used by current Algorithm ('trust-region-reflective'):
(Other available algorithms: 'levenberg-marquardt')

Set by user:
No options set by user.

Default:

Algorithm: 'trust-region-reflective'
DerivativeCheck: 'off'
.               .
.               .
.               .
.               .
TolX: 1.0000e-06

Show options not used by current Algorithm ('trust-region-reflective')

Set by user:
ScaleProblem: 'jacobian'

This gives the same result using dot notation:

opts = optimoptions ('lsqnonlin')
opts.ScaleProblem = 'Jacobian'

2) ... or to assign an option that does not exist in the specified solver or any solver?

This gives an error. Trying to set the option for SQP Algorithm, which is not used by lsqnonlin:

opts=optimoptions('lsqnonlin', 'MaxSQPIter', 100)
Error using optimoptions
'MaxSQPIter' is not an option for LSQNONLIN

using dot notation:

opts.MaxSQPIter = 100;
No public field MaxSQPIter exists for class optim.options.Lsqnonlin.

3) If options are transfered to a different solver with optimoptions, are there errors or warnings if the new solver does not have some of these options?

No errors or warnings. The options common to both solvers are copied. They could be options of different algorithms. For eg.

opts = optimoptions ('fmincon', 'Algorithm', 'sqp', 'TolX', 1e-10)
opts_lsq = optimoptions ('lsqnonlin', opts)

Options set in opts_lsq are:
PrecondBandWidth: 0
TolX: 1.0000e-10

The option PrecondBandwidth belongs to the trust-region algorithm of fmincon solver.
Another option copied from opts in opts_lsq belongs to the lev-mar algorithm of lsqnonlin. It is the stored option as mentioned in 1)

ScaleProblem: 'none'

4) Can options returned by optimoptions be used with optimset, and   vice versa?

This returns an error in both cases:

opts = optimoptions ('lsqnonlin','MaxIter',100);
opts = optimset (opts, 'TolX', 1e-6);
Error using optimset
Cannot use OPTIMSET to alter options created using OPTIMOPTIONS.
Use OPTIMOPTIONS instead.

opts = optimset ('TolX', 1e-6);
opts = optimoptions ('lsqnonlin', opts);
Error using optimoptions
Invalid option name specified. Provide a string (such as 'Display').

References:
[1] http://octave.1599824.n4.nabble.com/lsqnonlin-and-nonlin-residmin-tp4670540p4670569.html

### Jordi Gutiérrez Hermoso

#### Copyleft for Mailpile

I have two reasons for why Mailpile’s license should be AGPL:

1. Do it for everyone, not just for the developers.
2. Copyleft is most beneficial when you’re most unique.

# The AGPL is freedom for users

The AGPL ensures that the software will always stay free, forever, for everyone. It means nobody will ever have to contend with a Safari instead of Webkit, or with a Swift instead of LLVM. Copyleft can ensure that stays free open for you, and nobody can ever convey a version without source code to anyone. For a program like Mailpile, source code is especially important for studying its cryptographic features, as we have seen examples such as RSA Security back-dooring cryptography for the NSA. No user should ever contend with a non-free derivative version of Mailpile.

As developers, some of us don’t like this. We even have a word to express how much we don’t like it when our public work must be free: we call this “viral”, as if it were contagious or poisonous. We even pronounce threats, “if you force us to keep our code free, we will just not give you any code.”

Freedom is a give-and-take proposition. When developers say that a non-copyleft license say is “more free”, they mean that it’s selfishly more free for developers. This is at the expense of taking freedom away some day from users or other developers. This is what lack of copyleft allows. Mailpile development has been run on donations and the goodwill of the community. Copyleft can ensure that this community process will live on without exception.

Furthermore, remember that copyleft only acts if you convey the code to someone else. If you make private changes to a behind-the-firewall copy of Mailpile, you’re under no obligation to give those changes to anyone else. So developer selfishness can still happen, but it cannot be imposed upon the public at large.

Copyleft is more friendly, more egalitarian, more shareable. Not just for a few, but for all.

# Tactical advantage of copyleft

Think of two successful copylefted packages you may know of: Linux and GCC. Think of how uniquely positioned each one of them was when it came out. Unique positioning together with copyleft brings the most contributors to a project. When you as a selfish contributor have a wide range of choices and some are copyleft and some are not, you may selfishly gravitate towards non-copyleft choices. But when there is only one choice and its copyleft compells you to contribute back, then the copyleft overrides your selfishness, and your contributions follow.

Such was the case with Linux.

Linux thrived almost immediately with contributions from everyone, encouraged by the GPL. Linus himself has said that putting it under the GPL is one of the best things he’s ever done. Several contributions that would have otherwise not been freely given could have gone into the BSDs, as Apple built Mac OS X on top of FreeBSD code. When Linux was the only game in town and the BSDs were being held back by dubious lawsuits with AT&T over the ownership of Unix, the rights afforded by copyleft brought in scores of contributions from around this world. This eventually culminated in most tech major companies such as Intel, IBM, Oracle, and Samsung all contributing to it.

A similar story happened with GCC. Long ago, when every other compiler was simply awful (obscure error messages, crashes, bugs), GCC’s technical advantage and extra portability, gave it the most contributors. GCC today still enjoys these features, and it produces machine code for more computer architectures than any other compiler, free or not. It also receives many commercial contributions from the same large companies that require its functionality.

Now let’s consider Mailpile. It currently is uniquely positioned. There is no other free, self-hosted, web-based, featureful, and privacy-conscious email client out there. These unique set of features mesh well tactically with copyleft. Mailpile is also emergent, breaking into a new domain, increasing its unique positioning. It is the most likely contender for the post-Snowden email world.

Finally, there is one more tactical advantage of copyleft: it levels the playing field. This is another thing that Linus has said, “[…] you don’t want to get involved if you feel like your contributions would be somehow taken advantage of, but with the GPLv2, that simply was never an issue.” With Mailpile under the AGPL, your contributions will never be taken advantage of, and nobody will ever close up the code away from anyone else.

## May 31, 2015

### Asma Afzal

#### lsqnonlin wrapping nonlin_residmin

So the first week of GSoC is officially over.

I was working on lsqnonlin. My code is accessible here:

https://github.com/AsmaAfzal/octave_workspace/blob/master/lsqnonlin.m

[x, resnorm, residual, exitflag, output, lambda, jacobian]...
= lsqnonlin (fun, x0, lb, ub, options)

[x, resid, cvg, outp] = nonlin_residmin (f, x0, settings)

A recap of encountered problems:
1. output lambda- it wasn't previously returned from the backend [1]
2. options- In addition to optimset, Matlab uses optimoptions to set options specific to a certain optimization function. optimoptions
• Creates and modifies only the options that apply to a solver
• Shows your option choices and default values for a specific solver/algorithm
• Displays links for more information on solver options and other available solver algorithms [2].
octave currently does not have this functionality.  For more discussion on this, check [3]
Things to do:
1. check how user-specified jacobian is to be provided.
2. check for matrix/complex inputs.
3. come up with a plan for writing optimoptions in octave.
References:
[1] http://octave.1599824.n4.nabble.com/lsqnonlin-and-nonlin-residmin-tp4670540p4670557.html
[2] http://uk.mathworks.com/help/optim/ug/optimoptions-and-optimset.html
[3] http://octave.1599824.n4.nabble.com/lsqnonlin-and-nonlin-residmin-tp4670540p4670560.html

## May 28, 2015

### Juan Pablo Carbajal

#### El que busca encuentra

El foro de discusión tiene una nueva entrada: una pregunta de Florencia que está resolviendo el desafío nro. 1.5.

La pregunta de Floppy nos da pié para discutir sobre un tipo de funciones muy útiles. Funciones que nos permiten obtener la posición, dentro de un vector, de elementos que cumplen cierta condición lógica. Esto es muy similar a lo que hicimos en la clase ¿Verdadero o falso? cuando buscamos números pares.

## Find == Econtrar

Las operaciones lógicas generan matrices con valores verdadero o falso. Así, la siguiente desigualdad

tf = [-1,5,-3] > 0

Genera un vector de valores lógicos (verdadero o falso), en este caso [false,true,false]. Estos vectores lógicos son de muchísima utilidad: nos permiten filtrar matrices o buscar elementos que cumplan ciertas condiciones lógicas. Para realizar lo segundo usamos la función find.
La función find (encontrar, en inglés) nos devuelve la posición de aquellos elementos que corresponden al valor "verdadero". Por ejemplo:

find ([false, true, false])

devuelve un 2, porque el segundo elemento es un valor verdadero. En Octave todo aquello que no es cero o vacío se considera verdadero. Es decir que

find ([0,-3, 0])

también devuelve 2, porque el segundo elemento es el único no cero. Cuando varios elementos son verdaderos, la función find devuelve todas las posiciones correspondientes, es decir

find ([false, true, true])

devuelve [2, 3].
La función también acepta matrices como entradas y nos devuelve el índice lineal de los elementos verdaderos. En Octave las matrices están ordenadas por columnas (en Inglés se dice column-major order), es decir contamos de arriba para abajo y de izquierda a derecha. Por ejemplo en una matriz de 3x4 los indices lineales son

Es decir que si ejecutamos

find ([0 1 0; 0 0 1; 0 1 0])

obtenemos [4,6,8]
Para obtener las filas y columnas de los elementos verdaderos, tenemos que llamar a la función con dos argumentos de salida. Si ejecutamos

[i, j] = find ([0 1 0; 0 0 1; 0 1 0]);
[i,j]
ans =
   1   2
   3   2
   2   3

obtenemos en i las filas y en j las columnas de los valores verdaderos.

## ¿Es miembro?

Otra función muy útil es ismember (es miembro, en inglés). Esta función toma como entrada dos argumentos y nos dice si los elementos del primero están en el segundo. Solo voy a proveer un corto ejemplo y los invito a que lean la ayuda de la función (en Octave: help ismember) y pongan sus preguntas en el foro de discusión.
Supongamos que queremos saber si la letra "a" está presente en un string

x = "Hay letras a en esta frase?";

podemos ejecutar

tf = ismember ("a", x)
  tf =  1

Claro que sí. La función también se puede usar para obtener la posición de las letras buscadas. Para esto intercambiamos los argumentos de entrada y buscamos el string x en la letra "a".

pos = find (ismember (x,"a"))
  pos =
      2    9   12   20   24

Pregunta 1. ¿Puedes explicar el porqué de este truquillo?

La función ismember también puede darnos la posición de los elementos encontrados, pero cuando el elemento buscado se repite (como en nuestro caso), solo nos devuelve la última ocurrencia

[tf, pos] = ismember ("a", x)
  tf  = 1
  pos = 24

Esta funcionalidad es útil cuando no hay repeticiones y se ilustra mejor con el ejemplo en la ayuda de ismember

a = [3, 10, 1];
s = [0:9];
[tf, pos] = ismember (a, s)
 tf = [1, 0, 1]
 s_idx = [4, 0, 2]

Las funciones find e ismember son de mucha utilidad y tienen varios otros modos de uso. Lee la ayuda, explora y pregunta en el foro!

## May 27, 2015

### Piotr Held

#### Milestone

I am happy to announce that I completed the first two sections.

Nonlinear Noise Reduction
From this section I added:
• ghkss
Thus deprecating project.  It is important to note that data allocation in ghkss (only) is done via new/delete because of the way the function was originally written. In the future this will be replaced with OCTAVE_LOCAL_BUFFER (OLB) macro.

Closer examination has revealed another interesting function in this section: nrlazy, which according to the documentation is similar to lazy. Because of this similarity porting it has low priority.

Phase Space Representation
From this section I added:
mutualfalse_nearestpoincare
The function poincare was discovered when further studying the documentation and was omitted in the initial plan, but as it seems important in the documentation it was ported.

The function false_nearest is in a similar situation to ghkss, that is, new/delete is used instead of OLB. This will be improved on in the future.

Since corr does not need to be ported, this section is complete.

Nonlinear Prediction
This next section is well on the way as the following have been ported:
upoupoembedlzo_test
The function predict turned out to be essentially the same with lzo_test. This had not been verified so far, but according to the documentation they do essentially the same. The additional option that predict has (flag -v) can be easily replaced using GNU Octave's std when calling lzo_test with parameter r. Therefore porting predict will be most likely unnecessary.

The function upoembed is closely associated with upo. It takes the output of upo and creates a cell array of delay vectors for each orbit produced by upo. It was not mentioned in the original outline, but it is an important function for the package.

The state of upo is not optimal. The original implementation only supported input up to 1e6 data points. This might not be a big problem as calculating upo on a 1e4 henon map takes about 8 seconds, so a 1e6 would take about 800 seconds ~ 15 min. Changing this might be problematic as the main data in the FORTAN program is in a common block (it is stored in a global variable), which cannot contain arrays of variable dimensions. The authors of TISEAN chose 1e6, but because of how the program is written making it unlimited would not be trivial. It might be beneficial to lift this number to 1e8 or 1e9 but one must keep in mind that because of how the FORTRAN program is written it will always allocate a local array of the maximum size possible (currently 1e6 elements). If the maximum input length is lifted to 1e8 or 1e9 the size of the data allocated by Octave and the local copy that the FORTRAN program uses can be a sizable amount of memory to allocate. Moreover it is important to note that each data point will be a real*8 (not a real*4). This brings me to the next point.

FORTRAN data types
This topic has been problematic for me from the very beginning. I had trouble realizing just how the dummy variables work. After some research I found that one can pass -freal-4-real-8 and promote all real*4 to real*8. This is beneficial as the input into a FOTRAN program is passed as doubles. This caused serious issues as I needed to ensure that when I call any TISEAN or SLATEC function/subroutine I needed to use real*4 instead of real*8. The solution I originally used was to copy the input variables into local variables. Apart from eliminating potential bugs and improving code complexity the previously mentioned flag also allows the FORTRAN programs to have the same type of precision expected from GNU Octave programs.

Other things
I spent my time also on other things. I significantly changed the Makefiles, removed all compilation warnings and everywhere except for ghkss and false_nearest moved from new/delete to OCTAVE_LOCAL_BUFFER or some type of Array.

## May 26, 2015

### Juan Pablo Carbajal

#### ¿Verdadero o falso?

En el cole te enseñaron una cosa que llaman lógica formal (o solamente lógica). Si te gustó tenés suerte, probablemente tu profesor/a era muy buen@ y entendía el tema (¡yo tuve mucha suerte!). Lamentablemente la mayoría de los estudiantes con los que charlo odian la lógica. En esta clase voy a tratar de darles un ejemplo de porqué la lógica es super importante y que no es tan aburrida como muchos creen.

Esta clase puede ayudarte con los desafíos del nivel 1, si todavía no los resolviste.

## "La lógica no sirve para nada"

¡Cuantas veces habrás escuchado esa frase o alguna otra similar! Como la mayoría de las herramientas matemáticas, la lógica no es más que una formalización de algo que hacemos naturalmente. Voy a tomar un ejemplo mundano: seleccionar fruta o verdura cuando vamos de compras.

Hace unos días, en uno de mis viajes de aprovisionamiento al mercadito de la esquina, observaba con cierta admiración a una mujer mayor seleccionar las manzanas que ponía en su bolsa de las compras. Su meticulosidad era sorprendente y me preguntaba si esa mujer no habría sido una matemática en sus años de actividad laboral.
Cada vez que la señora juntaba una manzana, la miraba de muchos ángulos diferentes, la golpeaba suavemente con un dedo, la olía y finalmente decidía si la manzana volvía a la góndola o si se convertía en parte de su compra.
Podemos hacer un modelito sencillo del proceso de selección y pensar que la señora ejecutaba una función que tomaba como entrada los aspectos de la manzana bajo observación. Esta función evalúa tres condiciones y decide si la manzana se compra o se devuelve:
1. Tiene buena forma?
2. Suena bien?
3. Huele bien?
La respuesta a estas preguntas es si o no. Podemos re-escribir las preguntas como afirmaciones (o condiciones) y decidir si estas afirmaciones son verdaderas (en inglés: true) o falsa (en inglés: false):
1. La manzana tiene buena forma.
2. La manzana suena bien al ser golpeada suavemente.
3. La manzana huele bien.
Si la manzana en cuestión hace que todas estas afirmaciones sean verdaderas, entonces la compramos. De lo contrario, si cualquiera de estas afirmaciones es falsa, devolvemos la manzana a la góndola.

## Símbolos y operadores

En el ejemplo introduje los dos símbolos fundamentales de la lógica: true y false. Estos son los nombres que se utilizan en GNU Octave para estos valores, pero vos podés usar cualquier par de valores que te guste: (V,F); (0,1); (-1,1); (0,5); (blanco, negro); etc.. Para poder ejercer la lógica necesitamos saber como operar con estos valores.

Tomemos dos afirmaciones, una falsa y otra verdadera, para ejemplificar.
1. La afirmación verdadera es: la palabra "hacha" tiene 5 letras
2. La afirmación falsa:  La palabra "hacha" no tiene ninguna letra "h".
Verifiquemos los valores lógicos de estas afirmaciones en GNU Octave. El siguiente código guarda el string "hacha" en una variable llamada p (de palabra) y evalúa la 1ra afirmación.

p = "hacha";
# Cuenta la cantidad de letras
nLetras = length (p);
# Evalua la 1ra afirmacion
nLetras == 5

La primera linea de código guarda la palabra en la variable. La linea luego del primer comentario devuelve la longitud del string p, es decir la cantidad de letras, que guardamos en a variable nLetras (recuerda que las lineas que empiezan con el caracter # o %, son ignoradas por GNU Octave, son lineas de comentarios, y tu también puedes ignorarlas. Son solo ayuda). La última linea de código pregunta si nLetras es igual a 5. Notar que el operador == no asigna el valor a nLetras, sino que compara los valores y responde verdadero si estos son iguales (experimenta un poco con este operador!).
Si ejecutas esas líneas de código en una sesión de Octave obtendrás la respuesta

ans = 1

En Octave el valor true se representa con un 1 (ejecuta true en tu sesión de Octave para verificar) por lo tanto el valor de la 1ra afirmación es verdadero. Veamos el código de la 2da afirmación:

all (p != "h")

¿Simple, eh? Dejo como ejercicio entender que esta ocurriendo aquí. Lo que necesitas saber es que all en inglés significa "todo/todos" y que el operador != es lo opuesto a == (verifica desigualdad en vez de igualdad) ¿Qué ocurre cuando comparamos el string p con el caracter "h"?

La respuesta de Octave a esta linea de código es 0. Indicando que, efectivamente, la segunda afirmación es falsa.

Ok, tenemos una afirmación verdadera y una falsa, y vamos a guardar estos valores de verdad en un vector:

afirmacion = [true, false]

De esta forma afirmacion(1) es verdadera (true, 1) y afirmacion(2) es falsa (false, 0).

¿Qué dirías de la nueva afirmación más abajo?
• La palabra "hacha" tiene 5 letras y la palabra "hacha" no tiene ninguna letra "h"
¿Es verdadera o falsa?

El operador que hemos usado para construir esta nueva afirmación es el "y", en inglés "and", y que en lógica llamán conjunción. En Octave podemos utilizar la función and o el operador &, veamos:

# operador
afirmacion(1) & afirmacion(2)
# funcion
and (afirmacion(1), afirmacion(2))

Ejecuta este código ¿Estas de acuerdo con Octave sobre el valor de verdad de la nueva afirmación?
¿Cuál sería el resultado si las dos afirmaciones fuesen verdaderas?

Veamos otra afirmación
• La palabra "hacha" tiene 5 letras o la palabra "hacha" no tiene ninguna letra "h"
Para construir esta nueva afirmación utilizamos la disyunción inclusiva, el "o". Pero no el "o" exclusivo (que algunos escriben "ó" para que quede claro), sino un "o" que está contento si esto es verdad, si lo otro, o si ambos son verdad. Este operador en Octave se escribe | y la función es or:

# operador
afirmacion(1) | afirmacion(2)
# funcion
or (afirmacion(1), afirmacion(2))

Experimenta con este operador para entender como funciona.

Los operadores & y |, al igual que las respectivas funciones toman como entrada dos afirmaciones, son operadores binarios (en el sentido que toman 2 entradas). Otros operadores binarios que ya conocés son el + y el *, ambos toman dos números como entrada y devuelven el resultado.

Ejercicio 1: ¿Puedes construir el resultado de los operadores & y | para todas las entradas posibles?

Ejercicio 2: ¿Puedes escribir la afirmación que la señora de la historia evaluaba para seleccionar las manzanas?

## Ejemplo numérico

Ok, toda esta cháchara sobre lógica, pero ¿Cómo nos puede ser útil para los desafíos?

En los desafíos de nivel 1 es necesario filtrar o seleccionar ciertos números según sus propiedades. Por ejemplo ¿Cómo podemos seleccionar números pares?
Lo que sabemos es que un número par se escribe como

K = 2*N

donde N puede ser cualquier numero entero. Todo bien con esto, pero para cada número K que me dan, tendría que buscar un N, entre todos los enteros, que cumpla con esta condición...posible, pero medio pesado. De manera equivalente podemos decir que un número es par si al dividirlo en 2 el resto es cero. Es decir, un número par es divisible por 2. GNU Octave tiene una función que nos devuelve el resto de una división entera, la función rem (del inglés "remainder", que quiere decir "resto" o "sobrante") . Como ejemplo de uso tomemos la division de 7 en 2 partes. 7 se puede repartir entre 2 en 3 partes iguales, pero nos sobra 1. Veamos que dice a la función:

rem (7, 2)
ans = 1

El primer argumento es el numero a dividir (el dividendo) y el segundo argumento es el divisor.

Ahora vamos a utilizar esta función y nuestro conocimiento de lógica para seleccionar los números pares de una lista de números:

l = 0:3:15;
tf = rem (l,2) == 0
tf =
   1   0   1   0   1   0

La lista l contiene números enteros desde 0 hasta 15 en incrementos de a 3 y Octave nos dice que la afirmación "el número es divisble por 2" es verdadera para el primer número, falsa para el segundo, verdadera para el tercero y así sucesivamente.
¿Cuales son estos números? Claramente 0, 6 y 12. En Octave podemos obtener estos números utilizando el vector lógico al que hemos llamado tf. Prueba lo siguiente

l(tf)

Sigue experimentando con este tipo de ejercicios hasta que entiendas como funciona la cuestión. Si tienes preguntas o sugerencias no dudes en escribirlas en el foro de discusión o los comentarios.

Adjunto un archivo con todos los comandos que hemos utilizado en esta clase.

### Antonio Pino

#### On galleries and the beginning of summer

After the community bonding period and before starting today the coding period, I will briefly list the transformation that has undergone my initial proposal: from just implementing new algorithms and then add them to GNU Octave, to various modifications of GNU Octave itself so that Higham's toolboxes run smoothly and in the end add the new algorithms. Sticking to what I said before, I expect to be doing the modifications (e.g. new bugs, patches, toolboxes) most of the first half of the coding period. From there we aim to go as far as we can about matrix functions, I will do so. These changes are noted in the new time line.
During the community bonding period I have been setting up, you might have seen me at freenode. I have been becoming more and more acquainted with GNU Octave, and found out that the gallery function was broken, with unassigned variables and missing auxiliary functions. This function will prove useful to test matrix functions, because the eigenvalue decomposition strategy (if $A=VDV^{-1}$  then $f\left(A\right) = V f\left( D \right) V^{-1}$) yields a big error using the ill conditioned matrices gallery provides. Another example are the useful positive definite matrices that have a computable principal p-th root. gallery is indeed interesting for anyone looking for a matrix with a special characteristic to test an edge case of a function.
Besides, I also have had the chance to see a plethora of Matlab-style short circuit operators, looping with infinite ranges, and even weird undocumented functions like superiorfloat (it returns either "single" or "double" strings depending on the input). What let Carnë to point me to Undocumented Matlab blog, where they document its unsupported hidden underbelly. More quirks (and their solutions) on the next post.
A final thank goes to The Project in general and my mentors (Carnë and Mario) in particular for this opportunity. I hope everyone pleasantly codes their summer away!

## May 25, 2015

### A brief intro

First of all, let me introduce myself: I am Antonio Pino Robles—an Electronic Engineering student from the Basque Country—and I will be improving matrix functions in GNU Octave this summer, following Google Summer of Code program.
The idea behind this is quite simple: given a square matrix $M\in \mathbb{C}^{n \times n}$ and a function $f$, GNU Octave will compute $f\left(M\right)$. You may think of them as an extension to scalar functions, i.e. starting from $f:\mathbb{C}\rightarrow \mathbb{C}$ compute $f:\mathbb{C}^{n \times n}\rightarrow \mathbb{C}^{n \times n}$. Their implementation is quite different, though. (Check Golub and van Loan's book[0] and the Short Course by Higham and Lin[1] for further info.)
Let me note that matrix functions are already part of octave: expm, logmsqrtm in octave itself and funm,  trigonometric and hyperbolic matrix functions in the Linear-Algebra Octave-Forge package. There are also GPLed toolboxes by Nicholas J. Higham, namely the mctoolbox[2] and the mftoolbox[3]; furthermore, GPLed software from the NAMF group—led by N. J. Higham at The University of Manchester—is available as well.
Hence, on a first part octave will be modified in order to run the toolboxes—as they are—smoothly , and then the existing implementations will be improved by means of updating their algorithms.
Finally, for a more detailed description of the project please refer to my octave-wiki page:
http://wiki.octave.org/User:Antonio_Pino

Agur bero bat!

[0] G.H. Golub and C.F. Van Loan. Matrix Computations, 4th Edition. The Johns Hopkins University Press, Baltimore, USA, 2013.
[1] Nicholas J. Higham and Lin Lijing, Matrix Functions: A Short Course, preprint, (2013).
[2] N. J. Higham. The Matrix Computation Toolbox. http://www.ma.man.ac.uk/~higham/mctoolbox
[3] N. J. Higham. The Matrix Function Toolbox. http://www.ma.man.ac.uk/~higham/mftoolbox

## May 23, 2015

### Asma Afzal

#### Quadratic Programming quadprog, qp

Just a post to review the things I've learned over the past week:

Intuitive explanation of Lagrange multipliers [1]. Given a D-dimensional function to optimize and a set of a few equality constraints.
• The vector normal to the function should be a scaled version of the vector normal to the constraint function (In other words, both normals are parallel to each other). This scaling factor is the Lagrange multiplier corresponding a particular constraint.
• $\nabla f=\lambda_1\nabla g+\lambda_2 \nabla h$, where $g$ and $h$ are two constraint functions.

KKT Conditions: Generalized method of Lagrange multipliers to be applicable on Inequality constraints.  $g_i(x)-b_i\geq 0$

• Feasibility- $g_i(x^*)-b_i\geq 0$
• Stationarity- $\nabla f(x^*)-\sum\limits_i\lambda_i^*\nabla g_i(x^*)=0$
• Complementary Slackness $\lambda_i^*(g_i(x^*)-b_i)=0$
• Positive Lagrange multipliers $\lambda_i \geq 0, \forall i$

In case we obtain negative Lagrange multiplier, the constraint corresponding to the most negative multiplier is removed and the optimization is performed again until all multipliers are positive.

A bit about active set algorithm [2]:

• Possibility that none of the constraints are active or may be some are active. We only need to solve for equality constraints that are active at the optimum (binding).
• When we have an active set $S^*, x^* \in F$, where $F=\{x|A x^*\leq b\}$,$\lambda^* \geq0$, where $\lambda^*$ is the set of Lagrange multipliers for equality constraints $Ax=b$
Algorithm:
• Start at $x=x_0$ and initial active set
• Calculate $x^*_{EQP}, \lambda_{EQP}^*$ which minimizes the EQP defined by the current active set. Two possible outcomes:
1. $x^*_{EQP}$ is feasible  ($x_{EQP}^* \in F$). Set $x=x_{EQP}$ and check Lagrange mulitpliers $\lambda_{EQP}^*$. If positive, solution found! Otherwise, remove constraint with $\min(\lambda_{EQP}^*)$ and repeat.
2. $x^*_{EQP}$is infeasible. We move as far as possible along the line segment from $x_0$ to $x^*_{EQP}$ while staying feasible. Add to $S$ the constraint we encounter that prevents further progress. This is the blocking constraint.

Quadratic programming:

$\min\limits_{x}\frac{1}{2} x^THx + x^Tq$
s.t.
$A_{eq} x = b_{eq}$
$lb \leq x \leq ub$
$A_{lb} \leq A_{in}x \leq A_{ub}$

What qp.m does:

[xobjinfolambda] = qp (x0HqAeqbeqlbubA_lbA_inA_ub)
• Checks feasibility of initial guess $x_0$
• Checks size of inputs and that they make sense.
• Checks if bounds lb,ub too close or A_lb or A_ub too close. If they are very close then the inequality is treated as an equality constraint instead.
• Checks if any bound is set to Inf or -Inf. qp simply strikes it off.
• Calls backend solver __qp__ using null space active set algorithm.
The ordering of lambda

• quadprog returns Lagrange multipliers in a structure (with fields upper, lower, eqlin, ineqlin) and the multipliers corresponding to the constraints not provided are left empty.
• In qp, lambda is a column vector with Lagrange multipliers associated to the constraints in the following order: [equality constraints; lower bounds; upper bounds; other inequality constraints]
• The length of lambda vector output from qp depends on the number of different constraints provided as  input.
• Two issues in wrapping qp.m
1.  the order, i.e. the position of the bounds constraints within the inequality constraints) is not specified by qp.m. The code could change and the ordering too.
2. qp.m strips off the INF constraints before calling __qp__ but does not process the lambda (returned by __qp__) accordingly.
• Solution:
• If this order is "specified" then we could extract parts of lambda. Patch for Inf checks in qp output lambda will make things easier but is not critical.
References
[1] http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html
[2] https://www.youtube.com/user/ucmath352/videos

## May 20, 2015

### Mike Miller

#### Birthday Resolutions - Review

Last year on my birthday I decided to try setting some goals for self-improvement with a deadline of the following birthday. People typically set New Year’s resolutions for themselves, but I wanted to try something different. Partly because I’m a natural contrarian, but also because my birthday last year was unique and offered more than a few reasons for self-reflection. So with another birthday looming, it’s time now to review how this experiment worked out.

First, because this was a particularly notable birthday I had decided to hold myself to 10 resolutions. So this was almost doomed to failure from the beginning, if success means hitting all 10, which I didn’t. If I want to do this again next year, I should definitely go with a smaller set of goals to better set myself up for success. Obvious.

Some of my goals were broader than others, which made it harder to define a successful target to aim for. For example, my goal to attend more free software developer conferences (did) was a lot easier to define and complete than my goal to make more time for creative pursuits (didn’t).

Despite these problems, I like how this experiment turned out. I was able to accomplish about half of my 10 goals (for some definition of “accomplish”), and I’m not one to dwell on the other half that didn’t get done. I also like pinning personal goals to my birthday, rather than arbitrarily to the start of the Western calendar year. It reminds me to not only celebrate my birthday but to keep trying to improve from year to year.

Anyone else tried this? Any other non-traditional ideas for annual, or more frequent, resolutions and personal goals?

## May 15, 2015

### Piotr Held

#### Progress report

As I haven't had any significant roadblock or breakthroughs this week I wanted to give a little progress report on my work.
1. Added functionality
I have managed to add the following functions:

• mutual
• spectrum
• lazy
• delay
• pca
Along with their documentation, tests and a demo (for lazy). I was really happy that once I had produced some examples of how I want to port these functions the process of porting each one accelerated rapidly.
I am especially excited about the fact that I have now henon, delay, an equivalent of addnoise and project available as this allowed me to create a nice noise reduction demo for project (and for lazy, but the one for project is more impressive). Fig.1 sums those efforts up.
 Fig. 1 Noisy data and data cleaned up by project.
2. Functions found to be non-equivalent
I also spent a lot of my time (almost a week) researching which programs from TISEAN have a GNU Octave equivalent. Apart from my positive identifications, most of which I discussed in previous posts, I have made some negative ones. I found that both extrema and polynom have no GNU Octave equivalent.
There was a suggestion made that extrema might be similar to findpeaks from signal. The only problem is that findpeaks searches (and returns) all peaks, whereas extrema returns either minimums or maximums. It might be easier to implement it in Octave than to port, but this decision has not been made yet.
The latter program polynom was compared to detrend, polyfit and wpolyfit. The results were disappointing. All of the GNU Octave functions attempt to fit a polynomial onto the data, whereas polynom tries to make a "polynomial ansatz" for the data. The results are vastly different as can be seen on Fig. 2.
 Fig. 2 Comparison of original data (green), polyfit fit (red), and polynom prediction (blue).
Both programs were run to try to use a 4th order polynomial.

## May 14, 2015

### Asma Afzal

#### Nonlinear Regression and 'nlinfit'

In MATLAB, all three fucntions 'lsqnonlin', 'lsqcurvefit' and 'nlinfit' are used to perform non-linear curve fitting.

To better understand the differences and similarities in these functions, consider the model function:
$y= \beta_1+\beta_2 \text{exp}(-\beta_3x)$

We wish to estimate the $\beta=\{\beta_1,\beta_2,\beta_3\}$ for the set of independents {$x_i$} and observed values {$y_i$} such that the model fits the data.

Both 'nlinfit' and 'lsqcurvefit' are very similar as we can pass the regression function to compute the parameters. 'lsqnonlin' on the other hand, solves optimization problems of the type $min_{\beta} \sum_k f_k(\beta)^2$, so we cannot directly specify the regression function and instead, an error function has to be provided.  This is shown in the code below:

modelfun = @(b,x)(b(1)+b(2)*exp(-b(3)*x));
b = [1;3;2]; %actual
x = exprnd(2,100,1); %independents
y = modelfun(b,x) + normrnd(0,0.1,100,1); %noisy observation
beta0 = [2;2;2]; %guess
beta = nlinfit(x,y,modelfun,beta0)
beta = lsqcurvefit(modelfun,beta0,x,y)
beta = lsqnonlin(@(b)err_fun(b,x,y),beta0) %err_fun = modelfun-y

All three functions generate:

beta =

1.0071
3.0805
2.1418

Observations:
• lsqcurvefit is more superior in the sense that we can define the bounds for the design variable (unlike nlinfit) while inputting the observed values separately (unlike lsqnonlin).
• Nlinfit provides extra statistics such as covariance matrix of the fitted coefficients and information about error model.
• As an alternative to defining weights for the observed values in 'nlinfit', 'RobustWgtFtn' option can choose from different pre-defined weight functions for robust regression (with robust regression, fitting criterion is not as vulnerable to unusual data as least squares weighting function.)

References:

## May 12, 2015

### Mike Miller

#### Octave + Python: A New Hope

As a fan of both Python and Octave for numerical computing, and an active Octave developer, I’m always excited to hear about projects in either environment that create new capabilities or open up new ways of looking for solutions to problems. So I am especially excited about a new project that has the potential to bring Octave and Python much closer together and to give users of either tool full use of the other.

The broad goal of this project is to provide a two-way interface layer between Octave and Python. What does this mean specifically? Well, I expect a future version of Octave to have a function that will call Python functions, using an embedded Python runtime, with transparent conversion between native Octave types and Python / NumPy types. There will also be a Python module to do the inverse: allow Python code to call Octave functions, invoke an embedded Octave interpreter, and have automatic conversion between Python and Octave types.

The way in which the seeds of this project came together very quickly is really interesting, and what I want to describe in this post. The first was in a mailing list side discussion in late March about the appropriateness of Octave and Matlab for teaching numerical programming. It was mentioned that recent versions of Matlab have a calling interface to Python. For years they had provided a similar interface to Java, but I had no idea that Python was now an option for Matlab users. I filed that away for later.

Then there is the Octave symbolic package, which relies heavily on SymPy to do the actual symbolic computation, but interacts with Python and SymPy over a pipe. So that existing package would definitely benefit from having a Python interpreter embedded in Octave or in a loadable oct-file.

And finally there was a post in early April from fellow Octave developer JordiGH, who wrote:

I have a wild idea. I like Python, and I think Numpy and Scipy are a great tool. Interfacing Scipy with Octave is also a good thing. … I therefore propose to bring Pytave into Octave proper.

Pytave is an already-existing project which provides a Python module that can call Octave functions. It worked with older versions of Octave years ago, but has not kept up with the Octave API. It did work, and it does have a lot of useful code for converting between Octave and Python types, lots of good groundwork to start building from.

I’m not sure what led Jordi to think of this “wild idea” or share it with us, but it definitely inspired me to latch onto this project. The timing of his message, after the other previous uses and mentions of Python, and being just days before the start of my first PyCon experience, read to me like a call to action. This felt like a perfect confluence of events and ideas to bring Octave and Python together in a novel way.

So, I have already put some effort into this, and am planning to do some more. I hope that I (and any other interested contributors) will be able to make some real progress on this Octave-Python interface during this summer. I will share some more specifics about the project in a followup post soon.

Thoughts about this project? Interested in following our progress or contributing?

## May 07, 2015

### Piotr Held

#### The problem with 'spectrum'

The 'spectrum' function from TISEAN most likely needs to be rewritten in GNU Octave or there is no need for it. This is because linking to it does not seem like a good idea. This is because there is a suspicion it does not produce good results for some data inputs.

1. Where 'spectrum' works
First it is important to note that 'spectrum' from TISEAN is basically a GNU Octave 'abs(fft(:))' with additional data manipulation/adjustment. This additional work is not an elegant one-line solution, which might warrant designating a separate function that would translate the Octave respective function into a form similar to the output of 'spectrum'. Although this might not be necessary since the data obtained from the Octave function is very similar to 'spectrum' (Fig. 1).
 Fig. 1 Unadjusted data from Octave
After adjusting the data (which was done by analyzing the source code to determine what actions the TISEAN programs perform) it was possible to get a close fit with a small difference. An example of this type of adjustment is listed below (Fig. 2)
 Fig. 2 Adjusted data from Octave.

As it is not a one-line fix to convert ' abs (fft (:))' into a similar format as 'spectrum' it will not be shown in the post. It is available in the 'tests/spectrum/test_spectrum.m' function located on the tisean package repo (here).

2. Where the problem lies
The problem is that when 'spectrum' is used to create a step response its results vary substantially from what is produced by Octave. The way the data looks suggests that there is something wrong with 'spectrum'. The adjusted version is situated below.
For the most part, the data fits perfectly, but there seems to be a shadow on the bottom of the TISEAN data. If it is the case that there is a problem with 'spectrum' then its code should not be used in the future Octave package and should be rewritten or omitted (as similar results can be obtained from a simple Octave call).

## May 06, 2015

### Piotr Held

#### Finding 'histogram' in GNU Octave

Unlike 'corr' it is quite easy to find a representative for 'histogram' from TISEAN. It is 'hist' from GNU Octave. The data is almost the same with the exception that the TISEAN package normalizes by default so one needs to be careful when calling the respective functions. I will describe differences in the data and describe the differences in usage.

1. Data comparison
I have attached a comparison of the two data sets (from 'hist' and 'histogram' on one chart)
 Fig. 1 Comparison between 'hist' (Octave) and 'histogram' (TISEAN)
When  one analyses the data close there is a slight discrepancy between the value on the 40th and 41st bar. But not only is it slight, it basically means that both programs assigned a certain value to two different bins, which should not be a major problem. All told we can say that both of those functions perform the same task.

2. Usage comparison
As mentioned before, usage varies on both functions.
 $histogram amplitude.dat -b#n -o "amplitude_histogram.dat"  [nn, xx] = hist (amplitude, #n, 1); nn = transpose (nn); xx = transpose (xx) amplitude_hist = [xx, nn]; This way the data stored in 'amplitude_hist' is essentially the same with 'amplitude_histogram.dat'. #### Finding a 'corr' representative in Octave This article describes the methodology used to compare function from GNU Octave and the TISEAN package. To achieve the desired results the author assumes you have installed the TISEAN package (available here) and have downloaded amplitude.dat and have installed GNU Octave with the 'signal' package in version 1.3.0 or newer. 1. Comparison Procedure taken to receive results: 1. Generate amp_corr.dat using the TISEAN package 'corr' with the call:  '$ corr amplitude.dat -D5000 -o "amp_corr.dat"'
2. Generate similar autocorelation data using (in GNU Octave):
 'load amplitude.dat; [a,b] = xcorr(amplitude, 5000, 'coeff');'
Then to save the data you can use:
 'idx = [rows(amplitude):2*rows(amplitude)-1];   xcorr_res = a(idx);   save "xcorr_res.dat" xcorr_res'
There is a strong difference in the data. This might be because of the different methods used in both cases (as explained further in the methods used Section 2. Methods). Because of those differences the amplitudes of the data generated using 'xcorr' from 'signal' decreases linearly. Thus to compare the data the oscillation amplitude of the data generated by 'xcorr' must be amplified. This linear decrease was not proven but observed on the 'amplitude.dat' data.

When a linear correction is applied:
 'mult = rows (amplitude) ./ (rows (amplitude) - [0:rows(amplitude)-1]);    xcorr_tisean_res = mult .* xcorr_res'
 Fig. 1 Difference between xcorr_tisean_res and amp_corr

The resultant xcorr_tisean_res is close to the TISEAN 'corr' function, and the difference is smaller than 3% (see Fig. 1). The end of the data begins to change and this is most likely because there is no more data past 5000 and so the results vary. If a autocorrelation is calculated for less data (e.g.4500 instead of 5000) the difference is much less, as can be seen on the chart above.

Even better results can be obtained for different data. We can generate a different set using the TISEAN package
 '$ar-model -s5000 amplitude.dat -p10 -o "amp_ar.dat"'  When the process described above is applied to this new data set ('amp_ar.dat') the resulting difference between 'xcorr' and 'corr' is shown on Fig. 2.  Fig. 2 Difference between 'xcorr' and 'corr' on 'amp_ar.dat' Similarly to the previous case the data is the same for small ( < 4000) numbers but when they get close to the edge the difference becomes more pronounced. 2. Methods The way TISEAN calculates autocorrelation in the 'corr' program is by using estimation method. It is described here: http://en.wikipedia.org/wiki/Autocorrelation#Estimation On the other hand the 'xcorr' function from the signal package uses the FFT (Fast Fourier Transform) method (it is described in the same Wikipedia article: here) This difference in methodology is the cause of the difference in the data results between both functions. 3. Conclusions [edited] After more test we found 'corr' from TISEAN and 'xcorr' from 'signal' to perform the same autocorrelation and therefor it is not necessary to port it. It is important to note the different usage: $ corr amplitude.dat -Dn# -n -o "amplitude_tisean.dat"
 [data, lags] xcorr (amplitude, n#, 'unbiased')     data = data(find (lags >0))
Both of the usage noted above produce the same data.

It is important to note the '-n' in the calling of the TISEAN program. It mean the data is not normalized. You can achieve similar data even when calling 'corr' with normalization, but it is more tricky:

# Define the command

The final part is to just turn this on:

[alias]
wip = log --graph --rev=wip --template=wip

This defines the hg wip command to be an alias to hg log together with the parameters for displaying the graph, using the wip revset, and the wip template.

And voilà, fancy shmancy colourful hg command! Here is the complete addition to your ~/.hgrc all at once, for delicious copypasta:

[revsetalias]
wip = (parents(not public()) or not public() or . or head()) and (not obsolete() or unstable()^) and not closed()

[templates]
wip = '{label("log.branch", ifeq(branch, "default", "", branch))} {label("changeset.{phase}", rev)} {label("grep.user", author|user)}{label("log.tag", if(tags," {tags}"))} {bookmarks % "{ifeq(bookmark, currentbookmark, label('log.activebookmark', bookmark), label('log.bookmark', bookmark))} "}\n{label(ifcontains(rev, revset('parents()'), 'desc.here'),desc|firstline)}'

[extensions]
color=

[color]
mode=terminfo

#Custom colours
color.orange=202
color.lightyellow=191
color.darkorange=220
color.brightyellow=226

#Colours for each label
log.branch=cyan
log.summary=lightyellow
log.description=lightyellow
log.bookmark=green
log.tag=darkorange
log.activebookmark = green bold underline

changeset.public=orange bold
changeset.secret=blue bold
changeset.draft=brightyellow bold

desc.here=bold blue_background

[alias]
wip = log --graph --rev=wip --template=wip

# Acknowledgements

This command comes from ideas cobbled together from Steven Losh, Augie Fackler, and Sean Farley. They are all great contributors to Mercurial, and they have taught me so much! Thanks, guys!

## Dense Output

Specifying specific output times for the solution, should not affect the internal time steps that the solver uses. The basic idea of the Dense Output concept is to provide the solution at a given time $$s \in [t, t+dt]$$ with the same order of accuracy as the solutions computed at the internal time points by using suitable interpolation methods.
Up to now only linear interpolation was performed and this significantly lowered the accuracy if a higher order solver was used.
I then implemented a series of interpolation function:

• linear_interpolation:
x_out = linear_interpolation (t, x, t_out)
Given the time span $$t=[t_0, t_1]$$ and the function values $$x=[x_0, x_1]$$, it returns the linear interpolation value $$x_{out}$$ at the point $$t_{out}$$.
• quadratic_interpolation:
x_out = quadratic_interpolation (t, x, der, t_out)
Given the time span $$t=[t_0, t_1]$$, the function values $$x=[x_0, x_1]$$ and the derivative of the function at the point $$x_0$$, it returns the quadratic interpolation value $$x_{out}$$ at the point $$t_{out}$$.
• hermite_cubic_interpolation:
x_out = hermite_cubic_interpolation (t, x, der, t_out)
Given the time span $$t=[t_0, t_1]$$, the function values $$x=[x_0, x_1]$$ and the derivatives of the function at both points $$x_0$$ and $$x_1$$, it returns the 3rd order approximation $$x_{out}$$ at the point $$t_{out}$$ by performing Hermite interpolation.
• hermite_quartic_interpolation:
x_out = hermite_quartic_interpolation (t, x, der, t_out)
Given the time span $$t=[t_0, t_1]$$, the function values $$x=[x_0, x_{1/2}, x_1]$$ (where $$x_{1/2}$$ is the value of the function at the time $$t_0+dt/2$$) and the derivatives of the function at the extremes $$x0$$ and $$x1$$, it returns the 4th order approximation $$x_{out}$$ at the point $$t_{out}$$ by performing Hermite interpolation.
• dorpri_interpolation:
x_out = dorpri_interpolation (t, x, k, t_out)
This interpolation method is specific for the Dormand-Prince Runge-Kutta scheme. Given the time span $$t=[t_0, t_1]$$, the function value $$x=x_0$$ and the vector $$k$$ with the function evaluations required in the Dormand-Prince method, it returns the 4th order approximation $$x_{out}$$ at the point $$t_{out}$$. For more information on the method have a look at Hairer, Noersett, Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (pag. 191-193).
• hermite_quintic_interpolation:
x_out = hermite_quintic_interpolation (t, x, der, t_out)
Given the time span $$t=[t_0, t_1]$$, the function values $$x=[x_0, x_{1/2}, x_1]$$ and the derivatives of the function at each point, it returns the 5th order approximation $$x_{out}$$ at the point $$t_{out}$$ by performing Hermite interpolation.
These methods are then used to perform the Dense Output according to the order of the solver chosen. This is the piece of code in integrate_adaptive.m that performs the interpolation:
% if next tspan value is caught, update counterif( (z(end) == tspan(counter)) || (abs (z(end) - tspan(counter)) / ...    (max (abs (z(end)), abs (tspan(counter)))) < 8*eps) )  counter++;% if there is an element in time vector at which the solution is required% the program must compute this solution before going on with next stepselseif( vdirection*z(end) > vdirection*tspan(counter) )% initializing counter for the following cycle  i = 2;  while ( i <= length (z) )    % if next tspan value is caught, update counter    if( (counter <= k) && ...        ( (z(i) == tspan(counter)) || (abs (z(i) - tspan(counter)) / ...        (max (abs (z(i)), abs (tspan(counter)))) < 8*eps)) )      counter++;    endif        % else, loop until there are requested values inside this subinterval    while((counter <= k) && (vdirection*z(i) > vdirection*tspan(counter)))      % choose interpolation scheme according to order of the solver      switch order      case 1        u_interp = linear_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], tspan(counter));      case 2        if (~isempty (k_vals))          der = k_vals(1);        else          der = feval (func, z(i-1) , u(:,i-1), options.vfunarguments{:});        endif        u_interp = quadratic_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], der, tspan(counter));      case 3        % only ode23 - use k_vals        u_interp = hermite_cubic_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], [k_vals(:,1) k_vals(:,end)], tspan(counter));      case 4        % if ode45 used without local extrapolation this function doesn't require a new function evaluation.        u_interp = dorpri_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], k_vals, tspan(counter));      case 5        % ode45 with Dormand-Prince scheme:        % 4th order approximation of y in t+dt/2 as proposed by Shampine in Lawrence, Shampine, "Some Practical Runge-Kutta Formulas", 1986.        u_half = u(:,i-1) + 1/2*dt*((6025192743/30085553152)*k_vals(:,1) + (51252292925/65400821598)*k_vals(:,3) - (2691868925/45128329728)*k_vals(:,4) + (187940372067/1594534317056)*k_vals(:,5) - (1776094331/19743644256)*k_vals(:,6) + (11237099/235043384)*k_vals(:,7));        u_interp = hermite_quartic_interpolation ([z(i-1) z(i)], [u(:,i-1) u_half u(:,i)], [k_vals(:,1) k_vals(:,end)], tspan(counter));        % it is also possible to do a new function evaluation and the quintic hermite interpolator        %f_half = feval (func, t+1/2*dt, u_half, options.vfunarguments{:});        %u_interp = hermite_quintic_interpolation ([z(i-1) z(i)], [u(:,i-1) u_half u(:,i)], [k_vals(:,1) f_half k_vals(:,end)], tspan(counter));      otherwise        warning ('high order interpolation not yet implemented: using cubic iterpolation instead');        der(:,1) = feval (func, z(i-1) , u(:,i-1), options.vfunarguments{:});        der(:,2) = feval (func, z(i) , u(:,i), options.vfunarguments{:});        u_interp = hermite_cubic_interpolation ([z(i-1) z(i)], [u(:,i-1) u(:,i)], der, tspan(counter));      end      % add the interpolated value of the solution      u = [u(:,1:i-1), u_interp, u(:,i:end)];                  % add the time requested      z = [z(1:i-1);tspan(counter);z(i:end)];      % update counters      counter++;      i++;    endwhile    % if new time requested is not out of this interval    if ((counter <= k) && (vdirection*z(end) > vdirection*tspan(counter)))      % update the counter      i++;    else      % else, stop the cycle and go on with the next iteration      i = length (z) + 1;    endif  endwhileendif

It is important to notice that:

• The 1st order approximation doesn't require any additional function evaluation.
• The 2nd order approximation may require the evaluation of the function at the current time. This can be avoided if the stepper already returns that value.
• The only 3rd order solver implemented is ode23. The 3rd order approximation exploits the Runge-Kutta $$k$$ values to avoid further function evaluations.
• There are no 4th order schemes as yet implemented. However if ones were to use ode45 without local extrapolation then the dorpri_interpolation function can be used to obtain a 4th order approximation without any additional function evaluation. For any other 4th order scheme the hermite_quartic_interpolation function can be used.
• For the 5th order method ode45, Shampine proposes to obtain a 4th order approximation at the middle point and to use quartic interpolation. It is however possible to directly do quintic interpolation but this require an additional function evaluation without (according to Shampine) a significant improvement.
• For the higher order solvers (ode78), a suitable interpolator has not yet been implemented.
Finally, since I wrote the interpolation functions in such a way that they are independent of the number of output points requested, a possible improvement would be to compute all the values of the dense output between $$t$$ and $$t+dt$$ all at once instead of one value at a time.

## August 18, 2014

### Eduardo Fernández

#### Firm pencils down.

During this week I have been reorganizing all the code, docs and tests in a better way for integrating into Octave. As Rik kindly suggested, I decided to organize things this way:

• Inside libinterp/dldfcn directory I have created two files, __ichol__.cc and __ilu__.cc
•  Within those files there are the dld functions that implements the each of the algorithms. They are ment to be built-in functions and follows the __foo__.cc naming convention.

* __ilu__.cc: contains __ilu0__() , __iluc__() and __ilutp__()

* __ichol__.cc: contains __ichol0__() and __icholt__().

• I have moved all the tests from .cc files to .m scripts so no tests are performed for built-in functions.

The code is ready to be pulled from my repo to be reviewed :

https://edu159@bitbucket.org/edu159/octave-edu159

Practical examples: preconditioned conjugate gradient

It is interesting to show how preconditioning techniques can improve the convergency of some iterative solvers. In this case I am running a Matlab example using the Poisson matrix (that is positive definite) obtained with gallery() function. The scritp:

1. In this first case the convergency of pcg using ICHOL(0) algorithm, modified ICHOL(0) algorithm and without preconditioning are compared.

N = 100;
A = gallery ('Poisson', N);
b = ones (size (A, 1), 1);
tol = 1e-6; maxit = 100;
[x0, fl0, rr0, it0, rv0] = pcg (A, b, tol, maxit);
L1 = ichol (A);
[x1, fl1, rr1, it1, rv1] = pcg(A, b, tol, maxit, L1, L1');
opts.type = 'nofill'; opts.michol = 'on';
L2 = ichol (A, opts);
e = ones (size (A, 2), 1);
norm(A * e - L2 * (L2' * e))
[x2, fl2, rr2, it2, rv2] = pcg (A, b, tol, maxit, L2, L2');
semilogy (0:maxit, rv0 ./ norm (b), 'b.');
hold on;
semilogy (0:it1, rv1 ./ norm(b), 'r.');
semilogy (0:it2, rv2 ./ norm(b), 'k.');
xlabel ('iterations');
ylabel ('error');
legend ('No Preconditioner', 'IC(0)', 'MIC(0)');

 Octave

 Matlab

2. In this second part of the script what is compared is the      convergency of ICHOLT algorithm with different values of droptol.

L3 = ichol(A, struct('type', 'ict', 'droptol', 1e-1));
[x3, fl3, rr3, it3, rv3] = pcg (A, b, tol, maxit, L3, L3');
L4 = ichol (A, struct ('type', 'ict', 'droptol', 1e-2));
[x4, fl4, rr4, it4, rv4] = pcg (A, b, tol, maxit, L4, L4');
L5 = ichol (A, struct ('type', 'ict', 'droptol', 1e-3));
[x5, fl5, rr5, it5, rv5] = pcg (A, b, tol, maxit, L5, L5');
figure; semilogy (0:maxit, rv0 ./ norm (b), 'b-', 'linewidth', 2);
hold on;
semilogy (0:it3, rv3 ./ norm(b), 'b-.', 'linewidth', 2);
semilogy (0:it4, rv4 ./ norm(b), 'b--', 'linewidth', 2);
semilogy (0:it5, rv5 ./ norm(b), 'b:', 'linewidth', 2);
ylabel ('error');
xlabel ('iterations');
legend ('No Preconditioner', 'ICT(1e-1)', 'ICT(1e-2)', ...
'ICT(1e-3)', 'Location', 'SouthEast');

 Octave
 Matlab

As it can be seen Octave plots are the same as Matlab's ones.  Both lead to a decrease in the number of steps upt to convergence of the pcg method. ILU algorithms could also have been used here, but due to the simetry of the problem matrix ICHOL is faster.

Regards,

Eduardo

## FSAL - new stepper implementation

As stated in the previous post, the implementation of the steppers as it was did not allow the possibility to exploit the FSAL (First Same As Last) property of the Bogacki-Shampine algorithm (ode23) and of the Dormand-Prince algorithm (ode45).
The input and output arguments of the steppers have then be modified. As an example here is the runge_kutta_23 stepper:

function varargout = runge_kutta_23 (f, t, x, dt, varargin)  options = varargin{1};  k = zeros (size (x, 1), 4);  if (nargin == 5) % only the options are passed    k(:,1) = feval (f, t , x, options.vfunarguments{:});  elseif (nargin == 6) % both the options and the k values are passed    k(:,1) = varargin{2}(:,end); % FSAL property  endif  k(:,2) = feval (f, t + (1/2)*dt, x + dt*(1/2)*k(:,1), options.vfunarguments{:});  k(:,3) = feval (f, t + (3/4)*dt, x + dt*(3/4)*k(:,2), options.vfunarguments{:});  %# computing new time and new values for the unkwnowns  varargout{1} = t + dt; %t_next  varargout{2} = x + dt.*((2/9)*k(:,1) + (1/3)*k(:,2) + (4/9)*k(:,3)); % return the 3rd order approximation x_next  %# if the estimation of the error is required  if (nargout >= 3)    %# new solution to be compared with the previous one    k(:,4) = feval (f, t + dt, varargout{2}, options.vfunarguments{:});    varargout{3} = x + dt.*((7/24)*k(:,1) + (1/4)*k(:,2) + (1/3)*k(:,3) + (1/8)*k(:,4)); %x_est    varargout{4} = k;  endifendfunction

And the call within the solver becomes:
[s, y, y_est, k_vals] = stepper (func, z(end), u(:,end), dt, options, k_vals);

where k_vals has to be initialized for the first iteration as f(t, x).
This implementation will reduce the number of function evaluation for each step.

Furthermore, after some tests in MATLAB, the return values for the solution and the estimate in the  runge_kutta_23 and runge_kutta_45 steppers have been swapped to automatically perform local extrapolation. The MATLAB functions are in fact of order 3 and 5 respectively.

## Status of the code: bugfixes and new issues

### ODESET and ODEGET

• odeset and odeget functions have been slightly modified to be compliant with MATLAB. Each MATLAB option is present and all the options are tested. The coding style has been adapted to the GNU-Octave standard.
• ode_struct_value_check: this function has been introduced by Roberto in addition to odepkg_structue_check. The relation between the two functions has to be clarified: in particular it is necessary to understand if it is really necessary to have two different functions or one is sufficient.

### CHANGES TO THE STEPPERS

• The runge_kutta_78 stepper has been implemented.
• Two 4th order steppers have been implemented: runge_kutta_45_dopri (Dormand-Prince coefficients) and runge_kutta_45_fehlberg (Fehlberg coefficients).

### CHANGES TO THE SOLVERS

• ode78 solver has been updated to the new structure. It now exploits the runge_kutta_78 stepper.
• A series of tests has been added to each solver to check all the functionalities and the all options. This has made me possible to detect some bugs that have been corrected. In particular the adaptive timestep evaluation had some issues that lead to the use of too short timesteps. This has been corrected and now the algorithm proposed in [1] is used.
• Furthermore the current implementation uses linear interpolation to evaluate the solution at the user specified times. This leads to a considerable loss in accuracy and is not consistent with MATLAB (which guarantees the same order of accuracy of the scheme used). In [1] different methods are proposed for the dense output: these will be used as a reference for the implementation of a better solution.
• In the released version of odepkg some of the solvers perform local extrapolation, that is the higher-order estimate is chosen as the solution. With the new stepper structure, as it is now, this choice effects all the solvers. It have to be decided whether to perform it or not (MATLAB doesn't seem to do it, thus I suggest to avoid it).
• MATLAB implementation of ode45 uses the Dormand-Prince (DP) coefficients. In the released version of odepkg there exits two solvers: ode45 that uses the Fehlberg coefficients and ode54 that uses the DP coefficients. To be consistent with MATLAB, ode45 now uses the DP method. This makes the runge_kutta_45_fehlberg stepper and the ode54 solver, as it is now, redundant. Either their elimination or a change of the solver might be considered. However one of the advantages of DP coefficients is the possibility to reuse the last function evaluation at a given step as the first evaluation of the subsequent one. This is not easily done with the stepper structure introduced by Roberto.

### CHANGES TO THE OPTIONS

• InitialStep option has been modified to be MATLAB compatible (it must be a positive scalar).
• RelTol defalut value has been changed to 1e-3 instead of 1e-6 to be MATLAB compatible.
• MaxStep option has been implemented.
• NormControl option has been implemented.

### TODO

In addition to the general plan, a couple of issues need to be addressed:

• Clarify the relation between ode_struct_value_check and odepkg_structue_check.
• Decide if local extrapolation has to be used or not. My opinion (and the current implementation) is to avoid it to be compliant to what MATLAB seems to be doing.
• Solve the dense output problem in a way that guarantees the consistency with MATLAB.
• Consider if it's possible to reduce the number of function evaluation for the Dormand-Prince stepper (ode45) and the Bogacki-Shampine stepper (ode23) exploiting the FSAL property (first same as last).
• Decide if in the future releases of odepkg ode54 has to be removed or maybe changed to become a Fehlberg solver.

[1] E. Hairer, S.P. N{\o}rsett, G. Wanner, Solving Ordinary Differential Equations, 1993, Springer.

## August 12, 2014

### Eugenio Gianniti

#### SubDomain

As said in my previous post, a missing feature in fem-fenics was the marking of subdomains. Indeed, I proposed an example that needed a file generated with a run of the corresponding Python code, which is not, honestly, the best approach. In order to address this issue, these days I have implemented a new class, subdomain, which can be used to mark mesh entities. In the following I will describe how to use this new functionality. Here is the code:

pkg load fem-fenics msh

ufl start Subdomains
ufl fe = FiniteElement "(""CG"", triangle, 2)"
ufl u = TrialFunction (fe)
ufl v = TestFunction (fe)
ufl
ufl a0 = Coefficient (fe)
ufl a1 = Coefficient (fe)
ufl g_L = Coefficient (fe)
ufl g_R = Coefficient (fe)
ufl f = Coefficient (fe)
ufl
ufl a = "inner(a0*grad(u), grad(v))*dx(0) + inner(a1*grad(u), grad(v))*dx(1)"
ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1)
ufl end

# Create mesh and define function space
x = y = linspace (0, 1, 65);
[msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1));

V = FunctionSpace ("Subdomains", msh);

# Define boundary conditions
bc1 = DirichletBC (V, @(x, y) 5.0, facets, 2);
bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4);

# Define problem coefficients
a0 = Constant ("a0", 1.0);
a1 = Constant ("a1", 0.01);
g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2));
g_R = Constant ("g_R", 1.0);
f = Constant ("f", 1.0);

# Define subdomains - Here are the edits #
domains = MeshFunction ("dx", msh, 2, 0);
obstacle = SubDomain (@(x,y) (y >= 0.5) && (y <= 0.7) && ...
(x >= 0.2) && (x <= 1.0), false);
domains = mark (obstacle, domains, 1);

# Define variational form
a = BilinearForm ("Subdomains", V, V, a0, a1, domains);
L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains);

# Assemble system
[A, b] = assemble_system (a, L, bc1, bc2);
sol = A \ b;
u = Function ("u", V, sol);

# Save solution in VTK format
save (u, "subdomains");

# Plot solution
[X, Y] = meshgrid (x, y);
U = u (X, Y);
surf (X, Y, U);

As you can see, it is basically the same as in the previous post, except the line used to import the meshfunction. I wrote in the corresponding comment where the edits are to be found. Now the workflow comprises these steps: first of all, a meshfunction needs to be created, then a subdomain, in the end we should mark cells.

The call to MeshFunction is something new, since it is now possible to instantiate a meshfunction given a mesh, the required topological dimension and the value to initialise it with. Moreover, the optional label "dx" means that it can be used in calls to BilinearForm or LinearForm to supply markers for subsets of the integration domains. In the example, this instruction returns a meshfunction of dimension 2, which means it holds values associated with each triangle in the mesh, initialised to be 0 in every entry.

The subsequent instruction, instead, defines a subdomain, passing as arguments a function handle and a logical flag. The former will be the override of the dolfin::SubDomain::inside method, so it must return true for entities contained in the subset and false otherwise. In facts it checks whether the coordinates are inside the 2-interval defining the obstacle. The latter, instead, can be used to ask for a boundary subdomain, when set to true.

At last, mark is called to set the entries corresponding to cells inside the subdomain to 1, so that the returned meshfunction now represents the obstacle: after these lines, the variable named domains assumes value 1 on cells inside the obstacle region and 0 outside. Thus, it is now possible to solve a problem whose formulation entails subdomains entirely using fem-fenics.

## August 11, 2014

### Eduardo Fernández

#### Soft pencils down.

It's been quite long since I posted here due to some personal situations. Anyway to sum up: I have finished ilu and ichol functions as I have planned in the beginning with great results.

Things done after mid-term evaluation:
• Implementing ICHOLT and ICHOL0 algorithms.
• Fixing several bugs in ILU algorithms and introducing some enhancements for big sparse matrices with verly low densities.
The files involved in ichol, within the repository, are:
• src/icholt.cc
• src/ichol0.cc
• ichol.m
You can clone the code from the repo:
• https://edu159@bitbucket.org/edu159/gsoc2014-edu15

Before going into the details of the algorithms' implementation, I want to point out some details about how ichol behave in MATLAB.

1. In the real case the matrix must be symetric positive definite.  In the complex case the input matrix must be hermitian. That means: diagonal elements of the input and output matrix have to be non-zero, positive and real values. So that, at each iteration those conditions have to be fullfilled.
2. If ichol is called just as L = ichol (A), Matlab ignores complex numbers and only work with their real part. Using L = ichol (A, setup) call, complex numbers are considered. Seriusly I do not understand why they do that and I have not followed that behaviour. Anyway if  to be 100% compatible I must change that, it would be only a line of code extra.

#### Details of implementation

-->src/ichol0.cc

In this file is located the implementation of ICHOL(0) algorithm. The zero-pattern of the output matrix is the same as the input one so it is known from the beginning how much  memory is needed to be allocated. The milu = ['on'|'off'] parameter indicates whether the dropped elements are added to the pivot or not (that keeps the colum sumation).

I will show two examples, one that corresponds to a big matrix with a very low density and the one that used Kai last year in his blog.

Example 1:

A = gallery ('poisson', 500);
size (A)
ans =

250000   250000
tic; L = ichol (A); toc;

Elapsed time is 0.031718 seconds.
density = nnz (A) / (size (A)(1))^2
density =    1.9968e-05

norm (A - L*L', 'fro') / norm (A, 'fro')
ans =  0.0924207846384523

norm(A-(L*L').*spones(A),'fro')./norm(A,'fro')
ans =    2.28617974245061e-17

It can be seen that the product L*L' is quite different from A, but the product L*L' will match A on its pattern (that is expected for the ICHOL(0) algorithm. The execution time is just given to give an idea of how fast the code is. It is executed in a i7 2.4GHz.

Example 2:

This example is taken from that post, written by Kai the past year. He faced problems with the michol option, obtaining different results from Matlab.

input:
A = [ 0.37, -0.05,  -0.05,  -0.07;
-0.05,  0.116,  0.0,   -0.05;
-0.05,  0.0,    0.116, -0.05;
-0.07, -0.05,  -0.05,   0.202];

A = sparse (A);
opts.michol = 'on';
L = ichol (A, opts);

Octave:
ans =

0.60828   0.00000   0.00000   0.00000
-0.08220   0.32014   0.00000   0.00000
-0.08220   0.00000   0.32014   0.00000
-0.11508  -0.18573  -0.18573   0.34607

Matlab:
ans =

0.6083         0         0         0
-0.0822    0.3201         0         0
-0.0822         0    0.3201         0
-0.1151   -0.1857   -0.1857    0.3461

Works fine.

-->src/icholt.cc

This file contains the implementation of ICHOLT algorithm. In this case the final structure of the output matrix is unknown. Therefore, a policy should be adopted for allocating memory. After trying different ways of doing that I end up using that one:

// max_len is the maximun length of ridx and data arrays for the output sparse matrix.
max_len = sm.nnz ();
max_len += (0.1 * max_len) > n ? 0.1 * max_len : n;

What is done here is just to increment 10% of the actual size of the ridx and data internal arrays of the output sparse matrix. But only if that amount is larger than the dimension of the input matrix (n). In other case the increment in size is just n. That policy seems to work very well in every case I tested and do not slow down the process at all due to reallocations.

Example 3:

icholt accepts a parameter for controling the sparsity of the ouput matrix called droptol. If droptol = 0 then the complete factorization takes place. If we increase that value the output matrix will become more sparse as more elements will be dropped. Taking the same matrix than in example 1:

A = gallery ('poisson', 500);
opts.type= 'ict'

% Complete factorization
opts.droptol = 0;
tic;L = ichol(A, opts);toc;
Elapsed time is 46.0734 seconds.
norm (A - L*L', 'fro') / norm (A, 'fro')
ans =    7.8595e-16

% droptol = 1e-2
opts.droptol=1e-2
tic;L = ichol(A, opts);toc;
Elapsed time is 0.0650802 seconds.

norm (A - L*L', 'fro') / norm (A, 'fro')
ans =  0.016734

% droptol = 1e-3
opts.droptol=1e-3
tic;L = ichol(A, opts);toc;
Elapsed time is 0.183416 seconds.

norm (A - L*L', 'fro') / norm (A, 'fro')
ans =  0.0021773

% droptol = 1e-4
opts.droptol=1e-4
tic;L = ichol(A, opts);toc;
Elapsed time is 0.589693 seconds.
norm (A - L*L', 'fro') / norm (A, 'fro')
ans =    2.4820e-04

As it can be seen, the higher the droptol parameter is, the sparser the matrix become. That lead to less execution times but on the other hand a higher error is obtained in the factorization. The complete factorization obviously have practically no error. Cool.

#### Location of source files inside Octave core

Now I've finished with the development of the algorithms, the final step is to integrate them into Octave core. For doing so I will create a subrepo of the default Octave repository and add the files. I have chosen the location for the functions looking at the last year repository Kai set.

Location:
libinterp/dldfcn: ilutp.cc ilu0.cc iluc.cc ichol0.cc icholt.cc
scripts/sparse: ilu.m ichol.m

That is just a sugestion and should be revised and accepted by the maintainers.

#### Future contributions

There is a week left that I want to use it to start (and hopefully finish) the development of sprandsym extra parameters that Matlab have but Octave does not. As I submitted in the past a changeset for a similar functionality in sprand and sprandn, it will be much easier to implement for me.

Also I am interested in developing some sparse linear solvers like minres and lsqr that Octave lacks. They are tightly related to the preconditioners I have been working on, and would be nice if they could be assigned to me for developing them.

Regards,

Eduardo

## August 09, 2014

### Eugenio Gianniti

#### New features of meshfunction

As you may recall from my last post, for DirichletBC to work in parallel runs I had to implement a new class, meshfunction. However it was still quite unfinished, with no way for the user to create one, except extracting it from a mesh produced by the msh package, no description to display, no way to save it. These days I have been tackling this issue: while at it I wondered what one could do with meshfunction and found out that they can come in handy when you are dealing with obstacles.

At this link you can find a detailed explanation of the problem. It is a Poisson equation with variable diffusion coefficient on the unit square. Precisely, on [0.2, 1]x[0.5, 0.7] its value is 0.01, otherwise it is 1. The mentioned subset is the obstacle to diffusion, so we study its effect applying u = 0 on the y = 0 edge and u = 5 on y = 1. Here is the fem-fenics code:

pkg load fem-fenics mshufl start Subdomainsufl fe = FiniteElement "(""CG"", triangle, 2)"ufl u = TrialFunction (fe)ufl v = TestFunction (fe)uflufl a0 = Coefficient (fe)ufl a1 = Coefficient (fe)ufl g_L = Coefficient (fe)ufl g_R = Coefficient (fe)ufl f = Coefficient (fe)uflufl a = "inner(a0*grad (u), grad (v))*dx(0) + inner(a1*grad (u), grad (v))*dx(1)"ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1)ufl end# Create mesh and define function spacex = y = linspace (0, 1, 65);[msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1));V = FunctionSpace ("Subdomains", msh);# Define boundary conditionsbc1 = DirichletBC (V, @(x, y) 5.0, facets, 2);bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4);# Define problem coefficientsa0 = Constant ("a0", 1.0);a1 = Constant ("a1", 0.01);g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2));g_R = Constant ("g_R", 1.0);f = Constant ("f", 1.0);# Define subdomainsdomains = MeshFunction ("dx", msh, "cells.xdmf");# Define variational forma = BilinearForm ("Subdomains", V, V, a0, a1, domains);L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains);# Assemble system[A, b] = assemble_system (a, L, bc1, bc2);sol = A \ b;u = Function ("u", V, sol);# Save solution in VTK formatsave (u, "subdomains");# Plot solution[X, Y] = meshgrid (x, y);U = u (X, Y);surf (X, Y, U);

In the beginning there is the now familiar ufl block. As you might have noticed, subscripted measures appear in the definition of the bilinear form a and of the linear functional L. This is UFL notation for the integration on specific subsets of the computational domain. For instance, dx(1) is an integral over the subdomain marked with label 1, while ds(3) is an integral over the exterior edges marked with label 3. A third possibility, even if not used in this example, is to use dS for integrals on interior facets, which could be of use for interior penalty methods. Going back to the example, you can see that markers are used to enforce non-homogeneous Neumann conditions on the side edges and to assign the proper coefficient on the two subdomains.

After defining the problem in UFL language, there are instructions to define the mesh, the function space, the essential boundary conditions and all the coefficients involved. All such lines come from fem-fenics before this summer or have been described in my previous posts, so I will not cover them in detail. The same applies for the assembly, solve and all the output in the end of the script. The only note is that the very last lines will error out in parallel runs: point-wise evaluations in DOLFIN can be performed only on local cells, but with meshgrid we are providing to every process the whole domain.
 The computed solution
In between there are my latest efforts. At first, the brand new MeshFunction. With this, providing a mesh and a file name you can import a dolfin::MeshFunction. In this case it was saved in the XDMF format, here you can find the files needed to execute the script. DOLFIN uses this format for parallel input/output. It comprises a .h5 file storing data and a .xdmf with metadata useful to read the other one. The optional first argument is a string identifying the role of the returned meshfunction in the variational problem. In this case, with "dx" it will be searched for markers of the integrals on cells. All the previously mentioned measures are available, and "ds" is automatically attached to the meshfunction returned by Mesh. In the example this behaviour is exploited for the measure on edges.

Afterwards, the mesh functions are passed as arguments to BilinearForm and LinearForm, so that the markers are available to assemble the system. In addition to the usual parameters, such as the name of the imported UFL problem, the function spaces and the coefficients, it is now possible to provide mesh functions properly labeled and they will be used.

Currently fem-fenics allows for easily marking subdomains and exterior edges copying markers from the PDEtool representation returned by the functions of the msh package, which makes it quite tricky to properly identify the obstacle in the example. The approach used in the python interface to DOLFIN entails subclassing dolfin::Subdomain with the proper implementation of the inside method, then use an object of the derived class to mark a dolfin::MeshFunction. This could be an interesting feature to implement in the future also in fem-fenics.

## August 04, 2014

### Eugenio Gianniti

#### MPI parallelism

After quite a struggle, I have been able to obtain a working implementation of fem-fenics supporting MPI parallelism. Let's go through an example and highlight what has changed lately.

pkg load fem-fenics msh

ufl start Poisson
ufl element = FiniteElement '("Lagrange", triangle, 1)'
ufl u = TrialFunction (element)
ufl v = TestFunction (element)
ufl f = Coefficient (element)
ufl g = Coefficient (element)
ufl a = "inner (grad (u), grad (v))*dx"
ufl L = f*v*dx + g*v*ds
ufl end

# Create mesh and define function space
x = y = linspace (0, 1, 33);
[mesh, facets] = Mesh (msh2m_structured_mesh (x, y, 1, 1:4));

V = FunctionSpace ('Poisson', mesh);

# Define boundary condition
bc = DirichletBC (V, @(x, y) 0.0, facets, [2;4]);

f = Expression ('f', @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
g = Expression ('g', @(x,y) sin (5.0 * x));

a = BilinearForm ('Poisson', V, V);
L = LinearForm ('Poisson', V, f, g);

# Compute solution
[A, b] = assemble_system (a, L, bc);
sol = A \ b;
u = Function ('u', V, sol);

# Save solution in VTK format
save (u, 'poisson');

The basic structure has remained the same. DOLFIN boasts the capability to be run both in serial and in parallel execution without intervening on the code, so I did my best to have the same behaviour from fem-fenics. The Poisson.m m-file above can be run either as you usually would do with any other m-file, or from the command line with an invocation such as:

mpiexec -np 4 octave --eval Poisson

Now, how is this possible? In the beginning, with the ufl block, the variational problem is defined in UFL language, written to an .ufl file and compiled via FFC. Since IO is performed, ufl.m ensures that only process zero will open and write to the file. Moreover, a MPI barrier makes sure that no process will proceed before the .ufl file is imported.

As soon as the just-in-time compilation is over, there are two instructions to build the mesh, in this case on the unit square. For this, we rely on the msh package, which returns a PDE-tool-like representation of it. Mesh.oct must, then, convert it to DOLFIN internal representation and distribute it among processes. Here comes an issue: fem-fenics relies on markers present in the PDE-tool format to impose essential boundary conditions, and in serial runs dolfin::Mesh can store them, so that DirichletBC.oct needs just to know the boundary subset label. Unfortunately, this feature is not supported yet in parallel by the DOLFIN library, then Mesh.oct has been edited to return, if requested, also a meshfunction holding this information, in the example above facets. This way markers can be conveyed to DirichletBC.oct and boundary conditions can be applied on the correct edges.

Further intervention was needed for the assembly and solve phase. In assemble_system.oct both the matrix and the vector are assembled locally on each portion of the mesh and, afterwards, gathered on process zero and joined, so that the system can be solved with the backslash instruction of Octave. In order to allow output in VTK format, in Function.oct the solution is split up and properly distributed among processes, so that each one holds the portion of degrees of freedom related to its subdomain and to the neighbouring vertices. After save.oct has written the solution to poisson.pvd and its auxiliary files, it can be visualised with ParaView.